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This exercise is also part of the weekly exercises of week 37.\n", - "Explain what the terms mean and discuss their interpretations.\n", - "\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n", diff --git a/doc/LectureNotes/_build/html/_sources/project1.ipynb b/doc/LectureNotes/_build/html/_sources/project1.ipynb index 467bad587..e28db853b 100644 --- a/doc/LectureNotes/_build/html/_sources/project1.ipynb +++ b/doc/LectureNotes/_build/html/_sources/project1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "7424fea1", + "id": "ee871b85", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "c9ca1d90", + "id": "de647a13", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "3e4164b6", + "id": "f0094ea8", "metadata": { "editable": true }, @@ -63,7 +63,7 @@ }, { "cell_type": "markdown", - "id": "dd4dcdf5", + "id": "9d47e07f", "metadata": { "editable": true }, @@ -85,7 +85,7 @@ }, { "cell_type": "markdown", - "id": "71ddc478", + "id": "13b3c63c", "metadata": { "editable": true }, @@ -100,7 +100,7 @@ }, { "cell_type": "markdown", - "id": "d4184afb", + "id": "588eb391", "metadata": { "editable": true }, @@ -133,7 +133,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "62880fe6", + "id": "2b565e88", "metadata": { "collapsed": false, "editable": true @@ -185,7 +185,7 @@ }, { "cell_type": "markdown", - "id": "6f7e7c97", + "id": "5a80d8fa", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "7377112f", + "id": "2b8a75d3", "metadata": { "editable": true }, @@ -220,7 +220,7 @@ }, { "cell_type": "markdown", - "id": "72f83fe7", + "id": "de8e68ca", "metadata": { "editable": true }, @@ -232,7 +232,7 @@ }, { "cell_type": "markdown", - "id": "7a3578ff", + "id": "70a2795e", "metadata": { "editable": true }, @@ -244,7 +244,7 @@ }, { "cell_type": "markdown", - "id": "e0aef819", + "id": "07d74c94", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "c7f751af", + "id": "cb7a1688", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "c026a4bd", + "id": "0c273378", "metadata": { "editable": true }, @@ -295,7 +295,7 @@ }, { "cell_type": "markdown", - "id": "2635bcbd", + "id": "b6aa07e9", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "92892df2", + "id": "4c297a7f", "metadata": { "editable": true }, @@ -330,7 +330,7 @@ }, { "cell_type": "markdown", - "id": "35987aab", + "id": "9b65abbd", "metadata": { "editable": true }, @@ -346,7 +346,7 @@ }, { "cell_type": "markdown", - "id": "e61048c2", + "id": "9f2c97bc", "metadata": { "editable": true }, @@ -358,7 +358,7 @@ }, { "cell_type": "markdown", - "id": "5ef456d4", + "id": "1390caea", "metadata": { "editable": true }, @@ -369,7 +369,7 @@ }, { "cell_type": "markdown", - "id": "52e6fbad", + "id": "757a87d4", "metadata": { "editable": true }, @@ -381,7 +381,7 @@ }, { "cell_type": "markdown", - "id": "db6028a0", + "id": "cf182a2a", "metadata": { "editable": true }, @@ -393,7 +393,7 @@ }, { "cell_type": "markdown", - "id": "9477ad5e", + "id": "c7f15f3d", "metadata": { "editable": true }, @@ -405,7 +405,7 @@ }, { "cell_type": "markdown", - "id": "8c2aeccd", + "id": "70c8c4e9", "metadata": { "editable": true }, @@ -416,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "6a774b74", + "id": "3879a5b4", "metadata": { "editable": true }, @@ -428,7 +428,7 @@ }, { "cell_type": "markdown", - "id": "98fde9b7", + "id": "17191387", "metadata": { "editable": true }, @@ -441,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "87a4d11d", + "id": "a7e3a698", "metadata": { "editable": true }, @@ -453,7 +453,7 @@ }, { "cell_type": "markdown", - "id": "b4844f1e", + "id": "93fc4a12", "metadata": { "editable": true }, @@ -463,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "f7662ade", + "id": "85d336ad", "metadata": { "editable": true }, @@ -475,7 +475,7 @@ }, { "cell_type": "markdown", - "id": "1d61362e", + "id": "869ed780", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "74bbb4ab", + "id": "88502fe8", "metadata": { "editable": true }, @@ -519,7 +519,7 @@ }, { "cell_type": "markdown", - "id": "ac939724", + "id": "db4583da", "metadata": { "editable": true }, @@ -531,7 +531,7 @@ }, { "cell_type": "markdown", - "id": "4066d012", + "id": "da3a0dfc", "metadata": { "editable": true }, @@ -550,7 +550,7 @@ }, { "cell_type": "markdown", - "id": "7e5c5ef4", + "id": "07b53a45", "metadata": { "editable": true }, @@ -562,7 +562,7 @@ }, { "cell_type": "markdown", - "id": "690337b9", + "id": "1d8bc8d4", "metadata": { "editable": true }, @@ -576,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "92dabf3c", + "id": "7f06bfd3", "metadata": { "editable": true }, @@ -588,7 +588,7 @@ }, { "cell_type": "markdown", - "id": "7da90966", + "id": "5624047e", "metadata": { "editable": true }, @@ -598,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "33a81d54", + "id": "1b26ef35", "metadata": { "editable": true }, @@ -610,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "abb9bd28", + "id": "6319fcec", "metadata": { "editable": true }, @@ -620,24 +620,24 @@ }, { "cell_type": "markdown", - "id": "5b4668fd", + "id": "2253bb98", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", + "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$" ] }, { "cell_type": "markdown", - "id": "b453f8fc", + "id": "d08baf03", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", + "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 38.\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of the Franke function by\n", @@ -651,7 +651,7 @@ }, { "cell_type": "markdown", - "id": "6fe323b0", + "id": "d06c2f80", "metadata": { "editable": true }, @@ -676,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "5fd2f486", + "id": "2a957aee", "metadata": { "editable": true }, @@ -704,7 +704,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "e88e39ea", + "id": "1c0d6323", "metadata": { "collapsed": false, "editable": true @@ -716,7 +716,7 @@ }, { "cell_type": "markdown", - "id": "29817da1", + "id": "183f2666", "metadata": { "editable": true }, @@ -728,7 +728,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "560fc9f4", + "id": "3ccf3ab9", "metadata": { "collapsed": false, "editable": true @@ -754,7 +754,7 @@ }, { "cell_type": "markdown", - "id": "3fc05383", + "id": "cf6cdda9", "metadata": { "editable": true }, @@ -779,7 +779,7 @@ }, { "cell_type": "markdown", - "id": "bb21261e", + "id": "f28a3bdf", "metadata": { "editable": true }, @@ -793,7 +793,7 @@ }, { "cell_type": "markdown", - "id": "c5ab9dbf", + "id": "ed920879", "metadata": { "editable": true }, @@ -823,7 +823,7 @@ }, { "cell_type": "markdown", - "id": "7d0dbb14", + "id": "f2bef519", "metadata": { "editable": true }, @@ -845,7 +845,7 @@ }, { "cell_type": "markdown", - "id": "e301049d", + "id": "e1891d3f", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/_build/html/exercisesweek38.html b/doc/LectureNotes/_build/html/exercisesweek38.html index e79780177..f19a0b429 100644 --- a/doc/LectureNotes/_build/html/exercisesweek38.html +++ b/doc/LectureNotes/_build/html/exercisesweek38.html @@ -451,8 +451,8 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_ \]

