From 408d467cb16f7d7c9f5be862b535eaa35204373b Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 22 Oct 2020 05:08:30 +0200 Subject: [PATCH] test --- doc/pub/week43/html/._week43-bs000.html | 316 +- doc/pub/week43/html/._week43-bs001.html | 316 +- doc/pub/week43/html/._week43-bs002.html | 316 +- doc/pub/week43/html/._week43-bs003.html | 316 +- doc/pub/week43/html/._week43-bs004.html | 316 +- doc/pub/week43/html/._week43-bs005.html | 316 +- doc/pub/week43/html/._week43-bs006.html | 316 +- doc/pub/week43/html/._week43-bs007.html | 316 +- doc/pub/week43/html/._week43-bs008.html | 316 +- doc/pub/week43/html/._week43-bs009.html | 316 +- doc/pub/week43/html/._week43-bs010.html | 316 +- doc/pub/week43/html/._week43-bs011.html | 316 +- doc/pub/week43/html/._week43-bs012.html | 316 +- doc/pub/week43/html/._week43-bs013.html | 316 +- doc/pub/week43/html/._week43-bs014.html | 322 +- doc/pub/week43/html/._week43-bs015.html | 316 +- doc/pub/week43/html/._week43-bs016.html | 316 +- doc/pub/week43/html/._week43-bs017.html | 316 +- doc/pub/week43/html/._week43-bs018.html | 316 +- doc/pub/week43/html/._week43-bs019.html | 316 +- doc/pub/week43/html/._week43-bs020.html | 316 +- doc/pub/week43/html/._week43-bs021.html | 318 +- doc/pub/week43/html/._week43-bs022.html | 342 +- doc/pub/week43/html/._week43-bs023.html | 479 +- doc/pub/week43/html/._week43-bs024.html | 522 +- doc/pub/week43/html/._week43-bs025.html | 378 +- doc/pub/week43/html/._week43-bs026.html | 354 +- doc/pub/week43/html/week43-bs.html | 316 +- doc/pub/week43/html/week43-reveal.html | 2885 ++++------ doc/pub/week43/html/week43-solarized.html | 2920 ++++------ doc/pub/week43/html/week43.html | 2920 ++++------ doc/pub/week43/ipynb/ipynb-week43-src.tar.gz | Bin 190 -> 191 bytes doc/pub/week43/ipynb/week43.ipynb | 5403 ++++++++---------- doc/src/week43/week43.do.txt | 4442 +++++++------- 34 files changed, 12361 insertions(+), 15560 deletions(-) diff --git a/doc/pub/week43/html/._week43-bs000.html b/doc/pub/week43/html/._week43-bs000.html index ca91b3389..6340568db 100644 --- a/doc/pub/week43/html/._week43-bs000.html +++ b/doc/pub/week43/html/._week43-bs000.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - 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  • Backpropagation
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  • Correlation Function and Design/Feature Matrix
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • LLE
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Resources
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  • The network with one input layer, specified number of hidden layers, and one output layer
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  • Example: Population growth
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  • Setting up the problem
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  • The trial solution
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • Partial Differential Equations
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  • Type of problem
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  • Network requirements
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  • More details
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  • Example: The diffusion equation
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  • Defining the problem
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  • Setting up the network using Autograd
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  • Setting up the network using Autograd; The trial solution
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  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • The problem to solve for
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  • The analytical solution
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  • Resources on differential equations and deep learning
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  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Towards the PCA theorem
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
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  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
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  • Incremental PCA
    • +
  • Randomized PCA
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  • Kernel PCA
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  • LLE
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    • diff --git a/doc/pub/week43/html/._week43-bs001.html b/doc/pub/week43/html/._week43-bs001.html index bedd969fd..e202d2c1c 100644 --- a/doc/pub/week43/html/._week43-bs001.html +++ b/doc/pub/week43/html/._week43-bs001.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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    • -
  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
    • -
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • The trial solution
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • Program flow in TensorFlow - Execution phase
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  • Example: The diffusion equation
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    • +
  • Why the jacobian?
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    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
    • +
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    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
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    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
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    • +
  • Randomized PCA
    • +
  • Kernel PCA
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    • diff --git a/doc/pub/week43/html/._week43-bs002.html b/doc/pub/week43/html/._week43-bs002.html index 94446bb0b..19c0857d9 100644 --- a/doc/pub/week43/html/._week43-bs002.html +++ b/doc/pub/week43/html/._week43-bs002.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • LLE
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  • Other techniques
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  • The network with one input, hidden, and output layer
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  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Setting up the problem
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • Using TensorFlow to model logistic population growth
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  • Setting up the network using Autograd; The trial solution
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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    • -
  • Resources
    • +
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  • Example: Population growth
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  • Setting up the problem
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  • Using forward Euler to solve the ODE
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  • Partial Differential Equations
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  • Type of problem
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  • Network requirements
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  • Example: The diffusion equation
    • +
  • Defining the problem
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  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
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  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Friday, Principal Component Analysis
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
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  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs003.html b/doc/pub/week43/html/._week43-bs003.html index 89787d64b..1a10efdf0 100644 --- a/doc/pub/week43/html/._week43-bs003.html +++ b/doc/pub/week43/html/._week43-bs003.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Why the jacobian?
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    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
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    • +
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    • +
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    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
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    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
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    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
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    • +
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    • +
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    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
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    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
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  • Randomized PCA
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  • Kernel PCA
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    • diff --git a/doc/pub/week43/html/._week43-bs004.html b/doc/pub/week43/html/._week43-bs004.html index d4bebb52d..fc746cbef 100644 --- a/doc/pub/week43/html/._week43-bs004.html +++ b/doc/pub/week43/html/._week43-bs004.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Randomized PCA
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  • LLE
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  • Other techniques
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  • The network with one input, hidden, and output layer
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  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Setting up the problem
    • -
  • The trial solution
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  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
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  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
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  • The full program modeling logistic population growth using TensorFlow
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
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  • Solving the equation using Autograd
    • -
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  • Using gradient descent in TensorFlow to solve Poisson equation
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  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
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  • Partial Differential Equations
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  • Example: The diffusion equation
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  • Defining the problem
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  • Setting up the network using Autograd
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  • Setting up the network using Autograd; The trial solution
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  • Setting up the network using Autograd; The full program
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
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    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
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  • Setting up the problem
    • +
  • The trial solution
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  • Using forward Euler to solve the ODE
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • Comparing with a numerical scheme
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  • Partial Differential Equations
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  • Type of problem
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  • Network requirements
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    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Example: Solving the wave equation with Neural Networks
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  • The problem to solve for
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources on differential equations and deep learning
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  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Correlation Matrix with Pandas
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs005.html b/doc/pub/week43/html/._week43-bs005.html index d019b0ff9..10cc92a61 100644 --- a/doc/pub/week43/html/._week43-bs005.html +++ b/doc/pub/week43/html/._week43-bs005.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Gradient descent
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Why the jacobian?
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    • +
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    • +
  • Friday, Principal Component Analysis
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    • +
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    • +
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    • +
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    • +
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    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
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    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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    • diff --git a/doc/pub/week43/html/._week43-bs006.html b/doc/pub/week43/html/._week43-bs006.html index 2b783b971..625c20e3e 100644 --- a/doc/pub/week43/html/._week43-bs006.html +++ b/doc/pub/week43/html/._week43-bs006.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
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  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
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  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
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  • Incremental PCA
    • -
  • Randomized PCA
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  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
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  • Using gradient descent in TensorFlow to solve Poisson equation
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  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
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  • Example: The diffusion equation
    • -
  • Defining the problem
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  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
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  • Setting up the network using Autograd; The full program
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  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
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  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
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  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
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  • The trial solution
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources on differential equations and deep learning
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  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs007.html b/doc/pub/week43/html/._week43-bs007.html index 5511bc43e..386eeff90 100644 --- a/doc/pub/week43/html/._week43-bs007.html +++ b/doc/pub/week43/html/._week43-bs007.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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    • diff --git a/doc/pub/week43/html/._week43-bs008.html b/doc/pub/week43/html/._week43-bs008.html index 5084cc239..f1a122605 100644 --- a/doc/pub/week43/html/._week43-bs008.html +++ b/doc/pub/week43/html/._week43-bs008.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • PCA Proof continued
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  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • Solving the wave equation - the full program using Autograd
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  • Setting up the problem
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Proof of the PCA Theorem
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs009.html b/doc/pub/week43/html/._week43-bs009.html index e730066c0..3879b20cf 100644 --- a/doc/pub/week43/html/._week43-bs009.html +++ b/doc/pub/week43/html/._week43-bs009.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
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    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Towards the PCA theorem
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    • -
  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • LLE
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  • The network with one input, hidden, and output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • The analytical solution
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  • Resources
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  • The code for solving the ODE
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  • The network with one input layer, specified number of hidden layers, and one output layer
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  • Example: Population growth
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  • Using forward Euler to solve the ODE
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  • Partial Differential Equations
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  • Network requirements
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  • More details
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  • Example: The diffusion equation
    • +
  • Defining the problem
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  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
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  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Example: Solving the wave equation with Neural Networks
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
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  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs010.html b/doc/pub/week43/html/._week43-bs010.html index bcbac7d8a..096ab291c 100644 --- a/doc/pub/week43/html/._week43-bs010.html +++ b/doc/pub/week43/html/._week43-bs010.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
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    • -
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs011.html b/doc/pub/week43/html/._week43-bs011.html index 06666d918..a74838958 100644 --- a/doc/pub/week43/html/._week43-bs011.html +++ b/doc/pub/week43/html/._week43-bs011.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • LLE
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  • The network with one input, hidden, and output layer
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  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Setting up the problem
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources
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  • The network with one input layer, specified number of hidden layers, and one output layer
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  • Example: Population growth
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  • Setting up the problem
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  • The trial solution
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  • The program using Autograd
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  • Using forward Euler to solve the ODE
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • Partial Differential Equations
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  • Type of problem
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  • Network requirements
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  • More details
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  • Example: The diffusion equation
    • +
  • Defining the problem
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  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
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  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Example: Solving the wave equation with Neural Networks
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
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  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs012.html b/doc/pub/week43/html/._week43-bs012.html index 77e7e02ba..9f48880c3 100644 --- a/doc/pub/week43/html/._week43-bs012.html +++ b/doc/pub/week43/html/._week43-bs012.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  •    Compute the sample covariance
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  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
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    • diff --git a/doc/pub/week43/html/._week43-bs013.html b/doc/pub/week43/html/._week43-bs013.html index f018902a1..2c109787d 100644 --- a/doc/pub/week43/html/._week43-bs013.html +++ b/doc/pub/week43/html/._week43-bs013.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
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  • Correlation Matrix with Pandas
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Using forward Euler to solve the ODE
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources
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  • Setting up the problem
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  • Network requirements
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    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
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  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
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  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Example: Solving the wave equation with Neural Networks
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  • The problem to solve for
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  • Friday, Principal Component Analysis
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
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  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs014.html b/doc/pub/week43/html/._week43-bs014.html index c7ea6be1f..0b5138ed0 100644 --- a/doc/pub/week43/html/._week43-bs014.html +++ b/doc/pub/week43/html/._week43-bs014.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  •    Compute the sample covariance
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  • The specific equation to solve for
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  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -349,17 +315,17 @@ $$ $$

    -Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes +Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes $$ -\min_{P} c(\boldsymbol{x}, P) +\min_{P} C(\boldsymbol{x}, P) $$

    In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could also be expressed as $$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) +\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) $$

    diff --git a/doc/pub/week43/html/._week43-bs015.html b/doc/pub/week43/html/._week43-bs015.html index 45d12c8bd..94d3391c3 100644 --- a/doc/pub/week43/html/._week43-bs015.html +++ b/doc/pub/week43/html/._week43-bs015.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
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  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs016.html b/doc/pub/week43/html/._week43-bs016.html index 8852dbff1..5f233c75d 100644 --- a/doc/pub/week43/html/._week43-bs016.html +++ b/doc/pub/week43/html/._week43-bs016.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({
  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
    • -
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    • -
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    • -
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    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
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    • -
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  • Why the jacobian?
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    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
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    • +
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    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
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    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
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  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
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    • +
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    • +
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    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs017.html b/doc/pub/week43/html/._week43-bs017.html index a7983117c..1fa12636a 100644 --- a/doc/pub/week43/html/._week43-bs017.html +++ b/doc/pub/week43/html/._week43-bs017.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
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  • Writing our own PCA code
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  •    Compute the sample mean and center the data
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • Other techniques
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  • The network with one input, hidden, and output layer
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  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
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  • The full program modeling logistic population growth using TensorFlow
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
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  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
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  • Using gradient descent in TensorFlow to solve Poisson equation
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  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
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  • Partial Differential Equations
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  • Example: The diffusion equation
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  • Defining the problem
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  • Setting up the network using Autograd
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  • Setting up the network using Autograd; The trial solution
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  • Setting up the network using Autograd; The full program
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
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  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • Solving the equation using Autograd
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  • Comparing with a numerical scheme
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  • Partial Differential Equations
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  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
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  • Setting up the network using Autograd; The full program
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  • Example: Solving the wave equation with Neural Networks
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  • The problem to solve for
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  • The trial solution
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources on differential equations and deep learning
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  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
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  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs018.html b/doc/pub/week43/html/._week43-bs018.html index feef46ec1..4345d7abf 100644 --- a/doc/pub/week43/html/._week43-bs018.html +++ b/doc/pub/week43/html/._week43-bs018.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Gradient descent
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  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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    • -
  • LLE
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    • -
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  • Why the jacobian?
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    • +
  • Resources on differential equations and deep learning
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    • +
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    • +
  • Correlation Function and Design/Feature Matrix
    • +
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    • +
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    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
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    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs019.html b/doc/pub/week43/html/._week43-bs019.html index 0bfaa7d38..4c92be862 100644 --- a/doc/pub/week43/html/._week43-bs019.html +++ b/doc/pub/week43/html/._week43-bs019.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
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  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
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  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
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  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
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  • PCA and scikit-learn
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  • Incremental PCA
    • -
  • Randomized PCA
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  • Kernel PCA
    • -
  • LLE
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  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
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  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
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  • Using gradient descent in TensorFlow to solve Poisson equation
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  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
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  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
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  • Setting up the network using Autograd; The full program
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  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
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  • The specific equation to solve for
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  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
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  • The trial solution
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  • The analytical solution
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  • Solving the wave equation - the full program using Autograd
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  • Resources on differential equations and deep learning
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  • Friday, Principal Component Analysis
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  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
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  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/._week43-bs020.html b/doc/pub/week43/html/._week43-bs020.html index d74198748..509bafc33 100644 --- a/doc/pub/week43/html/._week43-bs020.html +++ b/doc/pub/week43/html/._week43-bs020.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
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  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
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    • -
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    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
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    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
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  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
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  • Setting up the problem
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  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
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    • -
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    • -
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  • The analytical solution
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  • The network with one input layer, specified number of hidden layers, and one output layer
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  • Example: Population growth
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  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
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    • +
  • Partial Differential Equations
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  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
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    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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    • diff --git a/doc/pub/week43/html/._week43-bs021.html b/doc/pub/week43/html/._week43-bs021.html index ba05f7867..baaaf2e57 100644 --- a/doc/pub/week43/html/._week43-bs021.html +++ b/doc/pub/week43/html/._week43-bs021.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; 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  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
    • -
  • The final step
    • -
  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • -
  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
    • -
  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -346,7 +312,7 @@ The next step is to decide how the parameters should be changed such that they m The chosen cost function for this problem is $$ -c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 $$

    diff --git a/doc/pub/week43/html/._week43-bs022.html b/doc/pub/week43/html/._week43-bs022.html index 20946615d..6e6640e9f 100644 --- a/doc/pub/week43/html/._week43-bs022.html +++ b/doc/pub/week43/html/._week43-bs022.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({

  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
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  • Classical PCA Theorem
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  • Proof of the PCA Theorem
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  • PCA Proof continued
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  • The final step
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  • Geometric Interpretation and link with Singular Value Decomposition
    • -
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • Other techniques
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  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
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  • Program flow in TensorFlow - Construction phase
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  • Program flow in TensorFlow - Execution phase
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    • -
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    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
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  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
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  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
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  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
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  • The problem to solve for
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  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -345,7 +311,7 @@ a direction where the cost function decreases goes to a minimum.

    In general, the update of some parameters \( \boldsymbol{\omega} \) given a cost -function defined by some weights \( \boldsymbol{\omega} \), \( c(\boldsymbol{x}, +function defined by some weights \( \boldsymbol{\omega} \), \( C(\boldsymbol{x}, \boldsymbol{\omega}) \), goes as follows: $$ @@ -356,31 +322,27 @@ $$ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance.

    -The value of \( \lambda \) decides how large steps the algorithm must take in the direction of $ \nabla_{\boldsymbol{\omega}} c(\boldsymbol{x}, \boldsymbol{\omega})$. -The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect to the elements in \( \boldsymbol{\omega \). +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$. +The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect +to the elements in \( \boldsymbol{\omega \).

    -In our case, we have to minimize the cost function \( c(\boldsymbol{x}, P) \) with respect to the two sets of weights and biases, that is for the hidden layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} } \) . +In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with +respect to the two sets of weights and biases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} +} \) .

    This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated by $$ \begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(\boldsymbol{x}, P) +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) \end{aligned} $$ -

    -In general, one could risk using a cost function having gradients that are cumbersome to derive analytically. -For our case, the cost functions are just the mean squared error. -One could employ an implementation of the back propagation for this case, but we will emphasis -on how one could use automatic differentiation in order to train the network. - -

    -However, it might be useful to know how automatic differentiation can be used, e.g through Autograd, in order to test an implementation. -

    diff --git a/doc/pub/week43/html/._week43-bs023.html b/doc/pub/week43/html/._week43-bs023.html index 34985eae4..34ba61dda 100644 --- a/doc/pub/week43/html/._week43-bs023.html +++ b/doc/pub/week43/html/._week43-bs023.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({

  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
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  • Covariance Matrix Examples
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  • Correlation Matrix
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  • Correlation Matrix with Pandas
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  • Correlation Matrix with Pandas and the Franke function
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
    • -
  • The final step
    • -
  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • -
  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
    • -
  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -337,26 +303,157 @@ MathJax.Hub.Config({ -

    Basic ideas of the Principal Component Analysis (PCA)

    +

    The code for solving the ODE

    -The principal component analysis deals with the problem of fitting a -low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than -the totaldimension \( D \) of the problem at hand (our data -set). Mathematically it can be formulated as a statistical problem or -a geometric problem. In our discussion of the theorem for the -classical PCA, we will stay with a statistical approach. This is also -what set the scene historically which for the PCA. + +

    import autograd.numpy as np
+from autograd import grad, elementwise_grad
+import autograd.numpy.random as npr
+from matplotlib import pyplot as plt
+
+def sigmoid(z):
+    return 1/(1 + np.exp(-z))
+
+# Assuming one input, hidden, and output layer
+def neural_network(params, x):
+
+    # Find the weights (including and biases) for the hidden and output layer.
+    # Assume that params is a list of parameters for each layer.
+    # The biases are the first element for each array in params,
+    # and the weights are the remaning elements in each array in params.
+
+    w_hidden = params[0]
+    w_output = params[1]
+
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+
+    ## Hidden layer:
+
+    # Add a row of ones to include bias
+    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
+
+    z_hidden = np.matmul(w_hidden, x_input)
+    x_hidden = sigmoid(z_hidden)
+
+    ## Output layer:
+
+    # Include bias:
+    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
+
+    z_output = np.matmul(w_output, x_hidden)
+    x_output = z_output
+
+    return x_output
+
+# The trial solution using the deep neural network:
+def g_trial(x,params, g0 = 10):
+    return g0 + x*neural_network(params,x)
+
+# The right side of the ODE:
+def g(x, g_trial, gamma = 2):
+    return -gamma*g_trial
+
+# The cost function:
+def cost_function(P, x):
+
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial(x,P)
+
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(neural_network,1)(P,x)
+
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial,0)(x,P)
+
+    # The right side of the ODE
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+
+    return cost_sum / np.size(err_sqr)
+
+# Solve the exponential decay ODE using neural network with one input, hidden, and output layer
+def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
+    ## Set up initial weights and biases
+
+    # For the hidden layer
+    p0 = npr.randn(num_neurons_hidden, 2 )
+
+    # For the output layer
+    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
+
+    P = [p0, p1]
+
+    print('Initial cost: %g'%cost_function(P, x))
+
+    ## Start finding the optimal weights using gradient descent
+
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_grad = grad(cost_function,0)
+
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P.
+        # The cost_grad consist now of two arrays;
+        # one for the gradient w.r.t P_hidden and
+        # one for the gradient w.r.t P_output
+        cost_grad =  cost_function_grad(P, x)
+
+        P[0] = P[0] - lmb * cost_grad[0]
+        P[1] = P[1] - lmb * cost_grad[1]
+
+    print('Final cost: %g'%cost_function(P, x))
+
+    return P
+
+def g_analytic(x, gamma = 2, g0 = 10):
+    return g0*np.exp(-gamma*x)
+
+# Solve the given problem
+if __name__ == '__main__':
+    # Set seed such that the weight are initialized
+    # with same weights and biases for every run.
+    npr.seed(15)
+
+    ## Decide the vales of arguments to the function to solve
+    N = 10
+    x = np.linspace(0, 1, N)
+
+    ## Set up the initial parameters
+    num_hidden_neurons = 10
+    num_iter = 10000
+    lmb = 0.001
+
+    # Use the network
+    P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+    # Print the deviation from the trial solution and true solution
+    res = g_trial(x,P)
+    res_analytical = g_analytic(x)
+
+    print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
+
+    # Plot the results
+    plt.figure(figsize=(10,10))
+
+    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
+    plt.plot(x, res_analytical)
+    plt.plot(x, res[0,:])
+    plt.legend(['analytical','nn'])
+    plt.xlabel('x')
+    plt.ylabel('g(x)')
+    plt.show()
+

    -We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) - -

    -

    @@ -337,64 +303,176 @@ MathJax.Hub.Config({ -

    Introducing the Covariance and Correlation functions

    +

    The network with one input layer, specified number of hidden layers, and one output layer

    -Before we discuss the PCA theorem, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities +It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.

    -Suppose we have defined two vectors -\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -$$ - -where for example -$$ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -With this definition and recalling that the variance is defined as -$$ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -$$ - -we can rewrite the covariance matrix as -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -$$ +The number of neurons within each hidden layer are given as a list of integers in the program below.

    -The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function -$$ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -$$ + +

    import autograd.numpy as np
+from autograd import grad, elementwise_grad
+import autograd.numpy.random as npr
+from matplotlib import pyplot as plt
 
-
    
-The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
-\in [-1,1] \). This avoids eventual problems with too large values. We
-can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
-and \( \boldsymbol{y} \) as
+def sigmoid(z):
+    return 1/(1 + np.exp(-z))
 
-$$
-\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
-                              \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
-             \end{bmatrix},
-$$
+# The neural network with one input layer and one output layer,
+# but with number of hidden layers specified by the user.
+def deep_neural_network(deep_params, x):
+    # N_hidden is the number of hidden layers
 
-
    
-In the above example this is the function we constructed using pandas.
+    N_hidden = np.size(deep_params) - 1 # -1 since params consists of
+                                        # parameters to all the hidden
+                                        # layers AND the output layer.
 
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+
+    # Due to multiple hidden layers, define a variable referencing to the
+    # output of the previous layer:
+    x_prev = x_input
+
+    ## Hidden layers:
+
+    for l in range(N_hidden):
+        # From the list of parameters P; find the correct weigths and bias for this layer
+        w_hidden = deep_params[l]
+
+        # Add a row of ones to include bias
+        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
+
+        z_hidden = np.matmul(w_hidden, x_prev)
+        x_hidden = sigmoid(z_hidden)
+
+        # Update x_prev such that next layer can use the output from this layer
+        x_prev = x_hidden
+
+    ## Output layer:
+
+    # Get the weights and bias for this layer
+    w_output = deep_params[-1]
+
+    # Include bias:
+    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
+
+    z_output = np.matmul(w_output, x_prev)
+    x_output = z_output
+
+    return x_output
+
+# The trial solution using the deep neural network:
+def g_trial_deep(x,params, g0 = 10):
+    return g0 + x*deep_neural_network(params, x)
+
+# The right side of the ODE:
+def g(x, g_trial, gamma = 2):
+    return -gamma*g_trial
+
+# The same cost function as before, but calls deep_neural_network instead.
+def cost_function_deep(P, x):
+
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial_deep(x,P)
+
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(deep_neural_network,1)(P,x)
+
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
+
+    # The right side of the ODE
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+
+    return cost_sum / np.size(err_sqr)
+
+# Solve the exponential decay ODE using neural network with one input and one output layer,
+# but with specified number of hidden layers from the user.
+def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
+    # num_hidden_neurons is now a list of number of neurons within each hidden layer
+
+    # The number of elements in the list num_hidden_neurons thus represents
+    # the number of hidden layers.
+
+    # Find the number of hidden layers:
+    N_hidden = np.size(num_neurons)
+
+    ## Set up initial weights and biases
+
+    # Initialize the list of parameters:
+    P = [None]*(N_hidden + 1) # + 1 to include the output layer
+
+    P[0] = npr.randn(num_neurons[0], 2 )
+    for l in range(1,N_hidden):
+        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
+
+    # For the output layer
+    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
+
+    print('Initial cost: %g'%cost_function_deep(P, x))
+
+    ## Start finding the optimal weights using gradient descent
+
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_deep_grad = grad(cost_function_deep,0)
+
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P.
+        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
+        # in the hidden layers and output layers evaluated at x.
+        cost_deep_grad =  cost_function_deep_grad(P, x)
+
+        for l in range(N_hidden+1):
+            P[l] = P[l] - lmb * cost_deep_grad[l]
+
+    print('Final cost: %g'%cost_function_deep(P, x))
+
+    return P
+
+def g_analytic(x, gamma = 2, g0 = 10):
+    return g0*np.exp(-gamma*x)
+
+# Solve the given problem
+if __name__ == '__main__':
+    npr.seed(15)
+
+    ## Decide the vales of arguments to the function to solve
+    N = 10
+    x = np.linspace(0, 1, N)
+
+    ## Set up the initial parameters
+    num_hidden_neurons = np.array([10,10])
+    num_iter = 10000
+    lmb = 0.001
+
+    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+    res = g_trial_deep(x,P)
+    res_analytical = g_analytic(x)
+
+    plt.figure(figsize=(10,10))
+
+    plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
+    plt.plot(x, res_analytical)
+    plt.plot(x, res[0,:])
+    plt.legend(['analytical','dnn'])
+    plt.ylabel('g(x)')
+    plt.show()
+

    diff --git a/doc/pub/week43/html/._week43-bs025.html b/doc/pub/week43/html/._week43-bs025.html index 86ab21fca..9948969e0 100644 --- a/doc/pub/week43/html/._week43-bs025.html +++ b/doc/pub/week43/html/._week43-bs025.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({

  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
    • -
  • The final step
    • -
  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • -
  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
    • -
  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -337,63 +303,27 @@ MathJax.Hub.Config({ -

    Correlation Function and Design/Feature Matrix

    +

    Example: Population growth

    -In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression -we defined the design/feature matrix \( \boldsymbol{X} \) as +A logistic model of population growth assumes that a population converges toward an equilibrium. +The population growth can be modeled by $$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - -with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the -entries \( n \) being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as -$$ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -$$ - -with a given vector -$$ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. +\begin{equation} \tag{10} + g'(t) = \alpha g(t)(A - g(t)) +\end{equation} $$

    -With these definitions, we can now rewrite our \( 2\times 2 \) -correaltion/covariance matrix in terms of a moe general design/feature -matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \) -covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \) +where \( g(t) \) is the population density at time \( t \), \( \alpha > 0 \) the growth rate and \( A > 0 \) is the maximum population number in the environment. +Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0 \) is some chosen constant. -$$ -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -$$ - -and the correlation matrix -$$ -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -$$ +

    +In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability +and high execution time (this might be more apparent in the examples solving PDEs), +a network has been constructed using TensorFlow also. +For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.

    diff --git a/doc/pub/week43/html/._week43-bs026.html b/doc/pub/week43/html/._week43-bs026.html index ad8e328ca..a9647bd00 100644 --- a/doc/pub/week43/html/._week43-bs026.html +++ b/doc/pub/week43/html/._week43-bs026.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({

  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
    • -
  • The final step
    • -
  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • -
  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
    • -
  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -337,46 +303,24 @@ MathJax.Hub.Config({ -

    Covariance Matrix Examples

    +

    Setting up the problem

    -The Numpy function np.cov calculates the covariance elements using -the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have -the exact mean values. The following simple function uses the -np.vstack function which takes each vector of dimension \( 1\times n \) -and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) +Here, we will model a population \( g(t) \) in an environment having carrying capacity \( A \). +The population follows the model $$ -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, +\begin{equation} \tag{11} +g'(t) = \alpha g(t)(A - g(t)) +\end{equation} $$

    -which in turn is converted into into the \( 2\times 2 \) covariance matrix -\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function. +where \( g(0) = g_0 \).

