From 7e96edbcf0ae34ffe9bd9dceecb3b58680228d3c Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 23 Sep 2021 08:34:01 +0200 Subject: [PATCH] update --- doc/pub/week38/html/._week38-bs000.html | 159 +++++----- doc/pub/week38/html/._week38-bs001.html | 159 +++++----- doc/pub/week38/html/._week38-bs002.html | 159 +++++----- doc/pub/week38/html/._week38-bs003.html | 159 +++++----- doc/pub/week38/html/._week38-bs004.html | 159 +++++----- doc/pub/week38/html/._week38-bs005.html | 159 +++++----- doc/pub/week38/html/._week38-bs006.html | 159 +++++----- doc/pub/week38/html/._week38-bs007.html | 159 +++++----- doc/pub/week38/html/._week38-bs008.html | 159 +++++----- doc/pub/week38/html/._week38-bs009.html | 159 +++++----- doc/pub/week38/html/._week38-bs010.html | 159 +++++----- doc/pub/week38/html/._week38-bs011.html | 159 +++++----- doc/pub/week38/html/._week38-bs012.html | 159 +++++----- doc/pub/week38/html/._week38-bs013.html | 161 +++++----- doc/pub/week38/html/._week38-bs014.html | 187 ++++++----- doc/pub/week38/html/._week38-bs015.html | 231 +++++++------- doc/pub/week38/html/._week38-bs016.html | 236 +++++++------- doc/pub/week38/html/._week38-bs017.html | 281 +++++++---------- doc/pub/week38/html/._week38-bs018.html | 226 +++++++------- doc/pub/week38/html/._week38-bs019.html | 215 +++++++------ doc/pub/week38/html/._week38-bs020.html | 252 ++++++++------- doc/pub/week38/html/._week38-bs021.html | 312 +++++++++++-------- doc/pub/week38/html/._week38-bs022.html | 288 ++++++----------- doc/pub/week38/html/._week38-bs023.html | 188 ++++++----- doc/pub/week38/html/._week38-bs024.html | 223 ++++++------- doc/pub/week38/html/._week38-bs025.html | 231 +++++++------- doc/pub/week38/html/._week38-bs026.html | 219 ++++++------- doc/pub/week38/html/._week38-bs027.html | 188 +++++------ doc/pub/week38/html/._week38-bs028.html | 189 ++++++----- doc/pub/week38/html/._week38-bs029.html | 193 ++++++------ doc/pub/week38/html/._week38-bs030.html | 192 ++++++------ doc/pub/week38/html/._week38-bs031.html | 189 ++++++----- doc/pub/week38/html/._week38-bs032.html | 231 ++++++++------ doc/pub/week38/html/._week38-bs033.html | 242 +++++++------- doc/pub/week38/html/._week38-bs034.html | 203 ++++++------ doc/pub/week38/html/._week38-bs035.html | 234 ++++++++------ doc/pub/week38/html/._week38-bs036.html | 231 ++++++-------- doc/pub/week38/html/._week38-bs037.html | 180 ++++++----- doc/pub/week38/html/._week38-bs038.html | 183 ++++++----- doc/pub/week38/html/._week38-bs039.html | 180 ++++++----- doc/pub/week38/html/._week38-bs040.html | 182 ++++++----- doc/pub/week38/html/._week38-bs041.html | 176 +++++------ doc/pub/week38/html/._week38-bs042.html | 181 +++++------ doc/pub/week38/html/week38-bs.html | 159 +++++----- doc/pub/week38/html/week38-reveal.html | 77 ++--- doc/pub/week38/html/week38-solarized.html | 80 ++--- doc/pub/week38/html/week38.html | 80 ++--- doc/pub/week38/ipynb/ipynb-week38-src.tar.gz | Bin 192 -> 193 bytes doc/pub/week38/ipynb/week38.ipynb | 84 ++--- doc/src/week38/week38.do.txt | 76 ++--- 50 files changed, 4303 insertions(+), 4744 deletions(-) diff --git a/doc/pub/week38/html/._week38-bs000.html b/doc/pub/week38/html/._week38-bs000.html index 406c60fa3..d5042016a 100644 --- a/doc/pub/week38/html/._week38-bs000.html +++ b/doc/pub/week38/html/._week38-bs000.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -466,7 +461,7 @@ MathJax.Hub.Config({
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  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -446,7 +441,7 @@ MathJax.Hub.Config({
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  • ...
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    • diff --git a/doc/pub/week38/html/._week38-bs002.html b/doc/pub/week38/html/._week38-bs002.html index bbcd424db..91c52aa41 100644 --- a/doc/pub/week38/html/._week38-bs002.html +++ b/doc/pub/week38/html/._week38-bs002.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
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    • +
  • Conjugate gradient method
    • +
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    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
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    • +
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    • +
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    • +
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    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -443,7 +438,7 @@ MathJax.Hub.Config({
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  • Taking out the mean
    • -
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  • Examples of likelihood functions used in logistic regression and nueral networks
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  • Two parameters
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  • Maximum likelihood
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  • The cost function rewritten
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  • Minimizing the cross entropy
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  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
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  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
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  • Revisiting our Logistic Regression case
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  • Solving using Newton-Raphson's method
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  • Brief reminder on Newton-Raphson's method
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    • +
  • The ideal
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  • The sensitiveness of the gradient descent
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    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
    • +
  • Simple program
    • +
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    • +
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    • +
  • Using gradient descent methods, limitations
    • @@ -497,7 +492,7 @@ $$
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    • diff --git a/doc/pub/week38/html/._week38-bs004.html b/doc/pub/week38/html/._week38-bs004.html index 48fda37a8..7c57ad552 100644 --- a/doc/pub/week38/html/._week38-bs004.html +++ b/doc/pub/week38/html/._week38-bs004.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
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  • Taking out the mean
    • -
  • More complicated Example: The Ising model
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  • Reformulating the problem to suit regression
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  • Linear regression
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  • Singular Value decomposition
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  • The one-dimensional Ising model
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  • Ridge regression
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  • LASSO regression
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  • Some selected properties
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  • The logistic function
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  • Examples of likelihood functions used in logistic regression and nueral networks
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  • Two parameters
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  • Maximum likelihood
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  • The cost function rewritten
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  • Minimizing the cross entropy
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  • A more compact expression
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  • Extending to more predictors
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  • Including more classes
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  • Friday September 24
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  • Wisconsin Cancer Data
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  • Discussing the correlation data
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  • Other measures in classification studies: Cancer Data again
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  • The ideal
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  • Friday September 25
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Revisiting some of our first Linear Regression Encounters
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Taking out the mean
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  • More complicated Example: The Ising model
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  • Reformulating the problem to suit regression
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  • Linear regression
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  • Singular Value decomposition
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  • The one-dimensional Ising model
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  • Ridge regression
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  • LASSO regression
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  • Performance as function of the regularization parameter
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  • Some selected properties
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  • Examples of likelihood functions used in logistic regression and nueral networks
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  • Two parameters
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  • Maximum likelihood
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  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -461,7 +456,7 @@ cross-validation (LOOCV).
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    • diff --git a/doc/pub/week38/html/._week38-bs005.html b/doc/pub/week38/html/._week38-bs005.html index 3546d505f..e7722ffe2 100644 --- a/doc/pub/week38/html/._week38-bs005.html +++ b/doc/pub/week38/html/._week38-bs005.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
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  • Taking out the mean
    • -
  • More complicated Example: The Ising model
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  • Reformulating the problem to suit regression
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  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
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  • Performance as function of the regularization parameter
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  • Logistic Regression
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  • Classification problems
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  • Optimization and Deep learning
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  • Linear classifier
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  • Some selected properties
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  • Simple example
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  • Plotting the mean value for each group
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  • The logistic function
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  • Examples of likelihood functions used in logistic regression and nueral networks
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  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
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  • More classes
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  • Friday September 24
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  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
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  • Discussing the correlation data
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  • Other measures in classification studies: Cancer Data again
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  • Brief reminder on Newton-Raphson's method
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  • Simple geometric interpretation
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  • Extending to more than one variable
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  • Steepest descent
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  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
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  • Convex functions
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  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
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    • -
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  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
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  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
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  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
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  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -473,7 +468,7 @@ $$
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    • -
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    • diff --git a/doc/pub/week38/html/._week38-bs006.html b/doc/pub/week38/html/._week38-bs006.html index 61671ce61..7efb0d77b 100644 --- a/doc/pub/week38/html/._week38-bs006.html +++ b/doc/pub/week38/html/._week38-bs006.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
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  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -464,7 +459,7 @@ For the various values of \( k \)
  • 15
  • 16
  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week38/html/._week38-bs007.html b/doc/pub/week38/html/._week38-bs007.html index a3b304fc9..15b115048 100644 --- a/doc/pub/week38/html/._week38-bs007.html +++ b/doc/pub/week38/html/._week38-bs007.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -543,7 +538,7 @@ plt.show()
  • 16
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    • -
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    • +
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  • »
    • diff --git a/doc/pub/week38/html/._week38-bs008.html b/doc/pub/week38/html/._week38-bs008.html index 2e94ceb3a..750e13d02 100644 --- a/doc/pub/week38/html/._week38-bs008.html +++ b/doc/pub/week38/html/._week38-bs008.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -470,7 +465,7 @@ from the library Scikit-Learn (when not shrinking \( \beta_0 \)) for the
  • 17
  • 18
  • ...
    • -
  • 92
    • +
  • 91
  • »
    • diff --git a/doc/pub/week38/html/._week38-bs009.html b/doc/pub/week38/html/._week38-bs009.html index 582a61dd7..5bc92cd2b 100644 --- a/doc/pub/week38/html/._week38-bs009.html +++ b/doc/pub/week38/html/._week38-bs009.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -479,7 +474,7 @@ This can clearly lead to problems in evaluating the cost/loss functions.
  • 18
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    • -
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    • diff --git a/doc/pub/week38/html/._week38-bs010.html b/doc/pub/week38/html/._week38-bs010.html index 814cd2d6d..7183ff19a 100644 --- a/doc/pub/week38/html/._week38-bs010.html +++ b/doc/pub/week38/html/._week38-bs010.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -473,7 +468,7 @@ y_pred = y_pred 19
  • 20
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    • diff --git a/doc/pub/week38/html/._week38-bs011.html b/doc/pub/week38/html/._week38-bs011.html index 1e85555ce..a2a36a0d1 100644 --- a/doc/pub/week38/html/._week38-bs011.html +++ b/doc/pub/week38/html/._week38-bs011.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -485,7 +480,7 @@ $$
  • 20
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week38/html/._week38-bs012.html b/doc/pub/week38/html/._week38-bs012.html index 21cab17d8..fd50ab60c 100644 --- a/doc/pub/week38/html/._week38-bs012.html +++ b/doc/pub/week38/html/._week38-bs012.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -511,7 +506,7 @@ $$
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    • diff --git a/doc/pub/week38/html/._week38-bs013.html b/doc/pub/week38/html/._week38-bs013.html index 5eee7cab0..981ac928b 100644 --- a/doc/pub/week38/html/._week38-bs013.html +++ b/doc/pub/week38/html/._week38-bs013.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -443,7 +438,7 @@ $$ $$

    -What does this mean for thus? And why do we insist on all this? Let us look at some examples. +What does this mean? And why do we insist on all this? Let us look at some examples.