Here the expected value \(\mathbb{E}\) is the sample value.

Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a -term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.

-

That is, show that

+term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise. +That is, show that

\[ \mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[y]+\mathrm{var}[\tilde{y}]+\sigma^2, @@ -465,10 +465,8 @@ term which measures the deviation from the true data and the mean value of the m

and

\[ -\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. +\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. \]
-

The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. -Explain what the terms mean and discuss their interpretations.

Explain what the terms mean and discuss their interpretations.

Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by studying the MSE value as function of the complexity of your model. Use ordinary least squares only.

diff --git a/doc/LectureNotes/_build/html/project1.html b/doc/LectureNotes/_build/html/project1.html index 80bd90f13..a43675e36 100644 --- a/doc/LectureNotes/_build/html/project1.html +++ b/doc/LectureNotes/_build/html/project1.html @@ -659,7 +659,7 @@ which polynomial fits the data best.

-
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20604/39730396.py:11: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
+
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_21682/39730396.py:11: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
   ax = fig.gca(projection='3d')
 
@@ -825,9 +825,9 @@ That is, show that

and

\[ -\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. +\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. \]
-

The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. +

The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 38. Explain what the terms mean and discuss their interpretations.

Perform then a bias-variance analysis of the Franke function by studying the MSE value as function of the complexity of your model.