    +In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \). - -

    # Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-

    diff --git a/doc/pub/week43/html/week43-bs.html b/doc/pub/week43/html/week43-bs.html index ca91b3389..6340568db 100644 --- a/doc/pub/week43/html/week43-bs.html +++ b/doc/pub/week43/html/week43-bs.html @@ -69,142 +69,111 @@ Automatically generated HTML file from DocOnce source ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -263,65 +232,62 @@ MathJax.Hub.Config({

  • Final technicalities IV
  • Backpropagation
  • Gradient descent
    • -
  • Basic ideas of the Principal Component Analysis (PCA)
    • -
  • Introducing the Covariance and Correlation functions
    • -
  • Correlation Function and Design/Feature Matrix
    • -
  • Covariance Matrix Examples
    • -
  • Correlation Matrix
    • -
  • Correlation Matrix with Pandas
    • -
  • Correlation Matrix with Pandas and the Franke function
    • -
  • Rewriting the Covariance and/or Correlation Matrix
    • -
  • Towards the PCA theorem
    • -
  • The Algorithm before the Theorem
    • -
  • Writing our own PCA code
    • -
  •    Compute the sample mean and center the data
    • -
  •    Compute the sample covariance
    • -
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • -
  • Classical PCA Theorem
    • -
  • Proof of the PCA Theorem
    • -
  • PCA Proof continued
    • -
  • The final step
    • -
  • Geometric Interpretation and link with Singular Value Decomposition
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • Back to the Cancer Data
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • -
  • The network with one input, hidden, and output layer
    • -
  • The network with one input layer, specified number of hidden layers, and one output layer output layer
    • -
  • Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme
    • -
  • Setting up the problem
    • -
  • The trial solution
    • -
  • The program using Autograd
    • -
  • Using forward Euler to solve the ODE
    • -
  • Using TensorFlow to model logistic population growth
    • -
  • The general program flow in TensorFlow
    • -
  • Program flow in TensorFlow - Construction phase
    • -
  • Program flow in TensorFlow - Execution phase
    • -
  • The full program modeling logistic population growth using TensorFlow
    • -
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • -
  • The specific equation to solve for
    • -
  • Solving the equation using Autograd
    • -
  • Comparing with a numerical scheme
    • -
  • Using gradient descent in TensorFlow to solve Poisson equation
    • -
  • Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation
    • -
  • Partial Differential Equations
    • -
  • Example: The diffusion equation
    • -
  • Defining the problem
    • -
  • Setting up the network using Autograd
    • -
  • Setting up the network using Autograd; The trial solution
    • -
  • Setting up the network using Autograd; The full program
    • -
  • Example: Solving the wave equation using Autograd and TensorFlow
    • -
  • The problem to solve for
    • -
  • The trial solution
    • -
  • The analytical solution
    • -
  • Solving the wave equation - the full program using Autograd
    • -
  • Solving the wave equation - the full program using TensorFlow
    • -
  • Resources
    • +
  • The code for solving the ODE
    • +
  • The network with one input layer, specified number of hidden layers, and one output layer
    • +
  • Example: Population growth
    • +
  • Setting up the problem
    • +
  • The trial solution
    • +
  • The program using Autograd
    • +
  • Using forward Euler to solve the ODE
    • +
  • Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow
    • +
  • The specific equation to solve for
    • +
  • Solving the equation using Autograd
    • +
  • Comparing with a numerical scheme
    • +
  • Partial Differential Equations
    • +
  • Type of problem
    • +
  • Network requirements
    • +
  • More details
    • +
  • Example: The diffusion equation
    • +
  • Defining the problem
    • +
  • Setting up the network using Autograd
    • +
  • Setting up the network using Autograd; The trial solution
    • +
  • Why the jacobian?
    • +
  • Setting up the network using Autograd; The full program
    • +
  • Example: Solving the wave equation with Neural Networks
    • +
  • The problem to solve for
    • +
  • The trial solution
    • +
  • The analytical solution
    • +
  • Solving the wave equation - the full program using Autograd
    • +
  • Resources on differential equations and deep learning
    • +
  • Friday, Principal Component Analysis
    • +
  • Basic ideas of the Principal Component Analysis (PCA)
    • +
  • Introducing the Covariance and Correlation functions
    • +
  • Correlation Function and Design/Feature Matrix
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • Rewriting the Covariance and/or Correlation Matrix
    • +
  • Towards the PCA theorem
    • +
  • The Algorithm before the Theorem
    • +
  • Writing our own PCA code
    • +
  •    Compute the sample mean and center the data
    • +
  •    Compute the sample covariance
    • +
  •    Diagonalize the sample covariance matrix to obtain the principal components
    • +
  • Classical PCA Theorem
    • +
  • Proof of the PCA Theorem
    • +
  • PCA Proof continued
    • +
  • The final step
    • +
  • Geometric Interpretation and link with Singular Value Decomposition
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • Back to the Cancer Data
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • diff --git a/doc/pub/week43/html/week43-reveal.html b/doc/pub/week43/html/week43-reveal.html index 4f7866ef3..0441ffdf7 100644 --- a/doc/pub/week43/html/week43-reveal.html +++ b/doc/pub/week43/html/week43-reveal.html @@ -481,11 +481,11 @@ $$

     

    -Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes +Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes

     
    $$ -\min_{P} c(\boldsymbol{x}, P) +\min_{P} C(\boldsymbol{x}, P) $$

     
    @@ -494,7 +494,7 @@ In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could al

     
    $$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) +\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) $$

     
    @@ -658,7 +658,7 @@ The chosen cost function for this problem is

     
    $$ -c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 $$

     
    @@ -679,7 +679,7 @@ a direction where the cost function decreases goes to a minimum.

    In general, the update of some parameters \( \boldsymbol{\omega} \) given a cost -function defined by some weights \( \boldsymbol{\omega} \), \( c(\boldsymbol{x}, +function defined by some weights \( \boldsymbol{\omega} \), \( C(\boldsymbol{x}, \boldsymbol{\omega}) \), goes as follows:

     
    @@ -692,11 +692,16 @@ $$ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance.

    -The value of \( \lambda \) decides how large steps the algorithm must take in the direction of $ \nabla_{\boldsymbol{\omega}} c(\boldsymbol{x}, \boldsymbol{\omega})$. -The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect to the elements in \( \boldsymbol{\omega \). +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$. +The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect +to the elements in \( \boldsymbol{\omega \).

    -In our case, we have to minimize the cost function \( c(\boldsymbol{x}, P) \) with respect to the two sets of weights and biases, that is for the hidden layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} } \) . +In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with +respect to the two sets of weights and biases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} +} \) .

    This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated by @@ -704,1133 +709,21 @@ This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated

     
    $$ \begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(\boldsymbol{x}, P) +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) \end{aligned} $$

     
    - -

    -In general, one could risk using a cost function having gradients that are cumbersome to derive analytically. -For our case, the cost functions are just the mean squared error. -One could employ an implementation of the back propagation for this case, but we will emphasis -on how one could use automatic differentiation in order to train the network. - -

    -However, it might be useful to know how automatic differentiation can be used, e.g through Autograd, in order to test an implementation.

    -

    Basic ideas of the Principal Component Analysis (PCA)

    - -

    -The principal component analysis deals with the problem of fitting a -low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than -the totaldimension \( D \) of the problem at hand (our data -set). Mathematically it can be formulated as a statistical problem or -a geometric problem. In our discussion of the theorem for the -classical PCA, we will stay with a statistical approach. This is also -what set the scene historically which for the PCA. - -

    -We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) - -

    -
    - - -
    -

    Introducing the Covariance and Correlation functions

    - -

    -Before we discuss the PCA theorem, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities - -

    -Suppose we have defined two vectors -\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -$$ -

     
    - -where for example -

     
    -$$ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ -

     
    - -With this definition and recalling that the variance is defined as -

     
    -$$ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -$$ -

     
    - -we can rewrite the covariance matrix as -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -$$ -

     
    - -

    -The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function - -

     
    -$$ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -$$ -

     
    - -

    -The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1] \). This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors \( \boldsymbol{x} \) -and \( \boldsymbol{y} \) as - -

     
    -$$ -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -$$ -

     
    - -

    -In the above example this is the function we constructed using pandas. -

    - - -
    -

    Correlation Function and Design/Feature Matrix

    - -

    -In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression -we defined the design/feature matrix \( \boldsymbol{X} \) as - -

     
    -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ -

     
    - -with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the -entries \( n \) being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as -

     
    -$$ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -$$ -

     
    - -with a given vector -

     
    -$$ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -$$ -

     
    - -

    -With these definitions, we can now rewrite our \( 2\times 2 \) -correaltion/covariance matrix in terms of a moe general design/feature -matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \) -covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \) - -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -$$ -

     
    - -and the correlation matrix -

     
    -$$ -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -$$ -

     
    -

    - - -
    -

    Covariance Matrix Examples

    - -

    -The Numpy function np.cov calculates the covariance elements using -the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have -the exact mean values. The following simple function uses the -np.vstack function which takes each vector of dimension \( 1\times n \) -and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) - -

     
    -$$ -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -$$ -

     
    - -

    -which in turn is converted into into the \( 2\times 2 \) covariance matrix -\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function. +

    The code for solving the ODE

    -

    # Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-
    -
    - - -
    -

    Correlation Matrix

    - -

    -The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). - -

    - - -

    import numpy as np
-n = 100
-# define two vectors                                                                                           
-x = np.random.random(size=n)
-y = 4+3*x+np.random.normal(size=n)
-#scaling the x and y vectors                                                                                   
-x = x - np.mean(x)
-y = y - np.mean(y)
-variance_x = np.sum(x@x)/n
-variance_y = np.sum(y@y)/n
-print(variance_x)
-print(variance_y)
-cov_xy = np.sum(x@y)/n
-cov_xx = np.sum(x@x)/n
-cov_yy = np.sum(y@y)/n
-C = np.zeros((2,2))
-C[0,0]= cov_xx/variance_x
-C[1,1]= cov_yy/variance_y
-C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
-C[1,0]= C[0,1]
-print(C)
-
    -

    -We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix. - -

    -The above procedure with numpy can be made more compact if we use pandas. -

    - - -
    -

    Correlation Matrix with Pandas

    - -

    -We whow here how we can set up the correlation matrix using pandas, as done in this simple code -

    - - -

    import numpy as np
-import pandas as pd
-n = 10
-x = np.random.normal(size=n)
-x = x - np.mean(x)
-y = 4+3*x+np.random.normal(size=n)
-y = y - np.mean(y)
-X = (np.vstack((x, y))).T
-print(X)
-Xpd = pd.DataFrame(X)
-print(Xpd)
-correlation_matrix = Xpd.corr()
-print(correlation_matrix)
-
    -

    -We expand this model to the Franke function discussed above. -

    - - -
    -

    Correlation Matrix with Pandas and the Franke function

    - -

    - - -

    # Common imports
-import numpy as np
-import pandas as pd
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 4
-N = 100
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-
-Xpd = pd.DataFrame(X)
-# subtract the mean values and set up the covariance matrix
-Xpd = Xpd - Xpd.mean()
-covariance_matrix = Xpd.cov()
-print(covariance_matrix)
-
    -

    -We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree \( n \)). - -

    -This means that the variance for these elements will be zero and will -cause problems when we set up the correlation matrix. We can simply -drop these elements and construct a correlation -matrix without these elements. -

    - - -
    -

    Rewriting the Covariance and/or Correlation Matrix

    - -

    -We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. -$$ -

     
    - -

    -To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \) -

     
    -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -$$ -

     
    - -

    -If we then compute the expectation value -

     
    -$$ -\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -$$ -

     
    - -which is just -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -$$ -

     
    - -where we wrote

     
    -$$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ -

     
    to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \). - -

    -It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). -

    - - -
    -

    Towards the PCA theorem

    - -

    -We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. -$$ -

     
    - -Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \). -These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \). - -

    -Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \). - -

    -That is we have -

     
    -$$ -\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, -$$ -

     
    - -since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S}^T \) from the left we have -

     
    -$$ -\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, -$$ -

     
    - -and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that - -

     
    -$$ -\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i. -$$ -

     
    - -

    -In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is -\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \). - -

    -The eigenvalues tell us then how much we need to stretch the -corresponding eigenvectors. Dimensions with large eigenvalues have -thus large variations (large variance) and define therefore useful -dimensions. The data points are more spread out in the direction of -these eigenvectors. Smaller eigenvalues mean on the other hand that -the corresponding eigenvectors are shrunk accordingly and the data -points are tightly bunched together and there is not much variation in -these specific directions. Hopefully then we could leave it out -dimensions where the eigenvalues are very small. If \( p \) is very large, -we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) -features/predictors. -

    - - -
    -

    The Algorithm before theorem

    - -

    -Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. - -

    -

     
    -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ -

     
    - - -

    -
    - - -
    -

    Writing our own PCA code

    - -

    -We will use a simple example first with two-dimensional data -drawn from a multivariate normal distribution with the following mean and covariance matrix: -

     
    -$$ -\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\ -2 & 2 -\end{bmatrix} -$$ -

     
    - -Note that the mean refers to each column of data. -We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from -this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \). - -

    -The following Python code aids in setting up the data and writing out the design matrix. -Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \). -

    - - -

    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-n = 10000
-mean = (-1, 2)
-cov = [[4, 2], [2, 2]]
-X = np.random.multivariate_normal(mean, cov, n)
-
    -

    -Now we are going to implement the PCA algorithm. We will break it down into various substeps. - -

    Compute the sample mean and center the data

    - -

    -The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is -

     
    -$$ -\mu_n = \frac{1}{n} \sum_{i=1}^n x_i -$$ -

     
    - -and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form -

     
    -$$ -\bar{x}_i = x_i - \mu_n. -$$ -

     
    - -When you are done with these steps, print out \( \mu_n \) to verify it is -close to \( \mu \) and plot your mean centered data to verify it is -centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above. -The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. -

    - - -

    df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-
    -

    -Alternatively, we could use the functions we discussed -earlier for scaling the data set. That is, we could have used the -StandardScaler function in Scikit-Learn, a function which ensures -that for each feature/predictor we study the mean value is zero and -the variance is one (every column in the design/feature matrix). You -would then not get the same results, since we divide by the -variance. The diagonal covariance matrix elements will then be one, -while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our -specific case. - -

    Compute the sample covariance

    - -

    -Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation -

     
    -$$ -\begin{equation*} -\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n) -\end{equation*} -$$ -

     
    - -where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \). -We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows -

    - - -

    print(df.cov())
-print(np.cov(X_centered.T))
-
    -

    -Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. -Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. -

    - - -

    # extract the relevant columns from the centered design matrix of dim n x 2
-x = X_centered[:,0]
-y = X_centered[:,1]
-Cov = np.zeros((2,2))
-Cov[0,1] = np.sum(x.T@y)/(n-1.0)
-Cov[0,0] = np.sum(x.T@x)/(n-1.0)
-Cov[1,1] = np.sum(y.T@y)/(n-1.0)
-Cov[1,0]= Cov[0,1]
-print("Centered covariance using own code")
-print(Cov)
-plt.plot(x, y, 'x')
-plt.axis('equal')
-plt.show()
-
    -

    -Depending on the number of points \( n \), we will get results that are close to the covariance values defined above. -The plot shows how the data are clustered around a line with slope close to one. Is this expected? - -

    Diagonalize the sample covariance matrix to obtain the principal components

    - -

    -Now we are ready to solve for the principal components! To do so we -diagonalize the sample covariance matrix \( \Sigma \). We can use the -function np.linalg.eig to do so. It will return the eigenvalues and -eigenvectors of \( \Sigma \). Once we have these we can perform the -following tasks: - -

    -

     
    -$$ -\begin{equation*} -x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0 -\end{equation*} -$$ -

     
    - -where \( v_0 \) is the first principal component. - -

    -Collecting all these steps we can write our own PCA function and -compare this with the functionality included in Scikit-Learn. - -

    -The code here outlines some of the elements we could include in the -analysis. Feel free to extend upon this in order to address the above -questions. - -

    - - -

    # diagonalize and obtain eigenvalues, not necessarily sorted
-EigValues, EigVectors = np.linalg.eig(Cov)
-# sort eigenvectors and eigenvalues
-#permute = EigValues.argsort()
-#EigValues = EigValues[permute]
-#EigVectors = EigVectors[:,permute]
-print("Eigenvalues of Covariance matrix")
-for i in range(2):
-    print(EigValues[i])
-FirstEigvector = EigVectors[:,0]
-SecondEigvector = EigVectors[:,1]
-print("First eigenvector")
-print(FirstEigvector)
-print("Second eigenvector")
-print(SecondEigvector)
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2Dsl = pca.fit_transform(X)
-print("Eigenvector of largest eigenvalue")
-print(pca.components_.T[:, 0])
-
    -

    -This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? -

    - - -
    -

    Classical PCA Theorem

    - -

    -We assume now that we have a design matrix \( \boldsymbol{X} \) which has been -centered as discussed above. For the sake of simplicity we skip the -overline symbol. The matrix is defined in terms of the various column -vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension -\( \boldsymbol{x}\in {\mathbb{R}}^{n} \). - -

    -We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \). We define the reconstruction error (which is similar to the mean squared error we have seen before) as -

     
    -$$ -J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2, -$$ -

     
    - -with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix -\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \). When doing PCA we want to reduce this dimensionality. - -

    -The PCA theorem states that minimizing the above reconstruction error -corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which -diagonalizes the empirical covariance(correlation) matrix. The optimal -low-dimensional encoding of the data is then given by a set of vectors -\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the -orthogonal projection of the data onto the columns spanned by the -eigenvectors of the covariance(correlations matrix). - -

    -The proof which follows will be updated by mid January 2020. -

    - - -
    -

    Proof of the PCA Theorem

    - -

    -To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as -

     
    -$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), -$$ -

     
    - -which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as -

     
    -$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). -$$ -

     
    - -Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that -

     
    -$$ -z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i, -$$ -

     
    - -where the vectors on the rhs are known. -

    - - -
    -

    PCA Proof continued

    - -

    -We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write -

     
    -$$ -J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2. -$$ -

     
    - -

    -We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by -

     
    -$$ -\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2, -$$ -

     
    - -since the expectation value of -

     
    -$$ -\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0, -$$ -

     
    - -where we have used the fact that our data are centered. - -

    -Recalling our definition of the covariance as -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T], -$$ -

     
    - -we have thus that -

     
    -$$ -\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0. -$$ -

     
    - -

    -We are almost there, we have obtained a relation between minimizing -the reconstruction error and the variance and the covariance -matrix. Minimizing the error is equivalent to maximizing the variance -of the projected data. -

    - - -
    -

    The final step

    - -

    -We could trivially maximize the variance of the projection (and -thereby minimize the error in the reconstruction function) by letting -the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we -want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a -Lagrange multiplier we can then in turn maximize - -

     
    -$$ -J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0). -$$ -

     
    - -Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain - -

     
    -$$ -\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0, -$$ -

     
    - -meaning that -

     
    -$$ -\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0. -$$ -

     
    - -The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is -

     
    -$$ -\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. -$$ -

     
    - -

    -If we want to maximize the variance (minimize the construction error) -we simply pick the eigenvector of the covariance matrix with the -largest eigenvalue. This establishes the link between the minimization -of the reconstruction function \( J \) in terms of an orthogonal matrix -and the maximization of the variance and thereby the covariance of our -observations encoded in the design/feature matrix \( \boldsymbol{X} \). - -

    -The proof -for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be -established by applying the above arguments and using the fact that -our basis of eigenvectors is orthogonal, see Murphy chapter -12.2. The -discussion in chapter 12.2 of Murphy's text has also a nice link with -the Singular Value Decomposition theorem. For categorical data, see -chapter 12.4 and discussion therein. - -

    -Additional part of the proof for the other eigenvectors will be added by mid January 2020. -

    - - -
    -

    Geometric Interpretation and link with Singular Value Decomposition

    - -

    -This material will be added by mid January 2020. -

    - - -
    -

    Principal Component Analysis

    - -

    -Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. -First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it. - -

    -The following Python code uses NumPy’s svd() function to obtain all the principal components of the -training set, then extracts the first two principal components. First we center the data using either pandas or our own code -

    - - -

    import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 vanilla matrix 
-rows = 10
-cols = 5
-X = np.random.randn(rows,cols)
-df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-display(df)
-
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-# Then check the difference between pandas and our own set up
-print(X_centered-df)
-#Now we do an SVD
-U, s, V = np.linalg.svd(X_centered)
-c1 = V.T[:, 0]
-c2 = V.T[:, 1]
-W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-print(X2D)
-
    -

    -PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering -the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t -forget to center the data first. - -

    -Once you have identified all the principal components, you can reduce the dimensionality of the dataset -down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components. -Selecting this hyperplane ensures that the projection will preserve as much variance as possible. -

    - - -

    W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-
    -
    - - -
    -

    PCA and scikit-learn

    - -

    -Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The -following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note -that it automatically takes care of centering the data): -

    - - -

    #thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D = pca.fit_transform(X)
-print(X2D)
-
    -

    -After fitting the PCA transformer to the dataset, you can access the principal components using the -components variable (note that it contains the PCs as horizontal vectors, so, for example, the first -principal component is equal to -

    - - -

    pca.components_.T[:, 0].
-
    -

    -Another very useful piece of information is the explained variance ratio of each principal component, -available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s -variance that lies along the axis of each principal component. -

    - - -
    -

    Back to the Cancer Data

    -We can now repeat the above but applied to real data, in this case our breast cancer data. -Here we compute performance scores on the training data using logistic regression. -

    - - -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-
-logreg = LogisticRegression()
-logreg.fit(X_train, y_train)
-print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
-# We scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Then perform again a log reg fit
-logreg.fit(X_train_scaled, y_train)
-print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D_train = pca.fit_transform(X_train_scaled)
-# and finally compute the log reg fit and the score on the training data	
-logreg.fit(X2D_train,y_train)
-print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
-
    -

    -We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. -

    - - -
    -

    More on the PCA

    - -

    -Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to -choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). -Unless, of course, you are reducing dimensionality for data visualization — in that case you will -generally want to reduce the dimensionality down to 2 or 3. -The following code computes PCA without reducing dimensionality, then computes the minimum number -of dimensions required to preserve 95% of the training set’s variance: -

    - - -

    pca = PCA()
-pca.fit(X)
-cumsum = np.cumsum(pca.explained_variance_ratio_)
-d = np.argmax(cumsum >= 0.95) + 1
-
    -

    -You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead -of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be -a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve: -

    - - -

    pca = PCA(n_components=0.95)
-X_reduced = pca.fit_transform(X)
-
    -
    - - -
    -

    Incremental PCA

    - -

    -One problem with the preceding implementation of PCA is that it requires the whole training set to fit in -memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have -been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch -at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new -instances arrive). -

    - - -
    -

    Randomized PCA

    - -

    -Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic -algorithm that quickly finds an approximation of the first d principal components. Its computational -complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the -previous algorithms when \( d \) is much smaller than \( n \). -

    - - -
    -

    Kernel PCA

    -
    - -

    -The kernel trick is a mathematical technique that implicitly maps instances into a -very high-dimensional space (called the feature space), enabling nonlinear classification and regression -with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature -space corresponds to a complex nonlinear decision boundary in the original space. -It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear -projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at -preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a -twisted manifold. -For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an -

    - - -

    from sklearn.decomposition import KernelPCA
-rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
-X_reduced = rbf_pca.fit_transform(X)
-
    - -
    -
    - - -
    -

    LLE

    - -

    -Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction -(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous -algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its -closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where -these local relationships are best preserved (more details shortly). -

    - - -
    -

    Other techniques

    - -

    -There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn. - -

    -Here are some of the most popular: - -

    -
    - - -
    -

    The network with one input, hidden, and output layer

    - -

    - - -

    # Autograd will be used for later, so the numpy wrapper for Autograd must be imported
-import autograd.numpy as np
+
    import autograd.numpy as np
 from autograd import grad, elementwise_grad
 import autograd.numpy.random as npr
 from matplotlib import pyplot as plt
@@ -1979,7 +872,7 @@ Here are some of the most popular:
 
 
 
    
-
    The network with one input layer, specified number of hidden layers, and one output layer  output layer 
    
+
    The network with one input layer, specified number of hidden layers, and one output layer 
    
 
 
    
 It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.
@@ -2153,7 +1046,7 @@ The number of neurons within each hidden layer are given as a list of integers i
 
 
 
    
-
    Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme 
    
+
    Example: Population growth 
    
 
 
    
 A logistic model of population growth assumes that a population converges toward an equilibrium.
@@ -2180,7 +1073,7 @@ For comparison, the forward Euler method has been implemented in order to see ho
 
 
 
    
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 
 
    
 Here, we will model a population \( g(t) \) in an environment having carrying capacity \( A \).
@@ -2203,7 +1096,9 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
 
 
 
    
-
    The trial solution 
    
+
    The trial solution 
    
+
+
    
 We will get a slightly different trial solution, as the boundary conditions are different
 compared to the case for exponential decay.
 
@@ -2231,7 +1126,7 @@ $$
 
 
 
    
-
    The program using Autograd 
    
+
    The program using Autograd 
    
 
 
    
 The network will be the similar as for the exponential decay example, but with some small modifications for our problem.
@@ -2407,7 +1302,7 @@ The network will be the similar as for the exponential decay example, but with s
 
 
 
    
-
    Using forward Euler to solve the ODE 
    
+
    Using forward Euler to solve the ODE 
    
 
 
    
 A straight-forward way of solving an ODE numerically, is to use Euler's method.
@@ -2543,180 +1438,11 @@ extending the program that uses the network using Autograd:
 
    
 
    
 Running the program gives
-
-
    
-
-
-
    Max absolute difference between Euler method and analytical: 0.011225
-Max absolute difference between deep neural network and analytical: 0.00424909
-
    -

    Using TensorFlow to model logistic population growth

    - -

    -TensorFlow is a library widely used in the machine learning community. -A neural network can be set up in a flexible manner, where various optimization algorithms are implemented and different types of networks can be used, making it easier to experiment on solving differential equations using neural networks. -

    - - -
    -

    The general program flow in TensorFlow

    - -

    -Usually, a program in TensorFlow is divided into two parts; the construction phase and the execution phase. -In the construction phase, the computational graph that TensorFlow uses to perform its calculations are set up. -In the execution phase, TensorFlow evaluates any procedure that was defined in the construction phase. - -

    Program flow in TensorFlow - Construction phase

    - -

    -Here, the architecture for the neural network will be set up, along with the cost function and an optimizer class used during training of the network. -Note that TensorFlow uses a different convention for the weighting done in each neuron in each layer within the network than in the implementation using Autograd. -The matrix-vector multiplication between the input from the previous layer and the weighting at the neuron at current layer in the program using Autograd, is the transpose of the convention used in TensorFlow. But it will not affect that much our construction, as TensorFlow takes care of most of the computations. The only thing we have to be aware of, is how the dimensions are for our inputs. - -

    Program flow in TensorFlow - Execution phase

    - -

    -The computation graph has been defined, and is ready to be evaluated. -In order to get access to the graph, it has to be initialized and be runned within a Session. - -

    The full program modeling logistic population growth using TensorFlow

    - -

    - - -

    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-# Set a seed to ensure getting the same results from every run
-tf.set_random_seed(4155)
-
-Nt = 10
-T = 1
-t = np.linspace(0,T, Nt)
-
-## The construction phase
-
-# Convert the values the trial solution is evaluated at to a tensor.
-t_tf = tf.convert_to_tensor(t.reshape(-1,1),dtype=tf.float64)
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(t.shape)),shape=(-1,1))
-
-# Define the parameters of the equation
-alpha = tf.constant(2.,dtype=tf.float64)
-A = tf.constant(1.,dtype=tf.float64)
-g0 = tf.constant(1.2,dtype=tf.float64)
-
-num_iter = 100000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [100,50,25]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = t_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = g0 + t_tf*dnn_output
-    d_g_trial = tf.gradients(g_trial,t_tf)
-
-    func = alpha*g_trial*(A - g_trial)
-    cost = tf.losses.mean_squared_error(zeros, d_g_trial[0] - func)
-
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-# Set up a referance to the result from the neural network:
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The training of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves for each iteration:
-        #if i % 1000 == 0:
-        #    print(cost.eval())
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-# Compare with analytical solution
-def get_parameters():
-    alpha = 2
-    A = 1
-    g0 = 1.2
-    return alpha, A, g0
-
-def g_analytic(t):
-    alpha,A, g0 = get_parameters()
-    return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
-
-g_analytical = g_analytic(t)
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(t, g_dnn_tf)
-plt.plot(t, g_analytical)
-
-plt.legend(['dnn, tensorflow', 'exact'])
-plt.xlabel('Time t')
-plt.ylabel('g(t)')
-
-plt.show()
-
    -
    - - -
    -

    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow

    +

    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow

    The Poisson equation for \( g(x) \) in one dimension is @@ -2747,17 +1473,11 @@ $$ This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used. The results from the networks can then be compared to the analytical solution. In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks. - -

    -There exists many different optimization methods implemented in TensorFlow. -In the examples program using TensorFlow, it could also be of interest to see how -the choice of an optimization method affects our results. -In the TensorFlow documentation about optimizers, a list over available optimization methods are shown.