    @@ -471,7 +466,7 @@ What does this mean for thus? And why do we insist on all this? Let us look at s

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    • diff --git a/doc/pub/week38/html/._week38-bs014.html b/doc/pub/week38/html/._week38-bs014.html index 63e354f36..3deb087c8 100644 --- a/doc/pub/week38/html/._week38-bs014.html +++ b/doc/pub/week38/html/._week38-bs014.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -439,6 +434,10 @@ Note also that we do not split the data into training and test. np.random.seed(2021) +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n + def fit_beta(X, y): return np.linalg.pinv(X.T @ X) @ X.T @ y @@ -466,6 +465,12 @@ clf = LinearRegression(fit_interceptprint(f"True beta: {true_beta}") print(f"Fitted beta: {beta}") print(f"Sklearn fitted beta: {clf.coef_}") +ypredictOwn = X @ beta +ypredictSKL = skl.predict(X) +print(f"MSE with intercept column: {intercept}") +print(MSE(y,ypredictOwn) +print(f"MSE with intercept column from SKL: {intercept}") +print(MSE(y,ypredictSKL) plt.figure() @@ -494,6 +499,12 @@ intercept = np. print(f"Fitted beta (wiothout intercept): {beta}") print(f"Sklearn intercept: {clf.intercept_}") print(f"Sklearn fitted beta (without intercept): {clf.coef_}") +ypredictOwn = X @ beta +ypredictSKL = skl.predict(X) +print(f"MSE with Manual intercept: {intercept}") +print(MSE(y,ypredictOwn) +print(f"MSE with Sklearn intercept: {clf.intercept_}") +print(MSE(y,ypredictSKL) plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)") plt.plot(x, clf.predict(X), "--", label="Sklearn (fit_intercept=True)") @@ -502,6 +513,18 @@ plt.legend() plt.show() +

    +The intercept is the value of our output/target variable +when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). + +

    +Printing the MSE, we see first that both methods give the same +MSE. However, changing the value of the intercept gives a larger or +smaller MSE, meaning that the MSE is penalized by the value of the +intercept! Setting the intercept to true in the Scikit-Learn +function or simply scaling our results, leads to a fit which is +independent of the specific value of the intercept. +

    @@ -528,7 +551,7 @@ plt.show()

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    • diff --git a/doc/pub/week38/html/._week38-bs015.html b/doc/pub/week38/html/._week38-bs015.html index 3e263491e..ac68c1ea3 100644 --- a/doc/pub/week38/html/._week38-bs015.html +++ b/doc/pub/week38/html/._week38-bs015.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -423,13 +418,10 @@ MathJax.Hub.Config({ -

    Another straight Line, the Intercept and Scaling again

    +

    Code Examples

    -As we said above, the intercept is the value of our output/target variable -when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). -Let us now study a case where we again fit a straight line including the intercept in the design matrix -using ordinary least squares. We are only interested in the MSE value here. +Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for yet another vanilla data set.

    @@ -444,10 +436,6 @@ using ordinary least squares. We are only interested in the MSE value here. n = np.size(y_model) return np.sum((y_data-y_model)**2)/n -def fit_beta(X, y): - return np.linalg.pinv(X.T @ X) @ X.T @ y - - # A seed just to ensure that the random numbers are the same for every run. # Useful for eventual debugging. @@ -455,36 +443,55 @@ np.random.= 100 x = np.random.rand(n) -y = 10.0+5*x +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) - -Maxpolydegree = 2 +Maxpolydegree = 20 X = np.zeros((n,Maxpolydegree)) - +#We include explicitely the intercept column for degree in range(Maxpolydegree): X[:,degree] = x**degree - - # We split the data in test and training data X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) -# Intercept is included in the design matrix -# own code first -OwnBeta = fit_beta(X_train, y_train) -# Scikit-Learn -skl = LinearRegression(fit_intercept=False).fit(X_train, y_train) -ypredictOwn = X_test @ OwnBeta -ypredictSKL = skl.predict(X_test) -print(MSE(y_test,ypredictOwn) -print(MSE(y_test,ypredictSKL) +p = Maxpolydegree +I = np.eye(p,p) +# Decide which values of lambda to use +nlambdas = 4 +MSEOwnRidgePredict = np.zeros(nlambdas) +MSERidgePredict = np.zeros(nlambdas) +lambdas = np.logspace(-4, 4, nlambdas) +for i in range(nlambdas): + lmb = lambdas[i] + OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train + # include lasso using Scikit-Learn + # Note: we include the intercept column and no scaling + RegRidge = linear_model.Ridge(lmb,fit_intercept=False) + RegRidge.fit(X_train,y_train) + # and then make the prediction + ytildeOwnRidge = X_train @ OwnRidgeBeta + ypredictOwnRidge = X_test @ OwnRidgeBeta + ytildeRidge = RegRidge.predict(X_train) + ypredictRidge = RegRidge.predict(X_test) + MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge) + MSERidgePredict[i] = MSE(y_test,ypredictRidge) + print("Beta values for own Ridge implementation") + print(OwnRidgeBeta) + print("Beta values for Scikit-Learn Ridge implementation") + print(RegRidge.coef_) +# Now plot the results +plt.figure() +plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test') +plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test') + +plt.xlabel('log10(lambda)') +plt.ylabel('MSE') +plt.legend() +plt.show()

    -Printing the MSE, we see first that both methods give the same -MSE. However, changing the value of the intercept gives a larger or -smaller MSE, meaning that the MSE is penalized by the value of the -intercept! Setting the intercept to true in the Scikit-Learn -function or simply scaling our results, leads to a fit which is -independent of the specific value of the intercept. +The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix. +We see that the results agree very well. What happens if we do not include the intercept in our fit? +Let us see how we can change this code by zero centering.

    @@ -512,7 +519,7 @@ independent of the specific value of the intercept.

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    • diff --git a/doc/pub/week38/html/._week38-bs016.html b/doc/pub/week38/html/._week38-bs016.html index 88ee61c24..f6be53d27 100644 --- a/doc/pub/week38/html/._week38-bs016.html +++ b/doc/pub/week38/html/._week38-bs016.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -423,11 +418,7 @@ MathJax.Hub.Config({ -

    Code Examples

    - -

    -Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for yet another vanilla data set. - +

    Taking out the mean

    @@ -436,68 +427,87 @@ Armed with this wisdom, we attempt first simply set the intercept eqault to F import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from sklearn import linear_model +from sklearn.preprocessing import StandardScaler def MSE(y_data,y_model): n = np.size(y_model) return np.sum((y_data-y_model)**2)/n - - # A seed just to ensure that the random numbers are the same for every run. # Useful for eventual debugging. -np.random.seed(3155) +np.random.seed(315) n = 100 x = np.random.rand(n) y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) Maxpolydegree = 20 -X = np.zeros((n,Maxpolydegree)) -#We include explicitely the intercept column -for degree in range(Maxpolydegree): - X[:,degree] = x**degree +X = np.zeros((n,Maxpolydegree-1)) + +for degree in range(1,Maxpolydegree): #No intercept column + X[:,degree-1] = x**(degree) + # We split the data in test and training data X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) -p = Maxpolydegree + + + + +#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable +X_train_mean = np.mean(X_train,axis=0) +#Center by removing mean from each feature +X_train_scaled = X_train - X_train_mean +X_test_scaled = X_test - X_train_mean +#The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered) +#Remove the intercept from the training data. +y_scaler = np.mean(y_train) +y_train_scaled = y_train - y_scaler + + +p = Maxpolydegree-1 I = np.eye(p,p) # Decide which values of lambda to use nlambdas = 4 MSEOwnRidgePredict = np.zeros(nlambdas) MSERidgePredict = np.zeros(nlambdas) -lambdas = np.logspace(-4, 4, nlambdas) + +lambdas = np.logspace(-4, 1, nlambdas) for i in range(nlambdas): lmb = lambdas[i] - OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train - # include lasso using Scikit-Learn - # Note: we include the intercept column and no scaling - RegRidge = linear_model.Ridge(lmb,fit_intercept=False) + OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled) + intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data + #Add intercept to prediction + ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ + #EQUIVALENT PREDICTION: + #Add intercept to prediction + ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler + print("Values for own Ridge prediction") + print(ypredictOwnRidge) + RegRidge = linear_model.Ridge(lmb) RegRidge.fit(X_train,y_train) - # and then make the prediction - ytildeOwnRidge = X_train @ OwnRidgeBeta - ypredictOwnRidge = X_test @ OwnRidgeBeta - ytildeRidge = RegRidge.predict(X_train) ypredictRidge = RegRidge.predict(X_test) + print("Values for SL Ridge prediction") + print(ypredictRidge) MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge) MSERidgePredict[i] = MSE(y_test,ypredictRidge) print("Beta values for own Ridge implementation") - print(OwnRidgeBeta) + print(OwnRidgeBeta) #Intercept is given by mean of target variable print("Beta values for Scikit-Learn Ridge implementation") print(RegRidge.coef_) + print('Intercept from own implementation:') + print(intercept_) + print('Intercept from Scikit-Learn Ridge implementation') + print(RegRidge.intercept_) + # Now plot the results plt.figure() -plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test') -plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test') - +plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test') +plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test') plt.xlabel('log10(lambda)') plt.ylabel('MSE') plt.legend() plt.show() -

    -The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix. -We see that the results agree very well. What happens if we do not include the intercept in our fit? -Let us see how we can change this code by zero centering. -

    @@ -524,7 +534,7 @@ Let us see how we can change this code by zero centering.