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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Exercises week 38","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques","Week 37: Statistical interpretations and Resampling Methods","Week 38: Logistic Regression and 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Exercises week 38","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques","Week 37: Statistical interpretations and Resampling Methods","Week 38: Logistic Regression and 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb index b7ce6dd2e..15327f1bf 100644 --- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "b2cf4bdf", + "id": "d8f7704f", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "43316fd6", + "id": "9e388b87", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "1f8db1a6", + "id": "80d7df2a", "metadata": { "editable": true }, @@ -45,7 +45,7 @@ }, { "cell_type": "markdown", - "id": "41067a3c", + "id": "80e18e80", "metadata": { "editable": true }, @@ -57,7 +57,7 @@ }, { "cell_type": "markdown", - "id": "68997f6b", + "id": "3a74059f", "metadata": { "editable": true }, @@ -76,7 +76,7 @@ }, { "cell_type": "markdown", - "id": "f7e45184", + "id": "1f92f219", "metadata": { "editable": true }, @@ -88,7 +88,7 @@ }, { "cell_type": "markdown", - "id": "4c15509e", + "id": "416cd63a", "metadata": { "editable": true }, @@ -97,13 +97,12 @@ "\n", "Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n", "term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.\n", - "\n", "That is, show that" ] }, { "cell_type": "markdown", - "id": "046ec084", + "id": "d65d4905", "metadata": { "editable": true }, @@ -115,7 +114,7 @@ }, { "cell_type": "markdown", - "id": "b2cc33db", + "id": "d102dec2", "metadata": { "editable": true }, @@ -125,7 +124,7 @@ }, { "cell_type": "markdown", - "id": "7ab1bc0e", + "id": "869dc9d9", "metadata": { "editable": true }, @@ -137,7 +136,7 @@ }, { "cell_type": "markdown", - "id": "a8da71db", + "id": "fee09cce", "metadata": { "editable": true }, @@ -147,26 +146,23 @@ }, { "cell_type": "markdown", - "id": "b74aa5e4", + "id": "b6fb6522", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", + "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$" ] }, { "cell_type": "markdown", - "id": "de559d80", + "id": "6fa010b9", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", - "Explain what the terms mean and discuss their interpretations.\n", - "\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n", diff --git a/doc/LectureNotes/_build/jupyter_execute/project1.ipynb b/doc/LectureNotes/_build/jupyter_execute/project1.ipynb index 19fe51876..e69ce0ab7 100644 --- a/doc/LectureNotes/_build/jupyter_execute/project1.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/project1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "7424fea1", + "id": "ee871b85", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "c9ca1d90", + "id": "de647a13", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "3e4164b6", + "id": "f0094ea8", "metadata": { "editable": true }, @@ -63,7 +63,7 @@ }, { "cell_type": "markdown", - "id": "dd4dcdf5", + "id": "9d47e07f", "metadata": { "editable": true }, @@ -85,7 +85,7 @@ }, { "cell_type": "markdown", - "id": "71ddc478", + "id": "13b3c63c", "metadata": { "editable": true }, @@ -100,7 +100,7 @@ }, { "cell_type": "markdown", - "id": "d4184afb", + "id": "588eb391", "metadata": { "editable": true }, @@ -133,7 +133,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "62880fe6", + "id": "2b565e88", "metadata": { "collapsed": false, "editable": true @@ -143,7 +143,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_20604/39730396.py:11: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n", + "/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_21682/39730396.py:11: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().