    -

    The specific equation to solve for

    +

    The specific equation to solve for

    Here, the function \( g(x) \) to solve for follows the equation @@ -2803,7 +1523,7 @@ $$

    -

    Solving the equation using Autograd

    +

    Solving the equation using Autograd

    @@ -2963,7 +1683,7 @@ $$

    -

    Comparing with a numerical scheme

    +

    Comparing with a numerical scheme

    The Poisson equation is possible to solve using Taylor series to approximate the second derivative. @@ -3279,170 +1999,9 @@ The max absolute difference between the analytical solution and numerical scheme

    -

    Using gradient descent in TensorFlow to solve Poisson equation

    -The program follows the similar idea as for the logistic population model. +

    Partial Differential Equations

    -What has changed, is what the cost function minimizes and the trial solution. - -

    - - -

    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-## Construction phase
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-tf.set_random_seed(4155)
-
-# Convert the values the trial solution is evaluated at to a tensor.
-Nx = 10
-x = np.linspace(0,1, Nx)
-x_tf = tf.convert_to_tensor(x.reshape(-1,1),dtype=tf.float64)
-
-
-num_iter = 10000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [20,10]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = x_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = x_tf*(1-x_tf)*dnn_output
-    d_g_trial = tf.gradients(g_trial,x_tf)
-    d2_g_trial = tf.gradients(d_g_trial,x_tf)
-
-    right_side = (3*x_tf + x_tf**2)*tf.exp(x_tf)
-
-    err = tf.square( -d2_g_trial[0] - right_side)
-    cost = tf.reduce_sum(err, name = 'cost')
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The traning of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-    writer = tf.summary.FileWriter("./output", sess.graph)
-    writer.close()
-
-# Evaluate the analytical function to compare with
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-g_analytical = g_analytic(x)
-
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(x, g_dnn_tf)
-plt.plot(x, g_analytical)
-
-plt.legend(['dnn, tensorflow','exact'])
-plt.xlabel('x')
-plt.ylabel('g(x)')
-
-plt.show()
-
    -
    - - -
    -

    Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation

    - -

    -We can see that the results using GradientDescentOptimizer seems to converge towards the analytical solution. -But there exists many other methods for optimization also, see the TensorFlow documentation on Optimizers. - -

    -Adam is an optimization algorithm that changes its learning rates accordingly to the function it tries to minimize for every iteration. -The algorithm is described in this paper. -How much an optimization algorithm has to say for the network to converge, could be interesting to experiment with. -Using the same TensorFlow program as before, the only change to do, is to replace the variable optimizer. - -

    -In the program that uses TensorFlow to solve for the Poisson equation, change the line - -

    - - -

    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-
    -

    -to - -

    - - -

    optimizer = tf.train.AdamOptimizer(learning_rate)
-
    -

    -The program using the Adam optimizer with a different initial learning rate yields indeed an interesting result: -

    - - -

    Max absolute difference between the analytical solution and solution from TensorFlow DNN: 7.11243e-05
-
    -
    - - -
    -

    Partial Differential Equations

    A partial differential equation (PDE) has a solution here the function is defined by multiple variables. The equation may involve all kinds of combinations of which variables the function is differentiated with respect to. @@ -3459,6 +2018,11 @@ $$

    where \( f \) is an expression involving all kinds of possible mixed derivatives of \( g(x_1,\dots,x_N) \) up to an order \( n \). In order for the solution to be unique, some additional conditions must also be given. +

    + + +
    +

    Type of problem

    The problem our network must solve for, is similar to the ODE case. @@ -3479,13 +2043,23 @@ The neural network \( N(x_1,\dots,x_N,P) \) has weights and biases described by

    The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensure that the output of \( N(x_1,\dots,x_N,P) \) is zero when \( g_t(x_1,\dots,x_N) \) is evaluated at the values of \( x_1,\dots,x_N \) where the given conditions must be satisfied. The function \( h_1(x_1,\dots,x_N) \) should alone make \( g_t(x_1,\dots,x_N) \) satisfy the conditions. +

    + + +
    +

    Network requirements

    -The network tries then the minimize the cost function following the same ideas as described for the ODE case, but now with more than one variables to consider. -The concept still remains the same; find a set of parameters \( P \) such that the expression \( f \) in (17) is as close to zero as possible. +The network tries then the minimize the cost function following the +same ideas as described for the ODE case, but now with more than one +variables to consider. The concept still remains the same; find a set +of parameters \( P \) such that the expression \( f \) in (17) is as +close to zero as possible.

    -As for the ODE case, the cost function is the mean squared error that the network must try to minimize. The cost function for the network to minimize is +As for the ODE case, the cost function is the mean squared error that +the network must try to minimize. The cost function for the network to +minimize is

     
    $$ @@ -3494,6 +2068,11 @@ c\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{ \end{equation*} $$

     
    +

    + + +
    +

    More details

    If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following: @@ -3518,7 +2097,7 @@ $$

    -

    Example: The diffusion equation

    +

    Example: The diffusion equation

    In one spatial dimension, the equation reads @@ -3547,7 +2126,7 @@ with \( u(x) \) being some given function.

    -

    Defining the problem

    +

    Defining the problem

    For this case, we want to find \( g(x,t) \) such that @@ -3583,13 +2162,16 @@ First, we will look into how Autograd could be used in a network tailored to sol

    -

    Setting up the network using Autograd

    +

    Setting up the network using Autograd

    -The only change to do here, is to extend our network such that functions of multiple parameters are correctly handled. -In this case we have two variables in our function to solve for, that is time \( t \) and position \( x \). -The variables will be represented by a one-dimensional array in the program. -The program will evaluate the network at each possible pair \( (x,t) \), given an array for the desired \( x \)-values and \( t \)-values to approximate the solution at. +The only change to do here, is to extend our network such that +functions of multiple parameters are correctly handled. In this case +we have two variables in our function to solve for, that is time \( t \) +and position \( x \). The variables will be represented by a +one-dimensional array in the program. The program will evaluate the +network at each possible pair \( (x,t) \), given an array for the desired +\( x \)-values and \( t \)-values to approximate the solution at.

    @@ -3643,8 +2225,13 @@ The program will evaluate the network at each possible pair \( (x,t) \), given a

    -

    Setting up the network using Autograd; The trial solution

    -The cost function must then iterate through the given arrays containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep neural network and the trial solution is evaluated at, and then finds the Jacobian of the trial solution. +

    Setting up the network using Autograd; The trial solution

    + +

    +The cost function must then iterate through the given arrays +containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep +neural network and the trial solution is evaluated at, and then finds +the Jacobian of the trial solution.

    A possible trial solution for this PDE is @@ -3667,18 +2254,33 @@ h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t $$

     
    since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \). +

    + + +
    +

    Why the jacobian?

    -The Jacobian is used because the program must find the derivative of the trial solution with respect to \( x \) and \( t \). +The Jacobian is used because the program must find the derivative of +the trial solution with respect to \( x \) and \( t \).

    -This gives the necessity of computing the Jacobian matrix, as we want to evaluate the gradient with respect to \( x \) and \( t \) (note that the Jacobian of a scalar-valued multivariate function is simply its gradient). +This gives the necessity of computing the Jacobian matrix, as we want +to evaluate the gradient with respect to \( x \) and \( t \) (note that the +Jacobian of a scalar-valued multivariate function is simply its +gradient).

    -In Autograd, the differentiation is by default done with respect to the first input argument of your Python function. Since the points is an array representing \( x \) and \( t \), the Jacobian is calculated using the values of \( x \) and \( t \). +In Autograd, the differentiation is by default done with respect to +the first input argument of your Python function. Since the points is +an array representing \( x \) and \( t \), the Jacobian is calculated using +the values of \( x \) and \( t \).

    -To find the second derivative with respect to \( x \) and \( t \), the Jacobian can be found for the second time. The result is a Hessian matrix, which is the matrix containing all the possible second order mixed derivatives of \( g(x,t) \). +To find the second derivative with respect to \( x \) and \( t \), the +Jacobian can be found for the second time. The result is a Hessian +matrix, which is the matrix containing all the possible second order +mixed derivatives of \( g(x,t) \).

    @@ -3724,7 +2326,9 @@ To find the second derivative with respect to \( x \) and \( t \), the Jacobian

    -

    Setting up the network using Autograd; The full program

    +

    Setting up the network using Autograd; The full program

    + +

    Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.

    @@ -3741,12 +2345,9 @@ A possible way to implement a neural network solving the PDE, is given below. Be aware, though, that it is fairly slow for the parameters used. A better result is possible, but requires more iterations, and thus longer time to complete. -

    -Using only 20 neurons in one hidden layer, the program managed to make the trial solution have the maximum absolute error of 0.0075. The execution time, however, was approximately one day and 14 hours on a computer having Intel i7-7560U 2.4 GHz CPU. -

    Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE. -Using TensorFlow in the next example sovling the wave equation, has a much better execution time. +Using TensorFlow results in a much better execution time. Try it!

    @@ -3979,7 +2580,7 @@ Using TensorFlow in the next example sovling the wave equation, has a much bette

    -

    Example: Solving the wave equation using Autograd and TensorFlow

    +

    Example: Solving the wave equation with Neural Networks

    The wave equation is @@ -4012,7 +2613,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati

    -

    The problem to solve for

    +

    The problem to solve for

    The wave equation to solve for, is @@ -4046,7 +2647,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi

    -

    The trial solution

    +

    The trial solution

    Setting up the network is done in similar matter as for the example of solving the diffusion equation. The only things we have to change, is the trial solution such that it satisfies the conditions from (20) and the cost function. @@ -4074,7 +2675,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
    -

    The analytical solution

    +

    The analytical solution

    The analytical solution for our specific problem, is @@ -4088,7 +2689,7 @@ $$

    -

    Solving the wave equation - the full program using Autograd

    +

    Solving the wave equation - the full program using Autograd

    @@ -4318,196 +2919,1124 @@ $$

    -

    Solving the wave equation - the full program using TensorFlow

    -As the program using Autograd is fairly slow, one could hope that using TensorFlow -could make a naive implementation faster, and more numerically robust. - -

    -In addition, having TensorFlow at hand, it could be easier to experiment with different -optimization algorithms, and other constructions of the network. - -

    -The following program solves the given wave equation much faster, - -

    - - -

    import tensorflow as tf
-import numpy as np
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-Nx = 10
-x_np = np.linspace(0,1,Nx)
-
-Nt = 10
-t_np = np.linspace(0,1,Nt)
-
-X,T = np.meshgrid(x_np, t_np)
-
-x = X.ravel()
-t = T.ravel()
-
-## The construction phase
-
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(x.shape)),shape=(-1,1))
-x = tf.reshape(tf.convert_to_tensor(x),shape=(-1,1))
-t = tf.reshape(tf.convert_to_tensor(t),shape=(-1,1))
-
-points = tf.concat([x,t],1)
-
-num_iter = 100000
-num_hidden_neurons = [90]
-
-X = tf.convert_to_tensor(X)
-T = tf.convert_to_tensor(T)
-
-
-with tf.variable_scope('dnn'):
-    num_hidden_layers = np.size(num_hidden_neurons)
-
-    previous_layer = points
-
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l],activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    dnn_output = tf.layers.dense(previous_layer, 1)
-
-
-def u(x):
-    return tf.sin(np.pi*x)
-
-def v(x):
-    return -np.pi*tf.sin(np.pi*x)
-
-with tf.name_scope('loss'):
-    g_trial = (1 - t**2)*u(x) + t*v(x) + x*(1-x)*t**2*dnn_output
-
-    g_trial_d2t =  tf.gradients(tf.gradients(g_trial,t),t)
-    g_trial_d2x = tf.gradients(tf.gradients(g_trial,x),x)
-
-    loss = tf.losses.mean_squared_error(zeros, g_trial_d2t[0] - g_trial_d2x[0])
-
-learning_rate = 0.01
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(loss)
-
-init = tf.global_variables_initializer()
-
-g_analytic = tf.sin(np.pi*x)*tf.cos(np.pi*t) - tf.sin(np.pi*x)*tf.sin(np.pi*t)
-g_dnn = None
-
-## The execution phase
-with tf.Session() as sess:
-    init.run()
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves during training
-        #if i % 100 == 0:
-        #    print(loss.eval())
-
-    g_analytic = g_analytic.eval()
-    g_dnn = g_trial.eval()
-
-
-## Compare with the analutical solution
-diff = np.abs(g_analytic - g_dnn)
-print('Max absolute difference between analytical solution and TensorFlow DNN = ',np.max(diff))
-
-G_analytic = g_analytic.reshape((Nt,Nx))
-G_dnn = g_dnn.reshape((Nt,Nx))
-
-diff = np.abs(G_analytic - G_dnn)
-
-# Plot the results
-
-X,T = np.meshgrid(x_np, t_np)
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-s = ax.plot_surface(X,T,G_dnn,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Analytical solution')
-s = ax.plot_surface(X,T,G_analytic,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Difference')
-s = ax.plot_surface(X,T,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-## Take some 3D slices
-
-indx1 = 0
-indx2 = int(Nt/2)
-indx3 = Nt-1
-
-t1 = t_np[indx1]
-t2 = t_np[indx2]
-t3 = t_np[indx3]
-
-# Slice the results from the DNN
-res1 = G_dnn[indx1,:]
-res2 = G_dnn[indx2,:]
-res3 = G_dnn[indx3,:]
-
-# Slice the analytical results
-res_analytical1 = G_analytic[indx1,:]
-res_analytical2 = G_analytic[indx2,:]
-res_analytical3 = G_analytic[indx3,:]
-
-# Plot the slices
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t1)
-plt.plot(x_np, res1)
-plt.plot(x_np,res_analytical1)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t2)
-plt.plot(x_np, res2)
-plt.plot(x_np,res_analytical2)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t3)
-plt.plot(x_np, res3)
-plt.plot(x_np,res_analytical3)
-plt.legend(['dnn','analytical'])
-
-plt.show()
-
    -

    -The program manages to find a solution having max absolute difference to the analytical -at approximately 0.0059, by just using some minutes! -It was found, by some testing, that one hidden layer with 90 neurons actually performed well. -

    - - -
    -

    Resources

    +

    Resources on differential equations and deep learning

    1. Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al

    2. Neural networks for solving differential equations by A. Honchar

    3. Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener
      4. -

    4. Introduction to Partial Differential Equations by A. Tveitio, R. Winther
      5. +

    5. Introduction to Partial Differential Equations by A. Tveito, R. Winther
    +
    +

    Friday, Principal Component Analysis

    +
    + + +
    +

    Basic ideas of the Principal Component Analysis (PCA)

    + +

    +The principal component analysis deals with the problem of fitting a +low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than +the totaldimension \( D \) of the problem at hand (our data +set). Mathematically it can be formulated as a statistical problem or +a geometric problem. In our discussion of the theorem for the +classical PCA, we will stay with a statistical approach. This is also +what set the scene historically which for the PCA. + +

    +We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) + +

      +

    • Each data point is determined by \( p \) extrinsic (measurement) variables
      • +

    • We may want to ask the following question: Are there fewer intrinsic variables (say \( d < < p \)) that still approximately describe the data?
      • +

    • If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.
      • +
    +
    + + +
    +

    Introducing the Covariance and Correlation functions

    + +

    +Before we discuss the PCA theorem, we need to remind ourselves about +the definition of the covariance and the correlation function. These are quantities + +

    +Suppose we have defined two vectors +\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ + \end{bmatrix}, +$$ +

     
    + +where for example +

     
    +$$ +\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). +$$ +

     
    + +With this definition and recalling that the variance is defined as +

     
    +$$ +\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, +$$ +

     
    + +we can rewrite the covariance matrix as +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ + \end{bmatrix}. +$$ +

     
    + +

    +The covariance takes values between zero and infinity and may thus +lead to problems with loss of numerical precision for particularly +large values. It is common to scale the covariance matrix by +introducing instead the correlation matrix defined via the so-called +correlation function + +

     
    +$$ +\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. +$$ +

     
    + +

    +The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] +\in [-1,1] \). This avoids eventual problems with too large values. We +can then define the correlation matrix for the two vectors \( \boldsymbol{x} \) +and \( \boldsymbol{y} \) as + +

     
    +$$ +\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ + \end{bmatrix}, +$$ +

     
    + +

    +In the above example this is the function we constructed using pandas. +

    + + +
    +

    Correlation Function and Design/Feature Matrix

    + +

    +In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression +we defined the design/feature matrix \( \boldsymbol{X} \) as + +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} +x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ +x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ +x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ +\dots & \dots & \dots & \dots \dots & \dots \\ +x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ +x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ +\end{bmatrix}, +$$ +

     
    + +with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the +entries \( n \) being the row elements. +We can rewrite the design/feature matrix in terms of its column vectors as +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, +$$ +

     
    + +with a given vector +

     
    +$$ +\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. +$$ +

     
    + +

    +With these definitions, we can now rewrite our \( 2\times 2 \) +correaltion/covariance matrix in terms of a moe general design/feature +matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \) +covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \) + +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} +\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ +\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ +\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ +\end{bmatrix}, +$$ +

     
    + +and the correlation matrix +

     
    +$$ +\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} +1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ +\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ +\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ +\end{bmatrix}, +$$ +

     
    +

    + + +
    +

    Covariance Matrix Examples

    + +

    +The Numpy function np.cov calculates the covariance elements using +the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have +the exact mean values. The following simple function uses the +np.vstack function which takes each vector of dimension \( 1\times n \) +and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) + +

     
    +$$ +\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ + x_1 & y_1 \\ + x_2 & y_2\\ + \dots & \dots \\ + x_{n-2} & y_{n-2}\\ + x_{n-1} & y_{n-1} & + \end{bmatrix}, +$$ +

     
    + +

    +which in turn is converted into into the \( 2\times 2 \) covariance matrix +\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy +function np.mean(x). We can also extract the eigenvalues of the +covariance matrix through the np.linalg.eig() function. + +

    + + +

    # Importing various packages
+import numpy as np
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
+
    +
    + + +
    +

    Correlation Matrix

    + +

    +The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). + +

    + + +

    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    +

    +We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix. + +

    +The above procedure with numpy can be made more compact if we use pandas. +

    + + +
    +

    Correlation Matrix with Pandas

    + +

    +We whow here how we can set up the correlation matrix using pandas, as done in this simple code +

    + + +

    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    +

    +We expand this model to the Franke function discussed above. +

    + + +
    +

    Correlation Matrix with Pandas and the Franke function

    + +

    + + +

    # Common imports
+import numpy as np
+import pandas as pd
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    +

    +We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree \( n \)). + +

    +This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements and construct a correlation +matrix without these elements. +

    + + +
    +

    Rewriting the Covariance and/or Correlation Matrix

    + +

    +We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. +$$ +

     
    + +

    +To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \) +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} +x_{00} & x_{01}\\ +x_{10} & x_{11}\\ +\end{bmatrix}=\begin{bmatrix} +\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ +\end{bmatrix}. +$$ +

     
    + +

    +If we then compute the expectation value +

     
    +$$ +\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix} +x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ +x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ +\end{bmatrix}, +$$ +

     
    + +which is just +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ + \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ + \end{bmatrix}, +$$ +

     
    + +where we wrote

     
    +$$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ +

     
    to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \). + +

    +It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). +

    + + +
    +

    Towards the PCA theorem

    + +

    +We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. +$$ +

     
    + +Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \). +These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \). + +

    +Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \). + +

    +That is we have +

     
    +$$ +\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, +$$ +

     
    + +since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S}^T \) from the left we have +

     
    +$$ +\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, +$$ +

     
    + +and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that + +

     
    +$$ +\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i. +$$ +

     
    + +

    +In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is +\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \). + +

    +The eigenvalues tell us then how much we need to stretch the +corresponding eigenvectors. Dimensions with large eigenvalues have +thus large variations (large variance) and define therefore useful +dimensions. The data points are more spread out in the direction of +these eigenvectors. Smaller eigenvalues mean on the other hand that +the corresponding eigenvectors are shrunk accordingly and the data +points are tightly bunched together and there is not much variation in +these specific directions. Hopefully then we could leave it out +dimensions where the eigenvalues are very small. If \( p \) is very large, +we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) +features/predictors. +

    + + +
    +

    The Algorithm before theorem

    + +

    +Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. + +

      +

    • Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.
      • +
    +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} +x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ +x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ +x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ +\dots & \dots & \dots & \dots \dots & \dots \\ +x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ +x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ +\end{bmatrix}, +$$ +

     
    + + +

      +

    • Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • +

    • Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).
      • +

    • Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • +

    • Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • +

    • Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • +
    +
    + + +
    +

    Writing our own PCA code

    + +

    +We will use a simple example first with two-dimensional data +drawn from a multivariate normal distribution with the following mean and covariance matrix: +

     
    +$$ +\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\ +2 & 2 +\end{bmatrix} +$$ +

     
    + +Note that the mean refers to each column of data. +We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from +this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \). + +

    +The following Python code aids in setting up the data and writing out the design matrix. +Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \). +

    + + +

    import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from IPython.display import display
+n = 10000
+mean = (-1, 2)
+cov = [[4, 2], [2, 2]]
+X = np.random.multivariate_normal(mean, cov, n)
+
    +

    +Now we are going to implement the PCA algorithm. We will break it down into various substeps. + +

    Compute the sample mean and center the data

    + +

    +The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is +

     
    +$$ +\mu_n = \frac{1}{n} \sum_{i=1}^n x_i +$$ +

     
    + +and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form +

     
    +$$ +\bar{x}_i = x_i - \mu_n. +$$ +

     
    + +When you are done with these steps, print out \( \mu_n \) to verify it is +close to \( \mu \) and plot your mean centered data to verify it is +centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above. +The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. +

    + + +

    df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+
    +

    +Alternatively, we could use the functions we discussed +earlier for scaling the data set. That is, we could have used the +StandardScaler function in Scikit-Learn, a function which ensures +that for each feature/predictor we study the mean value is zero and +the variance is one (every column in the design/feature matrix). You +would then not get the same results, since we divide by the +variance. The diagonal covariance matrix elements will then be one, +while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our +specific case. + +

    Compute the sample covariance

    + +

    +Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation +

     
    +$$ +\begin{equation*} +\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n) +\end{equation*} +$$ +

     
    + +where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \). +We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows +

    + + +

    print(df.cov())
+print(np.cov(X_centered.T))
+
    +

    +Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. +Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. +

    + + +

    # extract the relevant columns from the centered design matrix of dim n x 2
+x = X_centered[:,0]
+y = X_centered[:,1]
+Cov = np.zeros((2,2))
+Cov[0,1] = np.sum(x.T@y)/(n-1.0)
+Cov[0,0] = np.sum(x.T@x)/(n-1.0)
+Cov[1,1] = np.sum(y.T@y)/(n-1.0)
+Cov[1,0]= Cov[0,1]
+print("Centered covariance using own code")
+print(Cov)
+plt.plot(x, y, 'x')
+plt.axis('equal')
+plt.show()
+
    +

    +Depending on the number of points \( n \), we will get results that are close to the covariance values defined above. +The plot shows how the data are clustered around a line with slope close to one. Is this expected? + +

    Diagonalize the sample covariance matrix to obtain the principal components

    + +

    +Now we are ready to solve for the principal components! To do so we +diagonalize the sample covariance matrix \( \Sigma \). We can use the +function np.linalg.eig to do so. It will return the eigenvalues and +eigenvectors of \( \Sigma \). Once we have these we can perform the +following tasks: + +

      +

    • We compute the percentage of the total variance captured by the first principal component
      • +

    • We plot the mean centered data and lines along the first and second principal components
      • +

    • Then we project the mean centered data onto the first and second principal components, and plot the projected data.
      • +

    • Finally, we approximate the data as
      • +
    +

     
    +$$ +\begin{equation*} +x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0 +\end{equation*} +$$ +

     
    + +where \( v_0 \) is the first principal component. + +

    +Collecting all these steps we can write our own PCA function and +compare this with the functionality included in Scikit-Learn. + +

    +The code here outlines some of the elements we could include in the +analysis. Feel free to extend upon this in order to address the above +questions. + +

    + + +

    # diagonalize and obtain eigenvalues, not necessarily sorted
+EigValues, EigVectors = np.linalg.eig(Cov)
+# sort eigenvectors and eigenvalues
+#permute = EigValues.argsort()
+#EigValues = EigValues[permute]
+#EigVectors = EigVectors[:,permute]
+print("Eigenvalues of Covariance matrix")
+for i in range(2):
+    print(EigValues[i])
+FirstEigvector = EigVectors[:,0]
+SecondEigvector = EigVectors[:,1]
+print("First eigenvector")
+print(FirstEigvector)
+print("Second eigenvector")
+print(SecondEigvector)
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2Dsl = pca.fit_transform(X)
+print("Eigenvector of largest eigenvalue")
+print(pca.components_.T[:, 0])
+
    +

    +This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? +

    + + +
    +

    Classical PCA Theorem

    + +

    +We assume now that we have a design matrix \( \boldsymbol{X} \) which has been +centered as discussed above. For the sake of simplicity we skip the +overline symbol. The matrix is defined in terms of the various column +vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension +\( \boldsymbol{x}\in {\mathbb{R}}^{n} \). + +

    +We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \). We define the reconstruction error (which is similar to the mean squared error we have seen before) as +

     
    +$$ +J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2, +$$ +

     
    + +with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix +\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \). When doing PCA we want to reduce this dimensionality. + +

    +The PCA theorem states that minimizing the above reconstruction error +corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which +diagonalizes the empirical covariance(correlation) matrix. The optimal +low-dimensional encoding of the data is then given by a set of vectors +\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the +orthogonal projection of the data onto the columns spanned by the +eigenvectors of the covariance(correlations matrix). + +

    +The proof which follows will be updated by mid January 2020. +

    + + +
    +

    Proof of the PCA Theorem

    + +

    +To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as +

     
    +$$ +J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), +$$ +

     
    + +which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as +

     
    +$$ +J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). +$$ +

     
    + +Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that +

     
    +$$ +z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i, +$$ +

     
    + +where the vectors on the rhs are known. +

    + + +
    +

    PCA Proof continued

    + +

    +We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write +

     
    +$$ +J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2. +$$ +

     
    + +

    +We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by +

     
    +$$ +\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2, +$$ +

     
    + +since the expectation value of +

     
    +$$ +\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0, +$$ +

     
    + +where we have used the fact that our data are centered. + +

    +Recalling our definition of the covariance as +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T], +$$ +

     
    + +we have thus that +

     
    +$$ +\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0. +$$ +

     
    + +

    +We are almost there, we have obtained a relation between minimizing +the reconstruction error and the variance and the covariance +matrix. Minimizing the error is equivalent to maximizing the variance +of the projected data. +

    + + +
    +

    The final step

    + +

    +We could trivially maximize the variance of the projection (and +thereby minimize the error in the reconstruction function) by letting +the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we +want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by +\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a +Lagrange multiplier we can then in turn maximize + +

     
    +$$ +J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0). +$$ +

     
    + +Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain + +

     
    +$$ +\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0, +$$ +

     
    + +meaning that +

     
    +$$ +\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0. +$$ +

     
    + +The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is +

     
    +$$ +\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. +$$ +

     
    + +

    +If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function \( J \) in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix \( \boldsymbol{X} \). + +

    +The proof +for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see Murphy chapter +12.2. The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein. + +

    +Additional part of the proof for the other eigenvectors will be added by mid January 2020. +

    + + +
    +

    Geometric Interpretation and link with Singular Value Decomposition

    + +

    +This material will be added by mid January 2020. +

    + + +
    +

    Principal Component Analysis

    + +

    +Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. +First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it. + +

    +The following Python code uses NumPy’s svd() function to obtain all the principal components of the +training set, then extracts the first two principal components. First we center the data using either pandas or our own code +