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    • diff --git a/doc/pub/week38/html/._week38-bs017.html b/doc/pub/week38/html/._week38-bs017.html index dc97a19c0..81818dadc 100644 --- a/doc/pub/week38/html/._week38-bs017.html +++ b/doc/pub/week38/html/._week38-bs017.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -423,96 +418,60 @@ MathJax.Hub.Config({ -

    Taking out the mean

    +

    More complicated Example: The Ising model

    + +

    +The one-dimensional Ising model with nearest neighbor interaction, no +external field and a constant coupling constant \( J \) is given by + +$$ +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\tag{1} +\end{align} +$$ + +

    +where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins +in the system is determined by \( L \). For the one-dimensional system +there is no phase transition. + +

    +We will look at a system of \( L = 40 \) spins with a coupling constant of +\( J = 1 \). To get enough training data we will generate 10000 states +with their respective energies. + 

    import numpy as np
-import pandas as pd
 import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
 from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-from sklearn.preprocessing import StandardScaler
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
 
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(315)
+L = 40
+n = int(1e4)
 
-n = 100
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
 
-Maxpolydegree = 20
-X = np.zeros((n,Maxpolydegree-1))
+energies = np.zeros(n)
 
-for degree in range(1,Maxpolydegree): #No intercept column
-    X[:,degree-1] = x**(degree)
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-
-
-
-
-#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
-X_train_mean = np.mean(X_train,axis=0)
-#Center by removing mean from each feature
-X_train_scaled = X_train - X_train_mean 
-X_test_scaled = X_test - X_train_mean
-#The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-#Remove the intercept from the training data.
-y_scaler = np.mean(y_train)           
-y_train_scaled = y_train - y_scaler   
-
-
-p = Maxpolydegree-1
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 4
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-
-lambdas = np.logspace(-4, 1, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
-    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
-    #Add intercept to prediction
-    ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ 
-    #EQUIVALENT PREDICTION:
-    #Add intercept to prediction
-    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler 
-    print("Values for own Ridge prediction")
-    print(ypredictOwnRidge)
-    RegRidge = linear_model.Ridge(lmb)
-    RegRidge.fit(X_train,y_train)
-    ypredictRidge = RegRidge.predict(X_test)
-    print("Values for SL Ridge prediction")
-    print(ypredictRidge)
-    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
-    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
-    print("Beta values for own Ridge implementation")
-    print(OwnRidgeBeta) #Intercept is given by mean of target variable
-    print("Beta values for Scikit-Learn Ridge implementation")
-    print(RegRidge.coef_)
-    print('Intercept from own implementation:')
-    print(intercept_)
-    print('Intercept from Scikit-Learn Ridge implementation')
-    print(RegRidge.intercept_)
-
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
    +

    +Here we use ordinary least squares +regression to predict the energy for the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use linear regression to fit a value for +the coupling constant to achieve this. +

    @@ -539,7 +498,7 @@ plt.show()

  • 26
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  • ...
    • -
  • 92
    • +
  • 91
  • »
    • diff --git a/doc/pub/week38/html/._week38-bs018.html b/doc/pub/week38/html/._week38-bs018.html index 4c735a419..f0f4a3def 100644 --- a/doc/pub/week38/html/._week38-bs018.html +++ b/doc/pub/week38/html/._week38-bs018.html @@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source 2, None, 'linear-regression-code-intercept-handling-first'), - ('Another straight Line, the Intercept and Scaling again', - 2, - None, - 'another-straight-line-the-intercept-and-scaling-again'), ('Code Examples', 2, None, 'code-examples'), ('Taking out the mean', 2, None, 'taking-out-the-mean'), ('More complicated Example: The Ising model', @@ -331,83 +327,82 @@ MathJax.Hub.Config({
  • Further Manipulations
  • Wrapping it up
  • Linear Regression code, Intercept handling first
    • -
  • Another straight Line, the Intercept and Scaling again
    • -
  • Code Examples
    • -
  • Taking out the mean
    • -
  • More complicated Example: The Ising model
    • -
  • Reformulating the problem to suit regression
    • -
  • Linear regression
    • -
  • Singular Value decomposition
    • -
  • The one-dimensional Ising model
    • -
  • Ridge regression
    • -
  • LASSO regression
    • -
  • Performance as function of the regularization parameter
    • -
  • Finding the optimal value of \( \lambda \)
    • -
  • Logistic Regression
    • -
  • Classification problems
    • -
  • Optimization and Deep learning
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • Simple example
    • -
  • Plotting the mean value for each group
    • -
  • The logistic function
    • -
  • Examples of likelihood functions used in logistic regression and nueral networks
    • -
  • Two parameters
    • -
  • Maximum likelihood
    • -
  • The cost function rewritten
    • -
  • Minimizing the cross entropy
    • -
  • A more compact expression
    • -
  • Extending to more predictors
    • -
  • Including more classes
    • -
  • More classes
    • -
  • Friday September 24
    • -
  • Wisconsin Cancer Data
    • -
  • Using the correlation matrix
    • -
  • Discussing the correlation data
    • -
  • Other measures in classification studies: Cancer Data again
    • -
  • Optimization, the central part of any Machine Learning algortithm
    • -
  • Revisiting our Logistic Regression case
    • -
  • The equations to solve
    • -
  • Solving using Newton-Raphson's method
    • -
  • Brief reminder on Newton-Raphson's method
    • -
  • The equations
    • -
  • Simple geometric interpretation
    • -
  • Extending to more than one variable
    • -
  • Steepest descent
    • -
  • More on Steepest descent
    • -
  • The ideal
    • -
  • The sensitiveness of the gradient descent
    • -
  • Convex functions
    • -
  • Convex function
    • -
  • Conditions on convex functions
    • -
  • More on convex functions
    • -
  • Some simple problems
    • -
  • Friday September 25
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Revisiting some of our first Linear Regression Encounters
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • +
  • Code Examples
    • +
  • Taking out the mean
    • +
  • More complicated Example: The Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Singular Value decomposition
    • +
  • The one-dimensional Ising model
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • +
  • Logistic Regression
    • +
  • Classification problems
    • +
  • Optimization and Deep learning
    • +
  • Basics
    • +
  • Linear classifier
    • +
  • Some selected properties
    • +
  • Simple example
    • +
  • Plotting the mean value for each group
    • +
  • The logistic function
    • +
  • Examples of likelihood functions used in logistic regression and nueral networks
    • +
  • Two parameters
    • +
  • Maximum likelihood
    • +
  • The cost function rewritten
    • +
  • Minimizing the cross entropy
    • +
  • A more compact expression
    • +
  • Extending to more predictors
    • +
  • Including more classes
    • +
  • More classes
    • +
  • Friday September 24
    • +
  • Wisconsin Cancer Data
    • +
  • Using the correlation matrix
    • +
  • Discussing the correlation data
    • +
  • Other measures in classification studies: Cancer Data again
    • +
  • Optimization, the central part of any Machine Learning algortithm
    • +
  • Revisiting our Logistic Regression case
    • +
  • The equations to solve
    • +
  • Solving using Newton-Raphson's method
    • +
  • Brief reminder on Newton-Raphson's method
    • +
  • The equations
    • +
  • Simple geometric interpretation
    • +
  • Extending to more than one variable
    • +
  • Steepest descent
    • +
  • More on Steepest descent
    • +
  • The ideal
    • +
  • The sensitiveness of the gradient descent
    • +
  • Convex functions
    • +
  • Convex function
    • +
  • Conditions on convex functions
    • +
  • More on convex functions
    • +
  • Some simple problems
    • +
  • Friday September 25
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting some of our first Linear Regression Encounters
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • @@ -423,60 +418,53 @@ MathJax.Hub.Config({ -

    More complicated Example: The Ising model

    +

    Reformulating the problem to suit regression

    -The one-dimensional Ising model with nearest neighbor interaction, no -external field and a constant coupling constant \( J \) is given by +A more general form for the one-dimensional Ising model is $$ \begin{align} - H = -J \sum_{k}^L s_k s_{k + 1}, -\tag{1} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\tag{2} \end{align} $$

    -where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins -in the system is determined by \( L \). For the one-dimensional system -there is no phase transition. +Here we allow for interactions beyond the nearest neighbors and a state dependent +coupling constant. This latter expression can be formulated as +a matrix-product +$$ +\begin{align} + \boldsymbol{H} = \boldsymbol{X} J, +\tag{3} +\end{align} +$$

    -We will look at a system of \( L = 40 \) spins with a coupling constant of -\( J = 1 \). To get enough training data we will generate 10000 states -with their respective energies. +where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the +elements \( -J_{jk} \). This form of writing the energy fits perfectly +with the form utilized in linear regression, that is + +$$ +\begin{align} + \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}, +\tag{4} +\end{align} +$$ + +

    +We split the data in training and test data as discussed in the previous example

    -

    import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.axes_grid1 import make_axes_locatable
-import seaborn as sns
-import scipy.linalg as scl
-from sklearn.model_selection import train_test_split
-import tqdm
-sns.set(color_codes=True)
-cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
-
-L = 40
-n = int(1e4)
-
-spins = np.random.choice([-1, 1], size=(n, L))
-J = 1.0
-
-energies = np.zeros(n)
-
+
    X = np.zeros((n, L ** 2))
 for i in range(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
 
    
-
    
-Here we use ordinary least squares
-regression to predict the energy for the nearest neighbor
-one-dimensional Ising model on a ring, i.e., the endpoints wrap
-around. We will use linear regression to fit a value for
-the coupling constant to achieve this.
-
 
    
 
    
 
@@ -503,7 +491,7 @@ the coupling constant to achieve this.
   
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diff --git a/doc/pub/week38/html/._week38-bs019.html b/doc/pub/week38/html/._week38-bs019.html
index 83603bda7..29fa795a5 100644
--- a/doc/pub/week38/html/._week38-bs019.html
+++ b/doc/pub/week38/html/._week38-bs019.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,52 +418,50 @@ MathJax.Hub.Config({
 
 
 
-
    Reformulating the problem to suit regression 
    
+
    Linear regression 
    
 
 
    
-A more general form for the one-dimensional Ising model is
+In the ordinary least squares method we choose the cost function
 
 $$
 \begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{2}
+    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\tag{5}
 \end{align}
 $$
 
 
    
-Here we allow for interactions beyond the nearest neighbors and a state dependent
-coupling constant. This latter expression can be formulated as
-a matrix-product
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
 $$
-\begin{align}
-    \boldsymbol{H} = \boldsymbol{X} J,
-\tag{3}
-\end{align}
+    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
 $$
 
 
    
-where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, that is
-
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
-\tag{4}
-\end{align}
-$$
-
-
    
-We split the data in training and test data as discussed in the previous example
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
 
 
    
 
 
-
    X = np.zeros((n, L ** 2))
-for i in range(n):
-    X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
    X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
    
+
    
+
+
+
    def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+beta = ols_inv(X_train_own, y_train)
 
    
 
    
 
    
@@ -496,7 +489,7 @@ X_train, X_test, y_train, y_test = train_tes
   
  • 28
    • 
   
  • 29
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs020.html b/doc/pub/week38/html/._week38-bs020.html
index 9695ffc89..8c4fc9923 100644
--- a/doc/pub/week38/html/._week38-bs020.html
+++ b/doc/pub/week38/html/._week38-bs020.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,51 +418,90 @@ MathJax.Hub.Config({
 
 
 
-
    Linear regression 
    
+
    Singular Value decomposition 
    
 
 
