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Morten Hjorth-Jensen
2023-09-19 11:52:10 +02:00
parent 029b2427e3
commit c38c1c3bbb
14 changed files with 211 additions and 226 deletions
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doc/LectureNotes/_build/.doctrees/environment.pickle
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"\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"
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"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
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"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",
doc/LectureNotes/_build/html/_sources/project1.ipynb
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"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
]
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"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 @@
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doc/LectureNotes/_build/html/exercisesweek38.html
+3 -5
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@@ -451,8 +451,8 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_
\]</div>
<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
<p>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.</p>
<p>That is, show that</p>
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</p>
<div class="math notranslate nohighlight">
\[
\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
<p>and</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\]</div>
<p>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.</p>
<p>Explain what the terms mean and discuss their interpretations.</p>
<p>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.</p>
doc/LectureNotes/_build/html/project1.html
+3 -3
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@@ -659,7 +659,7 @@ which polynomial fits the data best.</p>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/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().
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/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=&#39;3d&#39;)
</pre></div>
</div>
@@ -825,9 +825,9 @@ That is, show that</p>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\tilde{y}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
\]</div>
<p>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.
<p>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.</p>
<p>Perform then a bias-variance analysis of the Franke function by
studying the MSE value as function of the complexity of your model.</p>
doc/LectureNotes/_build/html/searchindex.js
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doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb
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@@ -2,7 +2,7 @@
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@@ -27,7 +27,7 @@
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@@ -57,7 +57,7 @@
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@@ -76,7 +76,7 @@
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@@ -88,7 +88,7 @@
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@@ -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": {
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@@ -115,7 +114,7 @@
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"id": "d102dec2",
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@@ -125,7 +124,7 @@
},
{
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@@ -137,7 +136,7 @@
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@@ -147,26 +146,23 @@
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"source": [
"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
]
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"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",
doc/LectureNotes/_build/jupyter_execute/project1.ipynb
+53 -53
View File
@@ -2,7 +2,7 @@
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@@ -63,7 +63,7 @@
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@@ -85,7 +85,7 @@
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@@ -100,7 +100,7 @@
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{
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@@ -133,7 +133,7 @@
{
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"execution_count": 1,
"id": "62880fe6",
"id": "2b565e88",
"metadata": {
"collapsed": false,
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@@ -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 @@
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@@ -230,7 +230,7 @@
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@@ -243,7 +243,7 @@
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@@ -255,7 +255,7 @@
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@@ -267,7 +267,7 @@
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@@ -277,7 +277,7 @@
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@@ -289,7 +289,7 @@
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@@ -318,7 +318,7 @@
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@@ -336,7 +336,7 @@
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@@ -353,7 +353,7 @@
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@@ -369,7 +369,7 @@
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@@ -381,7 +381,7 @@
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@@ -392,7 +392,7 @@
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@@ -404,7 +404,7 @@
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@@ -416,7 +416,7 @@
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@@ -428,7 +428,7 @@
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@@ -439,7 +439,7 @@
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@@ -451,7 +451,7 @@
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@@ -464,7 +464,7 @@
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@@ -476,7 +476,7 @@
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@@ -542,7 +542,7 @@
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@@ -599,7 +599,7 @@
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@@ -611,7 +611,7 @@
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@@ -621,7 +621,7 @@
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@@ -633,7 +633,7 @@
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@@ -643,24 +643,24 @@
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{
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"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
]
},
{
"cell_type": "markdown",
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"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 @@
},
{
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@@ -699,7 +699,7 @@
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@@ -727,7 +727,7 @@
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@@ -751,7 +751,7 @@
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@@ -763,7 +763,7 @@
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@@ -789,7 +789,7 @@
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@@ -814,7 +814,7 @@
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@@ -828,7 +828,7 @@
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@@ -858,7 +858,7 @@
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@@ -880,7 +880,7 @@
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doc/LectureNotes/_build/jupyter_execute/project1.py
+2 -2
View File
@@ -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
doc/LectureNotes/exercisesweek38.ipynb
+14 -18
View File
@@ -2,7 +2,7 @@
"cells": [
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@@ -14,7 +14,7 @@
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@@ -88,7 +88,7 @@
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@@ -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"
]
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@@ -115,7 +114,7 @@
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@@ -137,7 +136,7 @@
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@@ -147,26 +146,23 @@
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"source": [
"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
]
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{
"cell_type": "markdown",
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"metadata": {
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"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",
doc/LectureNotes/project1.ipynb
+52 -52
View File
@@ -2,7 +2,7 @@
"cells": [
{
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@@ -14,7 +14,7 @@
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@@ -27,7 +27,7 @@
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@@ -63,7 +63,7 @@
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@@ -85,7 +85,7 @@
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@@ -100,7 +100,7 @@
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@@ -185,7 +185,7 @@
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@@ -207,7 +207,7 @@
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@@ -220,7 +220,7 @@
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@@ -232,7 +232,7 @@
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@@ -244,7 +244,7 @@
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@@ -254,7 +254,7 @@
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@@ -266,7 +266,7 @@
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@@ -295,7 +295,7 @@
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@@ -313,7 +313,7 @@
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@@ -330,7 +330,7 @@
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@@ -346,7 +346,7 @@
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@@ -358,7 +358,7 @@
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@@ -369,7 +369,7 @@
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@@ -381,7 +381,7 @@
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@@ -393,7 +393,7 @@
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@@ -405,7 +405,7 @@
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@@ -416,7 +416,7 @@
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@@ -428,7 +428,7 @@
},
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@@ -441,7 +441,7 @@
},
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@@ -453,7 +453,7 @@
},
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@@ -463,7 +463,7 @@
},
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@@ -475,7 +475,7 @@
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@@ -486,7 +486,7 @@
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@@ -519,7 +519,7 @@
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@@ -531,7 +531,7 @@
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@@ -550,7 +550,7 @@
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@@ -562,7 +562,7 @@
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@@ -576,7 +576,7 @@
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@@ -588,7 +588,7 @@
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@@ -598,7 +598,7 @@
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@@ -610,7 +610,7 @@
},
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@@ -620,24 +620,24 @@
},
{
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"source": [
"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$"
]
},
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"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 @@
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@@ -676,7 +676,7 @@
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@@ -704,7 +704,7 @@
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@@ -716,7 +716,7 @@
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@@ -728,7 +728,7 @@
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@@ -754,7 +754,7 @@
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@@ -779,7 +779,7 @@
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@@ -793,7 +793,7 @@
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@@ -823,7 +823,7 @@
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@@ -845,7 +845,7 @@
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