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<center><h1>Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis</h1></center> <!-- document title -->
<p>
<!-- author(s): Morten Hjorth-Jensen -->
<center>
<b>Morten Hjorth-Jensen</b> [1, 2]
</center>
<p>
<!-- institution(s) -->
<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 11, 2020</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec0">Why Linear Regression (aka Ordinary Least Squares and family) </h2>
<p>
Fitting a continuous function with linear parameterization in terms of the parameters \( \boldsymbol{\beta} \).
<ul>
<li> Method of choice for fitting a continuous function!</li>
<li> Gives an excellent introduction to central Machine Learning features with <b>understandable pedagogical</b> links to other methods like <b>Neural Networks</b>, <b>Support Vector Machines</b> etc</li>
<li> Analytical expression for the fitting parameters \( \boldsymbol{\beta} \)</li>
<li> Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more</li>
<li> Analytical relation with probabilistic interpretations</li>
<li> Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics</li>
<li> Easy to code! And links well with classification problems and logistic regression and neural networks</li>
<li> Allows for <b>easy</b> hands-on understanding of gradient descent methods</li>
<li> and many more features</li>
</ul>
For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec1">Regression analysis, overarching aims </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) and how it varies as function of another variable or a set of such variables \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \).
The first variable is called the <b>dependent</b>, the <b>outcome</b> or the <b>response</b> variable while the set of variables \( \boldsymbol{x} \) is called the independent variable, or the predictor variable or the explanatory variable.
<p>
A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
<ul>
<li> \( n \) cases \( i = 0, 1, 2, \dots, n-1 \)</li>
<li> Response (target, dependent or outcome) variable \( y_i \) with \( i = 0, 1, 2, \dots, n-1 \)</li>
<li> \( p \) so-called explanatory (independent or predictor) variables \( \boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}] \) with \( i = 0, 1, 2, \dots, n-1 \) and explanatory variables running from \( 0 \) to \( p-1 \). See below for more explicit examples.</li>
</ul>
The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec2">Regression analysis, overarching aims II </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Consider an experiment in which \( p \) characteristics of \( n \) samples are
measured. The data from this experiment, for various explanatory variables \( p \) are normally represented by a matrix
\( \mathbf{X} \).
<p>
The matrix \( \mathbf{X} \) is called the <em>design
matrix</em>. Additional information of the samples is available in the
form of \( \boldsymbol{y} \) (also as above). The variable \( \boldsymbol{y} \) is
generally referred to as the <em>response variable</em>. The aim of
regression analysis is to explain \( \boldsymbol{y} \) in terms of
\( \boldsymbol{X} \) through a functional relationship like \( y_i =
f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of
\( f(\cdot) \) is available, it is common to assume a linear relationship
between \( \boldsymbol{X} \) and \( \boldsymbol{y} \). This assumption gives rise to
the <em>linear regression model</em> where \( \boldsymbol{\beta} = [\beta_0, \ldots,
\beta_{p-1}]^{T} \) are the <em>regression parameters</em>.
<p>
Linear regression gives us a set of analytical equations for the parameters \( \beta_j \).
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec3">Examples </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \),
consider the model we discussed for describing nuclear binding energies.
<p>
There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.
Assuming
$$
BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},
$$
we have five predictors, that is the intercept, the \( A \) dependent term, the \( A^{2/3} \) term and the \( A^{-1/3} \) and \( A^{-1} \) terms.
This gives \( p=0,1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is a
\( p\times n \) matrix \( \boldsymbol{X} \).
<p>
Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the
so-called <a href="https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub" target="_blank">credit card default data from Taiwan</a>. The data set contains data on \( n=30000 \) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \( 24 \) such predictors or attributes leading to a design matrix of dimensionality \( 24 \times 30000 \). This is however a classification problem and we will come back to it when we discuss Logistic Regression.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec4">General linear models </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.