    
-In the ordinary least squares method we choose the cost function
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the singular
+value decomposition. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
 
+$$
+    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+
    
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+
    
+Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
 $$
 \begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
-\tag{5}
+    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{6}
 \end{align}
 $$
 
 
    
-We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
-This yields the expression for \( \boldsymbol{\beta} \) to be
-
-$$
-    \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
-$$
-
-
    
-which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
-an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
-intercept, i.e., a constant term, we must make sure that the
-first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
 
 
    
 
 
-
    X_train_own = np.concatenate(
-    (np.ones(len(X_train))[:, np.newaxis], X_train),
-    axis=1
-)
-X_test_own = np.concatenate(
-    (np.ones(len(X_test))[:, np.newaxis], X_test),
-    axis=1
-)
+
    def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
 
    
 
    
 
 
-
    def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
-    return scl.inv(x.T @ x) @ (x.T @ y)
-beta = ols_inv(X_train_own, y_train)
+
    beta = ols_svd(X_train_own,y_train)
 
    
+
    
+When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
+
+
    
+
+
+
    J = beta[1:].reshape(L, L)
+
    
+
    
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+
    
+
+
+
    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J, **cmap_args)
+plt.title("OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
    
+
    
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+
    
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
 
    
 
    
 
@@ -494,7 +528,7 @@ beta = ols_inv(X_train_own, y_train)
   
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    • 
   
  • 30
    • 
   
  • ...
    • 
-  
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    • 
+  
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  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs021.html b/doc/pub/week38/html/._week38-bs021.html
index 28a951dc3..10aaddbc0 100644
--- a/doc/pub/week38/html/._week38-bs021.html
+++ b/doc/pub/week38/html/._week38-bs021.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,74 +418,128 @@ MathJax.Hub.Config({
 
 
 
-
    Singular Value decomposition 
    
+
    The one-dimensional Ising model 
    
 
 
    
-Doing the inversion directly turns out to be a bad idea since the matrix
-\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the singular
-value decomposition. Using the definition of the Moore-Penrose
-pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+Let us bring back the Ising model again, but now with an additional
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data.  The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant \( J \) is
+given by
 
-$$
-    \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
-$$
-
-
    
-where the pseudoinverse of \( \boldsymbol{X} \) is given by
-
-$$
-    \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
-$$
-
-
    
-Using singular value decomposition we can decompose the matrix  \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
-where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
-where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
-\( \omega \) to
 $$
 \begin{align}
-    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
-\tag{6}
+    H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{7}
+\end{align}
+$$
+
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
+
+
    
+We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
+
+
    
+
+
+
    import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
    
+
    
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{8}
 \end{align}
 $$
 
 
    
-Note that solving this equation by actually doing the pseudoinverse
-(which is what we will do) is not a good idea as this operation scales
-as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
-general matrix. Instead, doing \( QR \)-factorization and solving the
-linear system as an equation would reduce this down to
-\( \mathcal{O}(n^2) \) operations.
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+$$
+\begin{align}
+    H = X J,
+\tag{9}
+\end{align}
+$$
+
+
    
+where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+$$
+\begin{align}
+    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
+\tag{10}
+\end{align}
+$$
+
+We organize the data as we did above
+
    
+
+
+
    X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
    
+
    
+We will do all fitting with Scikit-Learn,
 
 
    
 
 
-
    def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
-    u, s, v = scl.svd(x)
-    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
    clf = skl.LinearRegression().fit(X_train, y_train)
 
    
 
    
+When  extracting the \( J \)-matrix we make sure to remove the intercept
+
    
 
 
-
    beta = ols_svd(X_train_own,y_train)
+
    J_sk = clf.coef_.reshape(L, L)
 
    
 
    
-When extracting the \( J \)-matrix  we need to make sure that we remove the intercept, as is done here
-
-
    
-
-
-
    J = beta[1:].reshape(L, L)
-
    
-
    
-A way of looking at the coefficients in \( J \) is to plot the matrices as images.
-
+And then we plot the results
 
    
 
 
 
    fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J, **cmap_args)
-plt.title("OLS", fontsize=18)
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
 plt.xticks(fontsize=18)
 plt.yticks(fontsize=18)
 cb = fig.colorbar(im)
@@ -498,14 +547,7 @@ cb.ax.se
 plt.show()
 
    
 
    
-It is interesting to note that OLS
-considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
-valid matrix elements for \( J \).
-In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
-this problem can be removed, partly and only with Lasso regression.
-
-
    
-In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+The results perfectly with our previous discussion where we used our own code.
 
 
    
 
    
@@ -533,7 +575,7 @@ In this case our matrix inversion was actually possible. The obvious question no
   
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    • 
   
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    • 
   
  • ...
    • 
-  
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs022.html b/doc/pub/week38/html/._week38-bs022.html
index f69710e86..d1b4e1c9a 100644
--- a/doc/pub/week38/html/._week38-bs022.html
+++ b/doc/pub/week38/html/._week38-bs022.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,137 +418,38 @@ MathJax.Hub.Config({
 
 
 
-
    The one-dimensional Ising model 
    
+
    Ridge regression 
    
 
 
    
-Let us bring back the Ising model again, but now with an additional
-focus on Ridge and Lasso regression as well. We repeat some of the
-basic parts of the Ising model and the setup of the training and test
-data.  The one-dimensional Ising model with nearest neighbor
-interaction, no external field and a constant coupling constant \( J \) is
-given by
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a regularizer. This
+involves a new cost function which leads to a new estimate for the
+weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
+cost function is given by
 
 $$
 \begin{align}
-    H = -J \sum_{k}^L s_k s_{k + 1},
-\tag{7}
-\end{align}
-$$
-
-where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
-
-
    
-We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
-
-
    
-
-
-
    import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.axes_grid1 import make_axes_locatable
-import seaborn as sns
-import scipy.linalg as scl
-from sklearn.model_selection import train_test_split
-import sklearn.linear_model as skl
-import tqdm
-sns.set(color_codes=True)
-cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
-
-L = 40
-n = int(1e4)
-
-spins = np.random.choice([-1, 1], size=(n, L))
-J = 1.0
-
-energies = np.zeros(n)
-
-for i in range(n):
-    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
-
    
-
    
-A more general form for the one-dimensional Ising model is
-
-$$
-\begin{align}
-    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
-\tag{8}
+    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
+\tag{11}
 \end{align}
 $$
 
-
    
-Here we allow for interactions beyond the nearest neighbors and a more
-adaptive coupling matrix. This latter expression can be formulated as
-a matrix-product on the form
-$$
-\begin{align}
-    H = X J,
-\tag{9}
-\end{align}
-$$
-
-
    
-where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
-elements \( -J_{jk} \). This form of writing the energy fits perfectly
-with the form utilized in linear regression, viz.
-$$
-\begin{align}
-    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
-\tag{10}
-\end{align}
-$$
-
-We organize the data as we did above
 
    
 
 
-
    X = np.zeros((n, L ** 2))
-for i in range(n):
-    X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
-
-X_train_own = np.concatenate(
-    (np.ones(len(X_train))[:, np.newaxis], X_train),
-    axis=1
-)
-
-X_test_own = np.concatenate(
-    (np.ones(len(X_test))[:, np.newaxis], X_test),
-    axis=1
-)
-
    
-
    
-We will do all fitting with Scikit-Learn,
-
-
    
-
-
-
    clf = skl.LinearRegression().fit(X_train, y_train)
-
    
-
    
-When  extracting the \( J \)-matrix we make sure to remove the intercept
-
    
-
-
-
    J_sk = clf.coef_.reshape(L, L)
-
    
-
    
-And then we plot the results
-
    
-
-
-
    fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_sk, **cmap_args)
-plt.title("LinearRegression from Scikit-learn", fontsize=18)
+
    _lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
 plt.xticks(fontsize=18)
 plt.yticks(fontsize=18)
 cb = fig.colorbar(im)
 cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
 plt.show()
 
    
-
    
-The results perfectly with our previous discussion where we used our own code.
-
 
    
 
    
 
@@ -580,7 +476,7 @@ The results perfectly with our previous discussion where we used our own code.
   
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-  
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs023.html b/doc/pub/week38/html/._week38-bs023.html
index 7cf5de9eb..c7b61759f 100644
--- a/doc/pub/week38/html/._week38-bs023.html
+++ b/doc/pub/week38/html/._week38-bs023.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,31 +418,29 @@ MathJax.Hub.Config({
 
 
 
-
    Ridge regression 
    
+
    LASSO regression 
    
 
 
    
-Having explored the ordinary least squares we move on to ridge
-regression. In ridge regression we include a regularizer. This
-involves a new cost function which leads to a new estimate for the
-weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
-cost function is given by
+In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
 
 $$
 \begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
-\tag{11}
+    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
+\tag{12}
 \end{align}
 $$
 
+
    
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
+
 
    
 
 
-
    _lambda = 0.1
-clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
-J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+
    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
 fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_ridge_sk, **cmap_args)
-plt.title("Ridge from Scikit-learn", fontsize=18)
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
 plt.xticks(fontsize=18)
 plt.yticks(fontsize=18)
 cb = fig.colorbar(im)
@@ -455,6 +448,11 @@ cb.ax.se
 
 plt.show()
 
    
+
    
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+\( J_{j, j + 1} = -1 \).
+
 
    
 
    
 
@@ -481,7 +479,7 @@ plt.show()
   
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-  
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs024.html b/doc/pub/week38/html/._week38-bs024.html
index df865fe1c..7ac38c92b 100644
--- a/doc/pub/week38/html/._week38-bs024.html
+++ b/doc/pub/week38/html/._week38-bs024.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,40 +418,56 @@ MathJax.Hub.Config({
 
 
 
-
    LASSO regression 
    
+
    Performance as  function of the regularization parameter 
    
 
 
    
-In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
-
-$$
-\begin{align}
-    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
-\tag{12}
-\end{align}
-$$
-
-
    
-Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
+We see how the different models perform for a different set of values for \( \lambda \).
 
 
    
 
 
-
    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
-J_lasso_sk = clf_lasso.coef_.reshape(L, L)
-fig = plt.figure(figsize=(20, 14))
-im = plt.imshow(J_lasso_sk, **cmap_args)
-plt.title("Lasso from Scikit-learn", fontsize=18)
-plt.xticks(fontsize=18)
-plt.yticks(fontsize=18)
-cb = fig.colorbar(im)
-cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
    lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
 
 plt.show()
 
    
 
    
-It is quite striking how LASSO breaks the symmetry of the coupling
-constant as opposed to ridge and OLS. We get a sparse solution with
-\( J_{j, j + 1} = -1 \).
+We see that LASSO reaches a good solution for low
+values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
+much. Ridge is more stable over a larger range of values for
+\( \lambda \), but eventually also fades away.
 