    + + +

    import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 vanilla matrix 
+rows = 10
+cols = 5
+X = np.random.randn(rows,cols)
+df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+display(df)
+
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+# Then check the difference between pandas and our own set up
+print(X_centered-df)
+#Now we do an SVD
+U, s, V = np.linalg.svd(X_centered)
+c1 = V.T[:, 0]
+c2 = V.T[:, 1]
+W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+print(X2D)
+
    +

    +PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering +the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t +forget to center the data first. + +

    +Once you have identified all the principal components, you can reduce the dimensionality of the dataset +down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components. +Selecting this hyperplane ensures that the projection will preserve as much variance as possible. +

    + + +

    W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+
    +
    + + +
    +

    PCA and scikit-learn

    + +

    +Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The +following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note +that it automatically takes care of centering the data): +

    + + +

    #thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D = pca.fit_transform(X)
+print(X2D)
+
    +

    +After fitting the PCA transformer to the dataset, you can access the principal components using the +components variable (note that it contains the PCs as horizontal vectors, so, for example, the first +principal component is equal to +

    + + +

    pca.components_.T[:, 0].
+
    +

    +Another very useful piece of information is the explained variance ratio of each principal component, +available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s +variance that lies along the axis of each principal component. +

    + + +
    +

    Back to the Cancer Data

    +We can now repeat the above but applied to real data, in this case our breast cancer data. +Here we compute performance scores on the training data using logistic regression. +

    + + +

    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.linear_model import LogisticRegression
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+
+logreg = LogisticRegression()
+logreg.fit(X_train, y_train)
+print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
+# We scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Then perform again a log reg fit
+logreg.fit(X_train_scaled, y_train)
+print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D_train = pca.fit_transform(X_train_scaled)
+# and finally compute the log reg fit and the score on the training data	
+logreg.fit(X2D_train,y_train)
+print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
+
    +

    +We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. +

    + + +
    +

    More on the PCA

    + +

    +Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to +choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). +Unless, of course, you are reducing dimensionality for data visualization — in that case you will +generally want to reduce the dimensionality down to 2 or 3. +The following code computes PCA without reducing dimensionality, then computes the minimum number +of dimensions required to preserve 95% of the training set’s variance: +

    + + +

    pca = PCA()
+pca.fit(X)
+cumsum = np.cumsum(pca.explained_variance_ratio_)
+d = np.argmax(cumsum >= 0.95) + 1
+
    +

    +You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead +of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be +a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve: +

    + + +

    pca = PCA(n_components=0.95)
+X_reduced = pca.fit_transform(X)
+
    +
    + + +
    +

    Incremental PCA

    + +

    +One problem with the preceding implementation of PCA is that it requires the whole training set to fit in +memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have +been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch +at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new +instances arrive). +

    + + +
    +

    Randomized PCA

    + +

    +Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic +algorithm that quickly finds an approximation of the first d principal components. Its computational +complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the +previous algorithms when \( d \) is much smaller than \( n \). +

    + + +
    +

    Kernel PCA

    +
    + +

    +The kernel trick is a mathematical technique that implicitly maps instances into a +very high-dimensional space (called the feature space), enabling nonlinear classification and regression +with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature +space corresponds to a complex nonlinear decision boundary in the original space. +It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear +projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at +preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a +twisted manifold. +For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an +

    + + +

    from sklearn.decomposition import KernelPCA
+rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
+X_reduced = rbf_pca.fit_transform(X)
+
    + +
    +
    + + +
    +

    LLE

    + +

    +Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction +(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous +algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its +closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where +these local relationships are best preserved (more details shortly). +

    + + +
    +

    Other techniques

    + +

    +There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn. + +

    +Here are some of the most popular: + +

      +

    • Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • +

    • Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • +

    • t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • +

    • Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • +
    +
    + + diff --git a/doc/pub/week43/html/week43-solarized.html b/doc/pub/week43/html/week43-solarized.html index 513d67b3a..d561e511f 100644 --- a/doc/pub/week43/html/week43-solarized.html +++ b/doc/pub/week43/html/week43-solarized.html @@ -89,142 +89,111 @@ div { text-align: justify; text-justify: inter-word; } ('Final technicalities IV', 2, None, '___sec18'), ('Backpropagation', 2, None, '___sec19'), ('Gradient descent', 2, None, '___sec20'), - ('Basic ideas of the Principal Component Analysis (PCA)', - 2, - None, - '___sec21'), - ('Introducing the Covariance and Correlation functions', + ('The code for solving the ODE', 2, None, '___sec21'), + ('The network with one input layer, specified number of hidden ' + 'layers, and one output layer', 2, None, '___sec22'), - ('Correlation Function and Design/Feature Matrix', - 2, - None, - '___sec23'), - ('Covariance Matrix Examples', 2, None, '___sec24'), - ('Correlation Matrix', 2, None, '___sec25'), - ('Correlation Matrix with Pandas', 2, None, '___sec26'), - ('Correlation Matrix with Pandas and the Franke function', - 2, - None, - '___sec27'), - ('Rewriting the Covariance and/or Correlation Matrix', - 2, - None, - '___sec28'), - ('Towards the PCA theorem', 2, None, '___sec29'), - ('The Algorithm before the Theorem', 2, None, '___sec30'), - ('Writing our own PCA code', 2, None, '___sec31'), - ('Compute the sample mean and center the data', - 3, - None, - '___sec32'), - ('Compute the sample covariance', 3, None, '___sec33'), - ('Diagonalize the sample covariance matrix to obtain the ' - 'principal components', - 3, - None, - '___sec34'), - ('Classical PCA Theorem', 2, None, '___sec35'), - ('Proof of the PCA Theorem', 2, None, '___sec36'), - ('PCA Proof continued', 2, None, '___sec37'), - ('The final step', 2, None, '___sec38'), - ('Geometric Interpretation and link with Singular Value ' - 'Decomposition', - 2, - None, - '___sec39'), - ('Principal Component Analysis', 2, None, '___sec40'), - ('PCA and scikit-learn', 2, None, '___sec41'), - ('Back to the Cancer Data', 2, None, '___sec42'), - ('More on the PCA', 2, None, '___sec43'), - ('Incremental PCA', 2, None, '___sec44'), - ('Randomized PCA', 2, None, '___sec45'), - ('Kernel PCA', 2, None, '___sec46'), - ('LLE', 2, None, '___sec47'), - ('Other techniques', 2, None, '___sec48'), - ('The network with one input, hidden, and output layer', - 2, - None, - '___sec49'), - ('The network with one input layer, specified number of hidden ' - 'layers, and one output layer output layer', - 2, - None, - '___sec50'), - ('Example: Population growth, comparing Autograd, TensorFlow, ' - "and Euler's scheme", - 2, - None, - '___sec51'), - ('Setting up the problem', 2, None, '___sec52'), - ('The trial solution', 2, None, '___sec53'), - ('The program using Autograd', 2, None, '___sec54'), - ('Using forward Euler to solve the ODE', 2, None, '___sec55'), - ('Using TensorFlow to model logistic population growth', - 2, - None, - '___sec56'), - ('The general program flow in TensorFlow', 2, None, '___sec57'), - ('Program flow in TensorFlow - Construction phase', - 2, - None, - '___sec58'), - ('Program flow in TensorFlow - Execution phase', - 2, - None, - '___sec59'), - ('The full program modeling logistic population growth using ' - 'TensorFlow', - 2, - None, - '___sec60'), + ('Example: Population growth', 2, None, '___sec23'), + ('Setting up the problem', 2, None, '___sec24'), + ('The trial solution', 2, None, '___sec25'), + ('The program using Autograd', 2, None, '___sec26'), + ('Using forward Euler to solve the ODE', 2, None, '___sec27'), ('Example: Solving the one dimensional Poisson equation using ' 'Autograd and TensorFlow', 2, None, - '___sec61'), - ('The specific equation to solve for', 2, None, '___sec62'), - ('Solving the equation using Autograd', 2, None, '___sec63'), - ('Comparing with a numerical scheme', 2, None, '___sec64'), - ('Using gradient descent in TensorFlow to solve Poisson equation', - 2, - None, - '___sec65'), - ('Using a different optimization algorithm implemented in ' - 'TensorFlow to solve Poisson equation', - 2, - None, - '___sec66'), - ('Partial Differential Equations', 2, None, '___sec67'), - ('Example: The diffusion equation', 2, None, '___sec68'), - ('Defining the problem', 2, None, '___sec69'), - ('Setting up the network using Autograd', 2, None, '___sec70'), + '___sec28'), + ('The specific equation to solve for', 2, None, '___sec29'), + ('Solving the equation using Autograd', 2, None, '___sec30'), + ('Comparing with a numerical scheme', 2, None, '___sec31'), + ('Partial Differential Equations', 2, None, '___sec32'), + ('Type of problem', 2, None, '___sec33'), + ('Network requirements', 2, None, '___sec34'), + ('More details', 2, None, '___sec35'), + ('Example: The diffusion equation', 2, None, '___sec36'), + ('Defining the problem', 2, None, '___sec37'), + ('Setting up the network using Autograd', 2, None, '___sec38'), ('Setting up the network using Autograd; The trial solution', 2, None, - '___sec71'), + '___sec39'), + ('Why the jacobian?', 2, None, '___sec40'), ('Setting up the network using Autograd; The full program', 2, None, - '___sec72'), - ('Example: Solving the wave equation using Autograd and ' - 'TensorFlow', + '___sec41'), + ('Example: Solving the wave equation with Neural Networks', 2, None, - '___sec73'), - ('The problem to solve for', 2, None, '___sec74'), - ('The trial solution', 2, None, '___sec75'), - ('The analytical solution', 2, None, '___sec76'), + '___sec42'), + ('The problem to solve for', 2, None, '___sec43'), + ('The trial solution', 2, None, '___sec44'), + ('The analytical solution', 2, None, '___sec45'), ('Solving the wave equation - the full program using Autograd', 2, None, - '___sec77'), - ('Solving the wave equation - the full program using TensorFlow', + '___sec46'), + ('Resources on differential equations and deep learning', 2, None, - '___sec78'), - ('Resources', 2, None, '___sec79')]} + '___sec47'), + ('Friday, Principal Component Analysis', 2, None, '___sec48'), + ('Basic ideas of the Principal Component Analysis (PCA)', + 2, + None, + '___sec49'), + ('Introducing the Covariance and Correlation functions', + 2, + None, + '___sec50'), + ('Correlation Function and Design/Feature Matrix', + 2, + None, + '___sec51'), + ('Covariance Matrix Examples', 2, None, '___sec52'), + ('Correlation Matrix', 2, None, '___sec53'), + ('Correlation Matrix with Pandas', 2, None, '___sec54'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec55'), + ('Rewriting the Covariance and/or Correlation Matrix', + 2, + None, + '___sec56'), + ('Towards the PCA theorem', 2, None, '___sec57'), + ('The Algorithm before the Theorem', 2, None, '___sec58'), + ('Writing our own PCA code', 2, None, '___sec59'), + ('Compute the sample mean and center the data', + 3, + None, + '___sec60'), + ('Compute the sample covariance', 3, None, '___sec61'), + ('Diagonalize the sample covariance matrix to obtain the ' + 'principal components', + 3, + None, + '___sec62'), + ('Classical PCA Theorem', 2, None, '___sec63'), + ('Proof of the PCA Theorem', 2, None, '___sec64'), + ('PCA Proof continued', 2, None, '___sec65'), + ('The final step', 2, None, '___sec66'), + ('Geometric Interpretation and link with Singular Value ' + 'Decomposition', + 2, + None, + '___sec67'), + ('Principal Component Analysis', 2, None, '___sec68'), + ('PCA and scikit-learn', 2, None, '___sec69'), + ('Back to the Cancer Data', 2, None, '___sec70'), + ('More on the PCA', 2, None, '___sec71'), + ('Incremental PCA', 2, None, '___sec72'), + ('Randomized PCA', 2, None, '___sec73'), + ('Kernel PCA', 2, None, '___sec74'), + ('LLE', 2, None, '___sec75'), + ('Other techniques', 2, None, '___sec76')]} end of tocinfo --> @@ -563,17 +532,17 @@ $$ $$

    -Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes +Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes $$ -\min_{P} c(\boldsymbol{x}, P) +\min_{P} C(\boldsymbol{x}, P) $$

    In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could also be expressed as $$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) +\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) $$

    @@ -723,7 +692,7 @@ The next step is to decide how the parameters should be changed such that they m The chosen cost function for this problem is $$ -c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 $$

    @@ -743,7 +712,7 @@ a direction where the cost function decreases goes to a minimum.

    In general, the update of some parameters \( \boldsymbol{\omega} \) given a cost -function defined by some weights \( \boldsymbol{\omega} \), \( c(\boldsymbol{x}, +function defined by some weights \( \boldsymbol{\omega} \), \( C(\boldsymbol{x}, \boldsymbol{\omega}) \), goes as follows: $$ @@ -754,1062 +723,36 @@ $$ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance.

    -The value of \( \lambda \) decides how large steps the algorithm must take in the direction of $ \nabla_{\boldsymbol{\omega}} c(\boldsymbol{x}, \boldsymbol{\omega})$. -The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect to the elements in \( \boldsymbol{\omega \). +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$. +The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect +to the elements in \( \boldsymbol{\omega \).

    -In our case, we have to minimize the cost function \( c(\boldsymbol{x}, P) \) with respect to the two sets of weights and biases, that is for the hidden layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} } \) . +In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with +respect to the two sets of weights and biases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} +} \) .

    This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated by $$ \begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(\boldsymbol{x}, P) +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) \end{aligned} $$ -

    -In general, one could risk using a cost function having gradients that are cumbersome to derive analytically. -For our case, the cost functions are just the mean squared error. -One could employ an implementation of the back propagation for this case, but we will emphasis -on how one could use automatic differentiation in order to train the network. - -

    -However, it might be useful to know how automatic differentiation can be used, e.g through Autograd, in order to test an implementation. -











    -

    Basic ideas of the Principal Component Analysis (PCA)

    - -

    -The principal component analysis deals with the problem of fitting a -low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than -the totaldimension \( D \) of the problem at hand (our data -set). Mathematically it can be formulated as a statistical problem or -a geometric problem. In our discussion of the theorem for the -classical PCA, we will stay with a statistical approach. This is also -what set the scene historically which for the PCA. - -

    -We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) - -

      -
    • Each data point is determined by \( p \) extrinsic (measurement) variables
      • -
    • We may want to ask the following question: Are there fewer intrinsic variables (say \( d < < p \)) that still approximately describe the data?
      • -
    • If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.
      • -
    - -









    - -

    Introducing the Covariance and Correlation functions

    - -

    -Before we discuss the PCA theorem, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities - -

    -Suppose we have defined two vectors -\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -$$ - -where for example -$$ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -With this definition and recalling that the variance is defined as -$$ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -$$ - -we can rewrite the covariance matrix as -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -$$ - -

    -The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function - -$$ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -$$ - -

    -The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1] \). This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors \( \boldsymbol{x} \) -and \( \boldsymbol{y} \) as - -$$ -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -$$ - -

    -In the above example this is the function we constructed using pandas. - -

    -









    - -

    Correlation Function and Design/Feature Matrix

    - -

    -In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression -we defined the design/feature matrix \( \boldsymbol{X} \) as - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - -with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the -entries \( n \) being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as -$$ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -$$ - -with a given vector -$$ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -$$ - -

    -With these definitions, we can now rewrite our \( 2\times 2 \) -correaltion/covariance matrix in terms of a moe general design/feature -matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \) -covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \) - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -$$ - -and the correlation matrix -$$ -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -$$ - -

    -









    - -

    Covariance Matrix Examples

    - -

    -The Numpy function np.cov calculates the covariance elements using -the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have -the exact mean values. The following simple function uses the -np.vstack function which takes each vector of dimension \( 1\times n \) -and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) - -$$ -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -$$ - -

    -which in turn is converted into into the \( 2\times 2 \) covariance matrix -\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function. +

    The code for solving the ODE

    -

    # Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-
    -

    -









    - -

    Correlation Matrix

    - -

    -The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). - -

    - - -

    import numpy as np
-n = 100
-# define two vectors                                                                                           
-x = np.random.random(size=n)
-y = 4+3*x+np.random.normal(size=n)
-#scaling the x and y vectors                                                                                   
-x = x - np.mean(x)
-y = y - np.mean(y)
-variance_x = np.sum(x@x)/n
-variance_y = np.sum(y@y)/n
-print(variance_x)
-print(variance_y)
-cov_xy = np.sum(x@y)/n
-cov_xx = np.sum(x@x)/n
-cov_yy = np.sum(y@y)/n
-C = np.zeros((2,2))
-C[0,0]= cov_xx/variance_x
-C[1,1]= cov_yy/variance_y
-C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
-C[1,0]= C[0,1]
-print(C)
-
    -

    -We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix. - -

    -The above procedure with numpy can be made more compact if we use pandas. - -

    -









    - -

    Correlation Matrix with Pandas

    - -

    -We whow here how we can set up the correlation matrix using pandas, as done in this simple code -

    - - -

    import numpy as np
-import pandas as pd
-n = 10
-x = np.random.normal(size=n)
-x = x - np.mean(x)
-y = 4+3*x+np.random.normal(size=n)
-y = y - np.mean(y)
-X = (np.vstack((x, y))).T
-print(X)
-Xpd = pd.DataFrame(X)
-print(Xpd)
-correlation_matrix = Xpd.corr()
-print(correlation_matrix)
-
    -

    -We expand this model to the Franke function discussed above. - -

    -









    - -

    Correlation Matrix with Pandas and the Franke function

    - -

    - - -

    # Common imports
-import numpy as np
-import pandas as pd
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 4
-N = 100
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-
-Xpd = pd.DataFrame(X)
-# subtract the mean values and set up the covariance matrix
-Xpd = Xpd - Xpd.mean()
-covariance_matrix = Xpd.cov()
-print(covariance_matrix)
-
    -

    -We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree \( n \)). - -

    -This means that the variance for these elements will be zero and will -cause problems when we set up the correlation matrix. We can simply -drop these elements and construct a correlation -matrix without these elements. - -

    -









    - -

    Rewriting the Covariance and/or Correlation Matrix

    - -

    -We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. -$$ - -

    -To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \) -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -$$ - -

    -If we then compute the expectation value -$$ -\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -$$ - -which is just -$$ -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -$$ - -where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \). - -

    -It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). - -

    -









    - -

    Towards the PCA theorem

    - -

    -We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T]. -$$ - -Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \). -These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \). - -

    -Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \). - -

    -That is we have -$$ -\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, -$$ - -since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S}^T \) from the left we have -$$ -\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T, -$$ - -and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that - -$$ -\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i. -$$ - -

    -In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is -\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \). - -

    -The eigenvalues tell us then how much we need to stretch the -corresponding eigenvectors. Dimensions with large eigenvalues have -thus large variations (large variance) and define therefore useful -dimensions. The data points are more spread out in the direction of -these eigenvectors. Smaller eigenvalues mean on the other hand that -the corresponding eigenvectors are shrunk accordingly and the data -points are tightly bunched together and there is not much variation in -these specific directions. Hopefully then we could leave it out -dimensions where the eigenvalues are very small. If \( p \) is very large, -we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) -features/predictors. - -

    -









    - -

    The Algorithm before theorem

    - -

    -Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. - -

      -
    • Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.
      • -
    - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - - -
      -
    • Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • -
    • Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).
      • -
    • Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • -
    • Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • -
    • Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • -
    - -









    - -

    Writing our own PCA code

    - -

    -We will use a simple example first with two-dimensional data -drawn from a multivariate normal distribution with the following mean and covariance matrix: -$$ -\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\ -2 & 2 -\end{bmatrix} -$$ - -Note that the mean refers to each column of data. -We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from -this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \). - -

    -The following Python code aids in setting up the data and writing out the design matrix. -Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \). -

    - - -

    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-n = 10000
-mean = (-1, 2)
-cov = [[4, 2], [2, 2]]
-X = np.random.multivariate_normal(mean, cov, n)
-
    -

    -Now we are going to implement the PCA algorithm. We will break it down into various substeps. - -

    Compute the sample mean and center the data

    - -

    -The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is -$$ -\mu_n = \frac{1}{n} \sum_{i=1}^n x_i -$$ - -and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form -$$ -\bar{x}_i = x_i - \mu_n. -$$ - -When you are done with these steps, print out \( \mu_n \) to verify it is -close to \( \mu \) and plot your mean centered data to verify it is -centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above. -The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. -

    - - -

    df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-
    -

    -Alternatively, we could use the functions we discussed -earlier for scaling the data set. That is, we could have used the -StandardScaler function in Scikit-Learn, a function which ensures -that for each feature/predictor we study the mean value is zero and -the variance is one (every column in the design/feature matrix). You -would then not get the same results, since we divide by the -variance. The diagonal covariance matrix elements will then be one, -while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our -specific case. - -

    Compute the sample covariance

    - -

    -Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation -$$ -\begin{equation*} -\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n) -\end{equation*} -$$ - -where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \). -We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows -

    - - -

    print(df.cov())
-print(np.cov(X_centered.T))
-
    -

    -Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. -Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. -

    - - -

    # extract the relevant columns from the centered design matrix of dim n x 2
-x = X_centered[:,0]
-y = X_centered[:,1]
-Cov = np.zeros((2,2))
-Cov[0,1] = np.sum(x.T@y)/(n-1.0)
-Cov[0,0] = np.sum(x.T@x)/(n-1.0)
-Cov[1,1] = np.sum(y.T@y)/(n-1.0)
-Cov[1,0]= Cov[0,1]
-print("Centered covariance using own code")
-print(Cov)
-plt.plot(x, y, 'x')
-plt.axis('equal')
-plt.show()
-
    -

    -Depending on the number of points \( n \), we will get results that are close to the covariance values defined above. -The plot shows how the data are clustered around a line with slope close to one. Is this expected? - -

    Diagonalize the sample covariance matrix to obtain the principal components

    - -

    -Now we are ready to solve for the principal components! To do so we -diagonalize the sample covariance matrix \( \Sigma \). We can use the -function np.linalg.eig to do so. It will return the eigenvalues and -eigenvectors of \( \Sigma \). Once we have these we can perform the -following tasks: - -

      -
    • We compute the percentage of the total variance captured by the first principal component
      • -
    • We plot the mean centered data and lines along the first and second principal components
      • -
    • Then we project the mean centered data onto the first and second principal components, and plot the projected data.
      • -
    • Finally, we approximate the data as
      • -
    - -$$ -\begin{equation*} -x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0 -\end{equation*} -$$ - -where \( v_0 \) is the first principal component. - -

    -Collecting all these steps we can write our own PCA function and -compare this with the functionality included in Scikit-Learn. - -

    -The code here outlines some of the elements we could include in the -analysis. Feel free to extend upon this in order to address the above -questions. - -

    - - -

    # diagonalize and obtain eigenvalues, not necessarily sorted
-EigValues, EigVectors = np.linalg.eig(Cov)
-# sort eigenvectors and eigenvalues
-#permute = EigValues.argsort()
-#EigValues = EigValues[permute]
-#EigVectors = EigVectors[:,permute]
-print("Eigenvalues of Covariance matrix")
-for i in range(2):
-    print(EigValues[i])
-FirstEigvector = EigVectors[:,0]
-SecondEigvector = EigVectors[:,1]
-print("First eigenvector")
-print(FirstEigvector)
-print("Second eigenvector")
-print(SecondEigvector)
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2Dsl = pca.fit_transform(X)
-print("Eigenvector of largest eigenvalue")
-print(pca.components_.T[:, 0])
-
    -

    -This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? - -

    -









    - -

    Classical PCA Theorem

    - -

    -We assume now that we have a design matrix \( \boldsymbol{X} \) which has been -centered as discussed above. For the sake of simplicity we skip the -overline symbol. The matrix is defined in terms of the various column -vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension -\( \boldsymbol{x}\in {\mathbb{R}}^{n} \). - -

    -We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \). We define the reconstruction error (which is similar to the mean squared error we have seen before) as -$$ -J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2, -$$ - -with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix -\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \). When doing PCA we want to reduce this dimensionality. - -

    -The PCA theorem states that minimizing the above reconstruction error -corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which -diagonalizes the empirical covariance(correlation) matrix. The optimal -low-dimensional encoding of the data is then given by a set of vectors -\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the -orthogonal projection of the data onto the columns spanned by the -eigenvectors of the covariance(correlations matrix). - -

    -The proof which follows will be updated by mid January 2020. - -

    -









    - -

    Proof of the PCA Theorem

    - -

    -To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as -$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), -$$ - -which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as -$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). -$$ - -Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that -$$ -z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i, -$$ - -where the vectors on the rhs are known. - -

    -









    - -

    PCA Proof continued

    - -

    -We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write -$$ -J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2. -$$ - -

    -We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by -$$ -\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2, -$$ - -since the expectation value of -$$ -\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0, -$$ - -where we have used the fact that our data are centered. - -

    -Recalling our definition of the covariance as -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T], -$$ - -we have thus that -$$ -\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0. -$$ - -

    -We are almost there, we have obtained a relation between minimizing -the reconstruction error and the variance and the covariance -matrix. Minimizing the error is equivalent to maximizing the variance -of the projected data. - -

    -









    - -

    The final step

    - -

    -We could trivially maximize the variance of the projection (and -thereby minimize the error in the reconstruction function) by letting -the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we -want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a -Lagrange multiplier we can then in turn maximize - -$$ -J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0). -$$ - -Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain - -$$ -\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0, -$$ - -meaning that -$$ -\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0. -$$ - -The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is -$$ -\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. -$$ - -

    -If we want to maximize the variance (minimize the construction error) -we simply pick the eigenvector of the covariance matrix with the -largest eigenvalue. This establishes the link between the minimization -of the reconstruction function \( J \) in terms of an orthogonal matrix -and the maximization of the variance and thereby the covariance of our -observations encoded in the design/feature matrix \( \boldsymbol{X} \). - -

    -The proof -for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be -established by applying the above arguments and using the fact that -our basis of eigenvectors is orthogonal, see Murphy chapter -12.2. The -discussion in chapter 12.2 of Murphy's text has also a nice link with -the Singular Value Decomposition theorem. For categorical data, see -chapter 12.4 and discussion therein. - -

    -Additional part of the proof for the other eigenvectors will be added by mid January 2020. - -

    -









    - -

    Geometric Interpretation and link with Singular Value Decomposition

    - -

    -This material will be added by mid January 2020. - -

    -









    - -

    Principal Component Analysis

    - -

    -Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. -First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it. - -

    -The following Python code uses NumPy’s svd() function to obtain all the principal components of the -training set, then extracts the first two principal components. First we center the data using either pandas or our own code -

    - - -

    import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 vanilla matrix 
-rows = 10
-cols = 5
-X = np.random.randn(rows,cols)
-df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-display(df)
-
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-# Then check the difference between pandas and our own set up
-print(X_centered-df)
-#Now we do an SVD
-U, s, V = np.linalg.svd(X_centered)
-c1 = V.T[:, 0]
-c2 = V.T[:, 1]
-W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-print(X2D)
-
    -

    -PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering -the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t -forget to center the data first. - -

    -Once you have identified all the principal components, you can reduce the dimensionality of the dataset -down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components. -Selecting this hyperplane ensures that the projection will preserve as much variance as possible. -

    - - -

    W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-
    -

    - - -

    PCA and scikit-learn

    - -

    -Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The -following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note -that it automatically takes care of centering the data): -

    - - -

    #thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D = pca.fit_transform(X)
-print(X2D)
-
    -

    -After fitting the PCA transformer to the dataset, you can access the principal components using the -components variable (note that it contains the PCs as horizontal vectors, so, for example, the first -principal component is equal to -

    - - -

    pca.components_.T[:, 0].
-
    -

    -Another very useful piece of information is the explained variance ratio of each principal component, -available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s -variance that lies along the axis of each principal component. - -

    -









    - -

    Back to the Cancer Data

    -We can now repeat the above but applied to real data, in this case our breast cancer data. -Here we compute performance scores on the training data using logistic regression. -

    - - -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-
-logreg = LogisticRegression()
-logreg.fit(X_train, y_train)
-print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
-# We scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Then perform again a log reg fit
-logreg.fit(X_train_scaled, y_train)
-print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D_train = pca.fit_transform(X_train_scaled)
-# and finally compute the log reg fit and the score on the training data	
-logreg.fit(X2D_train,y_train)
-print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
-
    -

    -We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. - -

    -









    - -

    More on the PCA

    - -

    -Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to -choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). -Unless, of course, you are reducing dimensionality for data visualization — in that case you will -generally want to reduce the dimensionality down to 2 or 3. -The following code computes PCA without reducing dimensionality, then computes the minimum number -of dimensions required to preserve 95% of the training set’s variance: -

    - - -

    pca = PCA()
-pca.fit(X)
-cumsum = np.cumsum(pca.explained_variance_ratio_)
-d = np.argmax(cumsum >= 0.95) + 1
-
    -

    -You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead -of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be -a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve: -

    - - -

    pca = PCA(n_components=0.95)
-X_reduced = pca.fit_transform(X)
-
    -

    -









    - -

    Incremental PCA

    - -

    -One problem with the preceding implementation of PCA is that it requires the whole training set to fit in -memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have -been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch -at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new -instances arrive). - -

    -









    - -

    Randomized PCA

    - -

    -Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic -algorithm that quickly finds an approximation of the first d principal components. Its computational -complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the -previous algorithms when \( d \) is much smaller than \( n \). - -

    -









    - -

    Kernel PCA

    -
    - -

    - -

    -The kernel trick is a mathematical technique that implicitly maps instances into a -very high-dimensional space (called the feature space), enabling nonlinear classification and regression -with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature -space corresponds to a complex nonlinear decision boundary in the original space. -It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear -projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at -preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a -twisted manifold. -For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an -

    - - -

    from sklearn.decomposition import KernelPCA
-rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
-X_reduced = rbf_pca.fit_transform(X)
-
    - -
    - - -

    -









    - -

    LLE

    - -

    -Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction -(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous -algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its -closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where -these local relationships are best preserved (more details shortly). - -

    -









    - -

    Other techniques

    - -

    -There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn. - -

    -Here are some of the most popular: - -

      -
    • Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • -
    • Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • -
    • t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • -
    • Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • -
    - -









    - -

    The network with one input, hidden, and output layer

    - -

    - - -

    # Autograd will be used for later, so the numpy wrapper for Autograd must be imported
-import autograd.numpy as np
+
    import autograd.numpy as np
 from autograd import grad, elementwise_grad
 import autograd.numpy.random as npr
 from matplotlib import pyplot as plt
@@ -1957,7 +900,7 @@ Here are some of the most popular:
 
    
 









    
 
-
    The network with one input layer, specified number of hidden layers, and one output layer  output layer 
    
+
    The network with one input layer, specified number of hidden layers, and one output layer 
    
 
 
    
 It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.
@@ -2130,7 +1073,7 @@ The number of neurons within each hidden layer are given as a list of integers i
 
    
 









    
 
-
    Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme 
    
+
    Example: Population growth 
    
 
 
    
 A logistic model of population growth assumes that a population converges toward an equilibrium.
@@ -2155,7 +1098,7 @@ For comparison, the forward Euler method has been implemented in order to see ho
 
    
 









    
 
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 
 
    
 Here, we will model a population \( g(t) \) in an environment having carrying capacity \( A \).
@@ -2176,7 +1119,9 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
 
    
 









    
 
-
    The trial solution 
    
+
    The trial solution 
    
+
+
    
 We will get a slightly different trial solution, as the boundary conditions are different
 compared to the case for exponential decay.
 