<p>
Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \( y \) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \( n-1 \) with \( n \) points, that is
$$
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i,
$$
where \( \epsilon_i \) is the error in our approximation.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec5">Rewriting the fitting procedure as a linear algebra problem </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
For every set of values \( y_i,x_i \) we have thus the corresponding set of equations
$$
\begin{align*}
y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
\dots & \dots \\
y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec6">Rewriting the fitting procedure as a linear algebra problem, more details </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Defining the vectors
$$
\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
$$
and
$$
\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
$$
and
$$
\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
$$
and the design matrix
$$
\boldsymbol{X}=
\begin{bmatrix}
1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\
1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\
1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\
\dots& \dots &\dots& \dots & \dots &\dots\\
1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\
\end{bmatrix}
$$
we can rewrite our equations as
$$
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
$$
The above design matrix is called a <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix" target="_blank">Vandermonde matrix</a>.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec7">Generalizing the fitting procedure as a linear algebra problem </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
We are obviously not limited to the above polynomial expansions. We
could replace the various powers of \( x \) with elements of Fourier
series or instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j
x_i)} \), or time series or other orthogonal functions. For every set
of values \( y_i,x_i \) we can then generalize the equations to
$$
\begin{align*}
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
\dots & \dots \\
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
\dots & \dots \\
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
<p>
<b>Note that we have \( p=n \) here. The matrix is symmetric. This is generally not the case!</b>
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec8">Generalizing the fitting procedure as a linear algebra problem </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
We redefine in turn the matrix \( \boldsymbol{X} \) as
$$
\boldsymbol{X}=
\begin{bmatrix}
x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\
x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\
x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\
\dots& \dots &\dots& \dots & \dots &\dots\\
x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\
\end{bmatrix}
$$
and without loss of generality we rewrite again our equations as
$$
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
$$
The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\epsilon} \) and the parameter vector \( \boldsymbol{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec9">Optimizing our parameters </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
We have defined the matrix \( \boldsymbol{X} \) via the equations
$$
\begin{align*}
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
\dots & \dots \\
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
\dots & \dots \\
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
$$
<p>
As we noted above, we stayed with a system with the design matrix
\( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \), that is we have \( p=n \). For reasons to come later (algorithmic arguments) we will hereafter define
our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors refering to the column numbers and the entries \( n \) being the row elements.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec10">Our model for the nuclear binding energies </h2>
<p>
In our <a href="https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html" target="_blank">introductory notes</a> we looked at the so-called <a href="https://en.wikipedia.org/wiki/Semi-empirical_mass_formula" target="_blank">liquid drop model</a>. Let us remind ourselves about what we did by looking at the code.
<p>
We restate the parts of the code we are most interested in.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;MassEval2016.dat&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the experimental data with Pandas</span>
Masses <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_fwf(infile, usecols<span style="color: #666666">=</span>(<span style="color: #666666">2</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">6</span>,<span style="color: #666666">11</span>),
names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;N&#39;</span>, <span style="color: #BA2121">&#39;Z&#39;</span>, <span style="color: #BA2121">&#39;A&#39;</span>, <span style="color: #BA2121">&#39;Element&#39;</span>, <span style="color: #BA2121">&#39;Ebinding&#39;</span>),
widths<span style="color: #666666">=</span>(<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">1</span>,<span style="color: #666666">13</span>,<span style="color: #666666">11</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">2</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">1</span>,<span style="color: #666666">12</span>,<span style="color: #666666">11</span>,<span style="color: #666666">1</span>),
header<span style="color: #666666">=39</span>,
index_col<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by &#39;#&#39; in place of the decimal place, so</span>
<span style="color: #408080; font-style: italic"># the Ebinding column won&#39;t be numeric. Coerce to float and drop these entries.</span>
Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>dropna()
<span style="color: #408080; font-style: italic"># Convert from keV to MeV.</span>
Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>] <span style="color: #666666">/=</span> <span style="color: #666666">1000</span>
<span style="color: #408080; font-style: italic"># Group the DataFrame by nucleon number, A.</span>
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>groupby(<span style="color: #BA2121">&#39;A&#39;</span>)