\n", " ax = fig.gca(projection='3d')\n" ] }, @@ -208,7 +208,7 @@ }, { "cell_type": "markdown", - "id": "6f7e7c97", + "id": "5a80d8fa", "metadata": { "editable": true }, @@ -230,7 +230,7 @@ }, { "cell_type": "markdown", - "id": "7377112f", + "id": "2b8a75d3", "metadata": { "editable": true }, @@ -243,7 +243,7 @@ }, { "cell_type": "markdown", - "id": "72f83fe7", + "id": "de8e68ca", "metadata": { "editable": true }, @@ -255,7 +255,7 @@ }, { "cell_type": "markdown", - "id": "7a3578ff", + "id": "70a2795e", "metadata": { "editable": true }, @@ -267,7 +267,7 @@ }, { "cell_type": "markdown", - "id": "e0aef819", + "id": "07d74c94", "metadata": { "editable": true }, @@ -277,7 +277,7 @@ }, { "cell_type": "markdown", - "id": "c7f751af", + "id": "cb7a1688", "metadata": { "editable": true }, @@ -289,7 +289,7 @@ }, { "cell_type": "markdown", - "id": "c026a4bd", + "id": "0c273378", "metadata": { "editable": true }, @@ -318,7 +318,7 @@ }, { "cell_type": "markdown", - "id": "2635bcbd", + "id": "b6aa07e9", "metadata": { "editable": true }, @@ -336,7 +336,7 @@ }, { "cell_type": "markdown", - "id": "92892df2", + "id": "4c297a7f", "metadata": { "editable": true }, @@ -353,7 +353,7 @@ }, { "cell_type": "markdown", - "id": "35987aab", + "id": "9b65abbd", "metadata": { "editable": true }, @@ -369,7 +369,7 @@ }, { "cell_type": "markdown", - "id": "e61048c2", + "id": "9f2c97bc", "metadata": { "editable": true }, @@ -381,7 +381,7 @@ }, { "cell_type": "markdown", - "id": "5ef456d4", + "id": "1390caea", "metadata": { "editable": true }, @@ -392,7 +392,7 @@ }, { "cell_type": "markdown", - "id": "52e6fbad", + "id": "757a87d4", "metadata": { "editable": true }, @@ -404,7 +404,7 @@ }, { "cell_type": "markdown", - "id": "db6028a0", + "id": "cf182a2a", "metadata": { "editable": true }, @@ -416,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "9477ad5e", + "id": "c7f15f3d", "metadata": { "editable": true }, @@ -428,7 +428,7 @@ }, { "cell_type": "markdown", - "id": "8c2aeccd", + "id": "70c8c4e9", "metadata": { "editable": true }, @@ -439,7 +439,7 @@ }, { "cell_type": "markdown", - "id": "6a774b74", + "id": "3879a5b4", "metadata": { "editable": true }, @@ -451,7 +451,7 @@ }, { "cell_type": "markdown", - "id": "98fde9b7", + "id": "17191387", "metadata": { "editable": true }, @@ -464,7 +464,7 @@ }, { "cell_type": "markdown", - "id": "87a4d11d", + "id": "a7e3a698", "metadata": { "editable": true }, @@ -476,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "b4844f1e", + "id": "93fc4a12", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "f7662ade", + "id": "85d336ad", "metadata": { "editable": true }, @@ -498,7 +498,7 @@ }, { "cell_type": "markdown", - "id": "1d61362e", + "id": "869ed780", "metadata": { "editable": true }, @@ -509,7 +509,7 @@ }, { "cell_type": "markdown", - "id": "74bbb4ab", + "id": "88502fe8", "metadata": { "editable": true }, @@ -542,7 +542,7 @@ }, { "cell_type": "markdown", - "id": "ac939724", + "id": "db4583da", "metadata": { "editable": true }, @@ -554,7 +554,7 @@ }, { "cell_type": "markdown", - "id": "4066d012", + "id": "da3a0dfc", "metadata": { "editable": true }, @@ -573,7 +573,7 @@ }, { "cell_type": "markdown", - "id": "7e5c5ef4", + "id": "07b53a45", "metadata": { "editable": true }, @@ -585,7 +585,7 @@ }, { "cell_type": "markdown", - "id": "690337b9", + "id": "1d8bc8d4", "metadata": { "editable": true }, @@ -599,7 +599,7 @@ }, { "cell_type": "markdown", - "id": "92dabf3c", + "id": "7f06bfd3", "metadata": { "editable": true }, @@ -611,7 +611,7 @@ }, { "cell_type": "markdown", - "id": "7da90966", + "id": "5624047e", "metadata": { "editable": true }, @@ -621,7 +621,7 @@ }, { "cell_type": "markdown", - "id": "33a81d54", + "id": "1b26ef35", "metadata": { "editable": true }, @@ -633,7 +633,7 @@ }, { "cell_type": "markdown", - "id": "abb9bd28", + "id": "6319fcec", "metadata": { "editable": true }, @@ -643,24 +643,24 @@ }, { "cell_type": "markdown", - "id": "5b4668fd", + "id": "2253bb98", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", + "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$" ] }, { "cell_type": "markdown", - "id": "b453f8fc", + "id": "d08baf03", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", + "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 38.