From dc10d28620a18c4aa54a119800badd2f540234ba Mon Sep 17 00:00:00 2001 From: mhjensen Date: Wed, 19 Aug 2020 13:19:34 +0200 Subject: [PATCH] removed files --- doc/src/Projects/2020/Exercises/README.txt | 2 - doc/src/Projects/2020/Exercises/hw1-bs.html | 656 ------------------ doc/src/Projects/2020/Exercises/hw1.html | 530 -------------- doc/src/Projects/2020/Exercises/hw1.p.tex | 593 ---------------- doc/src/Projects/2020/Exercises/hw1.pdf | Bin 262445 -> 0 bytes doc/src/Projects/2020/Exercises/hw1.tex | 561 --------------- .../2020/Exercises/ipynb-hw1-src.tar.gz | Bin 189 -> 0 bytes 7 files changed, 2342 deletions(-) delete mode 100644 doc/src/Projects/2020/Exercises/README.txt delete mode 100644 doc/src/Projects/2020/Exercises/hw1-bs.html delete mode 100644 doc/src/Projects/2020/Exercises/hw1.html delete mode 100644 doc/src/Projects/2020/Exercises/hw1.p.tex delete mode 100644 doc/src/Projects/2020/Exercises/hw1.pdf delete mode 100644 doc/src/Projects/2020/Exercises/hw1.tex delete mode 100644 doc/src/Projects/2020/Exercises/ipynb-hw1-src.tar.gz diff --git a/doc/src/Projects/2020/Exercises/README.txt b/doc/src/Projects/2020/Exercises/README.txt deleted file mode 100644 index 3f77b8ba3..000000000 --- a/doc/src/Projects/2020/Exercises/README.txt +++ /dev/null @@ -1,2 +0,0 @@ -This IPython notebook hw1.ipynb does not require any additional -programs. diff --git a/doc/src/Projects/2020/Exercises/hw1-bs.html b/doc/src/Projects/2020/Exercises/hw1-bs.html deleted file mode 100644 index f89e1f3ab..000000000 --- a/doc/src/Projects/2020/Exercises/hw1-bs.html +++ /dev/null @@ -1,656 +0,0 @@ - - - - - - - - -Homework 1 Fall Semester 2020 - - - - - - - - - - - - - - - - - - - - - - - -
-
- - Homework 1 Fall Semester 2020 -
- -
- -
-
- - -
- -

 

 

 

- - - - - - -
-

Homework 1 Fall Semester 2020

- -

- - -

-Data Analysis and Machine Learning FYS-STK3155/FYS4155 -
- -

- - -

Department of Physics, University of Oslo, Norway
-
-

-

Aug 19, 2020

-
-

-

- -

Exercise, Setting up various Python environments

- -

-The first exercise here is of a mere technical art. We want you to have - -

- -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -IPython/Jupyter notebooks invaluable in your work. You can run R -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Fortran etc if you prefer. The focus in these lectures will be -on Python. - -

-If you have Python installed (we recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via pip as - -

    -
  1. pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow
    2. -
- -For Tensorflow, we recommend following the instructions in the text of -Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly - -

-We will come back to tensorflow later. - -

-For Python3, replace pip with pip3. - -

-For OSX users we recommend, after having installed Xcode, to -install brew. Brew allows for a seamless installation of additional -software via for example - -

    -
  1. brew install python3
    2. -
- -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use pip as well and simply install Python as - -
    -
  1. sudo apt-get install python3 (or python for Python2.7)
    2. -
- -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - - - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system conda. - - - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -

-We recommend using Anaconda. - -

- - -

Exercise 1: Our first Python encounter

- -

-This exercise has as its aim to write a small program which reads in data from a csv file on the equation of state for dense nuclear matter. The file is localized at https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv. Thereafter you will have to set up the design matrix \( \boldsymbol{X} \) for the \( n \) -datapoints and a polynomial of degree \( 3 \). The steps are: - -

- -We recommend looking at the examples in the regression slides. - -

- - -

- - -Solution. - -

-

- -

- - -

import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-# 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')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organized into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-X = np.zeros((len(Density),5))
-X[:,0] = 1
-X[:,1] = Density**(2.0/3.0)
-X[:,2] = Density
-X[:,3] = Density**(4.0/3.0)
-X[:,4] = Density**(5.0/3.0)
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
-