<span style="color: #408080; font-style: italic"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>apply(<span style="color: #008000; font-weight: bold">lambda</span> t: t[t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">==</span>t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">.</span>max()])
A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;A&#39;</span>]
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Z&#39;</span>]
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;N&#39;</span>]
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Element&#39;</span>]
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>]
<span style="color: #408080; font-style: italic"># Now we set up the design matrix X</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(A),<span style="color: #666666">5</span>))
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> A
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0/3.0</span>)
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0</span>)
<span style="color: #408080; font-style: italic"># Then nice printout using pandas</span>
DesignMatrix <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
DesignMatrix<span style="color: #666666">.</span>index <span style="color: #666666">=</span> A
DesignMatrix<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">&#39;1&#39;</span>, <span style="color: #BA2121">&#39;A&#39;</span>, <span style="color: #BA2121">&#39;A^(2/3)&#39;</span>, <span style="color: #BA2121">&#39;A^(-1/3)&#39;</span>, <span style="color: #BA2121">&#39;1/A&#39;</span>]
display(DesignMatrix)
</pre></div>
<p>
With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as
$$
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
$$
throughout these lectures.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec11">Optimizing our parameters, more details </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
With the above we use the design matrix to define the approximation \( \boldsymbol{\tilde{y}} \) via the unknown quantity \( \boldsymbol{\beta} \) as
$$
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
$$
and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parameterized values \( \tilde{y}_i \), namely
$$
C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
$$
or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as
$$
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
$$
This function is one possible way to define the so-called cost function.
<p>
It is also common to define
the function \( C \) as
$$
C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2,
$$
since when taking the first derivative with respect to the unknown parameters \( \beta \), the factor of \( 2 \) cancels out.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec12">Interpretations and optimizing our parameters </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
The function
$$
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\},
$$
can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value.
When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value
$$
y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i,
$$
<p>
where \( \langle y_i \rangle \) is the mean value. Keep in mind also that
till now we have treated \( y_i \) as the exact value. Normally, the
response (dependent or outcome) variable \( y_i \) the outcome of a
numerical experiment or another type of experiment and is thus only an
approximation to the true value. It is then always accompanied by an
error estimate, often limited to a statistical error estimate given by
the standard deviation discussed earlier. In the discussion here we
will treat \( y_i \) as our exact value for the response variable.
<p>
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( C(\boldsymbol{\beta}) \), that is we are going to solve the problem
$$
{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
$$
In practical terms it means we will require
$$
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
$$
which results in
$$
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
$$
or in a matrix-vector form as
$$
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right).
$$
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec13">Interpretations and optimizing our parameters </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
We can rewrite
$$
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right),
$$
as
$$
\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta},
$$
and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the solution
$$
\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
$$
<p>
We note also that since our design matrix is defined as \( \boldsymbol{X}\in
{\mathbb{R}}^{n\times p} \), the product \( \boldsymbol{X}^T\boldsymbol{X} \in
{\mathbb{R}}^{p\times p} \). In the above case we have that \( p \ll n \),
in our case \( p=5 \) meaning that we end up with inverting a small
\( 5\times 5 \) matrix. This is a rather common situation, in many cases we end up with low-dimensional
matrices to invert. The methods discussed here and for many other
supervised learning algorithms like classification with logistic
regression or support vector machines, exhibit dimensionalities which
allow for the usage of direct linear algebra methods such as <b>LU</b> decomposition or <b>Singular Value Decomposition</b> (SVD) for finding the inverse of the matrix
\( \boldsymbol{X}^T\boldsymbol{X} \).
</div>
<p>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<b>Small question</b>: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)? What kind of problems can we expect?
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec14">Some useful matrix and vector expressions </h2>
<p>
The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and
matrices as upper case boldfaced letters.
$$
\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b},
$$
$$
\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a},
$$
$$
\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T,
$$
$$
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
$$
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec15">Interpretations and optimizing our parameters </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
The residuals \( \boldsymbol{\epsilon} \) are in turn given by
$$
\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta},
$$
and with
$$
\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
$$
we have
$$
\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
$$
meaning that the solution for \( \boldsymbol{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.