@@ -2200,7 +1145,7 @@ $$
 
    
 









    
 
-
    The program using Autograd 
    
+
    The program using Autograd 
    
 
 
    
 The network will be the similar as for the exponential decay example, but with some small modifications for our problem.
@@ -2375,7 +1320,7 @@ The network will be the similar as for the exponential decay example, but with s
 
    
 









    
 
-
    Using forward Euler to solve the ODE 
    
+
    Using forward Euler to solve the ODE 
    
 
 
    
 A straight-forward way of solving an ODE numerically, is to use Euler's method.
@@ -2504,177 +1449,10 @@ extending the program that uses the network using Autograd:
 
    
 Running the program gives
 
-
    
-
-
-
    Max absolute difference between Euler method and analytical: 0.011225
-Max absolute difference between deep neural network and analytical: 0.00424909
-
    
 
    
 









    
 
-
    Using TensorFlow to model logistic population growth 
    
-
-
    
-TensorFlow is a library widely used in the machine learning community.
-A neural network can be set up in a flexible manner, where various optimization algorithms are implemented and different types of networks can be used, making it easier to experiment on solving differential equations using neural networks.
-
-
    
-









    
-
-
    The general program flow in TensorFlow 
    
-
-
    
-Usually, a program in TensorFlow is divided into two parts; the construction phase and the execution phase.
-In the construction phase, the computational graph that TensorFlow uses to perform its calculations are set up.
-In the execution phase, TensorFlow evaluates any procedure that was defined in the construction phase.
-
-
    Program flow in TensorFlow - Construction phase 
    
-
-
    
-Here, the architecture for the neural network will be set up, along with the cost function and an optimizer class used during training of the network.
-Note that TensorFlow uses a different convention for the weighting done in each neuron in each layer within the network than in the implementation using Autograd.
-The matrix-vector multiplication between the input from the previous layer and the weighting at the neuron at current layer in the program using Autograd, is the transpose of the convention used in TensorFlow. But it will not affect that much our construction, as TensorFlow takes care of most of the computations. The only thing we have to be aware of, is how the dimensions are for our inputs.
-
-
    Program flow in TensorFlow - Execution phase 
    
-
-
    
-The computation graph has been defined, and is ready to be evaluated.
-In order to get access to the graph, it has to be initialized and be runned within a Session.
-
-
    The full program modeling logistic population growth using TensorFlow 
    
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-# Set a seed to ensure getting the same results from every run
-tf.set_random_seed(4155)
-
-Nt = 10
-T = 1
-t = np.linspace(0,T, Nt)
-
-## The construction phase
-
-# Convert the values the trial solution is evaluated at to a tensor.
-t_tf = tf.convert_to_tensor(t.reshape(-1,1),dtype=tf.float64)
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(t.shape)),shape=(-1,1))
-
-# Define the parameters of the equation
-alpha = tf.constant(2.,dtype=tf.float64)
-A = tf.constant(1.,dtype=tf.float64)
-g0 = tf.constant(1.2,dtype=tf.float64)
-
-num_iter = 100000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [100,50,25]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = t_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = g0 + t_tf*dnn_output
-    d_g_trial = tf.gradients(g_trial,t_tf)
-
-    func = alpha*g_trial*(A - g_trial)
-    cost = tf.losses.mean_squared_error(zeros, d_g_trial[0] - func)
-
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-# Set up a referance to the result from the neural network:
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The training of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves for each iteration:
-        #if i % 1000 == 0:
-        #    print(cost.eval())
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-# Compare with analytical solution
-def get_parameters():
-    alpha = 2
-    A = 1
-    g0 = 1.2
-    return alpha, A, g0
-
-def g_analytic(t):
-    alpha,A, g0 = get_parameters()
-    return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
-
-g_analytical = g_analytic(t)
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(t, g_dnn_tf)
-plt.plot(t, g_analytical)
-
-plt.legend(['dnn, tensorflow', 'exact'])
-plt.xlabel('Time t')
-plt.ylabel('g(t)')
-
-plt.show()
-
    
-
    
-









    
-
-
    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow 
    
+
    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow 
    
 
 
    
 The Poisson equation for \( g(x) \) in one dimension is
@@ -2702,16 +1480,10 @@ This equation can be solved numerically using programs where e.g Autograd and Te
 The results from the networks can then be compared to the analytical solution.
 In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.
 
-
    
-There exists many different optimization methods implemented in TensorFlow.
-In the examples program using TensorFlow, it could also be of interest to see how
-the choice of an optimization method affects our results.
-In the TensorFlow documentation about optimizers, a list over available optimization methods are shown.
-
 
    
 









    
 
-
    The specific equation to solve for 
    
+
    The specific equation to solve for 
    
 
 
    
 Here, the function \( g(x) \) to solve for follows the equation
@@ -2749,7 +1521,7 @@ $$
 
    
 









    
 
-
    Solving the equation using Autograd 
    
+
    Solving the equation using Autograd 
    
 
 
    
 
@@ -2908,7 +1680,7 @@ $$
 
    
 









    
 
-
    Comparing with a numerical scheme 
    
+
    Comparing with a numerical scheme 
    
 
 
    
 The Poisson equation is possible to solve using Taylor series to approximate the second derivative.
@@ -3211,168 +1983,9 @@ The max absolute difference between the analytical solution and numerical scheme
 
    
 









    
 
-
    Using gradient descent in TensorFlow to solve Poisson equation 
    
-The program follows the similar idea as for the logistic population model.
+
    Partial Differential Equations 
    
 
 
    
-What has changed, is what the cost function minimizes and the trial solution.
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-## Construction phase
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-tf.set_random_seed(4155)
-
-# Convert the values the trial solution is evaluated at to a tensor.
-Nx = 10
-x = np.linspace(0,1, Nx)
-x_tf = tf.convert_to_tensor(x.reshape(-1,1),dtype=tf.float64)
-
-
-num_iter = 10000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [20,10]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = x_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = x_tf*(1-x_tf)*dnn_output
-    d_g_trial = tf.gradients(g_trial,x_tf)
-    d2_g_trial = tf.gradients(d_g_trial,x_tf)
-
-    right_side = (3*x_tf + x_tf**2)*tf.exp(x_tf)
-
-    err = tf.square( -d2_g_trial[0] - right_side)
-    cost = tf.reduce_sum(err, name = 'cost')
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The traning of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-    writer = tf.summary.FileWriter("./output", sess.graph)
-    writer.close()
-
-# Evaluate the analytical function to compare with
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-g_analytical = g_analytic(x)
-
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(x, g_dnn_tf)
-plt.plot(x, g_analytical)
-
-plt.legend(['dnn, tensorflow','exact'])
-plt.xlabel('x')
-plt.ylabel('g(x)')
-
-plt.show()
-
    
-
    
-









    
-
-
    Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation 
    
-
-
    
-We can see that the results using GradientDescentOptimizer seems to converge towards the analytical solution.
-But there exists many other methods for optimization also, see the TensorFlow documentation on Optimizers.
-
-
    
-Adam is an optimization algorithm that changes its learning rates accordingly to the function it tries to minimize for every iteration.
-The algorithm is described in this paper.
-How much an optimization algorithm has to say for the network to converge, could be interesting to experiment with.
-Using the same TensorFlow program as before, the only change to do, is to replace the variable optimizer.
-
-
    
-In the program that uses TensorFlow to solve for the Poisson equation, change the line
-
-
    
-
-
-
    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-
    
-
    
-to
-
-
    
-
-
-
    optimizer = tf.train.AdamOptimizer(learning_rate)
-
    
-
    
-The program using the Adam optimizer with a different initial learning rate yields indeed an interesting result:
-
    
-
-
-
    Max absolute difference between the analytical solution and solution from TensorFlow DNN: 7.11243e-05
-
    
-
    
-









    
-
-
    Partial Differential Equations 
    
 A partial differential equation (PDE) has a solution here the function is defined by multiple variables.
 The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.
 
@@ -3388,6 +2001,11 @@ $$
 
    
 where \( f \) is an expression involving all kinds of possible mixed derivatives of \( g(x_1,\dots,x_N) \) up to an order \( n \). In order for the solution to be unique, some additional conditions must also be given.
 
+
    
+









    
+
+
    Type of problem 
    
+
 
    
 The problem our network must solve for, is similar to the ODE case.
 We must have a trial solution \( g_t \) at hand.
@@ -3407,11 +2025,21 @@ The neural network \( N(x_1,\dots,x_N,P) \) has weights and biases described by
 The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensure that the output of \( N(x_1,\dots,x_N,P) \) is zero when \( g_t(x_1,\dots,x_N) \) is evaluated at the values of \( x_1,\dots,x_N \) where the given conditions must be satisfied. The function \( h_1(x_1,\dots,x_N) \) should alone make \( g_t(x_1,\dots,x_N) \) satisfy the conditions.
 
 
    
-The network tries then the minimize the cost function following the same ideas as described for the ODE case, but now with more than one variables to consider.
-The concept still remains the same; find a set of parameters \( P \) such that the expression \( f \) in \eqref{PDE} is as close to zero as possible.
+









    
+
+
    Network requirements 
    
 
 
    
-As for the ODE case, the cost function is the mean squared error that the network must try to minimize. The cost function for the network to minimize is
+The network tries then the minimize the cost function following the
+same ideas as described for the ODE case, but now with more than one
+variables to consider.  The concept still remains the same; find a set
+of parameters \( P \) such that the expression \( f \) in \eqref{PDE} is as
+close to zero as possible.
+
+
    
+As for the ODE case, the cost function is the mean squared error that
+the network must try to minimize. The cost function for the network to
+minimize is
 
 $$
 \begin{equation*}
@@ -3419,6 +2047,11 @@ c\left(x_1, \dots, x_N, P\right) = \left(  f\left(x_1, \, \dots \, , x_N, \frac{
 \end{equation*}
 $$
 
+
    
+









    
+
+
    More details 
    
+
 
    
 If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
 $$
@@ -3438,7 +2071,7 @@ $$
 
    
 









    
 
-
    Example: The diffusion equation 
    
+
    Example: The diffusion equation 
    
 
 
    
 In one spatial dimension, the equation reads
@@ -3463,7 +2096,7 @@ with \( u(x) \) being some given function.
 
    
 









    
 
-
    Defining the problem 
    
+
    Defining the problem 
    
 
 
    
 For this case, we want to find \( g(x,t) \) such that
@@ -3495,13 +2128,16 @@ First, we will look into how Autograd could be used in a network tailored to sol
 
    
 









    
 
-
    Setting up the network using Autograd 
    
+
    Setting up the network using Autograd 
    
 
 
    
-The only change to do here, is to extend our network such that functions of multiple parameters are correctly handled.
-In this case we have two variables in our function to solve for, that is time \( t \) and position \( x \).
-The variables will be represented by a one-dimensional array in the program.
-The program will evaluate the network at each possible pair \( (x,t) \), given an array for the desired \( x \)-values and \( t \)-values to approximate the solution at.
+The only change to do here, is to extend our network such that
+functions of multiple parameters are correctly handled.  In this case
+we have two variables in our function to solve for, that is time \( t \)
+and position \( x \).  The variables will be represented by a
+one-dimensional array in the program.  The program will evaluate the
+network at each possible pair \( (x,t) \), given an array for the desired
+\( x \)-values and \( t \)-values to approximate the solution at.
 
 
    
 
@@ -3554,8 +2190,13 @@ The program will evaluate the network at each possible pair \( (x,t) \), given a
 
    
 









    
 
-
    Setting up the network using Autograd; The trial solution 
    
-The cost function must then iterate through the given arrays containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep neural network and the trial solution is evaluated at, and then finds the Jacobian of the trial solution.
+
    Setting up the network using Autograd; The trial solution 
    
+
+
    
+The cost function must then iterate through the given arrays
+containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep
+neural network and the trial solution is evaluated at, and then finds
+the Jacobian of the trial solution.
 
 
    
 A possible trial solution for this PDE is
@@ -3576,16 +2217,31 @@ $$
 since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
 
 
    
-The Jacobian is used because the program must find the derivative of the trial solution with respect to \( x \) and \( t \).
+









    
+
+
    Why the jacobian? 
    
 
 
    
-This gives the necessity of computing the Jacobian matrix, as we want to evaluate the gradient with respect to \( x \) and \( t \) (note that the Jacobian of a scalar-valued multivariate function is simply its gradient).
+The Jacobian is used because the program must find the derivative of
+the trial solution with respect to \( x \) and \( t \).
 
 
    
-In Autograd, the differentiation is by default done with respect to the first input argument of your Python function. Since the points is an array representing \( x \) and \( t \), the Jacobian is calculated using the values of \( x \) and \( t \).
+This gives the necessity of computing the Jacobian matrix, as we want
+to evaluate the gradient with respect to \( x \) and \( t \) (note that the
+Jacobian of a scalar-valued multivariate function is simply its
+gradient).
 
 
    
-To find the second derivative with respect to \( x \) and \( t \), the Jacobian can be found for the second time. The result is a Hessian matrix, which is the matrix containing all the possible second order mixed derivatives of \( g(x,t) \).
+In Autograd, the differentiation is by default done with respect to
+the first input argument of your Python function. Since the points is
+an array representing \( x \) and \( t \), the Jacobian is calculated using
+the values of \( x \) and \( t \).
+
+
    
+To find the second derivative with respect to \( x \) and \( t \), the
+Jacobian can be found for the second time. The result is a Hessian
+matrix, which is the matrix containing all the possible second order
+mixed derivatives of \( g(x,t) \).
 
 
    
 
@@ -3630,7 +2286,9 @@ To find the second derivative with respect to \( x \) and \( t \), the Jacobian
 
    
 









    
 
-
    Setting up the network using Autograd; The full program 
    
+
    Setting up the network using Autograd; The full program 
    
+
+
    
 Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.
 
 
    
@@ -3645,12 +2303,9 @@ A possible way to implement a neural network solving the PDE, is given below.
 Be aware, though, that it is fairly slow for the parameters used.
 A better result is possible, but requires more iterations, and thus longer time to complete.
 
-
    
-Using only 20 neurons in one hidden layer, the program managed to make the trial solution have the maximum absolute error of 0.0075. The execution time, however, was approximately one day and 14 hours on a computer having Intel i7-7560U 2.4 GHz CPU.
-
 
    
 Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.
-Using TensorFlow in the next example sovling the wave equation, has a much better execution time.
+Using TensorFlow results in a much better execution time. Try it!
 
 
    
 
@@ -3882,7 +2537,7 @@ Using TensorFlow in the next example sovling the wave equation, has a much bette
 
    
 









    
 
-
    Example: Solving the wave equation using Autograd and TensorFlow 
    
+
    Example: Solving the wave equation with Neural Networks 
    
 
 
    
 The wave equation is
@@ -3911,7 +2566,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
 
    
 









    
 
-
    The problem to solve for 
    
+
    The problem to solve for 
    
 
 
    
 The wave equation to solve for, is
@@ -3941,7 +2596,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
 
    
 









    
 
-
    The trial solution 
    
+
    The trial solution 
    
 Setting up the network is done in similar matter as for the example of solving the diffusion equation.
 The only things we have to change, is the trial solution such that it satisfies the conditions from \eqref{condwave} and the cost function.
 
@@ -3965,7 +2620,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
 
    
 









    
 
-
    The analytical solution 
    
+
    The analytical solution 
    
 
 
    
 The analytical solution for our specific problem, is
@@ -3977,7 +2632,7 @@ $$
 
    
 









    
 
-
    Solving the wave equation - the full program using Autograd 
    
+
    Solving the wave equation - the full program using Autograd 
    
 
 
    
 
@@ -4206,194 +2861,1041 @@ $$
 
    
 









    
 
-
    Solving the wave equation - the full program using TensorFlow 
    
-As the program using Autograd is fairly slow, one could hope that using TensorFlow
-could make a naive implementation faster, and more numerically robust.
-
-
    
-In addition, having TensorFlow at hand, it could be easier to experiment with different
-optimization algorithms, and other constructions of the network.
-
-
    
-The following program solves the given wave equation much faster,
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-Nx = 10
-x_np = np.linspace(0,1,Nx)
-
-Nt = 10
-t_np = np.linspace(0,1,Nt)
-
-X,T = np.meshgrid(x_np, t_np)
-
-x = X.ravel()
-t = T.ravel()
-
-## The construction phase
-
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(x.shape)),shape=(-1,1))
-x = tf.reshape(tf.convert_to_tensor(x),shape=(-1,1))
-t = tf.reshape(tf.convert_to_tensor(t),shape=(-1,1))
-
-points = tf.concat([x,t],1)
-
-num_iter = 100000
-num_hidden_neurons = [90]
-
-X = tf.convert_to_tensor(X)
-T = tf.convert_to_tensor(T)
-
-
-with tf.variable_scope('dnn'):
-    num_hidden_layers = np.size(num_hidden_neurons)
-
-    previous_layer = points
-
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l],activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    dnn_output = tf.layers.dense(previous_layer, 1)
-
-
-def u(x):
-    return tf.sin(np.pi*x)
-
-def v(x):
-    return -np.pi*tf.sin(np.pi*x)
-
-with tf.name_scope('loss'):
-    g_trial = (1 - t**2)*u(x) + t*v(x) + x*(1-x)*t**2*dnn_output
-
-    g_trial_d2t =  tf.gradients(tf.gradients(g_trial,t),t)
-    g_trial_d2x = tf.gradients(tf.gradients(g_trial,x),x)
-
-    loss = tf.losses.mean_squared_error(zeros, g_trial_d2t[0] - g_trial_d2x[0])
-
-learning_rate = 0.01
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(loss)
-
-init = tf.global_variables_initializer()
-
-g_analytic = tf.sin(np.pi*x)*tf.cos(np.pi*t) - tf.sin(np.pi*x)*tf.sin(np.pi*t)
-g_dnn = None
-
-## The execution phase
-with tf.Session() as sess:
-    init.run()
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves during training
-        #if i % 100 == 0:
-        #    print(loss.eval())
-
-    g_analytic = g_analytic.eval()
-    g_dnn = g_trial.eval()
-
-
-## Compare with the analutical solution
-diff = np.abs(g_analytic - g_dnn)
-print('Max absolute difference between analytical solution and TensorFlow DNN = ',np.max(diff))
-
-G_analytic = g_analytic.reshape((Nt,Nx))
-G_dnn = g_dnn.reshape((Nt,Nx))
-
-diff = np.abs(G_analytic - G_dnn)
-
-# Plot the results
-
-X,T = np.meshgrid(x_np, t_np)
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-s = ax.plot_surface(X,T,G_dnn,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Analytical solution')
-s = ax.plot_surface(X,T,G_analytic,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Difference')
-s = ax.plot_surface(X,T,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-## Take some 3D slices
-
-indx1 = 0
-indx2 = int(Nt/2)
-indx3 = Nt-1
-
-t1 = t_np[indx1]
-t2 = t_np[indx2]
-t3 = t_np[indx3]
-
-# Slice the results from the DNN
-res1 = G_dnn[indx1,:]
-res2 = G_dnn[indx2,:]
-res3 = G_dnn[indx3,:]
-
-# Slice the analytical results
-res_analytical1 = G_analytic[indx1,:]
-res_analytical2 = G_analytic[indx2,:]
-res_analytical3 = G_analytic[indx3,:]
-
-# Plot the slices
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t1)
-plt.plot(x_np, res1)
-plt.plot(x_np,res_analytical1)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t2)
-plt.plot(x_np, res2)
-plt.plot(x_np,res_analytical2)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t3)
-plt.plot(x_np, res3)
-plt.plot(x_np,res_analytical3)
-plt.legend(['dnn','analytical'])
-
-plt.show()
-
    
-
    
-The program manages to find a solution having max absolute difference to the analytical
-at approximately 0.0059, by just using some minutes!
-It was found, by some testing, that one hidden layer with 90 neurons actually performed well.
-
-
    
-









    
-
-
    Resources 
    
+
    Resources on differential equations and deep learning 
    
 
 
      
 
    1.  Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al
      2. 
 
    2.  Neural networks for solving differential equations by A. Honchar
      3. 
 
    3.  Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener
      4. 
-
    4.  Introduction to Partial Differential Equations by A. Tveitio, R. Winther
      5. 
+
    5.  Introduction to Partial Differential Equations by A. Tveito, R. Winther
      6. 
 
    
 
+









    
+
+
    Friday, Principal Component Analysis 
    
+
+
    
+









    
+
+
    Basic ideas of the Principal Component Analysis (PCA) 
    
+
+
    
+The principal component analysis deals with the problem of fitting a
+low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than
+the totaldimension \( D \) of the problem at hand (our data
+set). Mathematically it can be formulated as a statistical problem or
+a geometric problem.  In our discussion of the theorem for the
+classical PCA, we will stay with a statistical approach. This is also
+what set the scene historically which for the PCA.
+
+
    
+We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) 
+
+
      
+
    •  Each data point is determined by \( p \) extrinsic (measurement) variables
      • 
+
    •  We may want to ask the following question: Are there fewer intrinsic variables (say \( d < < p \)) that still approximately describe the data?
      • 
+
    •  If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.
      •  
+
    
+
+









    
+
+
    Introducing the Covariance and Correlation functions  
    
+
+
    
+Before we discuss the PCA theorem, we need to remind ourselves about
+the definition of the covariance and the correlation function. These are quantities
+
+
    
+Suppose we have defined two vectors
+\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as 
+$$
+\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
+             \end{bmatrix},
+$$
+
+where for example
+$$
+\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
+$$
+
+With this definition and recalling that the variance is defined as
+$$
+\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
+$$
+
+we can rewrite the covariance matrix as 
+$$
+\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
+             \end{bmatrix}.
+$$
+
+
    
+The covariance takes values between zero and infinity and may thus
+lead to problems with loss of numerical precision for particularly
+large values. It is common to scale the covariance matrix by
+introducing instead the correlation matrix defined via the so-called
+correlation function
+
+$$
+\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
+$$
+
+
    
+The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
+\in [-1,1] \). This avoids eventual problems with too large values. We
+can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
+and \( \boldsymbol{y} \) as
+
+$$
+\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
+             \end{bmatrix},
+$$
+
+
    
+In the above example this is the function we constructed using pandas.
+
+
    
+









    
+
+
    Correlation Function and Design/Feature Matrix 
    
+
+
    
+In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression
+we defined the design/feature matrix \( \boldsymbol{X} \) as
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  refering to the column numbers and the
+entries \( n \) being the row elements.
+We can rewrite the design/feature matrix in terms of its column vectors as
+$$
+\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix},
+$$
+
+with a given vector
+$$
+\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
+$$
+
+
    
+With these definitions, we can now rewrite our \( 2\times 2 \)
+correaltion/covariance matrix in terms of a moe general design/feature
+matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \)
+covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \)
+
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
+\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
+\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
+\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & \mathrm{var}[\boldsymbol{x}_{p-1}]\\
+\end{bmatrix},
+$$
+
+and the correlation matrix
+$$
+\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
+1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
+\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1  & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
+\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & 1\\
+\end{bmatrix},
+$$
+
+
    
+









    
+
+
    Covariance Matrix Examples 
    
+
+
    
+The Numpy function np.cov calculates the covariance elements using
+the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
+the exact mean values.  The following simple function uses the
+np.vstack function which takes each vector of dimension \( 1\times n \)
+and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
+
+$$
+\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\
+                          x_1 & y_1 \\
+                          x_2 & y_2\\
+                          \dots & \dots \\
+                          x_{n-2} & y_{n-2}\\
+                          x_{n-1} & y_{n-1} & 
+             \end{bmatrix},
+$$
+
+
    
+which in turn is converted into into the \( 2\times 2 \) covariance matrix
+\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate
+the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
+function np.mean(x). We can also extract the eigenvalues of the
+covariance matrix through the np.linalg.eig() function.
+
+
    
+
+
+
    # Importing various packages
+import numpy as np
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
+
    
+
    
+









    
+
+
    Correlation Matrix  
    
+
+
    
+The previous example can be converted into the correlation matrix by
+simply scaling the matrix elements with the variances.  We should also
+subtract the mean values for each column. This leads to the following
+code which sets up the correlations matrix for the previous example in
+a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
+
+
    
+
+
+
    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    
+
    
+We see that the matrix elements along the diagonal are one as they
+should be and that the matrix is symmetric. Furthermore, diagonalizing
+this matrix we easily see that it is a positive definite matrix.
+
+
    
+The above procedure with numpy can be made more compact if we use pandas.
+
+
    
+









    
+
+
    Correlation Matrix with Pandas 
    
+
+
    
+We whow here how we can set up the correlation matrix using pandas, as done in this simple code
+
    
+
+
+
    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    
+
    
+We expand this model to the Franke function discussed above.
+
+
    
+









    
+
+
    Correlation Matrix with Pandas and the Franke function 
    
+
+
    
+
+
+
    # Common imports
+import numpy as np
+import pandas as pd
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    
+
    
+We note here that the covariance is zero for the first rows and
+columns since all matrix elements in the design matrix were set to one
+(we are fitting the function in terms of a polynomial of degree \( n \)).
+
+
    
+This means that the variance for these elements will be zero and will
+cause problems when we set up the correlation matrix.  We can simply
+drop these elements and construct a correlation
+matrix without these elements.
+
+
    
+









    
+
+
    Rewriting the Covariance and/or Correlation Matrix 
    
+
+
    
+We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as 
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
+$$
+
+
    
+To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{00} & x_{01}\\
+x_{10} & x_{11}\\
+\end{bmatrix}=\begin{bmatrix}
+\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
+\end{bmatrix}.
+$$
+
+
    
+If we then compute the expectation value
+$$
+\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix}
+x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
+x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
+\end{bmatrix},
+$$
+
+which is just 
+$$
+\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
+                              \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
+             \end{bmatrix},
+$$
+
+where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
+
+
    
+It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
+
+
    
+









    
+
+
    Towards the PCA theorem 
    
+
+
    
+We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
+$$
+
+Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \).
+These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
+
+
    
+Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
+
+
    
+That is we have
+$$
+\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
+$$
+
+since the matrix \( \boldsymbol{S} \) is not a data dependent matrix.   Multiplying with \( \boldsymbol{S}^T \) from the left we have
+$$
+\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
+$$
+
+and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
+
+$$
+\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
+$$
+
+
    
+In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
+\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
+
+
    
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If \( p \) is very large,
+we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
+features/predictors.
+
+
    
+









    
+
+
    The Algorithm before theorem 
    
+
+
    
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+
+
      
+
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
+
    
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+
+
      
+
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
+
    •  Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).
      • 
+
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
+
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
+
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
+
    
+
+









    
+
+
    Writing our own PCA code 
    
+
+
    
+We will use a simple example first with two-dimensional data
+drawn from a multivariate normal distribution with the following mean and covariance matrix:
+$$
+\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
+2 & 2
+\end{bmatrix}
+$$
+
+Note that the mean refers to each column of data. 
+We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from
+this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \).
+
+
    
+The following Python code aids in setting up the data and writing out the design matrix.
+Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
+
    
+
+
+
    import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from IPython.display import display
+n = 10000
+mean = (-1, 2)
+cov = [[4, 2], [2, 2]]
+X = np.random.multivariate_normal(mean, cov, n)
+
    
+
    
+Now we are going to implement the PCA algorithm. We will break it down into various substeps.
+
+
    Compute the sample mean and center the data 
    
+
+
    
+The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
+$$
+\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
+$$
+
+and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
+$$
+\bar{x}_i = x_i - \mu_n.
+$$
+
+When you are done with these steps, print out \( \mu_n \) to verify it is
+close to \( \mu \) and plot your mean centered data to verify it is
+centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above.
+The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. 
+
    
+
+
+
    df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+
    
+
    
+Alternatively, we could use the functions we discussed
+earlier for scaling the data set.  That is, we could have used the
+StandardScaler function in Scikit-Learn, a function which ensures
+that for each feature/predictor we study the mean value is zero and
+the variance is one (every column in the design/feature matrix).  You
+would then not get the same results, since we divide by the
+variance. The diagonal covariance matrix elements will then be one,
+while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
+specific case.
+
+
    Compute the sample covariance 
    
+
+
    
+Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
+$$
+\begin{equation*}
+\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
+\end{equation*}
+$$
+
+where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \).
+We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows
+
    
+
+
+
    print(df.cov())
+print(np.cov(X_centered.T))
+
    
+
    
+Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. 
+Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. 
+
    
+
+
+
    # extract the relevant columns from the centered design matrix of dim n x 2
+x = X_centered[:,0]
+y = X_centered[:,1]
+Cov = np.zeros((2,2))
+Cov[0,1] = np.sum(x.T@y)/(n-1.0)
+Cov[0,0] = np.sum(x.T@x)/(n-1.0)
+Cov[1,1] = np.sum(y.T@y)/(n-1.0)
+Cov[1,0]= Cov[0,1]
+print("Centered covariance using own code")
+print(Cov)
+plt.plot(x, y, 'x')
+plt.axis('equal')
+plt.show()
+
    
+
    
+Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
+The plot shows how the data are clustered around a line with slope close to one. Is this expected?
+
+
    Diagonalize the sample covariance matrix to obtain the principal components 
    
+
+
    
+Now we are ready to solve for the principal components! To do so we
+diagonalize the sample covariance matrix \( \Sigma \). We can use the
+function np.linalg.eig to do so. It will return the eigenvalues and
+eigenvectors of \( \Sigma \). Once we have these we can perform the 
+following tasks:
+
+
      
+
    •  We compute the percentage of the total variance captured by the first principal component
      • 
+
    •  We plot the mean centered data and lines along the first and second principal components
      • 
+
    •  Then we project the mean centered data onto the first and second principal components, and plot the projected data.
      •  
+
    •  Finally, we approximate the data as
      • 
+
    
+
+$$
+\begin{equation*}
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
+\end{equation*}
+$$
+
+where \( v_0 \) is the first principal component.
+
+
    
+Collecting all these steps we can write our own PCA function and
+compare this with the functionality included in Scikit-Learn.
+
+
    
+The code here outlines some of the elements we could include in the
+analysis. Feel free to extend upon this in order to address the above
+questions.
+
+
    
+
+
+
    # diagonalize and obtain eigenvalues, not necessarily sorted
+EigValues, EigVectors = np.linalg.eig(Cov)
+# sort eigenvectors and eigenvalues
+#permute = EigValues.argsort()
+#EigValues = EigValues[permute]
+#EigVectors = EigVectors[:,permute]
+print("Eigenvalues of Covariance matrix")
+for i in range(2):
+    print(EigValues[i])
+FirstEigvector = EigVectors[:,0]
+SecondEigvector = EigVectors[:,1]
+print("First eigenvector")
+print(FirstEigvector)
+print("Second eigenvector")
+print(SecondEigvector)
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2Dsl = pca.fit_transform(X)
+print("Eigenvector of largest eigenvalue")
+print(pca.components_.T[:, 0])
+
    
+
    
+This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code.  Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?
+
+
    
+









    
+
+
    Classical PCA Theorem   
    
+
+
    
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
+centered as discussed above. For the sake of simplicity we skip the
+overline symbol. The matrix is defined in terms of the various column
+vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension
+\( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+
    
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2,
+$$
+
+with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+
+
    
+The PCA theorem states that minimizing the above reconstruction error
+corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which
+diagonalizes the empirical covariance(correlation) matrix. The optimal
+low-dimensional encoding of the data is then given by a set of vectors
+\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the
+orthogonal projection of the data onto the columns spanned by the
+eigenvectors of the covariance(correlations matrix).
+
+
    
+The proof which follows will be updated by mid January 2020.
+
+
    
+









    
+
+
    Proof of the PCA Theorem  
    
+
+
    
+To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+
+where the vectors on the rhs are known.
+
+
    
+









    
+
+
    PCA Proof continued 
    
+
+
    
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
+$$
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
+$$
+
+
    
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+$$
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
+$$
+
+since the expectation value of
+$$
+\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0,
+$$
+
+where we have used the fact that our data are centered.
+
+
    
+Recalling our definition of the covariance as 
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+$$
+
+we have thus that
+$$
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+$$
+
+
    
+We are almost there, we have obtained a relation between minimizing
+the reconstruction error and the variance and the covariance
+matrix. Minimizing the error is equivalent to maximizing the variance
+of the projected data.
+
+
    
+









    
+
+
    The final step 
    
+
+
    
+We could trivially maximize the variance of the projection (and
+thereby minimize the error in the reconstruction function) by letting
+the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we
+want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by
+\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a
+Lagrange multiplier we can then in turn maximize
+
+$$
+J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0).
+$$
+
+Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain
+
+$$
+\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0,
+$$
+
+meaning that
+$$
+\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0.
+$$
+
+The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is
+$$
+\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0.
+$$
+
+
    
+If we want to maximize the variance (minimize the construction error)
+we simply pick the eigenvector of the covariance matrix with the
+largest eigenvalue. This establishes the link between the minimization
+of the reconstruction function \( J \) in terms of an orthogonal matrix
+and the maximization of the variance and thereby the covariance of our
+observations encoded in the design/feature matrix \( \boldsymbol{X} \).
+
+
    
+The proof
+for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be
+established by applying the above arguments and using the fact that
+our basis of eigenvectors is orthogonal, see Murphy chapter
+12.2.  The
+discussion in chapter 12.2 of Murphy's text has also a nice link with
+the Singular Value Decomposition theorem. For categorical data, see
+chapter 12.4 and discussion therein.
+
+
    
+Additional part of the proof for the other eigenvectors will be added by mid January 2020.
+
+
    
+









    
+
+
    Geometric Interpretation and link with Singular Value Decomposition 
    
+
+
    
+This material will be added by mid January 2020.
+
+
    
+









    
+
+
    Principal Component Analysis 
    
+
+
    
+Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
+First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
+
+
    
+The following Python code uses NumPy’s svd() function to obtain all the principal components of the
+training set, then extracts the first two principal components. First we center the data using either pandas or our own code
+
    
+
+
+
    import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 vanilla matrix 
+rows = 10
+cols = 5
+X = np.random.randn(rows,cols)
+df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+display(df)
+
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+# Then check the difference between pandas and our own set up
+print(X_centered-df)
+#Now we do an SVD
+U, s, V = np.linalg.svd(X_centered)
+c1 = V.T[:, 0]
+c2 = V.T[:, 1]
+W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+print(X2D)
+
    
+
    
+PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
+the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
+forget to center the data first.
+
+
    
+Once you have identified all the principal components, you can reduce the dimensionality of the dataset
+down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
+Selecting this hyperplane ensures that the projection will preserve as much variance as possible. 
+
    
+
+
+
    W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+
    
+
    
+
+
+
    PCA and scikit-learn 
    
+
+
    
+Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
+following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
+that it automatically takes care of centering the data):
+
    
+
+
+
    #thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D = pca.fit_transform(X)
+print(X2D)
+
    
+
    
+After fitting the PCA transformer to the dataset, you can access the principal components using the
+components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
+principal component is equal to 
+
    
+
+
+
    pca.components_.T[:, 0].
+
    
+
    
+Another very useful piece of information is the explained variance ratio of each principal component,
+available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
+variance that lies along the axis of each principal component.
+
+
    
+









    
+
+
    Back to the Cancer Data 
    
+We can now repeat the above but applied to real data, in this case our breast cancer data.
+Here we compute performance scores on the training data using logistic regression.
+
    
+
+
+
    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.linear_model import LogisticRegression
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+
+logreg = LogisticRegression()
+logreg.fit(X_train, y_train)
+print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
+# We scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Then perform again a log reg fit
+logreg.fit(X_train_scaled, y_train)
+print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D_train = pca.fit_transform(X_train_scaled)
+# and finally compute the log reg fit and the score on the training data	
+logreg.fit(X2D_train,y_train)
+print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
+
    
+
    
+We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.
+
+
    
+









    
+
+
    More on the PCA 
    
+
+
    
+Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
+choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
+Unless, of course, you are reducing dimensionality for data visualization — in that case you will
+generally want to reduce the dimensionality down to 2 or 3.
+The following code computes PCA without reducing dimensionality, then computes the minimum number
+of dimensions required to preserve 95% of the training set’s variance:
+
    
+
+
+
    pca = PCA()
+pca.fit(X)
+cumsum = np.cumsum(pca.explained_variance_ratio_)
+d = np.argmax(cumsum >= 0.95) + 1
+
    
+
    
+You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
+of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
+a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
+
    
+
+
+
    pca = PCA(n_components=0.95)
+X_reduced = pca.fit_transform(X)
+
    
+
    
+









    
+
+
    Incremental PCA 
    
+
+
    
+One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
+memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
+been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
+at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
+instances arrive).
+
+
    
+









    
+
+
    Randomized PCA 
    
+
+
    
+Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
+algorithm that quickly finds an approximation of the first d principal components. Its computational
+complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
+previous algorithms when \( d \) is much smaller than \( n \).
+
+
    
+









    
+
+
    Kernel PCA 
    
+
    
+
+
    
+
+
    
+The kernel trick is a mathematical technique that implicitly maps instances into a
+very high-dimensional space (called the feature space), enabling nonlinear classification and regression
+with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
+space corresponds to a complex nonlinear decision boundary in the original space.
+It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
+projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
+preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
+twisted manifold.
+For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
+
    
+
+
+
    from sklearn.decomposition import KernelPCA
+rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
+X_reduced = rbf_pca.fit_transform(X)
+
    
+
+
    
+
+
+
    
+









    
+
+
    LLE 
    
+
+
    
+Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
+(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
+algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
+closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
+these local relationships are best preserved (more details shortly).
+
+
    
+









    
+
+
    Other techniques 
    
+
+
    
+There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
+
+
    
+Here are some of the most popular:
+
+
      
+
    •  Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • 
+
    •  Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • 
+
    •  t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • 
+
    •  Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • 
+
    
+
 
 
 
diff --git a/doc/pub/week43/html/week43.html b/doc/pub/week43/html/week43.html
index 56e8899d8..b001118b9 100644
--- a/doc/pub/week43/html/week43.html
+++ b/doc/pub/week43/html/week43.html
@@ -94,142 +94,111 @@ div { text-align: justify; text-justify: inter-word; }
               ('Final technicalities IV', 2, None, '___sec18'),
               ('Backpropagation', 2, None, '___sec19'),
               ('Gradient descent', 2, None, '___sec20'),
-              ('Basic ideas of the Principal Component Analysis (PCA)',
-               2,
-               None,
-               '___sec21'),
-              ('Introducing the Covariance and Correlation functions',
+              ('The code for solving the ODE', 2, None, '___sec21'),
+              ('The network with one input layer, specified number of hidden '
+               'layers, and one output layer',
                2,
                None,
                '___sec22'),
-              ('Correlation Function and Design/Feature Matrix',
-               2,
-               None,
-               '___sec23'),
-              ('Covariance Matrix Examples', 2, None, '___sec24'),
-              ('Correlation Matrix', 2, None, '___sec25'),
-              ('Correlation Matrix with Pandas', 2, None, '___sec26'),
-              ('Correlation Matrix with Pandas and the Franke function',
-               2,
-               None,
-               '___sec27'),
-              ('Rewriting the Covariance and/or Correlation Matrix',
-               2,
-               None,
-               '___sec28'),
-              ('Towards the PCA theorem', 2, None, '___sec29'),
-              ('The Algorithm before the Theorem', 2, None, '___sec30'),
-              ('Writing our own PCA code', 2, None, '___sec31'),
-              ('Compute the sample mean and center the data',
-               3,
-               None,
-               '___sec32'),
-              ('Compute the sample covariance', 3, None, '___sec33'),
-              ('Diagonalize the sample covariance matrix to obtain the '
-               'principal components',
-               3,
-               None,
-               '___sec34'),
-              ('Classical PCA Theorem', 2, None, '___sec35'),
-              ('Proof of the PCA Theorem', 2, None, '___sec36'),
-              ('PCA Proof continued', 2, None, '___sec37'),
-              ('The final step', 2, None, '___sec38'),
-              ('Geometric Interpretation and link with Singular Value '
-               'Decomposition',
-               2,
-               None,
-               '___sec39'),
-              ('Principal Component Analysis', 2, None, '___sec40'),
-              ('PCA and scikit-learn', 2, None, '___sec41'),
-              ('Back to the Cancer Data', 2, None, '___sec42'),
-              ('More on the PCA', 2, None, '___sec43'),
-              ('Incremental PCA', 2, None, '___sec44'),
-              ('Randomized PCA', 2, None, '___sec45'),
-              ('Kernel PCA', 2, None, '___sec46'),
-              ('LLE', 2, None, '___sec47'),
-              ('Other techniques', 2, None, '___sec48'),
-              ('The network with one input, hidden, and output layer',
-               2,
-               None,
-               '___sec49'),
-              ('The network with one input layer, specified number of hidden '
-               'layers, and one output layer  output layer',
-               2,
-               None,
-               '___sec50'),
-              ('Example: Population growth, comparing Autograd, TensorFlow, '
-               "and Euler's scheme",
-               2,
-               None,
-               '___sec51'),
-              ('Setting up the problem', 2, None, '___sec52'),
-              ('The trial solution', 2, None, '___sec53'),
-              ('The program using Autograd', 2, None, '___sec54'),
-              ('Using forward Euler to solve the ODE', 2, None, '___sec55'),
-              ('Using TensorFlow to model logistic population growth',
-               2,
-               None,
-               '___sec56'),
-              ('The general program flow in TensorFlow', 2, None, '___sec57'),
-              ('Program flow in TensorFlow - Construction phase',
-               2,
-               None,
-               '___sec58'),
-              ('Program flow in TensorFlow - Execution phase',
-               2,
-               None,
-               '___sec59'),
-              ('The full program modeling logistic population growth using '
-               'TensorFlow',
-               2,
-               None,
-               '___sec60'),
+              ('Example: Population growth', 2, None, '___sec23'),
+              ('Setting up the problem', 2, None, '___sec24'),
+              ('The trial solution', 2, None, '___sec25'),
+              ('The program using Autograd', 2, None, '___sec26'),
+              ('Using forward Euler to solve the ODE', 2, None, '___sec27'),
               ('Example: Solving the one dimensional Poisson equation using '
                'Autograd and TensorFlow',
                2,
                None,
-               '___sec61'),
-              ('The specific equation to solve for', 2, None, '___sec62'),
-              ('Solving the equation using Autograd', 2, None, '___sec63'),
-              ('Comparing with a numerical scheme', 2, None, '___sec64'),
-              ('Using gradient descent in TensorFlow to solve Poisson equation',
-               2,
-               None,
-               '___sec65'),
-              ('Using a different optimization algorithm implemented in '
-               'TensorFlow to solve Poisson equation',
-               2,
-               None,
-               '___sec66'),
-              ('Partial Differential Equations', 2, None, '___sec67'),
-              ('Example: The diffusion equation', 2, None, '___sec68'),
-              ('Defining the problem', 2, None, '___sec69'),
-              ('Setting up the network using Autograd', 2, None, '___sec70'),
+               '___sec28'),
+              ('The specific equation to solve for', 2, None, '___sec29'),
+              ('Solving the equation using Autograd', 2, None, '___sec30'),
+              ('Comparing with a numerical scheme', 2, None, '___sec31'),
+              ('Partial Differential Equations', 2, None, '___sec32'),
+              ('Type of problem', 2, None, '___sec33'),
+              ('Network requirements', 2, None, '___sec34'),
+              ('More details', 2, None, '___sec35'),
+              ('Example: The diffusion equation', 2, None, '___sec36'),
+              ('Defining the problem', 2, None, '___sec37'),
+              ('Setting up the network using Autograd', 2, None, '___sec38'),
               ('Setting up the network using Autograd; The trial solution',
                2,
                None,
-               '___sec71'),
+               '___sec39'),
+              ('Why the jacobian?', 2, None, '___sec40'),
               ('Setting up the network using Autograd; The full program',
                2,
                None,
-               '___sec72'),
-              ('Example: Solving the wave equation using Autograd and '
-               'TensorFlow',
+               '___sec41'),
+              ('Example: Solving the wave equation with Neural Networks',
                2,
                None,
-               '___sec73'),
-              ('The problem to solve for', 2, None, '___sec74'),
-              ('The trial solution', 2, None, '___sec75'),
-              ('The analytical solution', 2, None, '___sec76'),
+               '___sec42'),
+              ('The problem to solve for', 2, None, '___sec43'),
+              ('The trial solution', 2, None, '___sec44'),
+              ('The analytical solution', 2, None, '___sec45'),
               ('Solving the wave equation - the full program using Autograd',
                2,
                None,
-               '___sec77'),
-              ('Solving the wave equation - the full program using TensorFlow',
+               '___sec46'),
+              ('Resources on differential equations and deep learning',
                2,
                None,
-               '___sec78'),
-              ('Resources', 2, None, '___sec79')]}
+               '___sec47'),
+              ('Friday, Principal Component Analysis', 2, None, '___sec48'),
+              ('Basic ideas of the Principal Component Analysis (PCA)',
+               2,
+               None,
+               '___sec49'),
+              ('Introducing the Covariance and Correlation functions',
+               2,
+               None,
+               '___sec50'),
+              ('Correlation Function and Design/Feature Matrix',
+               2,
+               None,
+               '___sec51'),
+              ('Covariance Matrix Examples', 2, None, '___sec52'),
+              ('Correlation Matrix', 2, None, '___sec53'),
+              ('Correlation Matrix with Pandas', 2, None, '___sec54'),
+              ('Correlation Matrix with Pandas and the Franke function',
+               2,
+               None,
+               '___sec55'),
+              ('Rewriting the Covariance and/or Correlation Matrix',
+               2,
+               None,
+               '___sec56'),
+              ('Towards the PCA theorem', 2, None, '___sec57'),
+              ('The Algorithm before the Theorem', 2, None, '___sec58'),
+              ('Writing our own PCA code', 2, None, '___sec59'),
+              ('Compute the sample mean and center the data',
+               3,
+               None,
+               '___sec60'),
+              ('Compute the sample covariance', 3, None, '___sec61'),
+              ('Diagonalize the sample covariance matrix to obtain the '
+               'principal components',
+               3,
+               None,
+               '___sec62'),
+              ('Classical PCA Theorem', 2, None, '___sec63'),
+              ('Proof of the PCA Theorem', 2, None, '___sec64'),
+              ('PCA Proof continued', 2, None, '___sec65'),
+              ('The final step', 2, None, '___sec66'),
+              ('Geometric Interpretation and link with Singular Value '
+               'Decomposition',
+               2,
+               None,
+               '___sec67'),
+              ('Principal Component Analysis', 2, None, '___sec68'),
+              ('PCA and scikit-learn', 2, None, '___sec69'),
+              ('Back to the Cancer Data', 2, None, '___sec70'),
+              ('More on the PCA', 2, None, '___sec71'),
+              ('Incremental PCA', 2, None, '___sec72'),
+              ('Randomized PCA', 2, None, '___sec73'),
+              ('Kernel PCA', 2, None, '___sec74'),
+              ('LLE', 2, None, '___sec75'),
+              ('Other techniques', 2, None, '___sec76')]}
 end of tocinfo -->
 
 
@@ -568,17 +537,17 @@ $$
 $$
 
 
    
-Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i  \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes
+Letting \( \boldsymbol{x} \) be a vector with elements \( x_i \) and \( C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i  \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes
 
 $$
-\min_{P} c(\boldsymbol{x}, P)
+\min_{P} C(\boldsymbol{x}, P)
 $$
 
 
    
 In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could also be expressed as
 
 $$
-\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\})
+\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\})
 $$
 
 
    
@@ -728,7 +697,7 @@ The next step is to decide how the parameters should be changed such that they m
 The chosen cost function for this problem is
 
 $$
-c(\boldsymbol{x}, P) = \frac{1}{N} \sum_i  \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2
+C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i  \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2
 $$
 
 
    
@@ -748,7 +717,7 @@ a direction where the cost function decreases goes to a minimum.
 
 
    
 In general, the update of some parameters \( \boldsymbol{\omega} \) given a cost
-function defined by some weights \( \boldsymbol{\omega} \), \( c(\boldsymbol{x},
+function defined by some weights \( \boldsymbol{\omega} \), \( C(\boldsymbol{x},
 \boldsymbol{\omega}) \), goes as follows:
 
 $$
@@ -759,1062 +728,36 @@ $$
 for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance.
 
 
    
-The value of \( \lambda \) decides how large steps the algorithm must take in the direction of $ \nabla_{\boldsymbol{\omega}} c(\boldsymbol{x}, \boldsymbol{\omega})$.
-The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect to the elements in \( \boldsymbol{\omega \).
+The value of \( \lambda \) decides how large steps the algorithm must take
+in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$.
+The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect
+to the elements in \( \boldsymbol{\omega \).
 
 
    
-In our case, we have to minimize the cost function \( c(\boldsymbol{x}, P) \) with respect to the two sets of weights and biases, that is for the hidden layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output} } \) .
+In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with
+respect to the two sets of weights and biases, that is for the hidden
+layer \( P_{\text{hidden} } \) and for the output layer \( P_{\text{output}
+} \) .
 
 
    
 This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is updated by
 
 $$
 \begin{aligned}
-P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(\boldsymbol{x}, P)  \\
-P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(\boldsymbol{x}, P)
+P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P)  \\
+P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P)
 \end{aligned}
 $$
 
-
    
-In general, one could risk using a cost function having gradients that are cumbersome to derive analytically.
-For our case, the cost functions are just the mean squared error.
-One could employ an implementation of the back propagation for this case, but we will emphasis
-on how one could use automatic differentiation in order to train the network.
-
-
    
-However, it might be useful to know how automatic differentiation can be used, e.g through Autograd, in order to test an implementation.
-
 
    
 









    
 
-
    Basic ideas of the Principal Component Analysis (PCA) 
    
-
-
    
-The principal component analysis deals with the problem of fitting a
-low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than
-the totaldimension \( D \) of the problem at hand (our data
-set). Mathematically it can be formulated as a statistical problem or
-a geometric problem.  In our discussion of the theorem for the
-classical PCA, we will stay with a statistical approach. This is also
-what set the scene historically which for the PCA.
-
-
    
-We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) 
-
-
      
-
    •  Each data point is determined by \( p \) extrinsic (measurement) variables
      • 
-
    •  We may want to ask the following question: Are there fewer intrinsic variables (say \( d < < p \)) that still approximately describe the data?
      • 
-
    •  If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.
      •  
-
    
-
-









    
-
-
    Introducing the Covariance and Correlation functions  
    
-
-
    
-Before we discuss the PCA theorem, we need to remind ourselves about
-the definition of the covariance and the correlation function. These are quantities
-
-
    
-Suppose we have defined two vectors
-\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as 
-$$
-\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
-                              \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
-             \end{bmatrix},
-$$
-
-where for example
-$$
-\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
-$$
-
-With this definition and recalling that the variance is defined as
-$$
-\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
-$$
-
-we can rewrite the covariance matrix as 
-$$
-\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
-                              \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
-             \end{bmatrix}.
-$$
-
-
    
-The covariance takes values between zero and infinity and may thus
-lead to problems with loss of numerical precision for particularly
-large values. It is common to scale the covariance matrix by
-introducing instead the correlation matrix defined via the so-called
-correlation function
-
-$$
-\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
-$$
-
-
    
-The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
-\in [-1,1] \). This avoids eventual problems with too large values. We
-can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
-and \( \boldsymbol{y} \) as
-
-$$
-\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
-                              \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
-             \end{bmatrix},
-$$
-
-
    
-In the above example this is the function we constructed using pandas.
-
-
    
-









    
-
-
    Correlation Function and Design/Feature Matrix 
    
-
-
    
-In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression
-we defined the design/feature matrix \( \boldsymbol{X} \) as
-
-$$
-\boldsymbol{X}=\begin{bmatrix}
-x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
-x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
-x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
-\dots & \dots & \dots & \dots \dots & \dots \\
-x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
-x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
-\end{bmatrix},
-$$
-
-with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  refering to the column numbers and the
-entries \( n \) being the row elements.
-We can rewrite the design/feature matrix in terms of its column vectors as
-$$
-\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix},
-$$
-
-with a given vector
-$$
-\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
-$$
-
-
    
-With these definitions, we can now rewrite our \( 2\times 2 \)
-correaltion/covariance matrix in terms of a moe general design/feature
-matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \)
-covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \)
-
-$$
-\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
-\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
-\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
-\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
-\dots & \dots & \dots & \dots & \dots & \dots \\
-\dots & \dots & \dots & \dots & \dots & \dots \\
-\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & \mathrm{var}[\boldsymbol{x}_{p-1}]\\
-\end{bmatrix},
-$$
-
-and the correlation matrix
-$$
-\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
-1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
-\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1  & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
-\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
-\dots & \dots & \dots & \dots & \dots & \dots \\
-\dots & \dots & \dots & \dots & \dots & \dots \\
-\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & 1\\
-\end{bmatrix},
-$$
-
-
    
-









    
-
-
    Covariance Matrix Examples 
    
-
-
    
-The Numpy function np.cov calculates the covariance elements using
-the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
-the exact mean values.  The following simple function uses the
-np.vstack function which takes each vector of dimension \( 1\times n \)
-and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
-
-$$
-\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\
-                          x_1 & y_1 \\
-                          x_2 & y_2\\
-                          \dots & \dots \\
-                          x_{n-2} & y_{n-2}\\
-                          x_{n-1} & y_{n-1} & 
-             \end{bmatrix},
-$$
-
-
    
-which in turn is converted into into the \( 2\times 2 \) covariance matrix
-\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate
-the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
-function np.mean(x). We can also extract the eigenvalues of the
-covariance matrix through the np.linalg.eig() function.
+
    The code for solving the ODE 
    
 
 
    
 
 
-
    # Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-
    
-
    
-









    
-
-
    Correlation Matrix  
    
-
-
    
-The previous example can be converted into the correlation matrix by
-simply scaling the matrix elements with the variances.  We should also
-subtract the mean values for each column. This leads to the following
-code which sets up the correlations matrix for the previous example in
-a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
-
-
    
-
-
-
    import numpy as np
-n = 100
-# define two vectors                                                                                           
-x = np.random.random(size=n)
-y = 4+3*x+np.random.normal(size=n)
-#scaling the x and y vectors                                                                                   
-x = x - np.mean(x)
-y = y - np.mean(y)
-variance_x = np.sum(x@x)/n
-variance_y = np.sum(y@y)/n
-print(variance_x)
-print(variance_y)
-cov_xy = np.sum(x@y)/n
-cov_xx = np.sum(x@x)/n
-cov_yy = np.sum(y@y)/n
-C = np.zeros((2,2))
-C[0,0]= cov_xx/variance_x
-C[1,1]= cov_yy/variance_y
-C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
-C[1,0]= C[0,1]
-print(C)
-
    