\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of the Franke function by\n", @@ -674,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "6fe323b0", + "id": "d06c2f80", "metadata": { "editable": true }, @@ -699,7 +699,7 @@ }, { "cell_type": "markdown", - "id": "5fd2f486", + "id": "2a957aee", "metadata": { "editable": true }, @@ -727,7 +727,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "e88e39ea", + "id": "1c0d6323", "metadata": { "collapsed": false, "editable": true @@ -751,7 +751,7 @@ }, { "cell_type": "markdown", - "id": "29817da1", + "id": "183f2666", "metadata": { "editable": true }, @@ -763,7 +763,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "560fc9f4", + "id": "3ccf3ab9", "metadata": { "collapsed": false, "editable": true @@ -789,7 +789,7 @@ }, { "cell_type": "markdown", - "id": "3fc05383", + "id": "cf6cdda9", "metadata": { "editable": true }, @@ -814,7 +814,7 @@ }, { "cell_type": "markdown", - "id": "bb21261e", + "id": "f28a3bdf", "metadata": { "editable": true }, @@ -828,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "c5ab9dbf", + "id": "ed920879", "metadata": { "editable": true }, @@ -858,7 +858,7 @@ }, { "cell_type": "markdown", - "id": "7d0dbb14", + "id": "f2bef519", "metadata": { "editable": true }, @@ -880,7 +880,7 @@ }, { "cell_type": "markdown", - "id": "e301049d", + "id": "e1891d3f", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/_build/jupyter_execute/project1.py b/doc/LectureNotes/_build/jupyter_execute/project1.py index 8342f068e..34615c06a 100644 --- a/doc/LectureNotes/_build/jupyter_execute/project1.py +++ b/doc/LectureNotes/_build/jupyter_execute/project1.py @@ -317,10 +317,10 @@ plt.show() # and # $$ -# \mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. +# \mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2. # $$ -# The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. +# The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 38. # Explain what the terms mean and discuss their interpretations. # # Perform then a bias-variance analysis of the Franke function by diff --git a/doc/LectureNotes/exercisesweek38.ipynb b/doc/LectureNotes/exercisesweek38.ipynb index 5f82b3d79..9f34b9108 100644 --- a/doc/LectureNotes/exercisesweek38.ipynb +++ b/doc/LectureNotes/exercisesweek38.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "b2cf4bdf", + "id": "d8f7704f", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "43316fd6", + "id": "9e388b87", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "1f8db1a6", + "id": "80d7df2a", "metadata": { "editable": true }, @@ -45,7 +45,7 @@ }, { "cell_type": "markdown", - "id": "41067a3c", + "id": "80e18e80", "metadata": { "editable": true }, @@ -57,7 +57,7 @@ }, { "cell_type": "markdown", - "id": "68997f6b", + "id": "3a74059f", "metadata": { "editable": true }, @@ -76,7 +76,7 @@ }, { "cell_type": "markdown", - "id": "f7e45184", + "id": "1f92f219", "metadata": { "editable": true }, @@ -88,7 +88,7 @@ }, { "cell_type": "markdown", - "id": "4c15509e", + "id": "416cd63a", "metadata": { "editable": true }, @@ -97,13 +97,12 @@ "\n", "Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n", "term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.\n", - "\n", "That is, show that" ] }, { "cell_type": "markdown", - "id": "046ec084", + "id": "d65d4905", "metadata": { "editable": true }, @@ -115,7 +114,7 @@ }, { "cell_type": "markdown", - "id": "b2cc33db", + "id": "d102dec2", "metadata": { "editable": true }, @@ -125,7 +124,7 @@ }, { "cell_type": "markdown", - "id": "7ab1bc0e", + "id": "869dc9d9", "metadata": { "editable": true }, @@ -137,7 +136,7 @@ }, { "cell_type": "markdown", - "id": "a8da71db", + "id": "fee09cce", "metadata": { "editable": true }, @@ -147,26 +146,23 @@ }, { "cell_type": "markdown", - "id": "b74aa5e4", + "id": "b6fb6522", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", + "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$" ] }, { "cell_type": "markdown", - "id": "de559d80", + "id": "6fa010b9", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", - "Explain what the terms mean and discuss their interpretations.