</div>
<p>
Let us now return to our nuclear binding energies and simply code the above equations.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec16">Own code for Ordinary Least Squares </h2>
<p>
It is rather straightforward to implement the matrix inversion and obtain the parameters \( \boldsymbol{\beta} \). After having defined the matrix \( \boldsymbol{X} \) we simply need to
write
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(X))<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(Energies)
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
ytilde <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
</pre></div>
<p>
Alternatively, you can use the least squares functionality in <b>Numpy</b> as
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>lstsq(X, Energies, rcond <span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>)[<span style="color: #666666">0</span>]
ytildenp <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(fit,X<span style="color: #666666">.</span>T)
</pre></div>
<p>
And finally we plot our fit with and compare with data
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>Masses[<span style="color: #BA2121">&#39;Eapprox&#39;</span>] <span style="color: #666666">=</span> ytilde
<span style="color: #408080; font-style: italic"># Generate a plot comparing the experimental with the fitted values values.</span>
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r&#39;$A = N + Z$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r&#39;$E_\mathrm</span><span style="color: #BB6688; font-weight: bold">{bind}</span><span style="color: #BA2121">\,/\mathrm</span><span style="color: #BB6688; font-weight: bold">{MeV}</span><span style="color: #BA2121">$&#39;</span>)
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">&#39;A&#39;</span>], Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Ame2016&#39;</span>)
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">&#39;A&#39;</span>], Masses[<span style="color: #BA2121">&#39;Eapprox&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&#39;m&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Fit&#39;</span>)
ax<span style="color: #666666">.</span>legend()
save_fig(<span style="color: #BA2121">&quot;Masses2016OLS&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec17">Adding error analysis and training set up </h2>
<p>
We can easily test our fit by computing the \( R2 \) score that we discussed in connection with the functionality of <b>Scikit-Learn</b> in the introductory slides.
Since we are not using <b>Scikit-Learn</b> here we can define our own \( R2 \) function as
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
</pre></div>
<p>
and we would be using it as
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000">print</span>(R2(Energies,ytilde))
</pre></div>
<p>
We can easily add our <b>MSE</b> score as
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
<span style="color: #008000">print</span>(MSE(Energies,ytilde))
</pre></div>
<p>
and finally the relative error as
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">RelativeError</span>(y_data,y_model):
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #008000">abs</span>((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">/</span>y_data)
<span style="color: #008000">print</span>(RelativeError(Energies, ytilde))
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec18">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Normally, the response (dependent or outcome) variable \( y_i \) is the
outcome of a numerical experiment or another type of experiment and is
thus only an approximation to the true value. It is then always
accompanied by an error estimate, often limited to a statistical error
estimate given by the standard deviation discussed earlier. In the
discussion here we will treat \( y_i \) as our exact value for the
response variable.
<p>
Introducing the standard deviation \( \sigma_i \) for each measurement
\( y_i \), we define now the \( \chi^2 \) function (omitting the \( 1/n \) term)
as
$$
\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
$$
where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec19">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\boldsymbol{\beta}) \) by requiring
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
$$
which results in
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
$$
or in a matrix-vector form as
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right).
$$
where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \boldsymbol{b} \) with elements \( b_i = y_i/\sigma_i \).
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec20">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
We can rewrite
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right),
$$
as
$$
\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta},
$$
and if the matrix \( \boldsymbol{A}^T\boldsymbol{A} \) is invertible we have the solution
$$
\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}.
$$
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec21">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
If we then introduce the matrix
$$
\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1},
$$
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \))
$$
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
$$
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as (we leave this as an exercise)
$$
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
$$
resulting in
$$
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
$$
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec22">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
$$
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
$$
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
$$
and
$$
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
$$
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec23">The \( \chi^2 \) function </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
For a linear fit (a first-order polynomial) we don't need to invert a matrix!!