-
    
-We see that the matrix elements along the diagonal are one as they
-should be and that the matrix is symmetric. Furthermore, diagonalizing
-this matrix we easily see that it is a positive definite matrix.
-
-
    
-The above procedure with numpy can be made more compact if we use pandas.
-
-
    
-









    
-
-
    Correlation Matrix with Pandas 
    
-
-
    
-We whow here how we can set up the correlation matrix using pandas, as done in this simple code
-
    
-
-
-
    import numpy as np
-import pandas as pd
-n = 10
-x = np.random.normal(size=n)
-x = x - np.mean(x)
-y = 4+3*x+np.random.normal(size=n)
-y = y - np.mean(y)
-X = (np.vstack((x, y))).T
-print(X)
-Xpd = pd.DataFrame(X)
-print(Xpd)
-correlation_matrix = Xpd.corr()
-print(correlation_matrix)
-
    
-
    
-We expand this model to the Franke function discussed above.
-
-
    
-









    
-
-
    Correlation Matrix with Pandas and the Franke function 
    
-
-
    
-
-
-
    # Common imports
-import numpy as np
-import pandas as pd
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 4
-N = 100
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-
-Xpd = pd.DataFrame(X)
-# subtract the mean values and set up the covariance matrix
-Xpd = Xpd - Xpd.mean()
-covariance_matrix = Xpd.cov()
-print(covariance_matrix)
-
    
-
    
-We note here that the covariance is zero for the first rows and
-columns since all matrix elements in the design matrix were set to one
-(we are fitting the function in terms of a polynomial of degree \( n \)).
-
-
    
-This means that the variance for these elements will be zero and will
-cause problems when we set up the correlation matrix.  We can simply
-drop these elements and construct a correlation
-matrix without these elements.
-
-
    
-









    
-
-
    Rewriting the Covariance and/or Correlation Matrix 
    
-
-
    
-We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as 
-$$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
-$$
-
-
    
-To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
-$$
-\boldsymbol{X}=\begin{bmatrix}
-x_{00} & x_{01}\\
-x_{10} & x_{11}\\
-\end{bmatrix}=\begin{bmatrix}
-\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
-\end{bmatrix}.
-$$
-
-
    
-If we then compute the expectation value
-$$
-\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix}
-x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
-x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
-\end{bmatrix},
-$$
-
-which is just 
-$$
-\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
-                              \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
-             \end{bmatrix},
-$$
-
-where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
-
-
    
-It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
-
-
    
-









    
-
-
    Towards the PCA theorem 
    
-
-
    
-We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
-$$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
-$$
-
-Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \).
-These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
-
-
    
-Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
-
-
    
-That is we have
-$$
-\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
-$$
-
-since the matrix \( \boldsymbol{S} \) is not a data dependent matrix.   Multiplying with \( \boldsymbol{S}^T \) from the left we have
-$$
-\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
-$$
-
-and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
-
-$$
-\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
-$$
-
-
    
-In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
-\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
-
-
    
-The eigenvalues tell us then how much we need to stretch the
-corresponding eigenvectors. Dimensions with large eigenvalues have
-thus large variations (large variance) and define therefore useful
-dimensions. The data points are more spread out in the direction of
-these eigenvectors.  Smaller eigenvalues mean on the other hand that
-the corresponding eigenvectors are shrunk accordingly and the data
-points are tightly bunched together and there is not much variation in
-these specific directions. Hopefully then we could leave it out
-dimensions where the eigenvalues are very small. If \( p \) is very large,
-we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
-features/predictors.
-
-
    
-









    
-
-
    The Algorithm before theorem 
    
-
-
    
-Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
-
-
      
-
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
-
    
-
-$$
-\boldsymbol{X}=\begin{bmatrix}
-x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
-x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
-x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
-\dots & \dots & \dots & \dots \dots & \dots \\
-x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
-x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
-\end{bmatrix},
-$$
-
-
-
      
-
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
-
    •  Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).
      • 
-
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
-
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
-
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
-
    
-
-









    
-
-
    Writing our own PCA code 
    
-
-
    
-We will use a simple example first with two-dimensional data
-drawn from a multivariate normal distribution with the following mean and covariance matrix:
-$$
-\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
-2 & 2
-\end{bmatrix}
-$$
-
-Note that the mean refers to each column of data. 
-We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from
-this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \).
-
-
    
-The following Python code aids in setting up the data and writing out the design matrix.
-Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
-
    
-
-
-
    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-n = 10000
-mean = (-1, 2)
-cov = [[4, 2], [2, 2]]
-X = np.random.multivariate_normal(mean, cov, n)
-
    
-
    
-Now we are going to implement the PCA algorithm. We will break it down into various substeps.
-
-
    Compute the sample mean and center the data 
    
-
-
    
-The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
-$$
-\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
-$$
-
-and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
-$$
-\bar{x}_i = x_i - \mu_n.
-$$
-
-When you are done with these steps, print out \( \mu_n \) to verify it is
-close to \( \mu \) and plot your mean centered data to verify it is
-centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above.
-The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. 
-
    
-
-
-
    df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-
    
-
    
-Alternatively, we could use the functions we discussed
-earlier for scaling the data set.  That is, we could have used the
-StandardScaler function in Scikit-Learn, a function which ensures
-that for each feature/predictor we study the mean value is zero and
-the variance is one (every column in the design/feature matrix).  You
-would then not get the same results, since we divide by the
-variance. The diagonal covariance matrix elements will then be one,
-while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
-specific case.
-
-
    Compute the sample covariance 
    
-
-
    
-Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
-$$
-\begin{equation*}
-\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
-\end{equation*}
-$$
-
-where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \).
-We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows
-
    
-
-
-
    print(df.cov())
-print(np.cov(X_centered.T))
-
    
-
    
-Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. 
-Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. 
-
    
-
-
-
    # extract the relevant columns from the centered design matrix of dim n x 2
-x = X_centered[:,0]
-y = X_centered[:,1]
-Cov = np.zeros((2,2))
-Cov[0,1] = np.sum(x.T@y)/(n-1.0)
-Cov[0,0] = np.sum(x.T@x)/(n-1.0)
-Cov[1,1] = np.sum(y.T@y)/(n-1.0)
-Cov[1,0]= Cov[0,1]
-print("Centered covariance using own code")
-print(Cov)
-plt.plot(x, y, 'x')
-plt.axis('equal')
-plt.show()
-
    
-
    
-Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
-The plot shows how the data are clustered around a line with slope close to one. Is this expected?
-
-
    Diagonalize the sample covariance matrix to obtain the principal components 
    
-
-
    
-Now we are ready to solve for the principal components! To do so we
-diagonalize the sample covariance matrix \( \Sigma \). We can use the
-function np.linalg.eig to do so. It will return the eigenvalues and
-eigenvectors of \( \Sigma \). Once we have these we can perform the 
-following tasks:
-
-
      
-
    •  We compute the percentage of the total variance captured by the first principal component
      • 
-
    •  We plot the mean centered data and lines along the first and second principal components
      • 
-
    •  Then we project the mean centered data onto the first and second principal components, and plot the projected data.
      •  
-
    •  Finally, we approximate the data as
      • 
-
    
-
-$$
-\begin{equation*}
-x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
-\end{equation*}
-$$
-
-where \( v_0 \) is the first principal component.
-
-
    
-Collecting all these steps we can write our own PCA function and
-compare this with the functionality included in Scikit-Learn.
-
-
    
-The code here outlines some of the elements we could include in the
-analysis. Feel free to extend upon this in order to address the above
-questions.
-
-
    
-
-
-
    # diagonalize and obtain eigenvalues, not necessarily sorted
-EigValues, EigVectors = np.linalg.eig(Cov)
-# sort eigenvectors and eigenvalues
-#permute = EigValues.argsort()
-#EigValues = EigValues[permute]
-#EigVectors = EigVectors[:,permute]
-print("Eigenvalues of Covariance matrix")
-for i in range(2):
-    print(EigValues[i])
-FirstEigvector = EigVectors[:,0]
-SecondEigvector = EigVectors[:,1]
-print("First eigenvector")
-print(FirstEigvector)
-print("Second eigenvector")
-print(SecondEigvector)
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2Dsl = pca.fit_transform(X)
-print("Eigenvector of largest eigenvalue")
-print(pca.components_.T[:, 0])
-
    
-
    
-This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code.  Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?
-
-
    
-









    
-
-
    Classical PCA Theorem   
    
-
-
    
-We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
-centered as discussed above. For the sake of simplicity we skip the
-overline symbol. The matrix is defined in terms of the various column
-vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension
-\( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
-
-
    
-We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
-$$
-J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2,
-$$
-
-with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
-\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
-
-
    
-The PCA theorem states that minimizing the above reconstruction error
-corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which
-diagonalizes the empirical covariance(correlation) matrix. The optimal
-low-dimensional encoding of the data is then given by a set of vectors
-\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the
-orthogonal projection of the data onto the columns spanned by the
-eigenvectors of the covariance(correlations matrix).
-
-
    
-The proof which follows will be updated by mid January 2020.
-
-
    
-









    
-
-
    Proof of the PCA Theorem  
    
-
-
    
-To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
-$$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
-$$
-
-which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
-$$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
-$$
-
-Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
-$$
-z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
-$$
-
-where the vectors on the rhs are known.
-
-
    
-









    
-
-
    PCA Proof continued 
    
-
-
    
-We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
-$$
-J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
-$$
-
-
    
-We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
-$$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
-$$
-
-since the expectation value of
-$$
-\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0,
-$$
-
-where we have used the fact that our data are centered.
-
-
    
-Recalling our definition of the covariance as 
-$$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
-$$
-
-we have thus that
-$$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
-$$
-
-
    
-We are almost there, we have obtained a relation between minimizing
-the reconstruction error and the variance and the covariance
-matrix. Minimizing the error is equivalent to maximizing the variance
-of the projected data.
-
-
    
-









    
-
-
    The final step 
    
-
-
    
-We could trivially maximize the variance of the projection (and
-thereby minimize the error in the reconstruction function) by letting
-the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we
-want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by
-\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a
-Lagrange multiplier we can then in turn maximize
-
-$$
-J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0).
-$$
-
-Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain
-
-$$
-\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0,
-$$
-
-meaning that
-$$
-\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0.
-$$
-
-The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is
-$$
-\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0.
-$$
-
-
    
-If we want to maximize the variance (minimize the construction error)
-we simply pick the eigenvector of the covariance matrix with the
-largest eigenvalue. This establishes the link between the minimization
-of the reconstruction function \( J \) in terms of an orthogonal matrix
-and the maximization of the variance and thereby the covariance of our
-observations encoded in the design/feature matrix \( \boldsymbol{X} \).
-
-
    
-The proof
-for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be
-established by applying the above arguments and using the fact that
-our basis of eigenvectors is orthogonal, see Murphy chapter
-12.2.  The
-discussion in chapter 12.2 of Murphy's text has also a nice link with
-the Singular Value Decomposition theorem. For categorical data, see
-chapter 12.4 and discussion therein.
-
-
    
-Additional part of the proof for the other eigenvectors will be added by mid January 2020.
-
-
    
-









    
-
-
    Geometric Interpretation and link with Singular Value Decomposition 
    
-
-
    
-This material will be added by mid January 2020.
-
-
    
-









    
-
-
    Principal Component Analysis 
    
-
-
    
-Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
-First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
-
-
    
-The following Python code uses NumPy’s svd() function to obtain all the principal components of the
-training set, then extracts the first two principal components. First we center the data using either pandas or our own code
-
    
-
-
-
    import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 vanilla matrix 
-rows = 10
-cols = 5
-X = np.random.randn(rows,cols)
-df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-display(df)
-
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-# Then check the difference between pandas and our own set up
-print(X_centered-df)
-#Now we do an SVD
-U, s, V = np.linalg.svd(X_centered)
-c1 = V.T[:, 0]
-c2 = V.T[:, 1]
-W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-print(X2D)
-
    
-
    
-PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
-the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
-forget to center the data first.
-
-
    
-Once you have identified all the principal components, you can reduce the dimensionality of the dataset
-down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
-Selecting this hyperplane ensures that the projection will preserve as much variance as possible. 
-
    
-
-
-
    W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
-
    
-
    
-
-
-
    PCA and scikit-learn 
    
-
-
    
-Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
-following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
-that it automatically takes care of centering the data):
-
    
-
-
-
    #thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D = pca.fit_transform(X)
-print(X2D)
-
    
-
    
-After fitting the PCA transformer to the dataset, you can access the principal components using the
-components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
-principal component is equal to 
-
    
-
-
-
    pca.components_.T[:, 0].
-
    
-
    
-Another very useful piece of information is the explained variance ratio of each principal component,
-available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
-variance that lies along the axis of each principal component.
-
-
    
-









    
-
-
    Back to the Cancer Data 
    
-We can now repeat the above but applied to real data, in this case our breast cancer data.
-Here we compute performance scores on the training data using logistic regression.
-
    
-
-
-
    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-
-logreg = LogisticRegression()
-logreg.fit(X_train, y_train)
-print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
-# We scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Then perform again a log reg fit
-logreg.fit(X_train_scaled, y_train)
-print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D_train = pca.fit_transform(X_train_scaled)
-# and finally compute the log reg fit and the score on the training data	
-logreg.fit(X2D_train,y_train)
-print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
-
    
-
    
-We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.
-
-
    
-









    
-
-
    More on the PCA 
    
-
-
    
-Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
-choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
-Unless, of course, you are reducing dimensionality for data visualization — in that case you will
-generally want to reduce the dimensionality down to 2 or 3.
-The following code computes PCA without reducing dimensionality, then computes the minimum number
-of dimensions required to preserve 95% of the training set’s variance:
-
    
-
-
-
    pca = PCA()
-pca.fit(X)
-cumsum = np.cumsum(pca.explained_variance_ratio_)
-d = np.argmax(cumsum >= 0.95) + 1
-
    
-
    
-You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
-of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
-a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
-
    
-
-
-
    pca = PCA(n_components=0.95)
-X_reduced = pca.fit_transform(X)
-
    
-
    
-









    
-
-
    Incremental PCA 
    
-
-
    
-One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
-memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
-been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
-at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
-instances arrive).
-
-
    
-









    
-
-
    Randomized PCA 
    
-
-
    
-Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
-algorithm that quickly finds an approximation of the first d principal components. Its computational
-complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
-previous algorithms when \( d \) is much smaller than \( n \).
-
-
    
-









    
-
-
    Kernel PCA 
    
-
    
-
-
    
-
-
    
-The kernel trick is a mathematical technique that implicitly maps instances into a
-very high-dimensional space (called the feature space), enabling nonlinear classification and regression
-with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
-space corresponds to a complex nonlinear decision boundary in the original space.
-It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
-projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
-preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
-twisted manifold.
-For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
-
    
-
-
-
    from sklearn.decomposition import KernelPCA
-rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
-X_reduced = rbf_pca.fit_transform(X)
-
    
-
-
    
-
-
-
    
-









    
-
-
    LLE 
    
-
-
    
-Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
-(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
-algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
-closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
-these local relationships are best preserved (more details shortly).
-
-
    
-









    
-
-
    Other techniques 
    
-
-
    
-There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
-
-
    
-Here are some of the most popular:
-
-
      
-
    •  Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • 
-
    •  Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • 
-
    •  t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • 
-
    •  Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • 
-
    
-
-









    
-
-
    The network with one input, hidden, and output layer 
    
-
-
    
-
-
-
    # Autograd will be used for later, so the numpy wrapper for Autograd must be imported
-import autograd.numpy as np
+
    import autograd.numpy as np
 from autograd import grad, elementwise_grad
 import autograd.numpy.random as npr
 from matplotlib import pyplot as plt
@@ -1962,7 +905,7 @@ Here are some of the most popular:
 
    
 









    
 
-
    The network with one input layer, specified number of hidden layers, and one output layer  output layer 
    
+
    The network with one input layer, specified number of hidden layers, and one output layer 
    
 
 
    
 It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.
@@ -2135,7 +1078,7 @@ The number of neurons within each hidden layer are given as a list of integers i
 
    
 









    
 
-
    Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme 
    
+
    Example: Population growth 
    
 
 
    
 A logistic model of population growth assumes that a population converges toward an equilibrium.
@@ -2160,7 +1103,7 @@ For comparison, the forward Euler method has been implemented in order to see ho
 
    
 









    
 
-
    Setting up the problem 
    
+
    Setting up the problem 
    
 
 
    
 Here, we will model a population \( g(t) \) in an environment having carrying capacity \( A \).
@@ -2181,7 +1124,9 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
 
    
 









    
 
-
    The trial solution 
    
+
    The trial solution 
    
+
+
    
 We will get a slightly different trial solution, as the boundary conditions are different
 compared to the case for exponential decay.
 
@@ -2205,7 +1150,7 @@ $$
 
    
 









    
 
-
    The program using Autograd 
    
+
    The program using Autograd 
    
 
 
    
 The network will be the similar as for the exponential decay example, but with some small modifications for our problem.
@@ -2380,7 +1325,7 @@ The network will be the similar as for the exponential decay example, but with s
 
    
 









    
 
-
    Using forward Euler to solve the ODE 
    
+
    Using forward Euler to solve the ODE 
    
 
 
    
 A straight-forward way of solving an ODE numerically, is to use Euler's method.
@@ -2509,177 +1454,10 @@ extending the program that uses the network using Autograd:
 
    
 Running the program gives
 
-
    
-
-
-
    Max absolute difference between Euler method and analytical: 0.011225
-Max absolute difference between deep neural network and analytical: 0.00424909
-
    
 
    
 









    
 
-
    Using TensorFlow to model logistic population growth 
    
-
-
    
-TensorFlow is a library widely used in the machine learning community.
-A neural network can be set up in a flexible manner, where various optimization algorithms are implemented and different types of networks can be used, making it easier to experiment on solving differential equations using neural networks.
-
-
    
-









    
-
-
    The general program flow in TensorFlow 
    
-
-
    
-Usually, a program in TensorFlow is divided into two parts; the construction phase and the execution phase.
-In the construction phase, the computational graph that TensorFlow uses to perform its calculations are set up.
-In the execution phase, TensorFlow evaluates any procedure that was defined in the construction phase.
-
-
    Program flow in TensorFlow - Construction phase 
    
-
-
    
-Here, the architecture for the neural network will be set up, along with the cost function and an optimizer class used during training of the network.
-Note that TensorFlow uses a different convention for the weighting done in each neuron in each layer within the network than in the implementation using Autograd.
-The matrix-vector multiplication between the input from the previous layer and the weighting at the neuron at current layer in the program using Autograd, is the transpose of the convention used in TensorFlow. But it will not affect that much our construction, as TensorFlow takes care of most of the computations. The only thing we have to be aware of, is how the dimensions are for our inputs.
-
-
    Program flow in TensorFlow - Execution phase 
    
-
-
    
-The computation graph has been defined, and is ready to be evaluated.
-In order to get access to the graph, it has to be initialized and be runned within a Session.
-
-
    The full program modeling logistic population growth using TensorFlow 
    
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-# Set a seed to ensure getting the same results from every run
-tf.set_random_seed(4155)
-
-Nt = 10
-T = 1
-t = np.linspace(0,T, Nt)
-
-## The construction phase
-
-# Convert the values the trial solution is evaluated at to a tensor.
-t_tf = tf.convert_to_tensor(t.reshape(-1,1),dtype=tf.float64)
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(t.shape)),shape=(-1,1))
-
-# Define the parameters of the equation
-alpha = tf.constant(2.,dtype=tf.float64)
-A = tf.constant(1.,dtype=tf.float64)
-g0 = tf.constant(1.2,dtype=tf.float64)
-
-num_iter = 100000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [100,50,25]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = t_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = g0 + t_tf*dnn_output
-    d_g_trial = tf.gradients(g_trial,t_tf)
-
-    func = alpha*g_trial*(A - g_trial)
-    cost = tf.losses.mean_squared_error(zeros, d_g_trial[0] - func)
-
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-# Set up a referance to the result from the neural network:
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The training of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves for each iteration:
-        #if i % 1000 == 0:
-        #    print(cost.eval())
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-# Compare with analytical solution
-def get_parameters():
-    alpha = 2
-    A = 1
-    g0 = 1.2
-    return alpha, A, g0
-
-def g_analytic(t):
-    alpha,A, g0 = get_parameters()
-    return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
-
-g_analytical = g_analytic(t)
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(t, g_dnn_tf)
-plt.plot(t, g_analytical)
-
-plt.legend(['dnn, tensorflow', 'exact'])
-plt.xlabel('Time t')
-plt.ylabel('g(t)')
-
-plt.show()
-
    
-
    
-









    
-
-
    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow 
    
+
    Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow 
    
 
 
    
 The Poisson equation for \( g(x) \) in one dimension is
@@ -2707,16 +1485,10 @@ This equation can be solved numerically using programs where e.g Autograd and Te
 The results from the networks can then be compared to the analytical solution.
 In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.
 
-
    
-There exists many different optimization methods implemented in TensorFlow.
-In the examples program using TensorFlow, it could also be of interest to see how
-the choice of an optimization method affects our results.
-In the TensorFlow documentation about optimizers, a list over available optimization methods are shown.
-
 
    
 









    
 
-
    The specific equation to solve for 
    
+
    The specific equation to solve for 
    
 
 
    
 Here, the function \( g(x) \) to solve for follows the equation
@@ -2754,7 +1526,7 @@ $$
 
    
 









    
 
-
    Solving the equation using Autograd 
    
+
    Solving the equation using Autograd 
    
 
 
    
 
@@ -2913,7 +1685,7 @@ $$
 
    
 









    
 
-
    Comparing with a numerical scheme 
    
+
    Comparing with a numerical scheme 
    
 
 
    
 The Poisson equation is possible to solve using Taylor series to approximate the second derivative.
@@ -3216,168 +1988,9 @@ The max absolute difference between the analytical solution and numerical scheme
 
    
 









    
 
-
    Using gradient descent in TensorFlow to solve Poisson equation 
    
-The program follows the similar idea as for the logistic population model.
+
    Partial Differential Equations 
    
 
 
    
-What has changed, is what the cost function minimizes and the trial solution.
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-import matplotlib.pyplot as plt
-## Construction phase
-
-# Just to reset the graph such that it is possible to rerun this in a
-# Jupyter cell without resetting the whole kernel.
-tf.reset_default_graph()
-
-tf.set_random_seed(4155)
-
-# Convert the values the trial solution is evaluated at to a tensor.
-Nx = 10
-x = np.linspace(0,1, Nx)
-x_tf = tf.convert_to_tensor(x.reshape(-1,1),dtype=tf.float64)
-
-
-num_iter = 10000
-
-# Define the number of neurons at each hidden layer
-num_hidden_neurons = [20,10]
-num_hidden_layers = np.size(num_hidden_neurons)
-
-# Construct the network.
-# tf.name_scope is used to group each step in the construction,
-# just for a more organized visualization in TensorBoard
-with tf.name_scope('dnn'):
-
-    # Input layer
-    previous_layer = x_tf
-
-    # Hidden layers
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    # Output layer
-    dnn_output = tf.layers.dense(previous_layer, 1, name='output')
-
-# Define the cost function
-with tf.name_scope('cost'):
-    g_trial = x_tf*(1-x_tf)*dnn_output
-    d_g_trial = tf.gradients(g_trial,x_tf)
-    d2_g_trial = tf.gradients(d_g_trial,x_tf)
-
-    right_side = (3*x_tf + x_tf**2)*tf.exp(x_tf)
-
-    err = tf.square( -d2_g_trial[0] - right_side)
-    cost = tf.reduce_sum(err, name = 'cost')
-
-# Choose the method to minimize the cost function, along with a learning rate
-learning_rate = 1e-2
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(cost)
-
-g_dnn_tf = None
-
-# Define a node that initializes all of the other nodes in the computational graph
-# used by TensorFlow:
-init = tf.global_variables_initializer()
-
-
-## Execution phase
-
-# Start a session where the graph defined from the construction phase can be evaluated at:
-
-with tf.Session() as sess:
-    # Initialize the whole graph
-    init.run()
-
-    # Evaluate the initial cost:
-    print('Initial cost: %g'%cost.eval())
-
-    # The traning of the network:
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-    # Training is done, and we have an approximate solution to the ODE
-    print('Final cost: %g'%cost.eval())
-
-    # Store the result
-    g_dnn_tf = g_trial.eval()
-
-    writer = tf.summary.FileWriter("./output", sess.graph)
-    writer.close()
-
-# Evaluate the analytical function to compare with
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-g_analytical = g_analytic(x)
-
-diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)
-
-print('\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))
-
-# Plot the result
-plt.figure(figsize=(10,10))
-
-plt.title('Numerical solutions of the ODE')
-
-plt.plot(x, g_dnn_tf)
-plt.plot(x, g_analytical)
-
-plt.legend(['dnn, tensorflow','exact'])
-plt.xlabel('x')
-plt.ylabel('g(x)')
-
-plt.show()
-
    
-
    
-









    
-
-
    Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation 
    
-
-
    
-We can see that the results using GradientDescentOptimizer seems to converge towards the analytical solution.
-But there exists many other methods for optimization also, see the TensorFlow documentation on Optimizers.
-
-
    
-Adam is an optimization algorithm that changes its learning rates accordingly to the function it tries to minimize for every iteration.
-The algorithm is described in this paper.
-How much an optimization algorithm has to say for the network to converge, could be interesting to experiment with.
-Using the same TensorFlow program as before, the only change to do, is to replace the variable optimizer.
-
-
    
-In the program that uses TensorFlow to solve for the Poisson equation, change the line
-
-
    
-
-
-
    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-
    
-
    
-to
-
-
    
-
-
-
    optimizer = tf.train.AdamOptimizer(learning_rate)
-
    
-
    
-The program using the Adam optimizer with a different initial learning rate yields indeed an interesting result:
-
    
-
-
-
    Max absolute difference between the analytical solution and solution from TensorFlow DNN: 7.11243e-05
-
    
-
    
-









    
-
-
    Partial Differential Equations 
    
 A partial differential equation (PDE) has a solution here the function is defined by multiple variables.
 The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.
 
@@ -3393,6 +2006,11 @@ $$
 
    
 where \( f \) is an expression involving all kinds of possible mixed derivatives of \( g(x_1,\dots,x_N) \) up to an order \( n \). In order for the solution to be unique, some additional conditions must also be given.
 
+
    
+









    
+
+
    Type of problem 
    
+
 
    
 The problem our network must solve for, is similar to the ODE case.
 We must have a trial solution \( g_t \) at hand.
@@ -3412,11 +2030,21 @@ The neural network \( N(x_1,\dots,x_N,P) \) has weights and biases described by
 The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensure that the output of \( N(x_1,\dots,x_N,P) \) is zero when \( g_t(x_1,\dots,x_N) \) is evaluated at the values of \( x_1,\dots,x_N \) where the given conditions must be satisfied. The function \( h_1(x_1,\dots,x_N) \) should alone make \( g_t(x_1,\dots,x_N) \) satisfy the conditions.
 
 
    
-The network tries then the minimize the cost function following the same ideas as described for the ODE case, but now with more than one variables to consider.
-The concept still remains the same; find a set of parameters \( P \) such that the expression \( f \) in \eqref{PDE} is as close to zero as possible.
+









    
+
+
    Network requirements 
    
 
 
    
-As for the ODE case, the cost function is the mean squared error that the network must try to minimize. The cost function for the network to minimize is
+The network tries then the minimize the cost function following the
+same ideas as described for the ODE case, but now with more than one
+variables to consider.  The concept still remains the same; find a set
+of parameters \( P \) such that the expression \( f \) in \eqref{PDE} is as
+close to zero as possible.
+
+
    
+As for the ODE case, the cost function is the mean squared error that
+the network must try to minimize. The cost function for the network to
+minimize is
 
 $$
 \begin{equation*}
@@ -3424,6 +2052,11 @@ c\left(x_1, \dots, x_N, P\right) = \left(  f\left(x_1, \, \dots \, , x_N, \frac{
 \end{equation*}
 $$
 
+
    
+









    
+
+
    More details 
    
+
 
    
 If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
 $$
@@ -3443,7 +2076,7 @@ $$
 
    
 









    
 
-
    Example: The diffusion equation 
    
+
    Example: The diffusion equation 
    
 
 
    
 In one spatial dimension, the equation reads
@@ -3468,7 +2101,7 @@ with \( u(x) \) being some given function.
 