 
    
 
    
@@ -484,7 +495,7 @@ constant as opposed to ridge and OLS. We get a sparse solution with
   
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diff --git a/doc/pub/week38/html/._week38-bs025.html b/doc/pub/week38/html/._week38-bs025.html
index 02f3d0474..3b5c8f105 100644
--- a/doc/pub/week38/html/._week38-bs025.html
+++ b/doc/pub/week38/html/._week38-bs025.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,56 +418,54 @@ MathJax.Hub.Config({
 
 
 
-
    Performance as  function of the regularization parameter 
    
+
    Finding the optimal value of \( \lambda \) 
    
 
 
    
-We see how the different models perform for a different set of values for \( \lambda \).
+To determine which value of \( \lambda \) is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
 
 
    
 
 
-
    lambdas = np.logspace(-4, 5, 10)
+
    fig = plt.figure(figsize=(20, 14))
 
-train_errors = {
-    "ols_sk": np.zeros(lambdas.size),
-    "ridge_sk": np.zeros(lambdas.size),
-    "lasso_sk": np.zeros(lambdas.size)
+colors = {
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
 }
 
-test_errors = {
-    "ols_sk": np.zeros(lambdas.size),
-    "ridge_sk": np.zeros(lambdas.size),
-    "lasso_sk": np.zeros(lambdas.size)
-}
-
-plot_counter = 1
-
-fig = plt.figure(figsize=(32, 54))
-
-for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
-    for key, method in zip(
-        ["ols_sk", "ridge_sk", "lasso_sk"],
-        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
-    ):
-        method = method.fit(X_train, y_train)
-
-        train_errors[key][i] = method.score(X_train, y_train)
-        test_errors[key][i] = method.score(X_test, y_test)
-
-        omega = method.coef_.reshape(L, L)
-
-        plt.subplot(10, 5, plot_counter)
-        plt.imshow(omega, **cmap_args)
-        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
-        plot_counter += 1
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
 
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
 plt.show()
 
    
 
    
-We see that LASSO reaches a good solution for low
-values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
-much. Ridge is more stable over a larger range of values for
-\( \lambda \), but eventually also fades away.
+From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of \( \lambda \).
 
 
    
 
    
@@ -500,7 +493,7 @@ much. Ridge is more stable over a larger range of values for
   
  • 34
    • 
   
  • 35
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs026.html b/doc/pub/week38/html/._week38-bs026.html
index 79e3730df..8019c1d72 100644
--- a/doc/pub/week38/html/._week38-bs026.html
+++ b/doc/pub/week38/html/._week38-bs026.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,56 +416,22 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Finding the optimal value of \( \lambda \) 
    
+
    Logistic Regression 
    
 
 
    
-To determine which value of \( \lambda \) is best we plot the accuracy of
-the models when predicting the training and the testing set. We expect
-the accuracy of the training set to be quite good, but if the accuracy
-of the testing set is much lower this tells us that we might be
-subject to an overfit model. The ideal scenario is an accuracy on the
-testing set that is close to the accuracy of the training set.
-
-
    
-
-
-
    fig = plt.figure(figsize=(20, 14))
-
-colors = {
-    "ols_sk": "r",
-    "ridge_sk": "y",
-    "lasso_sk": "c"
-}
-
-for key in train_errors:
-    plt.semilogx(
-        lambdas,
-        train_errors[key],
-        colors[key],
-        label="Train {0}".format(key),
-        linewidth=4.0
-    )
-
-for key in test_errors:
-    plt.semilogx(
-        lambdas,
-        test_errors[key],
-        colors[key] + "--",
-        label="Test {0}".format(key),
-        linewidth=4.0
-    )
-plt.legend(loc="best", fontsize=18)
-plt.xlabel(r"$\lambda$", fontsize=18)
-plt.ylabel(r"$R^2$", fontsize=18)
-plt.tick_params(labelsize=18)
-plt.show()
-
    
-
    
-From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
-achieves a very good accuracy on the test set. This by far surpasses the
-other models for all values of \( \lambda \).
+In linear regression our main interest was centered on learning the
+coefficients of a functional fit (say a polynomial) in order to be
+able to predict the response of a continuous variable on some unseen
+data. The fit to the continuous variable \( y_i \) is based on some
+independent variables \( \boldsymbol{x}_i \). Linear regression resulted in
+analytical expressions for standard ordinary Least Squares or Ridge
+regression (in terms of matrices to invert) for several quantities,
+ranging from the variance and thereby the confidence intervals of the
+parameters \( \boldsymbol{\beta} \) to the mean squared error. If we can invert
+the product of the design matrices, linear regression gives then a
+simple recipe for fitting our data.
 
 
    
 
    
@@ -498,7 +459,7 @@ other models for all values of \( \lambda \).
   
  • 35
    • 
   
  • 36
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs027.html b/doc/pub/week38/html/._week38-bs027.html
index f18d73644..ec191791b 100644
--- a/doc/pub/week38/html/._week38-bs027.html
+++ b/doc/pub/week38/html/._week38-bs027.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,20 +418,25 @@ MathJax.Hub.Config({
 
 
 
-
    Logistic Regression 
    
+
    Classification problems 
    
 
 
    
-In linear regression our main interest was centered on learning the
-coefficients of a functional fit (say a polynomial) in order to be
-able to predict the response of a continuous variable on some unseen
-data. The fit to the continuous variable \( y_i \) is based on some
-independent variables \( \boldsymbol{x}_i \). Linear regression resulted in
-analytical expressions for standard ordinary Least Squares or Ridge
-regression (in terms of matrices to invert) for several quantities,
-ranging from the variance and thereby the confidence intervals of the
-parameters \( \boldsymbol{\beta} \) to the mean squared error. If we can invert
-the product of the design matrices, linear regression gives then a
-simple recipe for fitting our data.
+Classification problems, however, are concerned with outcomes taking
+the form of discrete variables (i.e. categories). We may for example,
+on the basis of DNA sequencing for a number of patients, like to find
+out which mutations are important for a certain disease; or based on
+scans of various patients' brains, figure out if there is a tumor or
+not; or given a specific physical system, we'd like to identify its
+state, say whether it is an ordered or disordered system (typical
+situation in solid state physics); or classify the status of a
+patient, whether she/he has a stroke or not and many other similar
+situations.
+
+
    
+The most common situation we encounter when we apply logistic
+regression is that of two possible outcomes, normally denoted as a
+binary outcome, true or false, positive or negative, success or
+failure etc.
 
 
    
 
    
@@ -464,7 +464,7 @@ simple recipe for fitting our data.
   
  • 36
    • 
   
  • 37
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs028.html b/doc/pub/week38/html/._week38-bs028.html
index 6d5f0796b..c17ab46b3 100644
--- a/doc/pub/week38/html/._week38-bs028.html
+++ b/doc/pub/week38/html/._week38-bs028.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,27 +416,25 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Classification problems 
    
+
    Optimization and Deep learning 
    
 
 
    
-Classification problems, however, are concerned with outcomes taking
-the form of discrete variables (i.e. categories). We may for example,
-on the basis of DNA sequencing for a number of patients, like to find
-out which mutations are important for a certain disease; or based on
-scans of various patients' brains, figure out if there is a tumor or
-not; or given a specific physical system, we'd like to identify its
-state, say whether it is an ordered or disordered system (typical
-situation in solid state physics); or classify the status of a
-patient, whether she/he has a stroke or not and many other similar
-situations.
+Logistic regression will also serve as our stepping stone towards
+neural network algorithms and supervised deep learning. For logistic
+learning, the minimization of the cost function leads to a non-linear
+equation in the parameters \( \boldsymbol{\beta} \). The optimization of the
+problem calls therefore for minimization algorithms. This forms the
+bottle neck of all machine learning algorithms, namely how to find
+reliable minima of a multi-variable function. This leads us to the
+family of gradient descent methods. The latter are the working horses
+of basically all modern machine learning algorithms.
 
 
    
-The most common situation we encounter when we apply logistic
-regression is that of two possible outcomes, normally denoted as a
-binary outcome, true or false, positive or negative, success or
-failure etc.
+We note also that many of the topics discussed here on logistic 
+regression are also commonly used in modern supervised Deep Learning
+models, as we will see later.
 
 
    
 
    
@@ -469,7 +462,7 @@ failure etc.
   
  • 37
    • 
   
  • 38
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs029.html b/doc/pub/week38/html/._week38-bs029.html
index d2e947ca1..1633e4aa3 100644
--- a/doc/pub/week38/html/._week38-bs029.html
+++ b/doc/pub/week38/html/._week38-bs029.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,25 +416,31 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Optimization and Deep learning 
    
+
    Basics 
    
 
 
    
-Logistic regression will also serve as our stepping stone towards
-neural network algorithms and supervised deep learning. For logistic
-learning, the minimization of the cost function leads to a non-linear
-equation in the parameters \( \boldsymbol{\beta} \). The optimization of the
-problem calls therefore for minimization algorithms. This forms the
-bottle neck of all machine learning algorithms, namely how to find
-reliable minima of a multi-variable function. This leads us to the
-family of gradient descent methods. The latter are the working horses
-of basically all modern machine learning algorithms.
+We consider the case where the dependent variables, also called the
+responses or the outcomes, \( y_i \) are discrete and only take values
+from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
 
 
    
-We note also that many of the topics discussed here on logistic 
-regression are also commonly used in modern supervised Deep Learning
-models, as we will see later.
+The goal is to predict the
+output classes from the design matrix \( \boldsymbol{X}\in\mathbb{R}^{n\times p} \)
+made of \( n \) samples, each of which carries \( p \) features or predictors. The
+primary goal is to identify the classes to which new unseen samples
+belong.
+
+
    
+Let us specialize to the case of two classes only, with outputs
+\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
+credit card user that could default or not on her/his credit card
+debt. That is
+
+$$
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\  1 & \mathrm{yes} \end{bmatrix}.
+$$
 
 
    
 
    
@@ -467,7 +468,7 @@ models, as we will see later.
   
  • 38
    • 
   
  • 39
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs030.html b/doc/pub/week38/html/._week38-bs030.html
index 25f02bff3..ecae8af76 100644
--- a/doc/pub/week38/html/._week38-bs030.html
+++ b/doc/pub/week38/html/._week38-bs030.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,32 +416,29 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Basics 
    
+
    Linear classifier 
    
 
 
    
-We consider the case where the dependent variables, also called the
-responses or the outcomes, \( y_i \) are discrete and only take values
-from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
+Before moving to the logistic model, let us try to use our linear
+regression model to classify these two outcomes. We could for example
+fit a linear model to the default case if \( y_i > 0.5 \) and the no
+default case \( y_i \leq 0.5 \).
 