Defining
$$
\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2},
$$
$$
\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2},
$$
$$
\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right),
$$
$$
\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2},
$$
$$
\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2},
$$
<p>
we obtain
$$
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
$$
$$
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
$$
<p>
This approach (different linear and non-linear regression) suffers
often from both being underdetermined and overdetermined in the
unknown coefficients \( \beta_i \). A better approach is to use the
Singular Value Decomposition (SVD) method discussed below. Or using
Lasso and Ridge regression. See below.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec24">Fitting an Equation of State for Dense Nuclear Matter </h2>
<p>
Before we continue, let us introduce yet another example. We are going to fit the
nuclear equation of state using results from many-body calculations.
The equation of state we have made available here, as function of
density, has been derived using modern nucleon-nucleon potentials with
<a href="https://www.sciencedirect.com/science/article/pii/S0370157399001106" target="_blank">the addition of three-body
forces</a>. This
time the file is presented as a standard <b>csv</b> file.
<p>
The beginning of the Python code here is similar to what you have seen
before, with the same initializations and declarations. We use also
<b>pandas</b> again, rather extensively in order to organize our data.
<p>
The difference now is that we use <b>Scikit-Learn's</b> regression tools
instead of our own matrix inversion implementation. Furthermore, we
sneak in <b>Ridge</b> regression (to be discussed below) which includes a
hyperparameter \( \lambda \), also to be explained below.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec25">The code </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">4</span>))
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
<span style="color: #408080; font-style: italic"># We use now Scikit-Learn&#39;s linear regressor and ridge regressor</span>
<span style="color: #408080; font-style: italic"># OLS part</span>
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
ytilde <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eols&#39;</span>] <span style="color: #666666">=</span> ytilde
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, ytilde))
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
<span style="color: #408080; font-style: italic"># The Ridge regression with a hyperparameter lambda = 0.1</span>
_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X, Energies)
yridge <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eridge&#39;</span>] <span style="color: #666666">=</span> yridge
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, yridge))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, yridge))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, yridge))
<span style="color: #008000">print</span>(clf_ridge<span style="color: #666666">.</span>coef_, clf_ridge<span style="color: #666666">.</span>intercept_)
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r&#39;$\rho[\mathrm</span><span style="color: #BB6688; font-weight: bold">{fm}</span><span style="color: #BA2121">^{-3}]$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r&#39;Energy per particle&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Theoretical data&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eols&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&#39;m&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;OLS&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eridge&#39;</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">&#39;g&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Ridge $\lambda = 0.1$&#39;</span>)
ax<span style="color: #666666">.</span>legend()
save_fig(<span style="color: #BA2121">&quot;EoSfitting&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
The above simple polynomial in density \( \rho \) gives an excellent fit
to the data.
<p>
We note also that there is a small deviation between the
standard OLS and the Ridge regression at higher densities. We discuss this in more detail
below.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec26">Splitting our Data in Training and Test data </h2>
<p>
It is normal in essentially all Machine Learning studies to split the
data in a training set and a test set (sometimes also an additional
validation set). <b>Scikit-Learn</b> has an own function for this. There
is no explicit recipe for how much data should be included as training
data and say test data. An accepted rule of thumb is to use
approximately \( 2/3 \) to \( 4/5 \) of the data as training data. We will
postpone a discussion of this splitting to the end of these notes and
our discussion of the so-called <b>bias-variance</b> tradeoff. Here we
limit ourselves to repeat the above equation of state fitting example
but now splitting the data into a training set and a test set.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organized into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">5</span>))
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">5.0/3.0</span>)
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, Energies, test_size<span style="color: #666666">=0.2</span>)
<span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X_train<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(X_train))<span style="color: #666666">.</span>dot(X_train<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y_train)
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
ytilde <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> beta
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Training R2&quot;</span>)
<span style="color: #008000">print</span>(R2(y_train,ytilde))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Training MSE&quot;</span>)
<span style="color: #008000">print</span>(MSE(y_train,ytilde))
ypredict <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> beta
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test R2&quot;</span>)
<span style="color: #008000">print</span>(R2(y_test,ypredict))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test MSE&quot;</span>)
<span style="color: #008000">print</span>(MSE(y_test,ypredict))
</pre></div>
<p>
<!-- !split -->
<h2 id="___sec27">The Boston housing data example </h2>
<p>
The Boston housing
data set was originally a part of UCI Machine Learning Repository
and has been removed now. The data set is now included in <b>Scikit-Learn</b>'s
library. There are 506 samples and 13 feature (predictor) variables
in this data set. The objective is to predict the value of prices of
the house using the features (predictors) listed here.