-

-
-

- -

- - -

- - -

- - -

Exercise 2: making your own data and exploring scikit-learn

- -

-We will generate our own dataset for a function \( y(x) \) where \( x \in [0,1] \) and defined by random numbers computed with the uniform distribution. The function \( y \) is a quadratic polynomial in \( x \) with added stochastic noise according to the normal distribution \( \cal {N}(0,1) \). -The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points). -

- - -

x = np.random.rand(100,1)
-y = 2.0+5*x*x+0.1*np.random.randn(100,1)
-
-
    -
  1. Write your own code (following the examples under the regression slides) for computing the parametrization of the data set fitting a second-order polynomial.
    2. -
  2. Use thereafter scikit-learn (see again the examples in the regression slides) and compare with your own code.
    3. -
  3. Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
    4. -
- -$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -$$ - -and the \( R^2 \) score function. -If \( \tilde{\hat{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as -$$ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -$$ - -where we have defined the mean value of \( \hat{y} \) as -$$ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -$$ - -You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. -Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. - -

- - -

- - -Solution. - -

-

- -

-The code here is an example of where we define our own design matrix and fit parameters \( \beta \). -

- - -

import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),3))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x**2
-# 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)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(beta)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
-

-

-
-

- -

- - -

- - -

- - -

Exercise 3: mean values and variances in linear regression

- -

-This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). - -

-The assumption we have made is -that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \) -which describes our data -$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ - -

-We then approximate this function with our model from the solution of the linear regression equations (ordinary least squares OLS), that is our -function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we minimized \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), with -$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -$$ - -The matrix \( \boldsymbol{X} \) is the so-called design matrix. - -

-a) -Show that the expectation value of \( \boldsymbol{y} \) for a given element \( i \) -$$ -\begin{align*} -\mathbb{E}(y_i) & =\mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -and that -its variance is -$$ -\begin{align*} \mbox{Var}(y_i) & = \sigma^2. -\end{align*} -$$ - -Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \). - -

- - -

- - -Solution. - -

-

- -

-We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) -$$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -while -its variance is -$$ -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -$$ - -Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD). - -

-

-
-

- -

- - -

-b) -With the OLS expressions for the parameters \( \boldsymbol{\beta} \) show that -$$ -\mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}. -$$ - -

- - -

- - -Solution. - -

-

- -$$ -\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -$$ - -This means that the estimator of the regression parameters is unbiased. - -

-

-
-

- -

- - -

-c) -Show finally that the variance of \( \boldsymbol{\beta} \) is -$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. -\end{eqnarray*} -$$ - -

- - -

- - -Solution. - -

-

- -

-The variance of \( \boldsymbol{\beta} \) is -$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -$$ - -

-where we have used that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the -variance of the estimate of the \( j \)-th regression coefficient: -\( \boldsymbol{\sigma}^2 (\hat{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} } \). This may be used to -construct a confidence interval for the estimates. - -

-In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters \( \boldsymbol{\beta} \) and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. - -

-

-
-

- -

- - -

- - -

- -

- - -
- - - - - - - -
- © 1999-2020, "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html". Released under CC Attribution-NonCommercial 4.0 license -
- - - - - - diff --git a/doc/src/Projects/2020/Exercises/hw1.html b/doc/src/Projects/2020/Exercises/hw1.html deleted file mode 100644 index 8fda53760..000000000 --- a/doc/src/Projects/2020/Exercises/hw1.html +++ /dev/null @@ -1,530 +0,0 @@ - - - - - - - - -Homework 1 Fall Semester 2020 - - - - - - - - - - - - - - - - - - - - - - - -

Homework 1 Fall Semester 2020

- -

- - -

-Data Analysis and Machine Learning FYS-STK3155/FYS4155 -
- -

- - -

Department of Physics, University of Oslo, Norway
-
-

-

Aug 19, 2020

-
- -

Exercise, Setting up various Python environments

- -

-The first exercise here is of a mere technical art. We want you to have - -

- -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -IPython/Jupyter notebooks invaluable in your work. You can run R -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Fortran etc if you prefer. The focus in these lectures will be -on Python. - -

-If you have Python installed (we recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via pip as - -

    -
  1. pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow
    2. -
- -For Tensorflow, we recommend following the instructions in the text of -Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly - -

-We will come back to tensorflow later. - -

-For Python3, replace pip with pip3. - -

-For OSX users we recommend, after having installed Xcode, to -install brew. Brew allows for a seamless installation of additional -software via for example - -