 
    
-The goal is to predict the
-output classes from the design matrix \( \boldsymbol{X}\in\mathbb{R}^{n\times p} \)
-made of \( n \) samples, each of which carries \( p \) features or predictors. The
-primary goal is to identify the classes to which new unseen samples
-belong.
-
-
    
-Let us specialize to the case of two classes only, with outputs
-\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
-credit card user that could default or not on her/his credit card
-debt. That is
-
+We would then have our 
+weighted linear combination, namely 
 $$
-y_i = \begin{bmatrix} 0 & \mathrm{no}\\  1 & \mathrm{yes} \end{bmatrix}.
+\begin{equation}
+\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} +  \boldsymbol{\epsilon},
+\tag{13}
+\end{equation}
 $$
 
+where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \boldsymbol{X} \) is our
+\( n\times p \) design matrix and \( \boldsymbol{\beta} \) represents our estimators/predictors.
+
 
    
 
    
 
@@ -473,7 +465,7 @@ $$
   
  • 39
    • 
   
  • 40
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs031.html b/doc/pub/week38/html/._week38-bs031.html
index 5e191feaf..806addd4a 100644
--- a/doc/pub/week38/html/._week38-bs031.html
+++ b/doc/pub/week38/html/._week38-bs031.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,26 +418,24 @@ MathJax.Hub.Config({
 
 
 
-
    Linear classifier 
    
+
    Some selected properties 
    
 
 
    
-Before moving to the logistic model, let us try to use our linear
-regression model to classify these two outcomes. We could for example
-fit a linear model to the default case if \( y_i > 0.5 \) and the no
-default case \( y_i \leq 0.5 \).
+The main problem with our function is that it takes values on the
+entire real axis. In the case of logistic regression, however, the
+labels \( y_i \) are discrete variables. A typical example is the credit
+card data discussed below here, where we can set the state of
+defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
+in the data set (see the full example below).
 
 
    
-We would then have our 
-weighted linear combination, namely 
-$$
-\begin{equation}
-\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} +  \boldsymbol{\epsilon},
-\tag{13}
-\end{equation}
-$$
-
-where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \boldsymbol{X} \) is our
-\( n\times p \) design matrix and \( \boldsymbol{\beta} \) represents our estimators/predictors.
+One simple way to get a discrete output is to have sign
+functions that map the output of a linear regressor to values \( \{0,1\} \),
+\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. 
+We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
+literature. This model is extremely simple. However, in many cases it is more
+favorable to use a ``soft" classifier that outputs
+the probability of a given category. This leads us to the logistic function.
 
 
    
 
    
@@ -470,7 +463,7 @@ where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \b
   
  • 40
    • 
   
  • 41
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
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    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs032.html b/doc/pub/week38/html/._week38-bs032.html
index 784425453..d4bbd8380 100644
--- a/doc/pub/week38/html/._week38-bs032.html
+++ b/doc/pub/week38/html/._week38-bs032.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,25 +418,69 @@ MathJax.Hub.Config({
 
 
 
-
    Some selected properties 
    
+
    Simple example 
    
 
 
    
-The main problem with our function is that it takes values on the
-entire real axis. In the case of logistic regression, however, the
-labels \( y_i \) are discrete variables. A typical example is the credit
-card data discussed below here, where we can set the state of
-defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
-in the data set (see the full example below).
+The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput  is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
 
 
    
-One simple way to get a discrete output is to have sign
-functions that map the output of a linear regressor to values \( \{0,1\} \),
-\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. 
-We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
-literature. This model is extremely simple. However, in many cases it is more
-favorable to use a ``soft" classifier that outputs
-the probability of a given category. This leads us to the logistic function.
 
+
+
    # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+from IPython.display import display
+from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("chddata.csv"),'r')
+
+# Read the chd data as  csv file and organize the data into arrays with age group, age, and chd
+chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
+chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
+output = chd['CHD']
+age = chd['Age']
+agegroup = chd['Agegroup']
+numberID  = chd['ID'] 
+display(chd)
+
+plt.scatter(age, output, marker='o')
+plt.axis([18,70.0,-0.1, 1.2])
+plt.xlabel(r'Age')
+plt.ylabel(r'CHD')
+plt.title(r'Age distribution and Coronary heart disease')
+plt.show()
+
    
 
    
 
    
 
@@ -468,7 +507,7 @@ the probability of a given category. This leads us to the logistic function.
   
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    • 
   
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    • 
   
  • ...
    • 
-  
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    • 
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    • 
   
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs033.html b/doc/pub/week38/html/._week38-bs033.html
index a22a99d32..aae61091f 100644
--- a/doc/pub/week38/html/._week38-bs033.html
+++ b/doc/pub/week38/html/._week38-bs033.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,69 +418,42 @@ MathJax.Hub.Config({
 
 
 
-
    Simple example 
    
+
    Plotting the mean value for each group 
    
 
 
    
-The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput  is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
+What we could attempt however is to plot the mean value for each group.
 
 
    
 
 
-
    # Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.model_selection import train_test_split
-from sklearn.utils import resample
-from sklearn.metrics import mean_squared_error
-from IPython.display import display
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("chddata.csv"),'r')
-
-# Read the chd data as  csv file and organize the data into arrays with age group, age, and chd
-chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
-chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
-output = chd['CHD']
-age = chd['Age']
-agegroup = chd['Agegroup']
-numberID  = chd['ID'] 
-display(chd)
-
-plt.scatter(age, output, marker='o')
-plt.axis([18,70.0,-0.1, 1.2])
-plt.xlabel(r'Age')
-plt.ylabel(r'CHD')
-plt.title(r'Age distribution and Coronary heart disease')
+
    agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
+group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
+plt.plot(group, agegroupmean, "r-")
+plt.axis([0,9,0, 1.0])
+plt.xlabel(r'Age group')
+plt.ylabel(r'CHD mean values')
+plt.title(r'Mean values for each age group')
 plt.show()
 
    
+
    
+We are now trying to find a function \( f(y\vert x) \), that is a function which gives us an expected value for the output \( y \) with a given input \( x \).
+In standard linear regression with a linear dependence on \( x \), we would write this in terms of our model
+$$
+f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
+$$
+
+
    
+This expression implies however that \( f(y_i\vert x_i) \) could take any
+value from minus infinity to plus infinity. If we however let
+\( f(y\vert y) \) be represented by the mean value, the above example
+shows us that we can constrain the function to take values between
+zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
+at our last curve we see also that it has an S-shaped form. This leads
+us to a very popular model for the function \( f \), namely the so-called
+Sigmoid function or logistic model. We will consider this function as
+representing the probability for finding a value of \( y_i \) with a given
+\( x_i \).
+
 
    
 
    
 
@@ -512,7 +480,7 @@ plt.show()
   
  • 42
    • 
   
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    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs034.html b/doc/pub/week38/html/._week38-bs034.html
index b4df67c5e..850014b15 100644
--- a/doc/pub/week38/html/._week38-bs034.html
+++ b/doc/pub/week38/html/._week38-bs034.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,41 +418,25 @@ MathJax.Hub.Config({
 
 
 
-
    Plotting the mean value for each group 
    
+
    The logistic function 
    
 
 
    
-What we could attempt however is to plot the mean value for each group.
-
-
    
-
-
-
    agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
-group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
-plt.plot(group, agegroupmean, "r-")
-plt.axis([0,9,0, 1.0])
-plt.xlabel(r'Age group')
-plt.ylabel(r'CHD mean values')
-plt.title(r'Mean values for each age group')
-plt.show()
-
    
-
    
-We are now trying to find a function \( f(y\vert x) \), that is a function which gives us an expected value for the output \( y \) with a given input \( x \).
-In standard linear regression with a linear dependence on \( x \), we would write this in terms of our model
+Another widely studied model, is the so-called 
+perceptron model, which is an example of a "hard classification" model. We
+will encounter this model when we discuss neural networks as
+well. Each datapoint is deterministically assigned to a category (i.e
+\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
+classifier that outputs the probability of a given category rather
+than a single value. For example, given \( x_i \), the classifier
+outputs the probability of being in a category \( k \).  Logistic regression
+is the most common example of a so-called soft classifier. In logistic
+regression, the probability that a data point \( x_i \)
+belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, 
 $$
-f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
 $$
 
-
    
-This expression implies however that \( f(y_i\vert x_i) \) could take any
-value from minus infinity to plus infinity. If we however let
-\( f(y\vert y) \) be represented by the mean value, the above example
-shows us that we can constrain the function to take values between
-zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
-at our last curve we see also that it has an S-shaped form. This leads
-us to a very popular model for the function \( f \), namely the so-called
-Sigmoid function or logistic model. We will consider this function as
-representing the probability for finding a value of \( y_i \) with a given
-\( x_i \).
+Note that \( 1-p(t)= p(-t) \).
 
 
    
 
    
@@ -485,7 +464,7 @@ representing the probability for finding a value of \( y_i \) with a given
   
  • 43
    • 
   
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    • 
   
  • ...
    • 
-  
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    • 
+  
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    • 
   
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs035.html b/doc/pub/week38/html/._week38-bs035.html
index e39c002e8..4acbc04d7 100644
--- a/doc/pub/week38/html/._week38-bs035.html
+++ b/doc/pub/week38/html/._week38-bs035.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,26 +418,69 @@ MathJax.Hub.Config({
 
 
 
-
    The logistic function 
    
+
    Examples of likelihood functions used in logistic regression and nueral networks 
    
 
 
    
-Another widely studied model, is the so-called 
-perceptron model, which is an example of a "hard classification" model. We
-will encounter this model when we discuss neural networks as
-well. Each datapoint is deterministically assigned to a category (i.e
-\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
-classifier that outputs the probability of a given category rather
-than a single value. For example, given \( x_i \), the classifier
-outputs the probability of being in a category \( k \).  Logistic regression
-is the most common example of a so-called soft classifier. In logistic
-regression, the probability that a data point \( x_i \)
-belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, 
-$$
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-$$
+The following code plots the logistic function, the step function and other functions we will encounter from here and on.
 
-Note that \( 1-p(t)= p(-t) \).
+
    
 
+
+
    """The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""tanh Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()
+
    
 
    
 
    
 
@@ -469,7 +507,7 @@ Note that \( 1-p(t)= p(-t) \).
   
