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adding pdf

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Morten Hjorth-Jensen
2021-09-09 07:23:59 +02:00
parent 1cd540f93d
commit 6766f68da6
7 changed files with 228 additions and 42 deletions
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doc/pub/week36/html/week36-bs.html
+31 -29
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@@ -124,6 +124,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'maximum-likelihood-estimation-mle'),
('A new Cost Function', 2, None, 'a-new-cost-function'),
('Friday September 10', 2, None, 'friday-september-10'),
('Why resampling methods', 2, None, 'why-resampling-methods'),
('Resampling methods', 2, None, 'resampling-methods'),
@@ -280,34 +281,35 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._week36-bs028.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs029.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs030.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs031.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs035.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs037.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs034.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs035.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs036.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs037.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs038.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs039.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs040.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs041.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs042.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs043.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs044.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs045.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs046.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs047.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs048.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs049.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs050.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs051.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs052.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs053.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs054.html#summing-up" style="font-size: 80%;">Summing up</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs055.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs056.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs057.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs058.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs031.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs032.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs036.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs038.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs035.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs036.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs037.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs038.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs039.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs040.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs041.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs042.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs043.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs044.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs045.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs046.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs047.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs048.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs049.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs050.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs051.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs052.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs053.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs054.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs055.html#summing-up" style="font-size: 80%;">Summing up</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs056.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs057.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs058.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs059.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
</ul>
</li>
@@ -366,7 +368,7 @@ MathJax.Hub.Config({
<li><a href="._week36-bs008.html">9</a></li>
<li><a href="._week36-bs009.html">10</a></li>
<li><a href="">...</a></li>
<li><a href="._week36-bs058.html">59</a></li>
<li><a href="._week36-bs059.html">60</a></li>
<li><a href="._week36-bs001.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/week36/html/week36-reveal.html
+37 -3
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@@ -1102,7 +1102,7 @@ p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{
$$
<p>&nbsp;<br>
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{beta} \).
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{\beta} \).
<p>
Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \( \boldsymbol{y} \) as the product of the single events, that is we have
@@ -1146,10 +1146,44 @@ is equivalent to the maximization/minimization of the function itself.
<section>
<h2 id="friday-september-10">Friday September 10 </h2>
<h2 id="a-new-cost-function">A new Cost Function </h2>
<p>
More text will be added here.
We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
<p>&nbsp;<br>
$$
C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})},
$$
<p>&nbsp;<br>
which becomes
<p>&nbsp;<br>
$$
C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
$$
<p>&nbsp;<br>
<p>
Taking the derivative of the <em>new</em> cost function with respect to the parameters \( \beta \) we recognize our familiar OLS equation, namely
<p>&nbsp;<br>
$$
\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
$$
<p>&nbsp;<br>
which leads to
<p>&nbsp;<br>
$$
\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}!
$$
<p>&nbsp;<br>
</section>
<section>
<h2 id="friday-september-10">Friday September 10 </h2>
</section>
doc/pub/week36/html/week36-solarized.html
+30 -3
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@@ -144,6 +144,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'maximum-likelihood-estimation-mle'),
('A new Cost Function', 2, None, 'a-new-cost-function'),
('Friday September 10', 2, None, 'friday-september-10'),
('Why resampling methods', 2, None, 'why-resampling-methods'),
('Resampling methods', 2, None, 'resampling-methods'),
@@ -1084,7 +1085,7 @@ $$
p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
$$
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{beta} \).
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{\beta} \).
<p>
Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \( \boldsymbol{y} \) as the product of the single events, that is we have
@@ -1126,10 +1127,36 @@ is equivalent to the maximization/minimization of the function itself.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="friday-september-10">Friday September 10 </h2>
<h2 id="a-new-cost-function">A new Cost Function </h2>
<p>
More text will be added here.
We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
$$
C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})},
$$
which becomes
$$
C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
$$
<p>
Taking the derivative of the <em>new</em> cost function with respect to the parameters \( \beta \) we recognize our familiar OLS equation, namely
$$
\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
$$
which leads to
$$
\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}!
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="friday-september-10">Friday September 10 </h2>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/week36/html/week36.html
+30 -3
View File
@@ -149,6 +149,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'maximum-likelihood-estimation-mle'),
('A new Cost Function', 2, None, 'a-new-cost-function'),
('Friday September 10', 2, None, 'friday-september-10'),
('Why resampling methods', 2, None, 'why-resampling-methods'),
('Resampling methods', 2, None, 'resampling-methods'),
@@ -1089,7 +1090,7 @@ $$
p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
$$
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{beta} \).
which reads as finding the likelihood of an event \( y_i \) given the input variables \( \boldsymbol{X} \) and the parameters (to be determined) \( \boldsymbol{\beta} \).