<p>
The features/predictors are
<ol>
<li> CRIM: Per capita crime rate by town</li>
<li> ZN: Proportion of residential land zoned for lots over 25000 square feet</li>
<li> INDUS: Proportion of non-retail business acres per town</li>
<li> CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)</li>
<li> NOX: Nitric oxide concentration (parts per 10 million)</li>
<li> RM: Average number of rooms per dwelling</li>
<li> AGE: Proportion of owner-occupied units built prior to 1940</li>
<li> DIS: Weighted distances to five Boston employment centers</li>
<li> RAD: Index of accessibility to radial highways</li>
<li> TAX: Full-value property tax rate per USD10000</li>
<li> B: \( 1000(Bk - 0.63)^2 \), where \( Bk \) is the proportion of [people of African American descent] by town</li>
<li> LSTAT: Percentage of lower status of the population</li>
<li> MEDV: Median value of owner-occupied homes in USD 1000s</li>
</ol>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec28">Housing data, the code </h2>
We start by importing the libraries
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
</pre></div>
<p>
and load the Boston Housing DataSet from <b>Scikit-Learn</b>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_boston
boston_dataset <span style="color: #666666">=</span> load_boston()
<span style="color: #408080; font-style: italic"># boston_dataset is a dictionary</span>
<span style="color: #408080; font-style: italic"># let&#39;s check what it contains</span>
boston_dataset<span style="color: #666666">.</span>keys()
</pre></div>
<p>
Then we invoke Pandas
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>boston <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(boston_dataset<span style="color: #666666">.</span>data, columns<span style="color: #666666">=</span>boston_dataset<span style="color: #666666">.</span>feature_names)
boston<span style="color: #666666">.</span>head()
boston[<span style="color: #BA2121">&#39;MEDV&#39;</span>] <span style="color: #666666">=</span> boston_dataset<span style="color: #666666">.</span>target
</pre></div>
<p>
and preprocess the data
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># check for missing values in all the columns</span>
boston<span style="color: #666666">.</span>isnull()<span style="color: #666666">.</span>sum()
</pre></div>
<p>
We can then visualize the data
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># set the size of the figure</span>
sns<span style="color: #666666">.</span>set(rc<span style="color: #666666">=</span>{<span style="color: #BA2121">&#39;figure.figsize&#39;</span>:(<span style="color: #666666">11.7</span>,<span style="color: #666666">8.27</span>)})
<span style="color: #408080; font-style: italic"># plot a histogram showing the distribution of the target values</span>
sns<span style="color: #666666">.</span>distplot(boston[<span style="color: #BA2121">&#39;MEDV&#39;</span>], bins<span style="color: #666666">=30</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
It is now useful to look at the correlation matrix
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># compute the pair wise correlation for all columns </span>
correlation_matrix <span style="color: #666666">=</span> boston<span style="color: #666666">.</span>corr()<span style="color: #666666">.</span>round(<span style="color: #666666">2</span>)
<span style="color: #408080; font-style: italic"># use the heatmap function from seaborn to plot the correlation matrix</span>
<span style="color: #408080; font-style: italic"># annot = True to print the values inside the square</span>
sns<span style="color: #666666">.</span>heatmap(data<span style="color: #666666">=</span>correlation_matrix, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
</pre></div>
<p>
From the above coorelation plot we can see that <b>MEDV</b> is strongly correlated to <b>LSTAT</b> and <b>RM</b>. We see also that <b>RAD</b> and <b>TAX</b> are stronly correlated, but we don't include this in our features together to avoid multi-colinearity
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">5</span>))
features <span style="color: #666666">=</span> [<span style="color: #BA2121">&#39;LSTAT&#39;</span>, <span style="color: #BA2121">&#39;RM&#39;</span>]
target <span style="color: #666666">=</span> boston[<span style="color: #BA2121">&#39;MEDV&#39;</span>]
<span style="color: #008000; font-weight: bold">for</span> i, col <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(features):
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(features) , i<span style="color: #666666">+1</span>)
x <span style="color: #666666">=</span> boston[col]