\n", - "\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n", diff --git a/doc/LectureNotes/project1.ipynb b/doc/LectureNotes/project1.ipynb index 467bad587..e28db853b 100644 --- a/doc/LectureNotes/project1.ipynb +++ b/doc/LectureNotes/project1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "7424fea1", + "id": "ee871b85", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "c9ca1d90", + "id": "de647a13", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "3e4164b6", + "id": "f0094ea8", "metadata": { "editable": true }, @@ -63,7 +63,7 @@ }, { "cell_type": "markdown", - "id": "dd4dcdf5", + "id": "9d47e07f", "metadata": { "editable": true }, @@ -85,7 +85,7 @@ }, { "cell_type": "markdown", - "id": "71ddc478", + "id": "13b3c63c", "metadata": { "editable": true }, @@ -100,7 +100,7 @@ }, { "cell_type": "markdown", - "id": "d4184afb", + "id": "588eb391", "metadata": { "editable": true }, @@ -133,7 +133,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "62880fe6", + "id": "2b565e88", "metadata": { "collapsed": false, "editable": true @@ -185,7 +185,7 @@ }, { "cell_type": "markdown", - "id": "6f7e7c97", + "id": "5a80d8fa", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "7377112f", + "id": "2b8a75d3", "metadata": { "editable": true }, @@ -220,7 +220,7 @@ }, { "cell_type": "markdown", - "id": "72f83fe7", + "id": "de8e68ca", "metadata": { "editable": true }, @@ -232,7 +232,7 @@ }, { "cell_type": "markdown", - "id": "7a3578ff", + "id": "70a2795e", "metadata": { "editable": true }, @@ -244,7 +244,7 @@ }, { "cell_type": "markdown", - "id": "e0aef819", + "id": "07d74c94", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "c7f751af", + "id": "cb7a1688", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "c026a4bd", + "id": "0c273378", "metadata": { "editable": true }, @@ -295,7 +295,7 @@ }, { "cell_type": "markdown", - "id": "2635bcbd", + "id": "b6aa07e9", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "92892df2", + "id": "4c297a7f", "metadata": { "editable": true }, @@ -330,7 +330,7 @@ }, { "cell_type": "markdown", - "id": "35987aab", + "id": "9b65abbd", "metadata": { "editable": true }, @@ -346,7 +346,7 @@ }, { "cell_type": "markdown", - "id": "e61048c2", + "id": "9f2c97bc", "metadata": { "editable": true }, @@ -358,7 +358,7 @@ }, { "cell_type": "markdown", - "id": "5ef456d4", + "id": "1390caea", "metadata": { "editable": true }, @@ -369,7 +369,7 @@ }, { "cell_type": "markdown", - "id": "52e6fbad", + "id": "757a87d4", "metadata": { "editable": true }, @@ -381,7 +381,7 @@ }, { "cell_type": "markdown", - "id": "db6028a0", + "id": "cf182a2a", "metadata": { "editable": true }, @@ -393,7 +393,7 @@ }, { "cell_type": "markdown", - "id": "9477ad5e", + "id": "c7f15f3d", "metadata": { "editable": true }, @@ -405,7 +405,7 @@ }, { "cell_type": "markdown", - "id": "8c2aeccd", + "id": "70c8c4e9", "metadata": { "editable": true }, @@ -416,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "6a774b74", + "id": "3879a5b4", "metadata": { "editable": true }, @@ -428,7 +428,7 @@ }, { "cell_type": "markdown", - "id": "98fde9b7", + "id": "17191387", "metadata": { "editable": true }, @@ -441,7 +441,7 @@ }, { "cell_type": "markdown", - "id": "87a4d11d", + "id": "a7e3a698", "metadata": { "editable": true }, @@ -453,7 +453,7 @@ }, { "cell_type": "markdown", - "id": "b4844f1e", + "id": "93fc4a12", "metadata": { "editable": true }, @@ -463,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "f7662ade", + "id": "85d336ad", "metadata": { "editable": true }, @@ -475,7 +475,7 @@ }, { "cell_type": "markdown", - "id": "1d61362e", + "id": "869ed780", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "74bbb4ab", + "id": "88502fe8", "metadata": { "editable": true }, @@ -519,7 +519,7 @@ }, { "cell_type": "markdown", - "id": "ac939724", + "id": "db4583da", "metadata": { "editable": true }, @@ -531,7 +531,7 @@ }, { "cell_type": "markdown", - "id": "4066d012", + "id": "da3a0dfc", "metadata": { "editable": true }, @@ -550,7 +550,7 @@ }, { "cell_type": "markdown", - "id": "7e5c5ef4", + "id": "07b53a45", "metadata": { "editable": true }, @@ -562,7 +562,7 @@ }, { "cell_type": "markdown", - "id": "690337b9", + "id": "1d8bc8d4", "metadata": { "editable": true }, @@ -576,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "92dabf3c", + "id": "7f06bfd3", "metadata": { "editable": true }, @@ -588,7 +588,7 @@ }, { "cell_type": "markdown", - "id": "7da90966", + "id": "5624047e", "metadata": { "editable": true }, @@ -598,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "33a81d54", + "id": "1b26ef35", "metadata": { "editable": true }, @@ -610,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "abb9bd28", + "id": "6319fcec", "metadata": { "editable": true }, @@ -620,24 +620,24 @@ }, { "cell_type": "markdown", - "id": "5b4668fd", + "id": "2253bb98", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", + "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$" ] }, { "cell_type": "markdown", - "id": "b453f8fc", + "id": "d08baf03", "metadata": { "editable": true }, "source": [ - "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37.\n", + "The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 38.\n", "Explain what the terms mean and discuss their interpretations.\n", "\n", "Perform then a bias-variance analysis of the Franke function by\n", @@ -651,7 +651,7 @@ }, { "cell_type": "markdown", - "id": "6fe323b0", + "id": "d06c2f80", "metadata": { "editable": true }, @@ -676,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "5fd2f486", + "id": "2a957aee", "metadata": { "editable": true }, @@ -704,7 +704,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "e88e39ea", + "id": "1c0d6323", "metadata": { "collapsed": false, "editable": true @@ -716,7 +716,7 @@ }, { "cell_type": "markdown", - "id": "29817da1", + "id": "183f2666", "metadata": { "editable": true }, @@ -728,7 +728,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "560fc9f4", + "id": "3ccf3ab9", "metadata": { "collapsed": false, "editable": true @@ -754,7 +754,7 @@ }, { "cell_type": "markdown", - "id": "3fc05383", + "id": "cf6cdda9", "metadata": { "editable": true }, @@ -779,7 +779,7 @@ }, { "cell_type": "markdown", - "id": "bb21261e", + "id": "f28a3bdf", "metadata": { "editable": true }, @@ -793,7 +793,7 @@ }, { "cell_type": "markdown", - "id": "c5ab9dbf", + "id": "ed920879", "metadata": { "editable": true }, @@ -823,7 +823,7 @@ }, { "cell_type": "markdown", - "id": "7d0dbb14", + "id": "f2bef519", "metadata": { "editable": true }, @@ -845,7 +845,7 @@ }, { "cell_type": "markdown", - "id": "e301049d", + "id": "e1891d3f", "metadata": { "editable": true }, diff --git a/doc/src/week38/exercisesweek38.do.txt b/doc/src/week38/exercisesweek38.do.txt index 58e43b517..a04146c53 100644 --- a/doc/src/week38/exercisesweek38.do.txt +++ b/doc/src/week38/exercisesweek38.do.txt @@ -39,7 +39,6 @@ Here the expected value $\mathbb{E}$ is the sample value. Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise. - That is, show that !bt \[ @@ -55,11 +54,10 @@ with and !bt \[ -\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2. +\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\bm{y}}-\mathbb{E}\left[\bm{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2. \] !et -The answer to this exercise should be included in the theory part of the report. This exercise is also part of the weekly exercises of week 37. -Explain what the terms mean and discuss their interpretations. + Explain what the terms mean and discuss their interpretations. @@ -76,3 +74,4 @@ You can follow the code example in the jupyter-book at URL:"https://compphysics. See also the whiteboard notes from week 37 at URL:"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf" +