    -
  1. brew install python3
    2. -
- -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use pip as well and simply install Python as - -
    -
  1. sudo apt-get install python3 (or python for Python2.7)
    2. -
- -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - - - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system conda. - - - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -

-We recommend using Anaconda. - -

- - -

Exercise 1: Our first Python encounter

- -

-This exercise has as its aim to write a small program which reads in data from a csv file on the equation of state for dense nuclear matter. The file is localized at https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv. Thereafter you will have to set up the design matrix \( \boldsymbol{X} \) for the \( n \) -datapoints and a polynomial of degree \( 3 \). The steps are: - -

- -We recommend looking at the examples in the regression slides. - -

- -Solution. -

- - -

import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-# 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')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organized into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-X = np.zeros((len(Density),5))
-X[:,0] = 1
-X[:,1] = Density**(2.0/3.0)
-X[:,2] = Density
-X[:,3] = Density**(4.0/3.0)
-X[:,4] = Density**(5.0/3.0)
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
-

- - -

- - -

- - -

Exercise 2: making your own data and exploring scikit-learn

- -

-We will generate our own dataset for a function \( y(x) \) where \( x \in [0,1] \) and defined by random numbers computed with the uniform distribution. The function \( y \) is a quadratic polynomial in \( x \) with added stochastic noise according to the normal distribution \( \cal {N}(0,1) \). -The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points). -

- - -

x = np.random.rand(100,1)
-y = 2.0+5*x*x+0.1*np.random.randn(100,1)
-
-
    -
  1. Write your own code (following the examples under the regression slides) for computing the parametrization of the data set fitting a second-order polynomial.
    2. -
  2. Use thereafter scikit-learn (see again the examples in the regression slides) and compare with your own code.
    3. -
  3. Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
    4. -
- -$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -$$ - -and the \( R^2 \) score function. -If \( \tilde{\hat{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as -$$ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -$$ - -where we have defined the mean value of \( \hat{y} \) as -$$ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -$$ - -You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. -Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. - -

- -Solution. -The code here is an example of where we define our own design matrix and fit parameters \( \beta \). -

- - -

import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),3))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x**2
-# 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)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(beta)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
-

- - -

- - -

- - -

Exercise 3: mean values and variances in linear regression

- -

-This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). - -

-The assumption we have made is -that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \) -which describes our data -$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ - -

-We then approximate this function with our model from the solution of the linear regression equations (ordinary least squares OLS), that is our -function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we minimized \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), with -$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -$$ - -The matrix \( \boldsymbol{X} \) is the so-called design matrix. - -

-a) -Show that the expectation value of \( \boldsymbol{y} \) for a given element \( i \) -$$ -\begin{align*} -\mathbb{E}(y_i) & =\mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -and that -its variance is -$$ -\begin{align*} \mbox{Var}(y_i) & = \sigma^2. -\end{align*} -$$ - -Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \). - -

- -Solution. -We can calculate the expectation value of \( \boldsymbol{y} \) for a given element \( i \) -$$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -while -its variance is -$$ -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -$$ - -Hence, \( y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2) \), that is \( \boldsymbol{y} \) follows a normal distribution with -mean value \( \boldsymbol{X}\boldsymbol{\beta} \) and variance \( \sigma^2 \) (not be confused with the singular values of the SVD). - -

- - -

-b) -With the OLS expressions for the parameters \( \boldsymbol{\beta} \) show that -$$ -\mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta}. -$$ - -

- -Solution. -$$ -\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -$$ - -This means that the estimator of the regression parameters is unbiased. - -

- - -

-c) -Show finally that the variance of \( \boldsymbol{\beta} \) is -$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. -\end{eqnarray*} -$$ - -

- -Solution. -The variance of \( \boldsymbol{\beta} \) is -$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -$$ - -

-where we have used that \( \mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn} \). From \( \mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1} \), one obtains an estimate of the -variance of the estimate of the \( j \)-th regression coefficient: -\( \boldsymbol{\sigma}^2 (\hat{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} } \). This may be used to -construct a confidence interval for the estimates. - -

-In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters \( \boldsymbol{\beta} \) and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. - -