y <span style="color: #666666">=</span> target
plt<span style="color: #666666">.</span>scatter(x, y, marker<span style="color: #666666">=</span><span style="color: #BA2121">&#39;o&#39;</span>)
plt<span style="color: #666666">.</span>title(col)
plt<span style="color: #666666">.</span>xlabel(col)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;MEDV&#39;</span>)
</pre></div>
<p>
Now we start training our model
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(np<span style="color: #666666">.</span>c_[boston[<span style="color: #BA2121">&#39;LSTAT&#39;</span>], boston[<span style="color: #BA2121">&#39;RM&#39;</span>]], columns <span style="color: #666666">=</span> [<span style="color: #BA2121">&#39;LSTAT&#39;</span>,<span style="color: #BA2121">&#39;RM&#39;</span>])
Y <span style="color: #666666">=</span> boston[<span style="color: #BA2121">&#39;MEDV&#39;</span>]
</pre></div>
<p>
We split the data into training and test sets
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #408080; font-style: italic"># splits the training and test data set in 80% : 20%</span>
<span style="color: #408080; font-style: italic"># assign random_state to any value.This ensures consistency.</span>
X_train, X_test, Y_train, Y_test <span style="color: #666666">=</span> train_test_split(X, Y, test_size <span style="color: #666666">=</span> <span style="color: #666666">0.2</span>, random_state<span style="color: #666666">=5</span>)
<span style="color: #008000">print</span>(X_train<span style="color: #666666">.</span>shape)
<span style="color: #008000">print</span>(X_test<span style="color: #666666">.</span>shape)
<span style="color: #008000">print</span>(Y_train<span style="color: #666666">.</span>shape)
<span style="color: #008000">print</span>(Y_test<span style="color: #666666">.</span>shape)
</pre></div>
<p>
Then we use the linear regression functionality from <b>Scikit-Learn</b>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score
lin_model <span style="color: #666666">=</span> LinearRegression()
lin_model<span style="color: #666666">.</span>fit(X_train, Y_train)
<span style="color: #408080; font-style: italic"># model evaluation for training set</span>
y_train_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_train)
rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_train, y_train_predict)))
r2 <span style="color: #666666">=</span> r2_score(Y_train, y_train_predict)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The model performance for training set&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;--------------------------------------&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;RMSE is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(rmse))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;R2 score is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(r2))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># model evaluation for testing set</span>
y_test_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_test)
<span style="color: #408080; font-style: italic"># root mean square error of the model</span>
rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_test, y_test_predict)))
<span style="color: #408080; font-style: italic"># r-squared score of the model</span>
r2 <span style="color: #666666">=</span> r2_score(Y_test, y_test_predict)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The model performance for testing set&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;--------------------------------------&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;RMSE is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(rmse))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;R2 score is </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(r2))
</pre></div>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># plotting the y_test vs y_pred</span>
<span style="color: #408080; font-style: italic"># ideally should have been a straight line</span>
plt<span style="color: #666666">.</span>scatter(Y_test, y_test_predict)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec29">Reducing the number of degrees of freedom, overarching view </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Many Machine Learning problems involve thousands or even millions of
features for each training instance. Not only does this make training
extremely slow, it can also make it much harder to find a good
solution, as we will see. This problem is often referred to as the
curse of dimensionality. Fortunately, in real-world problems, it is
often possible to reduce the number of features considerably, turning
an intractable problem into a tractable one.