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    • 
   
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    • 
   
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    • 
-  
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diff --git a/doc/pub/week38/html/._week38-bs036.html b/doc/pub/week38/html/._week38-bs036.html
index a53aa23bb..e6c59bd28 100644
--- a/doc/pub/week38/html/._week38-bs036.html
+++ b/doc/pub/week38/html/._week38-bs036.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,69 +418,25 @@ MathJax.Hub.Config({
 
 
 
-
    Examples of likelihood functions used in logistic regression and nueral networks 
    
+
    Two parameters 
    
 
 
    
-The following code plots the logistic function, the step function and other functions we will encounter from here and on.
+We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities 
+$$
+\begin{align*}
+p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
+\end{align*}
+$$
+
+where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
 
 
    
+Note that we used
+$$
+p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
+$$
 
-
-
    """The sigmoid function (or the logistic curve) is a
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""tanh Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.tanh(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('tanh function')
-
-plt.show()
-
    
 
    
 
    
 
@@ -512,7 +463,7 @@ plt.show()
   
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    • 
   
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-  
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diff --git a/doc/pub/week38/html/._week38-bs037.html b/doc/pub/week38/html/._week38-bs037.html
index 0a66ee898..63291841f 100644
--- a/doc/pub/week38/html/._week38-bs037.html
+++ b/doc/pub/week38/html/._week38-bs037.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,25 +416,26 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Two parameters 
    
+
    Maximum likelihood 
    
 
 
    
-We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities 
+In order to define the total likelihood for all possible outcomes from a  
+dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
+\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. 
+We aim thus at maximizing 
+the probability of seeing the observed data. We can then approximate the 
+likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is 
 $$
 \begin{align*}
-p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
-p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
+P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
 \end{align*}
 $$
 
-where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
-
-
    
-Note that we used
+from which we obtain the log-likelihood and our cost/loss function
 $$
-p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
+\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
 $$
 
 
    
@@ -468,7 +464,7 @@ $$
   
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    • 
   
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    • 
 
 
diff --git a/doc/pub/week38/html/._week38-bs038.html b/doc/pub/week38/html/._week38-bs038.html
index fedb97b7a..58c4b249a 100644
--- a/doc/pub/week38/html/._week38-bs038.html
+++ b/doc/pub/week38/html/._week38-bs038.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -421,28 +416,26 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Maximum likelihood 
    
+
    The cost function rewritten 
    
 
 
    
-In order to define the total likelihood for all possible outcomes from a  
-dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
-\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. 
-We aim thus at maximizing 
-the probability of seeing the observed data. We can then approximate the 
-likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is 
+Reordering the logarithms, we can rewrite the cost/loss function as
 $$
-\begin{align*}
-P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
-\end{align*}
+\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
 $$
 
-from which we obtain the log-likelihood and our cost/loss function
+
    
+The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
+Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
 $$
-\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
+\mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
 $$
 
+This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, 
+in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression.
+
 
    
 
    
 
@@ -469,7 +462,7 @@ $$
   
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    • 
   
  • 48
    • 
   
  • ...
    • 
-  
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+  
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diff --git a/doc/pub/week38/html/._week38-bs039.html b/doc/pub/week38/html/._week38-bs039.html
index 7d67584f0..e177ad1ab 100644
--- a/doc/pub/week38/html/._week38-bs039.html
+++ b/doc/pub/week38/html/._week38-bs039.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,23 +418,24 @@ MathJax.Hub.Config({
 
 
 
-
    The cost function rewritten 
    
+
    Minimizing the cross entropy 
    
 
 
    
-Reordering the logarithms, we can rewrite the cost/loss function as
-$$
-\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
-$$
+The cross entropy is a convex function of the weights \( \boldsymbol{\beta} \) and,
+therefore, any local minimizer is a global minimizer.
 
 
    
-The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
-Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
+Minimizing this
+cost function with respect to the two parameters \( \beta_0 \) and \( \beta_1 \) we obtain
+
 $$
-\mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n  \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n  \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
 $$
 
-This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, 
-in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression.
+and 
+$$
+\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n  \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+$$
 
 
    
 
    
@@ -467,7 +463,7 @@ in practice we often supplement the cross-entropy with additional regularization
   
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    • 
   
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diff --git a/doc/pub/week38/html/._week38-bs040.html b/doc/pub/week38/html/._week38-bs040.html
index 0f0d15c2d..6010937b7 100644
--- a/doc/pub/week38/html/._week38-bs040.html
+++ b/doc/pub/week38/html/._week38-bs040.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,23 +418,24 @@ MathJax.Hub.Config({
 
 
 
-
    Minimizing the cross entropy 
    
+
    A more compact expression 
    
 
 
    
-The cross entropy is a convex function of the weights \( \boldsymbol{\beta} \) and,
-therefore, any local minimizer is a global minimizer.
+Let us now define a vector \( \boldsymbol{y} \) with \( n \) elements \( y_i \), an
+\( n\times p \) matrix \( \boldsymbol{X} \) which contains the \( x_i \) values and a
+vector \( \boldsymbol{p} \) of fitted probabilities \( p(y_i\vert x_i,\boldsymbol{\beta}) \). We can rewrite in a more compact form the first
+derivative of cost function as
+
+$$
+\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right). 
+$$
 
 
    
-Minimizing this
-cost function with respect to the two parameters \( \beta_0 \) and \( \beta_1 \) we obtain
+If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements 
+\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as 
 
 $$
-\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n  \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
-$$
-
-and 
-$$
-\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n  \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}. 
 $$
 
 
    
@@ -468,7 +464,7 @@ $$
   
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diff --git a/doc/pub/week38/html/._week38-bs041.html b/doc/pub/week38/html/._week38-bs041.html
index ba527bc21..1024a0957 100644
--- a/doc/pub/week38/html/._week38-bs041.html
+++ b/doc/pub/week38/html/._week38-bs041.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,24 +418,17 @@ MathJax.Hub.Config({
 
 
 
-
    A more compact expression 
    
+
    Extending to more predictors 
    
 
 
    
-Let us now define a vector \( \boldsymbol{y} \) with \( n \) elements \( y_i \), an
-\( n\times p \) matrix \( \boldsymbol{X} \) which contains the \( x_i \) values and a
-vector \( \boldsymbol{p} \) of fitted probabilities \( p(y_i\vert x_i,\boldsymbol{\beta}) \). We can rewrite in a more compact form the first
-derivative of cost function as
-
+Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
 $$
-\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right). 
+\log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
 $$
 
-
    
-If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements 
-\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as 
-
+Here we defined \( \boldsymbol{x}=[1,x_1,x_2,\dots,x_p] \) and \( \boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
 $$
-\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}. 
+p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
 $$
 
 
    
@@ -469,7 +457,7 @@ $$
   
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diff --git a/doc/pub/week38/html/._week38-bs042.html b/doc/pub/week38/html/._week38-bs042.html
index df0ea302a..6e8f03803 100644
--- a/doc/pub/week38/html/._week38-bs042.html
+++ b/doc/pub/week38/html/._week38-bs042.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -423,19 +418,31 @@ MathJax.Hub.Config({
 
 
 
-
    Extending to more predictors 
    
+
    Including more classes 
    
 
 
    
-Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
+Till now we have mainly focused on two classes, the so-called binary
+system. Suppose we wish to extend to \( K \) classes.  Let us for the sake
+of simplicity assume we have only two predictors. We have then following model
+
 $$
-\log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
+\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
 $$
 
-Here we defined \( \boldsymbol{x}=[1,x_1,x_2,\dots,x_p] \) and \( \boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
+and 
 $$
-p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
+\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
 $$
 
+and so on till the class \( C=K-1 \) class
+$$
+\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
+$$
+
+
    
+and the model is specified in term of \( K-1 \) so-called log-odds or
+logit transformations.
+
 
    
 
    
 
@@ -462,7 +469,7 @@ $$
   
  • 51
    • 
   
  • 52
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/week38-bs.html b/doc/pub/week38/html/week38-bs.html
index 406c60fa3..d5042016a 100644
--- a/doc/pub/week38/html/week38-bs.html
+++ b/doc/pub/week38/html/week38-bs.html
@@ -81,10 +81,6 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -331,83 +327,82 @@ MathJax.Hub.Config({
       
  • Further Manipulations
    • 
       
  • Wrapping it up
    • 
       
  • Linear Regression code, Intercept handling first
    • 
-      
  • Another straight Line, the Intercept and Scaling again
    • 
-      
  • Code Examples
    • 
-      
  • Taking out the mean
    • 
-      
  • More complicated Example: The Ising model
    • 
-      
  • Reformulating the problem to suit regression
    • 
-      
  • Linear regression
    • 
-      
  • Singular Value decomposition
    • 
-      
  • The one-dimensional Ising model
    • 
-      
  • Ridge regression
    • 
-      
  • LASSO regression
    • 
-      
  • Performance as  function of the regularization parameter
    • 
-      
  • Finding the optimal value of \( \lambda \)
    • 
-      
  • Logistic Regression
    • 
-      
  • Classification problems
    • 
-      
  • Optimization and Deep learning
    • 
-      
  • Basics
    • 
-      
  • Linear classifier
    • 
-      
  • Some selected properties
    • 
-      
  • Simple example
    • 
-      
  • Plotting the mean value for each group
    • 
-      
  • The logistic function
    • 
-      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
-      
  • Two parameters
    • 
-      
  • Maximum likelihood
    • 
-      
  • The cost function rewritten
    • 
-      
  • Minimizing the cross entropy
    • 
-      
  • A more compact expression
    • 
-      
  • Extending to more predictors
    • 
-      
  • Including more classes
    • 
-      
  • More classes
    • 
-      
  • Friday September 24
    • 
-      
  • Wisconsin Cancer Data
    • 
-      
  • Using the correlation matrix
    • 
-      
  • Discussing the correlation data
    • 
-      
  • Other measures in classification studies: Cancer Data  again
    • 
-      
  • Optimization, the central part of any Machine Learning algortithm
    • 
-      
  • Revisiting our Logistic Regression case
    • 
-      
  • The equations to solve
    • 
-      
  • Solving using Newton-Raphson's method
    • 
-      
  • Brief reminder on Newton-Raphson's method
    • 
-      
  • The equations
    • 
-      
  • Simple geometric interpretation
    • 
-      
  • Extending to more than one variable
    • 
-      
  • Steepest descent
    • 
-      
  • More on Steepest descent
    • 
-      
  • The ideal
    • 
-      
  • The sensitiveness of the gradient descent
    • 
-      
  • Convex functions
    • 
-      
  • Convex function
    • 
-      
  • Conditions on convex functions
    • 
-      
  • More on convex functions
    • 
-      
  • Some simple problems
    • 
-      
  • Friday September 25
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Revisiting some of our first Linear Regression Encounters
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
+      
  • Code Examples
    • 
+      
  • Taking out the mean
    • 
+      
  • More complicated Example: The Ising model
    • 
+      
  • Reformulating the problem to suit regression
    • 
+      
  • Linear regression
    • 
+      
  • Singular Value decomposition
    • 
+      
  • The one-dimensional Ising model
    • 
+      
  • Ridge regression
    • 
+      
  • LASSO regression
    • 
+      
  • Performance as  function of the regularization parameter
    • 
+      
  • Finding the optimal value of \( \lambda \)
    • 
+      
  • Logistic Regression
    • 
+      
  • Classification problems
    • 
+      
  • Optimization and Deep learning
    • 
+      
  • Basics
    • 
+      
  • Linear classifier
    • 
+      
  • Some selected properties
    • 
+      
  • Simple example
    • 
+      
  • Plotting the mean value for each group
    • 
+      
  • The logistic function
    • 
+      
  • Examples of likelihood functions used in logistic regression and nueral networks
    • 
+      
  • Two parameters
    • 
+      
  • Maximum likelihood
    • 
+      
  • The cost function rewritten
    • 
+      
  • Minimizing the cross entropy
    • 
+      
  • A more compact expression
    • 
+      
  • Extending to more predictors
    • 
+      
  • Including more classes
    • 
+      
  • More classes
    • 
+      
  • Friday September 24
    • 
+      
  • Wisconsin Cancer Data
    • 
+      
  • Using the correlation matrix
    • 
+      
  • Discussing the correlation data
    • 
+      
  • Other measures in classification studies: Cancer Data  again
    • 
+      
  • Optimization, the central part of any Machine Learning algortithm
    • 
+      
  • Revisiting our Logistic Regression case
    • 
+      
  • The equations to solve
    • 
+      
  • Solving using Newton-Raphson's method
    • 
+      
  • Brief reminder on Newton-Raphson's method
    • 
+      
  • The equations
    • 
+      
  • Simple geometric interpretation
    • 
+      
  • Extending to more than one variable
    • 
+      
  • Steepest descent
    • 
+      
  • More on Steepest descent
    • 
+      
  • The ideal
    • 
+      
  • The sensitiveness of the gradient descent
    • 
+      
  • Convex functions
    • 
+      
  • Convex function
    • 
+      
  • Conditions on convex functions
    • 
+      
  • More on convex functions
    • 
+      
  • Some simple problems
    • 
+      
  • Friday September 25
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting some of our first Linear Regression Encounters
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
 