- - -

- - - - - -

- © 1999-2020, "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html". Released under CC Attribution-NonCommercial 4.0 license -
- - - - - - diff --git a/doc/src/Projects/2020/Exercises/hw1.p.tex b/doc/src/Projects/2020/Exercises/hw1.p.tex deleted file mode 100644 index 00f94739c..000000000 --- a/doc/src/Projects/2020/Exercises/hw1.p.tex +++ /dev/null @@ -1,593 +0,0 @@ -%% -%% Automatically generated file from DocOnce source -%% (https://github.com/hplgit/doconce/) -%% -%% -% #ifdef PTEX2TEX_EXPLANATION -%% -%% The file follows the ptex2tex extended LaTeX format, see -%% ptex2tex: http://code.google.com/p/ptex2tex/ -%% -%% Run -%% ptex2tex myfile -%% or -%% doconce ptex2tex myfile -%% -%% to turn myfile.p.tex into an ordinary LaTeX file myfile.tex. -%% (The ptex2tex program: http://code.google.com/p/ptex2tex) -%% Many preprocess options can be added to ptex2tex or doconce ptex2tex -%% -%% ptex2tex -DMINTED myfile -%% doconce ptex2tex myfile envir=minted -%% -%% ptex2tex will typeset code environments according to a global or local -%% .ptex2tex.cfg configure file. doconce ptex2tex will typeset code -%% according to options on the command line (just type doconce ptex2tex to -%% see examples). 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Released under CC Attribution-NonCommercial 4.0 license}} - \renewcommand{\footrulewidth}{0mm} - \renewcommand{\headrulewidth}{0mm} -} - -\pagestyle{fancy} - - -% prevent orhpans and widows -\clubpenalty = 10000 -\widowpenalty = 10000 - -\newenvironment{doconceexercise}{}{} -\newcounter{doconceexercisecounter} - - -% ------ header in subexercises ------ -%\newcommand{\subex}[1]{\paragraph{#1}} -%\newcommand{\subex}[1]{\par\vspace{1.7mm}\noindent{\bf #1}\ \ } -\makeatletter -% 1.5ex is the spacing above the header, 0.5em the spacing after subex title -\newcommand\subex{\@startsection{paragraph}{4}{\z@}% - {1.5ex\@plus1ex \@minus.2ex}% - {-0.5em}% - {\normalfont\normalsize\bfseries}} -\makeatother - - -% --- end of standard preamble for documents --- - - -% insert custom LaTeX commands... - -\raggedbottom -\makeindex -\usepackage[totoc]{idxlayout} % for index in the toc -\usepackage[nottoc]{tocbibind} % for references/bibliography in the toc - -%-------------------- end preamble ---------------------- - -\begin{document} - -% matching end for #ifdef PREAMBLE -% #endif - -\newcommand{\exercisesection}[1]{\subsection*{#1}} - - -% ------------------- main content ---------------------- - - - -% ----------------- title ------------------------- - -\thispagestyle{empty} - -\begin{center} -{\LARGE\bf -\begin{spacing}{1.25} -Homework 1 Fall Semester 2020 -\end{spacing} -} -\end{center} - -% ----------------- author(s) ------------------------- - -\begin{center} -{\bf \href{{http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html}}{Data Analysis and Machine Learning FYS-STK3155/FYS4155}} -\end{center} - - \begin{center} -% List of all institutions: -\centerline{{\small Department of Physics, University of Oslo, Norway}} -\end{center} - -% ----------------- end author(s) ------------------------- - -% --- begin date --- -\begin{center} -Aug 19, 2020 -\end{center} -% --- end date --- - -\vspace{1cm} - - -\subsection{Exercise, Setting up various Python environments} - -The first exercise here is of a mere technical art. We want you to have -\begin{itemize} -\item git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo \href{{https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html}}{GitHub facilities}. - -\item Install various Python packages -\end{itemize} - -\noindent -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -IPython/Jupyter notebooks invaluable in your work. You can run \textbf{R} -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Fortran etc if you prefer. The focus in these lectures will be -on Python. - -If you have Python installed (we recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via \textbf{pip} as - -\begin{enumerate} -\item pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow -\end{enumerate} - -\noindent -For \textbf{Tensorflow}, we recommend following the instructions in the text of -\href{{http://shop.oreilly.com/product/0636920052289.do}}{Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly} - -We will come back to \textbf{tensorflow} later. - -For Python3, replace \textbf{pip} with \textbf{pip3}. - -For OSX users we recommend, after having installed Xcode, to -install \textbf{brew}. Brew allows for a seamless installation of additional -software via for example - -\begin{enumerate} -\item brew install python3 -\end{enumerate} - -\noindent -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use \textbf{pip} as well and simply install Python as - -\begin{enumerate} -\item sudo apt-get install python3 (or python for Python2.7) -\end{enumerate} - -\noindent -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -\begin{itemize} -\item \href{{https://docs.anaconda.com/}}{Anaconda}, -\end{itemize} - -\noindent -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system \textbf{conda}. - -\begin{itemize} -\item \href{{https://www.enthought.com/product/canopy/}}{Enthought canopy} -\end{itemize} - -\noindent -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -We recommend using \textbf{Anaconda}. - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection{Exercise \thedoconceexercisecounter: Our first Python encounter} - - -This exercise has as its aim to write a small program which reads in data from a \textbf{csv} file on the equation of state for dense nuclear matter. The file is localized at \href{{https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv}}{\nolinkurl{https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv}}. Thereafter you will have to set up the design matrix $\bm{X}$ for the $n$ -datapoints and a polynomial of degree $3$. The steps are: -\begin{itemize} -\item Write a Python code which reads the in the above mentioned file. - -\item Use for example \textbf{pandas} to order your data and find out how many data points there are. - -\item Set thereafter up the design matrix with dimensionality $n\times p$ where $p=4$ and where you have defined a polynomial of degree $p-1=3$. Print the matrix and check that the numbers are correct. -\end{itemize} - -\noindent -We recommend looking at the examples in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/Regression/html/Regression-bs.html}}{regression slides}. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -\bpycod -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split -# 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') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organized into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),5)) -X[:,0] = 1 -X[:,1] = Density**(2.0/3.0) -X[:,2] = Density -X[:,3] = Density**(4.0/3.0) -X[:,4] = Density**(5.0/3.0) -# We split the data in test and training data -X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) -\epycod - -% --- end solution of exercise --- - -\end{doconceexercise} -% --- end exercise --- - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection{Exercise \thedoconceexercisecounter: making your own data and exploring scikit-learn} - - -We will generate our own dataset for a function $y(x)$ where $x \in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\cal {N}(0,1)$. -The following simple Python instructions define our $x$ and $y$ values (with 100 data points). -\bpycod -x = np.random.rand(100,1) -y = 2.0+5*x*x+0.1*np.random.randn(100,1) -\epycod - -\begin{enumerate} -\item Write your own code (following the examples under the \href{{https://compphysics.github.io/MachineLearningECT/doc/pub/Day1/html/Day1-bs.html}}{regression slides}) for computing the parametrization of the data set fitting a second-order polynomial. - -\item Use thereafter \textbf{scikit-learn} (see again the examples in the regression slides) and compare with your own code. - -\item Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as -\end{enumerate} - -\noindent -\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -\] -and the $R^2$ score function. -If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as -\[ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -\] -where we have defined the mean value of $\hat{y}$ as -\[ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -\] -You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. -Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -The code here is an example of where we define our own design matrix and fit parameters $\beta$. -\bpycod -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -x = np.random.rand(100) -y = 2.0+5*x*x+0.1*np.random.randn(100) - - -# The design matrix now as function of a given polynomial -X = np.zeros((len(x),3)) -X[:,0] = 1.0 -X[:,1] = x -X[:,2] = x**2 -# 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) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train -print(beta) -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) -\epycod - -% --- end solution of exercise --- - -\end{doconceexercise} -% --- end exercise --- - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection{Exercise \thedoconceexercisecounter: mean values and variances in linear regression} - - -This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). - -The assumption we have made is -that there exists a function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$ -which describes our data -\[ -\bm{y} = f(\bm{x})+\bm{\varepsilon} -\] - -We then approximate this function with our model from the solution of the linear regression equations (ordinary least squares OLS), that is our -function $f$ is approximated by $\bm{\tilde{y}}$ where we minimized $(\bm{y}-\bm{\tilde{y}})^2$, with -\[ -\bm{\tilde{y}} = \bm{X}\bm{\beta}. -\] -The matrix $\bm{X}$ is the so-called design matrix. - - -\subex{a)} -Show that the expectation value of $\bm{y}$ for a given element $i$ -\begin{align*} -\mathbb{E}(y_i) & =\mathbf{X}_{i, \ast} \, \beta, -\end{align*} -and that -its variance is -\begin{align*} \mbox{Var}(y_i) & = \sigma^2. -\end{align*} -Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\beta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with -mean value $\bm{X}\bm{\beta}$ and variance $\sigma^2$. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -We can calculate the expectation value of $\bm{y}$ for a given element $i$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \bm{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -while -its variance is -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \bm{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \bm{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\beta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with -mean value $\bm{X}\bm{\beta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD). - -% --- end solution of exercise --- - -\subex{b)} -With the OLS expressions for the parameters $\bm{\beta}$ show that -\[ -\mathbb{E}(\bm{\beta}) = \bm{\beta}. -\] - - -% --- begin solution of exercise --- -\paragraph{Solution.} -\[ -\mathbb{E}(\bm{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\bm{\beta}=\bm{\beta}. -\] -This means that the estimator of the regression parameters is unbiased. - -% --- end solution of exercise --- - -\subex{c)} -Show finally that the variance of $\bm{\beta}$ is -\begin{eqnarray*} -\mbox{Var}(\bm{\beta}) & = & \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. -\end{eqnarray*} - - -% --- begin solution of exercise --- -\paragraph{Solution.} -The variance of $\bm{\beta}$ is -\begin{eqnarray*} -\mbox{Var}(\bm{\beta}) & = & \mathbb{E} \{ [\bm{\beta} - \mathbb{E}(\bm{\beta})] [\bm{\beta} - \mathbb{E}(\bm{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \bm{\beta} \, \bm{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \bm{\beta} \bm{\beta}^T -\\ -& = & \bm{\beta} \, \bm{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} - -where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\bm{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the -variance of the estimate of the $j$-th regression coefficient: -$\bm{\sigma}^2 (\hat{\beta}_j ) = \bm{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }$. This may be used to -construct a confidence interval for the estimates. - - -In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters $\bm{\beta}$ and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. - -% --- end solution of exercise --- - - - - -\end{doconceexercise} -% --- end exercise --- - - -% ------------------- end of main content --------------- - -% #ifdef PREAMBLE -\end{document} -% #endif - diff --git a/doc/src/Projects/2020/Exercises/hw1.pdf b/doc/src/Projects/2020/Exercises/hw1.pdf deleted file mode 100644 index 061218b9058814e8beaf596a0233bb71c68c3686..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 262445 zcma&NLy#^^v~63q?fS~LZQI^u+qP}nwrzWt&0V%#_rEV*@7%`8Rz@~6R%4DCW05I} ziqkRCv%!!pERU|kFf#%e0ro~#Fuc4l46>$n<}MZhW>zLf!2fq(7{o1YTuhw+4B|G1 zE~cWU#`Y$rF#P;5&Mr=-hPE&so3WZ(PFtKPzVjLt4^67E;HUtP?Dotn&g6|N6Y7^` 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Released under CC Attribution-NonCommercial 4.0 license}} - \renewcommand{\footrulewidth}{0mm} - \renewcommand{\headrulewidth}{0mm} -} - -\pagestyle{fancy} - - -% prevent orhpans and widows -\clubpenalty = 10000 -\widowpenalty = 10000 - -\newenvironment{doconceexercise}{}{} -\newcounter{doconceexercisecounter} - - -% ------ header in subexercises ------ -%\newcommand{\subex}[1]{\paragraph{#1}} -%\newcommand{\subex}[1]{\par\vspace{1.7mm}\noindent{\bf #1}\ \ } -\makeatletter -% 1.5ex is the spacing above the header, 0.5em the spacing after subex title -\newcommand\subex{\@startsection*{paragraph}{4}{\z@}% - {1.5ex\@plus1ex \@minus.2ex}% - {-0.5em}% - {\normalfont\normalsize\bfseries}} -\makeatother - - -% --- end of standard preamble for documents --- - - -% insert custom LaTeX commands... - -\raggedbottom -\makeindex -\usepackage[totoc]{idxlayout} % for index in the toc -\usepackage[nottoc]{tocbibind} % for references/bibliography in the toc - -%-------------------- end preamble ---------------------- - -\begin{document} - -% matching end for #ifdef PREAMBLE - -\newcommand{\exercisesection}[1]{\subsection*{#1}} - - -% ------------------- main content ---------------------- - - - -% ----------------- title ------------------------- - -\thispagestyle{empty} - -\begin{center} -{\LARGE\bf -\begin{spacing}{1.25} -Homework 1 Fall Semester 2020 -\end{spacing} -} -\end{center} - -% ----------------- author(s) ------------------------- - -\begin{center} -{\bf \href{{http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html}}{Data Analysis and Machine Learning FYS-STK3155/FYS4155}} -\end{center} - - \begin{center} -% List of all institutions: -\centerline{{\small Department of Physics, University of Oslo, Norway}} -\end{center} - -% ----------------- end author(s) ------------------------- - -% --- begin date --- -\begin{center} -Aug 19, 2020 -\end{center} -% --- end date --- - -\vspace{1cm} - - -\subsection*{Exercise, Setting