<p>
Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \( \boldsymbol{y} \) as the product of the single events, that is we have
@@ -1131,10 +1132,36 @@ is equivalent to the maximization/minimization of the function itself.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="friday-september-10">Friday September 10 </h2>
<h2 id="a-new-cost-function">A new Cost Function </h2>
<p>
More text will be added here.
We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
$$
C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i\vert \boldsymbol{X};\boldsymbol{\beta})},
$$
which becomes
$$
C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
$$
<p>
Taking the derivative of the <em>new</em> cost function with respect to the parameters \( \beta \) we recognize our familiar OLS equation, namely
$$
\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
$$
which leads to
$$
\hat{\boldsymbol{\beta}}_{mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right^{-1}\boldsymbol{X}^T\boldsymbol{y}!
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="friday-september-10">Friday September 10 </h2>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/week36/ipynb/ipynb-week36-src.tar.gz
BIN
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Binary file not shown.
doc/pub/week36/ipynb/week36.ipynb
+67 -2
View File
@@ -1408,7 +1408,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"which reads as finding the likelihood of an event $y_i$ given the input variables $\\boldsymbol{X}$ and the parameters (to be determined) $\\boldsymbol{beta}$.\n",
"which reads as finding the likelihood of an event $y_i$ given the input variables $\\boldsymbol{X}$ and the parameters (to be determined) $\\boldsymbol{\\beta}$.\n",
"\n",
"Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\\boldsymbol{y}$ as the product of the single events, that is we have"
]
@@ -1453,10 +1453,75 @@
"\n",
"\n",
"\n",
"## A new Cost Function\n",
"\n",
"We could now define a new cost function to minimize, namely the negative logarithm of the above PDF"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta}=-\\log{\\prod_{i=0}^{n-1}p(y_i\\vert \\boldsymbol{X};\\boldsymbol{\\beta})}=-\\sum_{i=0}^{n-1}\\log{p(y_i\\vert \\boldsymbol{X};\\boldsymbol{\\beta})},\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"which becomes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"C(\\boldsymbol{\\beta}=\\frac{n}{2}\\log{2\\pi\\sigma^2}+\\frac{\\vert\\vert (\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\vert\\vert_2^2}{2\\sigma^2}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Taking the derivative of the *new* cost function with respect to the parameters $\\beta$ we recognize our familiar OLS equation, namely"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right) =0,\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"which leads to"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\hat{\\boldsymbol{\\beta}}_{mathrm{OLS}}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right^{-1}\\boldsymbol{X}^T\\boldsymbol{y}!\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Friday September 10\n",
"\n",
"More text will be added here.\n",
"\n",
"\n",
"## Why resampling methods\n",
"\n",
doc/src/week36/week36.do.txt
+33 -2
View File
@@ -791,7 +791,7 @@ We define this distribution as
p(y_i\vert \bm{X};\bm{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\bm{X}_{i,*}\bm{\beta})^2}{2\sigma^2}\right]},
\]
!et
which reads as finding the likelihood of an event $y_i$ given the input variables $\bm{X}$ and the parameters (to be determined) $\bm{beta}$.
which reads as finding the likelihood of an event $y_i$ given the input variables $\bm{X}$ and the parameters (to be determined) $\bm{\beta}$.
Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\bm{y}$ as the product of the single events, that is we have
@@ -829,11 +829,42 @@ is equivalent to the maximization/minimization of the function itself.
!split
===== A new Cost Function =====
We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
!bt
\[
C(\bm{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i\vert \bm{X};\bm{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i\vert \bm{X};\bm{\beta})},
\]
!et
which becomes
!bt
\[
C(\bm{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\bm{y}-\bm{X}\bm{\beta})\vert\vert_2^2}{2\sigma^2}.
\]
!et
Taking the derivative of the *new* cost function with respect to the parameters $\beta$ we recognize our familiar OLS equation, namely
!bt
\[
\bm{X}^T\left(\bm{y}-\bm{X}\bm{\beta}\right) =0,
\]
!et
which leads to
!bt
\[
\hat{\bm{\beta}}_{mathrm{OLS}}=\left(\bm{X}^T\bm{X}\right^{-1}\bm{X}^T\bm{y}!
\]
!et
!split
===== Friday September 10 =====
More text will be added here.
!split
===== Why resampling methods =====