         
       
@@ -466,7 +461,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 92
    • 
+  
  • 91
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week38/html/week38-reveal.html b/doc/pub/week38/html/week38-reveal.html
index 758544922..da9264ad6 100644
--- a/doc/pub/week38/html/week38-reveal.html
+++ b/doc/pub/week38/html/week38-reveal.html
@@ -666,7 +666,7 @@ $$
 
     
    
 
 
    
-What does this mean for thus? And why do we insist on all this? Let us look at some examples.
+What does this mean? And why do we insist on all this? Let us look at some examples.
 
 
 
@@ -687,6 +687,10 @@ Note also that we do not split the data into training and test.
 
 np.random.seed(2021)
 
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
 
 def fit_beta(X, y):
     return np.linalg.pinv(X.T @ X) @ X.T @ y
@@ -714,6 +718,12 @@ clf = LinearRegression(fit_intercept=print(f"True beta: {true_beta}")
 print(f"Fitted beta: {beta}")
 print(f"Sklearn fitted beta: {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with intercept column from SKL: {intercept}")
+print(MSE(y,ypredictSKL)
 
 
 plt.figure()
@@ -742,6 +752,12 @@ intercept = np.mean(y_offset - X_offset @ beta)
 print(f"Fitted beta (wiothout intercept): {beta}")
 print(f"Sklearn intercept: {clf.intercept_}")
 print(f"Sklearn fitted beta (without intercept): {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with Sklearn intercept: {clf.intercept_}")
+print(MSE(y,ypredictSKL)
 
 plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
 plt.plot(x, clf.predict(X), "--", label="Sklearn (fit_intercept=True)")
@@ -750,65 +766,10 @@ plt.legend()
 
 plt.show()
 
    
-
-
-
-
    
-
    Another straight Line, the Intercept and Scaling again 
    
-
 
    
-As we said above, the intercept is the value of our output/target variable
-when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). 
-Let us now study a case where we again fit a straight line including the intercept in the design matrix
-using ordinary least squares. We are only interested in the MSE value here.
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
 
-
    
-
-
-
    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-def fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = 10.0+5*x
-
-
-Maxpolydegree = 2
-X = np.zeros((n,Maxpolydegree))
-
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# Intercept is included in the design matrix
-# own code first
-OwnBeta = fit_beta(X_train, y_train)
-# Scikit-Learn
-skl = LinearRegression(fit_intercept=False).fit(X_train, y_train)
-ypredictOwn = X_test @ OwnBeta
-ypredictSKL = skl.predict(X_test)
-print(MSE(y_test,ypredictOwn)
-print(MSE(y_test,ypredictSKL)
-
    
 
    
 Printing the MSE, we see first that both methods give the same
 MSE. However, changing the value of the intercept gives a larger or
diff --git a/doc/pub/week38/html/week38-solarized.html b/doc/pub/week38/html/week38-solarized.html
index 38dcb81ba..c0fabbfa9 100644
--- a/doc/pub/week38/html/week38-solarized.html
+++ b/doc/pub/week38/html/week38-solarized.html
@@ -101,10 +101,6 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -804,7 +800,7 @@ $$
 $$
 
 
    
-What does this mean for thus? And why do we insist on all this? Let us look at some examples.
+What does this mean? And why do we insist on all this? Let us look at some examples.
 
 
    
 









    
@@ -825,6 +821,10 @@ Note also that we do not split the data into training and test.
 
 np.random.seed(2021)
 
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
 
 def fit_beta(X, y):
     return np.linalg.pinv(X.T @ X) @ X.T @ y
@@ -852,6 +852,12 @@ clf = LinearRegression(fit_intercept=print(f"True beta: {true_beta}")
 print(f"Fitted beta: {beta}")
 print(f"Sklearn fitted beta: {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with intercept column from SKL: {intercept}")
+print(MSE(y,ypredictSKL)
 
 
 plt.figure()
@@ -880,6 +886,12 @@ intercept = np.mean(y_offset - X_offset @ beta)
 print(f"Fitted beta (wiothout intercept): {beta}")
 print(f"Sklearn intercept: {clf.intercept_}")
 print(f"Sklearn fitted beta (without intercept): {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with Sklearn intercept: {clf.intercept_}")
+print(MSE(y,ypredictSKL)
 
 plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
 plt.plot(x, clf.predict(X), "--", label="Sklearn (fit_intercept=True)")
@@ -889,63 +901,9 @@ plt.legend()
 plt.show()
 
    
 
    
-









    
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
 
-
    Another straight Line, the Intercept and Scaling again 
    
-
-
    
-As we said above, the intercept is the value of our output/target variable
-when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). 
-Let us now study a case where we again fit a straight line including the intercept in the design matrix
-using ordinary least squares. We are only interested in the MSE value here.
-
-
    
-
-
-
    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-def fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = 10.0+5*x
-
-
-Maxpolydegree = 2
-X = np.zeros((n,Maxpolydegree))
-
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# Intercept is included in the design matrix
-# own code first
-OwnBeta = fit_beta(X_train, y_train)
-# Scikit-Learn
-skl = LinearRegression(fit_intercept=False).fit(X_train, y_train)
-ypredictOwn = X_test @ OwnBeta
-ypredictSKL = skl.predict(X_test)
-print(MSE(y_test,ypredictOwn)
-print(MSE(y_test,ypredictSKL)
-
    
 
    
 Printing the MSE, we see first that both methods give the same
 MSE. However, changing the value of the intercept gives a larger or
diff --git a/doc/pub/week38/html/week38.html b/doc/pub/week38/html/week38.html
index b66923b1a..15a917992 100644
--- a/doc/pub/week38/html/week38.html
+++ b/doc/pub/week38/html/week38.html
@@ -106,10 +106,6 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'linear-regression-code-intercept-handling-first'),
-              ('Another straight Line, the Intercept and Scaling again',
-               2,
-               None,
-               'another-straight-line-the-intercept-and-scaling-again'),
               ('Code Examples', 2, None, 'code-examples'),
               ('Taking out the mean', 2, None, 'taking-out-the-mean'),
               ('More complicated Example: The Ising model',
@@ -809,7 +805,7 @@ $$
 $$
 
 
    
-What does this mean for thus? And why do we insist on all this? Let us look at some examples.
+What does this mean? And why do we insist on all this? Let us look at some examples.
 
 
    
 









    
@@ -830,6 +826,10 @@ Note also that we do not split the data into training and test.
 
 np.random.seed(2021)
 
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
 
 def fit_beta(X, y):
     return np.linalg.pinv(X.T @ X) @ X.T @ y
@@ -857,6 +857,12 @@ clf = LinearRegression(fit_interceptprint(f"True beta: {true_beta}")
 print(f"Fitted beta: {beta}")
 print(f"Sklearn fitted beta: {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with intercept column from SKL: {intercept}")
+print(MSE(y,ypredictSKL)
 
 
 plt.figure()
@@ -885,6 +891,12 @@ intercept = np.
 print(f"Fitted beta (wiothout intercept): {beta}")
 print(f"Sklearn intercept: {clf.intercept_}")
 print(f"Sklearn fitted beta (without intercept): {clf.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept: {intercept}")
+print(MSE(y,ypredictOwn)
+print(f"MSE with Sklearn intercept: {clf.intercept_}")
+print(MSE(y,ypredictSKL)
 
 plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
 plt.plot(x, clf.predict(X), "--", label="Sklearn (fit_intercept=True)")
@@ -894,63 +906,9 @@ plt.legend()
 plt.show()
 
    
 
    
-









    
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
 
-
    Another straight Line, the Intercept and Scaling again 
    
-
-
    
-As we said above, the intercept is the value of our output/target variable
-when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case). 
-Let us now study a case where we again fit a straight line including the intercept in the design matrix
-using ordinary least squares. We are only interested in the MSE value here.
-
-
    
-
-
-
    import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-def fit_beta(X, y):
-    return np.linalg.pinv(X.T @ X) @ X.T @ y
-
-
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-n = 100
-x = np.random.rand(n)
-y = 10.0+5*x
-
-
-Maxpolydegree = 2
-X = np.zeros((n,Maxpolydegree))
-
-for degree in range(Maxpolydegree):
-    X[:,degree] = x**degree
-
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# Intercept is included in the design matrix
-# own code first
-OwnBeta = fit_beta(X_train, y_train)
-# Scikit-Learn
-skl = LinearRegression(fit_intercept=False).fit(X_train, y_train)
-ypredictOwn = X_test @ OwnBeta
-ypredictSKL = skl.predict(X_test)
-print(MSE(y_test,ypredictOwn)
-print(MSE(y_test,ypredictSKL)
-
    
 
    
 Printing the MSE, we see first that both methods give the same
 MSE. However, changing the value of the intercept gives a larger or
diff --git a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz
index 0d38a1c71460e8196452d8c4a122828a23ed364d..822c4fc4f1629aff7b51ffe6b26d72f907d68f94 100644
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