up various Python environments} - -The first exercise here is of a mere technical art. We want you to have -\begin{itemize} -\item git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo \href{{https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html}}{GitHub facilities}. - -\item Install various Python packages -\end{itemize} - -\noindent -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -IPython/Jupyter notebooks invaluable in your work. You can run \textbf{R} -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Fortran etc if you prefer. The focus in these lectures will be -on Python. - -If you have Python installed (we recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via \textbf{pip} as - -\begin{enumerate} -\item pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow -\end{enumerate} - -\noindent -For \textbf{Tensorflow}, we recommend following the instructions in the text of -\href{{http://shop.oreilly.com/product/0636920052289.do}}{Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly} - -We will come back to \textbf{tensorflow} later. - -For Python3, replace \textbf{pip} with \textbf{pip3}. - -For OSX users we recommend, after having installed Xcode, to -install \textbf{brew}. Brew allows for a seamless installation of additional -software via for example - -\begin{enumerate} -\item brew install python3 -\end{enumerate} - -\noindent -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use \textbf{pip} as well and simply install Python as - -\begin{enumerate} -\item sudo apt-get install python3 (or python for Python2.7) -\end{enumerate} - -\noindent -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -\begin{itemize} -\item \href{{https://docs.anaconda.com/}}{Anaconda}, -\end{itemize} - -\noindent -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system \textbf{conda}. - -\begin{itemize} -\item \href{{https://www.enthought.com/product/canopy/}}{Enthought canopy} -\end{itemize} - -\noindent -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -We recommend using \textbf{Anaconda}. - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection*{Exercise \thedoconceexercisecounter: Our first Python encounter} - - -This exercise has as its aim to write a small program which reads in data from a \textbf{csv} file on the equation of state for dense nuclear matter. The file is localized at \href{{https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv}}{\nolinkurl{https://github.com/mhjensen/MachineLearningMSU-FRIB2020/blob/master/doc/pub/Regression/ipynb/datafiles/EoS.csv}}. Thereafter you will have to set up the design matrix $\bm{X}$ for the $n$ -datapoints and a polynomial of degree $3$. The steps are: -\begin{itemize} -\item Write a Python code which reads the in the above mentioned file. - -\item Use for example \textbf{pandas} to order your data and find out how many data points there are. - -\item Set thereafter up the design matrix with dimensionality $n\times p$ where $p=4$ and where you have defined a polynomial of degree $p-1=3$. Print the matrix and check that the numbers are correct. -\end{itemize} - -\noindent -We recommend looking at the examples in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/Regression/html/Regression-bs.html}}{regression slides}. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -\begin{print} -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split -# 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') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organized into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),5)) -X[:,0] = 1 -X[:,1] = Density**(2.0/3.0) -X[:,2] = Density -X[:,3] = Density**(4.0/3.0) -X[:,4] = Density**(5.0/3.0) -# We split the data in test and training data -X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) -\end{print} - -% --- end solution of exercise --- - -\end{doconceexercise} -% --- end exercise --- - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection*{Exercise \thedoconceexercisecounter: making your own data and exploring scikit-learn} - - -We will generate our own dataset for a function $y(x)$ where $x \in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\cal {N}(0,1)$. -The following simple Python instructions define our $x$ and $y$ values (with 100 data points). -\begin{print} -x = np.random.rand(100,1) -y = 2.0+5*x*x+0.1*np.random.randn(100,1) -\end{print} - -\begin{enumerate} -\item Write your own code (following the examples under the \href{{https://compphysics.github.io/MachineLearningECT/doc/pub/Day1/html/Day1-bs.html}}{regression slides}) for computing the parametrization of the data set fitting a second-order polynomial. - -\item Use thereafter \textbf{scikit-learn} (see again the examples in the regression slides) and compare with your own code. - -\item Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as -\end{enumerate} - -\noindent -\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -\] -and the $R^2$ score function. -If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as -\[ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -\] -where we have defined the mean value of $\hat{y}$ as -\[ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -\] -You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. -Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -The code here is an example of where we define our own design matrix and fit parameters $\beta$. -\begin{print} -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -x = np.random.rand(100) -y = 2.0+5*x*x+0.1*np.random.randn(100) - - -# The design matrix now as function of a given polynomial -X = np.zeros((len(x),3)) -X[:,0] = 1.0 -X[:,1] = x -X[:,2] = x**2 -# 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) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train -print(beta) -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) -\end{print} - -% --- end solution of exercise --- - -\end{doconceexercise} -% --- end exercise --- - - - - -% --- begin exercise --- -\begin{doconceexercise} -\refstepcounter{doconceexercisecounter} - -\exercisesection*{Exercise \thedoconceexercisecounter: mean values and variances in linear regression} - - -This exercise deals with various mean values ad variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}). - -The assumption we have made is -that there exists a function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$ -which describes our data -\[ -\bm{y} = f(\bm{x})+\bm{\varepsilon} -\] - -We then approximate this function with our model from the solution of the linear regression equations (ordinary least squares OLS), that is our -function $f$ is approximated by $\bm{\tilde{y}}$ where we minimized $(\bm{y}-\bm{\tilde{y}})^2$, with -\[ -\bm{\tilde{y}} = \bm{X}\bm{\beta}. -\] -The matrix $\bm{X}$ is the so-called design matrix. - - -\subex{a)} -Show that the expectation value of $\bm{y}$ for a given element $i$ -\begin{align*} -\mathbb{E}(y_i) & =\mathbf{X}_{i, \ast} \, \beta, -\end{align*} -and that -its variance is -\begin{align*} \mbox{Var}(y_i) & = \sigma^2. -\end{align*} -Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\beta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with -mean value $\bm{X}\bm{\beta}$ and variance $\sigma^2$. - - -% --- begin solution of exercise --- -\paragraph{Solution.} -We can calculate the expectation value of $\bm{y}$ for a given element $i$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \bm{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -while -its variance is -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \bm{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \bm{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\beta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with -mean value $\bm{X}\bm{\beta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD). - -% --- end solution of exercise --- - -\subex{b)} -With the OLS expressions for the parameters $\bm{\beta}$ show that -\[ -\mathbb{E}(\bm{\beta}) = \bm{\beta}. -\] - - -% --- begin solution of exercise --- -\paragraph{Solution.} -\[ -\mathbb{E}(\bm{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\bm{\beta}=\bm{\beta}. -\] -This means that the estimator of the regression parameters is unbiased. - -% --- end solution of exercise --- - -\subex{c)} -Show finally that the variance of $\bm{\beta}$ is -\begin{eqnarray*} -\mbox{Var}(\bm{\beta}) & = & \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}. -\end{eqnarray*} - - -% --- begin solution of exercise --- -\paragraph{Solution.} -The variance of $\bm{\beta}$ is -\begin{eqnarray*} -\mbox{Var}(\bm{\beta}) & = & \mathbb{E} \{ [\bm{\beta} - \mathbb{E}(\bm{\beta})] [\bm{\beta} - \mathbb{E}(\bm{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \bm{\beta} \, \bm{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \bm{\beta} \bm{\beta}^T -\\ -& = & \bm{\beta} \, \bm{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} - -where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\bm{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the -variance of the estimate of the $j$-th regression coefficient: -$\bm{\sigma}^2 (\hat{\beta}_j ) = \bm{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }$. This may be used to -construct a confidence interval for the estimates. - - -In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters $\bm{\beta}$ and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. - -% --- end solution of exercise --- - - - - -\end{doconceexercise} -% --- end exercise --- - - -% ------------------- end of main content --------------- - -\end{document} - diff --git a/doc/src/Projects/2020/Exercises/ipynb-hw1-src.tar.gz b/doc/src/Projects/2020/Exercises/ipynb-hw1-src.tar.gz deleted file mode 100644 index 398dcd682934857c1f815c80d75403e8393996cf..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 189 zcmV;u07CyCiwFQw3O!!{1MSaC3c@fD2H>uHia9~aq_4GL7cK-5FObsIL~T-&6z%Qp z19YX}qKJ@h^E1OR%zEl%5ht!JS8wNrvAX`SXMP2q`6muSTG;G*C$s{k9WS*$gScTN+g_DWX>P|p r7=G=f;lNZ6tg57wTCq#m8htdZu~GQzPdv}_yl*@Jl0=aL00;m8w4GLD