    
 









    
 
-
    Defining the problem 
    
+
    Defining the problem 
    
 
 
    
 For this case, we want to find \( g(x,t) \) such that
@@ -3500,13 +2133,16 @@ First, we will look into how Autograd could be used in a network tailored to sol
 
    
 









    
 
-
    Setting up the network using Autograd 
    
+
    Setting up the network using Autograd 
    
 
 
    
-The only change to do here, is to extend our network such that functions of multiple parameters are correctly handled.
-In this case we have two variables in our function to solve for, that is time \( t \) and position \( x \).
-The variables will be represented by a one-dimensional array in the program.
-The program will evaluate the network at each possible pair \( (x,t) \), given an array for the desired \( x \)-values and \( t \)-values to approximate the solution at.
+The only change to do here, is to extend our network such that
+functions of multiple parameters are correctly handled.  In this case
+we have two variables in our function to solve for, that is time \( t \)
+and position \( x \).  The variables will be represented by a
+one-dimensional array in the program.  The program will evaluate the
+network at each possible pair \( (x,t) \), given an array for the desired
+\( x \)-values and \( t \)-values to approximate the solution at.
 
 
    
 
@@ -3559,8 +2195,13 @@ The program will evaluate the network at each possible pair \( (x,t) \), given a
 
    
 









    
 
-
    Setting up the network using Autograd; The trial solution 
    
-The cost function must then iterate through the given arrays containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep neural network and the trial solution is evaluated at, and then finds the Jacobian of the trial solution.
+
    Setting up the network using Autograd; The trial solution 
    
+
+
    
+The cost function must then iterate through the given arrays
+containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep
+neural network and the trial solution is evaluated at, and then finds
+the Jacobian of the trial solution.
 
 
    
 A possible trial solution for this PDE is
@@ -3581,16 +2222,31 @@ $$
 since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
 
 
    
-The Jacobian is used because the program must find the derivative of the trial solution with respect to \( x \) and \( t \).
+









    
+
+
    Why the jacobian? 
    
 
 
    
-This gives the necessity of computing the Jacobian matrix, as we want to evaluate the gradient with respect to \( x \) and \( t \) (note that the Jacobian of a scalar-valued multivariate function is simply its gradient).
+The Jacobian is used because the program must find the derivative of
+the trial solution with respect to \( x \) and \( t \).
 
 
    
-In Autograd, the differentiation is by default done with respect to the first input argument of your Python function. Since the points is an array representing \( x \) and \( t \), the Jacobian is calculated using the values of \( x \) and \( t \).
+This gives the necessity of computing the Jacobian matrix, as we want
+to evaluate the gradient with respect to \( x \) and \( t \) (note that the
+Jacobian of a scalar-valued multivariate function is simply its
+gradient).
 
 
    
-To find the second derivative with respect to \( x \) and \( t \), the Jacobian can be found for the second time. The result is a Hessian matrix, which is the matrix containing all the possible second order mixed derivatives of \( g(x,t) \).
+In Autograd, the differentiation is by default done with respect to
+the first input argument of your Python function. Since the points is
+an array representing \( x \) and \( t \), the Jacobian is calculated using
+the values of \( x \) and \( t \).
+
+
    
+To find the second derivative with respect to \( x \) and \( t \), the
+Jacobian can be found for the second time. The result is a Hessian
+matrix, which is the matrix containing all the possible second order
+mixed derivatives of \( g(x,t) \).
 
 
    
 
@@ -3635,7 +2291,9 @@ To find the second derivative with respect to \( x \) and \( t \), the Jacobian
 
    
 









    
 
-
    Setting up the network using Autograd; The full program 
    
+
    Setting up the network using Autograd; The full program 
    
+
+
    
 Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.
 
 
    
@@ -3650,12 +2308,9 @@ A possible way to implement a neural network solving the PDE, is given below.
 Be aware, though, that it is fairly slow for the parameters used.
 A better result is possible, but requires more iterations, and thus longer time to complete.
 
-
    
-Using only 20 neurons in one hidden layer, the program managed to make the trial solution have the maximum absolute error of 0.0075. The execution time, however, was approximately one day and 14 hours on a computer having Intel i7-7560U 2.4 GHz CPU.
-
 
    
 Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.
-Using TensorFlow in the next example sovling the wave equation, has a much better execution time.
+Using TensorFlow results in a much better execution time. Try it!
 
 
    
 
@@ -3887,7 +2542,7 @@ Using TensorFlow in the next example sovling the wave equation, has a much bette
 
    
 









    
 
-
    Example: Solving the wave equation using Autograd and TensorFlow 
    
+
    Example: Solving the wave equation with Neural Networks 
    
 
 
    
 The wave equation is
@@ -3916,7 +2571,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
 
    
 









    
 
-
    The problem to solve for 
    
+
    The problem to solve for 
    
 
 
    
 The wave equation to solve for, is
@@ -3946,7 +2601,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
 
    
 









    
 
-
    The trial solution 
    
+
    The trial solution 
    
 Setting up the network is done in similar matter as for the example of solving the diffusion equation.
 The only things we have to change, is the trial solution such that it satisfies the conditions from \eqref{condwave} and the cost function.
 
@@ -3970,7 +2625,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
 
    
 









    
 
-
    The analytical solution 
    
+
    The analytical solution 
    
 
 
    
 The analytical solution for our specific problem, is
@@ -3982,7 +2637,7 @@ $$
 
    
 









    
 
-
    Solving the wave equation - the full program using Autograd 
    
+
    Solving the wave equation - the full program using Autograd 
    
 
 
    
 
@@ -4211,194 +2866,1041 @@ $$
 
    
 









    
 
-
    Solving the wave equation - the full program using TensorFlow 
    
-As the program using Autograd is fairly slow, one could hope that using TensorFlow
-could make a naive implementation faster, and more numerically robust.
-
-
    
-In addition, having TensorFlow at hand, it could be easier to experiment with different
-optimization algorithms, and other constructions of the network.
-
-
    
-The following program solves the given wave equation much faster,
-
-
    
-
-
-
    import tensorflow as tf
-import numpy as np
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-Nx = 10
-x_np = np.linspace(0,1,Nx)
-
-Nt = 10
-t_np = np.linspace(0,1,Nt)
-
-X,T = np.meshgrid(x_np, t_np)
-
-x = X.ravel()
-t = T.ravel()
-
-## The construction phase
-
-zeros = tf.reshape(tf.convert_to_tensor(np.zeros(x.shape)),shape=(-1,1))
-x = tf.reshape(tf.convert_to_tensor(x),shape=(-1,1))
-t = tf.reshape(tf.convert_to_tensor(t),shape=(-1,1))
-
-points = tf.concat([x,t],1)
-
-num_iter = 100000
-num_hidden_neurons = [90]
-
-X = tf.convert_to_tensor(X)
-T = tf.convert_to_tensor(T)
-
-
-with tf.variable_scope('dnn'):
-    num_hidden_layers = np.size(num_hidden_neurons)
-
-    previous_layer = points
-
-    for l in range(num_hidden_layers):
-        current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l],activation=tf.nn.sigmoid)
-        previous_layer = current_layer
-
-    dnn_output = tf.layers.dense(previous_layer, 1)
-
-
-def u(x):
-    return tf.sin(np.pi*x)
-
-def v(x):
-    return -np.pi*tf.sin(np.pi*x)
-
-with tf.name_scope('loss'):
-    g_trial = (1 - t**2)*u(x) + t*v(x) + x*(1-x)*t**2*dnn_output
-
-    g_trial_d2t =  tf.gradients(tf.gradients(g_trial,t),t)
-    g_trial_d2x = tf.gradients(tf.gradients(g_trial,x),x)
-
-    loss = tf.losses.mean_squared_error(zeros, g_trial_d2t[0] - g_trial_d2x[0])
-
-learning_rate = 0.01
-with tf.name_scope('train'):
-    optimizer = tf.train.GradientDescentOptimizer(learning_rate)
-    traning_op = optimizer.minimize(loss)
-
-init = tf.global_variables_initializer()
-
-g_analytic = tf.sin(np.pi*x)*tf.cos(np.pi*t) - tf.sin(np.pi*x)*tf.sin(np.pi*t)
-g_dnn = None
-
-## The execution phase
-with tf.Session() as sess:
-    init.run()
-    for i in range(num_iter):
-        sess.run(traning_op)
-
-        # If one desires to see how the cost function behaves during training
-        #if i % 100 == 0:
-        #    print(loss.eval())
-
-    g_analytic = g_analytic.eval()
-    g_dnn = g_trial.eval()
-
-
-## Compare with the analutical solution
-diff = np.abs(g_analytic - g_dnn)
-print('Max absolute difference between analytical solution and TensorFlow DNN = ',np.max(diff))
-
-G_analytic = g_analytic.reshape((Nt,Nx))
-G_dnn = g_dnn.reshape((Nt,Nx))
-
-diff = np.abs(G_analytic - G_dnn)
-
-# Plot the results
-
-X,T = np.meshgrid(x_np, t_np)
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-s = ax.plot_surface(X,T,G_dnn,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Analytical solution')
-s = ax.plot_surface(X,T,G_analytic,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-fig = plt.figure(figsize=(10,10))
-ax = fig.gca(projection='3d')
-ax.set_title('Difference')
-s = ax.plot_surface(X,T,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
-ax.set_xlabel('Time $t$')
-ax.set_ylabel('Position $x$');
-
-## Take some 3D slices
-
-indx1 = 0
-indx2 = int(Nt/2)
-indx3 = Nt-1
-
-t1 = t_np[indx1]
-t2 = t_np[indx2]
-t3 = t_np[indx3]
-
-# Slice the results from the DNN
-res1 = G_dnn[indx1,:]
-res2 = G_dnn[indx2,:]
-res3 = G_dnn[indx3,:]
-
-# Slice the analytical results
-res_analytical1 = G_analytic[indx1,:]
-res_analytical2 = G_analytic[indx2,:]
-res_analytical3 = G_analytic[indx3,:]
-
-# Plot the slices
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t1)
-plt.plot(x_np, res1)
-plt.plot(x_np,res_analytical1)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t2)
-plt.plot(x_np, res2)
-plt.plot(x_np,res_analytical2)
-plt.legend(['dnn','analytical'])
-
-plt.figure(figsize=(10,10))
-plt.title("Computed solutions at time = %g"%t3)
-plt.plot(x_np, res3)
-plt.plot(x_np,res_analytical3)
-plt.legend(['dnn','analytical'])
-
-plt.show()
-
    
-
    
-The program manages to find a solution having max absolute difference to the analytical
-at approximately 0.0059, by just using some minutes!
-It was found, by some testing, that one hidden layer with 90 neurons actually performed well.
-
-
    
-









    
-
-
    Resources 
    
+
    Resources on differential equations and deep learning 
    
 
 
      
 
    1.  Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al
      2. 
 
    2.  Neural networks for solving differential equations by A. Honchar
      3. 
 
    3.  Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener
      4. 
-
    4.  Introduction to Partial Differential Equations by A. Tveitio, R. Winther
      5. 
+
    5.  Introduction to Partial Differential Equations by A. Tveito, R. Winther
      6. 
 
    
 
+









    
+
+
    Friday, Principal Component Analysis 
    
+
+
    
+









    
+
+
    Basic ideas of the Principal Component Analysis (PCA) 
    
+
+
    
+The principal component analysis deals with the problem of fitting a
+low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than
+the totaldimension \( D \) of the problem at hand (our data
+set). Mathematically it can be formulated as a statistical problem or
+a geometric problem.  In our discussion of the theorem for the
+classical PCA, we will stay with a statistical approach. This is also
+what set the scene historically which for the PCA.
+
+
    
+We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition) 
+
+
      
+
    •  Each data point is determined by \( p \) extrinsic (measurement) variables
      • 
+
    •  We may want to ask the following question: Are there fewer intrinsic variables (say \( d < < p \)) that still approximately describe the data?
      • 
+
    •  If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.
      •  
+
    
+
+









    
+
+
    Introducing the Covariance and Correlation functions  
    
+
+
    
+Before we discuss the PCA theorem, we need to remind ourselves about
+the definition of the covariance and the correlation function. These are quantities
+
+
    
+Suppose we have defined two vectors
+\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as 
+$$
+\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
+             \end{bmatrix},
+$$
+
+where for example
+$$
+\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
+$$
+
+With this definition and recalling that the variance is defined as
+$$
+\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
+$$
+
+we can rewrite the covariance matrix as 
+$$
+\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
+             \end{bmatrix}.
+$$
+
+
    
+The covariance takes values between zero and infinity and may thus
+lead to problems with loss of numerical precision for particularly
+large values. It is common to scale the covariance matrix by
+introducing instead the correlation matrix defined via the so-called
+correlation function
+
+$$
+\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
+$$
+
+
    
+The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
+\in [-1,1] \). This avoids eventual problems with too large values. We
+can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
+and \( \boldsymbol{y} \) as
+
+$$
+\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
+                              \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
+             \end{bmatrix},
+$$
+
+
    
+In the above example this is the function we constructed using pandas.
+
+
    
+









    
+
+
    Correlation Function and Design/Feature Matrix 
    
+
+
    
+In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression
+we defined the design/feature matrix \( \boldsymbol{X} \) as
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  refering to the column numbers and the
+entries \( n \) being the row elements.
+We can rewrite the design/feature matrix in terms of its column vectors as
+$$
+\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix},
+$$
+
+with a given vector
+$$
+\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
+$$
+
+
    
+With these definitions, we can now rewrite our \( 2\times 2 \)
+correaltion/covariance matrix in terms of a moe general design/feature
+matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \)
+covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \)
+
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
+\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
+\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1]  & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
+\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & \mathrm{var}[\boldsymbol{x}_{p-1}]\\
+\end{bmatrix},
+$$
+
+and the correlation matrix
+$$
+\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
+1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1]  & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
+\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1  & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
+\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots & \dots \\
+\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0]   & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}]  & \dots & \dots  & 1\\
+\end{bmatrix},
+$$
+
+
    
+









    
+
+
    Covariance Matrix Examples 
    
+
+
    
+The Numpy function np.cov calculates the covariance elements using
+the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
+the exact mean values.  The following simple function uses the
+np.vstack function which takes each vector of dimension \( 1\times n \)
+and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
+
+$$
+\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\
+                          x_1 & y_1 \\
+                          x_2 & y_2\\
+                          \dots & \dots \\
+                          x_{n-2} & y_{n-2}\\
+                          x_{n-1} & y_{n-1} & 
+             \end{bmatrix},
+$$
+
+
    
+which in turn is converted into into the \( 2\times 2 \) covariance matrix
+\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate
+the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
+function np.mean(x). We can also extract the eigenvalues of the
+covariance matrix through the np.linalg.eig() function.
+
+
    
+
+
+
    # Importing various packages
+import numpy as np
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
+
    
+
    
+









    
+
+
    Correlation Matrix  
    
+
+
    
+The previous example can be converted into the correlation matrix by
+simply scaling the matrix elements with the variances.  We should also
+subtract the mean values for each column. This leads to the following
+code which sets up the correlations matrix for the previous example in
+a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
+
+
    
+
+
+
    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    
+
    
+We see that the matrix elements along the diagonal are one as they
+should be and that the matrix is symmetric. Furthermore, diagonalizing
+this matrix we easily see that it is a positive definite matrix.
+
+
    
+The above procedure with numpy can be made more compact if we use pandas.
+
+
    
+









    
+
+
    Correlation Matrix with Pandas 
    
+
+
    
+We whow here how we can set up the correlation matrix using pandas, as done in this simple code
+
    
+
+
+
    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    
+
    
+We expand this model to the Franke function discussed above.
+
+
    
+









    
+
+
    Correlation Matrix with Pandas and the Franke function 
    
+
+
    
+
+
+
    # Common imports
+import numpy as np
+import pandas as pd
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    
+
    
+We note here that the covariance is zero for the first rows and
+columns since all matrix elements in the design matrix were set to one
+(we are fitting the function in terms of a polynomial of degree \( n \)).
+
+
    
+This means that the variance for these elements will be zero and will
+cause problems when we set up the correlation matrix.  We can simply
+drop these elements and construct a correlation
+matrix without these elements.
+
+
    
+









    
+
+
    Rewriting the Covariance and/or Correlation Matrix 
    
+
+
    
+We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as 
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
+$$
+
+
    
+To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{00} & x_{01}\\
+x_{10} & x_{11}\\
+\end{bmatrix}=\begin{bmatrix}
+\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
+\end{bmatrix}.
+$$
+
+
    
+If we then compute the expectation value
+$$
+\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix}
+x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
+x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
+\end{bmatrix},
+$$
+
+which is just 
+$$
+\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
+                              \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
+             \end{bmatrix},
+$$
+
+where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
+
+
    
+It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
+
+
    
+









    
+
+
    Towards the PCA theorem 
    
+
+
    
+We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
+$$
+
+Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \).
+These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
+
+
    
+Assume also that there is a transformation \( \boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
+
+
    
+That is we have
+$$
+\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
+$$
+
+since the matrix \( \boldsymbol{S} \) is not a data dependent matrix.   Multiplying with \( \boldsymbol{S}^T \) from the left we have
+$$
+\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
+$$
+
+and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
+
+$$
+\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
+$$
+
+
    
+In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
+\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
+
+
    
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If \( p \) is very large,
+we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
+features/predictors.
+
+
    
+









    
+
+
    The Algorithm before theorem 
    
+
+
    
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+
+
      
+
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
+
    
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+
+
      
+
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
+
    •  Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).
      • 
+
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
+
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
+
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
+
    
+
+









    
+
+
    Writing our own PCA code 
    
+
+
    
+We will use a simple example first with two-dimensional data
+drawn from a multivariate normal distribution with the following mean and covariance matrix:
+$$
+\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
+2 & 2
+\end{bmatrix}
+$$
+
+Note that the mean refers to each column of data. 
+We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from
+this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \).
+
+
    
+The following Python code aids in setting up the data and writing out the design matrix.
+Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
+
    
+
+
+
    import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from IPython.display import display
+n = 10000
+mean = (-1, 2)
+cov = [[4, 2], [2, 2]]
+X = np.random.multivariate_normal(mean, cov, n)
+
    
+
    
+Now we are going to implement the PCA algorithm. We will break it down into various substeps.
+
+
    Compute the sample mean and center the data 
    
+
+
    
+The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
+$$
+\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
+$$
+
+and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
+$$
+\bar{x}_i = x_i - \mu_n.
+$$
+
+When you are done with these steps, print out \( \mu_n \) to verify it is
+close to \( \mu \) and plot your mean centered data to verify it is
+centered at the origin! Compare your code with the functionality from Scikit-Learn discussed above.
+The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function. 
+
    
+
+
+
    df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+
    
+
    
+Alternatively, we could use the functions we discussed
+earlier for scaling the data set.  That is, we could have used the
+StandardScaler function in Scikit-Learn, a function which ensures
+that for each feature/predictor we study the mean value is zero and
+the variance is one (every column in the design/feature matrix).  You
+would then not get the same results, since we divide by the
+variance. The diagonal covariance matrix elements will then be one,
+while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
+specific case.
+
+
    Compute the sample covariance 
    
+
+
    
+Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
+$$
+\begin{equation*}
+\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
+\end{equation*}
+$$
+
+where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \).
+We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows
+
    
+
+
+
    print(df.cov())
+print(np.cov(X_centered.T))
+
    
+
    
+Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. 
+Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix. 
+
    
+
+
+
    # extract the relevant columns from the centered design matrix of dim n x 2
+x = X_centered[:,0]
+y = X_centered[:,1]
+Cov = np.zeros((2,2))
+Cov[0,1] = np.sum(x.T@y)/(n-1.0)
+Cov[0,0] = np.sum(x.T@x)/(n-1.0)
+Cov[1,1] = np.sum(y.T@y)/(n-1.0)
+Cov[1,0]= Cov[0,1]
+print("Centered covariance using own code")
+print(Cov)
+plt.plot(x, y, 'x')
+plt.axis('equal')
+plt.show()
+
    
+
    
+Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
+The plot shows how the data are clustered around a line with slope close to one. Is this expected?
+
+
    Diagonalize the sample covariance matrix to obtain the principal components 
    
+
+
    
+Now we are ready to solve for the principal components! To do so we
+diagonalize the sample covariance matrix \( \Sigma \). We can use the
+function np.linalg.eig to do so. It will return the eigenvalues and
+eigenvectors of \( \Sigma \). Once we have these we can perform the 
+following tasks:
+
+
      
+
    •  We compute the percentage of the total variance captured by the first principal component
      • 
+
    •  We plot the mean centered data and lines along the first and second principal components
      • 
+
    •  Then we project the mean centered data onto the first and second principal components, and plot the projected data.
      •  
+
    •  Finally, we approximate the data as
      • 
+
    
+
+$$
+\begin{equation*}
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
+\end{equation*}
+$$
+
+where \( v_0 \) is the first principal component.
+
+
    
+Collecting all these steps we can write our own PCA function and
+compare this with the functionality included in Scikit-Learn.
+
+
    
+The code here outlines some of the elements we could include in the
+analysis. Feel free to extend upon this in order to address the above
+questions.
+
+
    
+
+
+
    # diagonalize and obtain eigenvalues, not necessarily sorted
+EigValues, EigVectors = np.linalg.eig(Cov)
+# sort eigenvectors and eigenvalues
+#permute = EigValues.argsort()
+#EigValues = EigValues[permute]
+#EigVectors = EigVectors[:,permute]
+print("Eigenvalues of Covariance matrix")
+for i in range(2):
+    print(EigValues[i])
+FirstEigvector = EigVectors[:,0]
+SecondEigvector = EigVectors[:,1]
+print("First eigenvector")
+print(FirstEigvector)
+print("Second eigenvector")
+print(SecondEigvector)
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2Dsl = pca.fit_transform(X)
+print("Eigenvector of largest eigenvalue")
+print(pca.components_.T[:, 0])
+
    
+
    
+This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code.  Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?
+
+
    
+









    
+
+
    Classical PCA Theorem   
    
+
+
    
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
+centered as discussed above. For the sake of simplicity we skip the
+overline symbol. The matrix is defined in terms of the various column
+vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension
+\( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+
    
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2,
+$$
+
+with \( \overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+
+
    
+The PCA theorem states that minimizing the above reconstruction error
+corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which
+diagonalizes the empirical covariance(correlation) matrix. The optimal
+low-dimensional encoding of the data is then given by a set of vectors
+\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the
+orthogonal projection of the data onto the columns spanned by the
+eigenvectors of the covariance(correlations matrix).
+
+
    
+The proof which follows will be updated by mid January 2020.
+
+
    
+









    
+
+
    Proof of the PCA Theorem  
    
+
+
    
+To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+
+where the vectors on the rhs are known.
+
+
    
+









    
+
+
    PCA Proof continued 
    
+
+
    
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
+$$
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
+$$
+
+
    
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+$$
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
+$$
+
+since the expectation value of
+$$
+\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0,
+$$
+
+where we have used the fact that our data are centered.
+
+
    
+Recalling our definition of the covariance as 
+$$
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+$$
+
+we have thus that
+$$
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+$$
+
+
    
+We are almost there, we have obtained a relation between minimizing
+the reconstruction error and the variance and the covariance
+matrix. Minimizing the error is equivalent to maximizing the variance
+of the projected data.
+
+
    
+









    
+
+
    The final step 
    
+
+
    
+We could trivially maximize the variance of the projection (and
+thereby minimize the error in the reconstruction function) by letting
+the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we
+want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by
+\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a
+Lagrange multiplier we can then in turn maximize
+
+$$
+J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0).
+$$
+
+Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain
+
+$$
+\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0,
+$$
+
+meaning that
+$$
+\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0.
+$$
+
+The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is
+$$
+\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0.
+$$
+
+
    
+If we want to maximize the variance (minimize the construction error)
+we simply pick the eigenvector of the covariance matrix with the
+largest eigenvalue. This establishes the link between the minimization
+of the reconstruction function \( J \) in terms of an orthogonal matrix
+and the maximization of the variance and thereby the covariance of our
+observations encoded in the design/feature matrix \( \boldsymbol{X} \).
+
+
    
+The proof
+for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be
+established by applying the above arguments and using the fact that
+our basis of eigenvectors is orthogonal, see Murphy chapter
+12.2.  The
+discussion in chapter 12.2 of Murphy's text has also a nice link with
+the Singular Value Decomposition theorem. For categorical data, see
+chapter 12.4 and discussion therein.
+
+
    
+Additional part of the proof for the other eigenvectors will be added by mid January 2020.
+
+
    
+









    
+
+
    Geometric Interpretation and link with Singular Value Decomposition 
    
+
+
    
+This material will be added by mid January 2020.
+
+
    
+









    
+
+
    Principal Component Analysis 
    
+
+
    
+Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
+First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
+
+
    
+The following Python code uses NumPy’s svd() function to obtain all the principal components of the
+training set, then extracts the first two principal components. First we center the data using either pandas or our own code
+
    
+
+
+
    import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 vanilla matrix 
+rows = 10
+cols = 5
+X = np.random.randn(rows,cols)
+df = pd.DataFrame(X)
+# Pandas does the centering for us
+df = df -df.mean()
+display(df)
+
+# we center it ourselves
+X_centered = X - X.mean(axis=0)
+# Then check the difference between pandas and our own set up
+print(X_centered-df)
+#Now we do an SVD
+U, s, V = np.linalg.svd(X_centered)
+c1 = V.T[:, 0]
+c2 = V.T[:, 1]
+W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+print(X2D)
+
    
+
    
+PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
+the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
+forget to center the data first.
+
+
    
+Once you have identified all the principal components, you can reduce the dimensionality of the dataset
+down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
+Selecting this hyperplane ensures that the projection will preserve as much variance as possible. 
+
    
+
+
+
    W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
+
    
+
    
+
+
+
    PCA and scikit-learn 
    
+
+
    
+Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
+following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
+that it automatically takes care of centering the data):
+
    
+
+
+
    #thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D = pca.fit_transform(X)
+print(X2D)
+
    
+
    
+After fitting the PCA transformer to the dataset, you can access the principal components using the
+components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
+principal component is equal to 
+
    
+
+
+
    pca.components_.T[:, 0].
+
    
+
    
+Another very useful piece of information is the explained variance ratio of each principal component,
+available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
+variance that lies along the axis of each principal component.
+
+
    
+









    
+
+
    Back to the Cancer Data 
    
+We can now repeat the above but applied to real data, in this case our breast cancer data.
+Here we compute performance scores on the training data using logistic regression.
+
    
+
+
+
    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.linear_model import LogisticRegression
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+
+logreg = LogisticRegression()
+logreg.fit(X_train, y_train)
+print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
+# We scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Then perform again a log reg fit
+logreg.fit(X_train_scaled, y_train)
+print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
+#thereafter we do a PCA with Scikit-learn
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D_train = pca.fit_transform(X_train_scaled)
+# and finally compute the log reg fit and the score on the training data	
+logreg.fit(X2D_train,y_train)
+print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
+
    
+
    
+We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.
+
+
    
+









    
+
+
    More on the PCA 
    
+
+
    
+Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
+choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
+Unless, of course, you are reducing dimensionality for data visualization — in that case you will
+generally want to reduce the dimensionality down to 2 or 3.
+The following code computes PCA without reducing dimensionality, then computes the minimum number
+of dimensions required to preserve 95% of the training set’s variance:
+
    
+
+
+
    pca = PCA()
+pca.fit(X)
+cumsum = np.cumsum(pca.explained_variance_ratio_)
+d = np.argmax(cumsum >= 0.95) + 1
+
    
+
    
+You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
+of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
+a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
+
    
+
+
+
    pca = PCA(n_components=0.95)
+X_reduced = pca.fit_transform(X)
+
    
+
    
+









    
+
+
    Incremental PCA 
    
+
+
    
+One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
+memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
+been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
+at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
+instances arrive).
+
+
    
+









    
+
+
    Randomized PCA 
    
+
+
    
+Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
+algorithm that quickly finds an approximation of the first d principal components. Its computational
+complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
+previous algorithms when \( d \) is much smaller than \( n \).
+
+
    
+









    
+
+
    Kernel PCA 
    
+
    
+
+
    
+
+
    
+The kernel trick is a mathematical technique that implicitly maps instances into a
+very high-dimensional space (called the feature space), enabling nonlinear classification and regression
+with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
+space corresponds to a complex nonlinear decision boundary in the original space.
+It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
+projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
+preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
+twisted manifold.
+For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
+
    
+
+
+
    from sklearn.decomposition import KernelPCA
+rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
+X_reduced = rbf_pca.fit_transform(X)
+
    
+
+
    
+
+
+
    
+









    
+
+
    LLE 
    
+
+
    
+Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
+(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
+algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
+closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
+these local relationships are best preserved (more details shortly).
+
+
    
+









    
+
+
    Other techniques 
    
+
+
    
+There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
+
+
    
+Here are some of the most popular:
+
+
      
+
    •  Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • 
+
    •  Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • 
+
    •  t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • 
+
    •  Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • 
+
    
+
 
 
 
diff --git a/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz b/doc/pub/week43/ipynb/ipynb-week43-src.tar.gz
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