<p>
Later we will discuss some of the most popular dimensionality reduction
techniques: the principal component analysis (PCA), Kernel PCA, and
Locally Linear Embedding (LLE).
<p>
Principal component analysis and its various variants deal with the
problem of fitting a low-dimensional <a href="https://en.wikipedia.org/wiki/Affine_space" target="_blank">affine
subspace</a> to a set of of
data points in a high-dimensional space. With its family of methods it
is one of the most used tools in data modeling, compression and
visualization.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec30">Preprocessing our data </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Before we proceed however, we will discuss how to preprocess our
data. Till now and in connection with our previous examples we have
not met so many cases where we are too sensitive to the scaling of our
data. Normally the data may need a rescaling and/or may be sensitive
to extreme values. Scaling the data renders our inputs much more
suitable for the algorithms we want to employ.
<p>
<b>Scikit-Learn</b> has several functions which allow us to rescale the
data, normally resulting in much better results in terms of various
accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b>
ensures that for each feature/predictor we study the mean value is
zero and the variance is one (every column in the design/feature
matrix). This scaling has the drawback that it does not ensure that
we have a particular maximum or minimum in our data set. Another
function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which
ensures that all features are exactly between \( 0 \) and \( 1 \). The
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec31">More preprocessing </h2>
<p>
The <b>Normalizer</b> scales each data
point such that the feature vector has a euclidean length of one. In other words, it
projects a data point on the circle (or sphere in the case of higher dimensions) with a
radius of 1. This means every data point is scaled by a different number (by the
inverse of it&#8217;s length).
This normalization is often used when only the direction (or angle) of the data matters,
not the length of the feature vector.
<p>
The <b>RobustScaler</b> works similarly to the StandardScaler in that it
ensures statistical properties for each feature that guarantee that
they are on the same scale. However, the RobustScaler uses the median
and quartiles, instead of mean and variance. This makes the
RobustScaler ignore data points that are very different from the rest
(like measurement errors). These odd data points are also called
outliers, and might often lead to trouble for other scaling
techniques.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec32">Simple preprocessing examples, Franke function and regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> MinMaxScaler, StandardScaler, Normalizer
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">FrankeFunction</span>(x,y):
term1 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">0.25*</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>))
term2 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>((<span style="color: #666666">9*</span>x<span style="color: #666666">+1</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">/49.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.1*</span>(<span style="color: #666666">9*</span>y<span style="color: #666666">+1</span>))
term3 <span style="color: #666666">=</span> <span style="color: #666666">0.5*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-7</span>)<span style="color: #666666">**2/4.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-3</span>)<span style="color: #666666">**2</span>))
term4 <span style="color: #666666">=</span> <span style="color: #666666">-0.2*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-4</span>)<span style="color: #666666">**2</span> <span style="color: #666666">-</span> (<span style="color: #666666">9*</span>y<span style="color: #666666">-7</span>)<span style="color: #666666">**2</span>)
<span style="color: #008000; font-weight: bold">return</span> term1 <span style="color: #666666">+</span> term2 <span style="color: #666666">+</span> term3 <span style="color: #666666">+</span> term4
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_X</span>(x, y, n ):
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #008000">len</span>(x<span style="color: #666666">.</span>shape) <span style="color: #666666">&gt;</span> <span style="color: #666666">1</span>:
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(x)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(y)
N <span style="color: #666666">=</span> <span style="color: #008000">len</span>(x)
l <span style="color: #666666">=</span> <span style="color: #008000">int</span>((n<span style="color: #666666">+1</span>)<span style="color: #666666">*</span>(n<span style="color: #666666">+2</span>)<span style="color: #666666">/2</span>) <span style="color: #408080; font-style: italic"># Number of elements in beta</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones((N,l))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n<span style="color: #666666">+1</span>):
q <span style="color: #666666">=</span> <span style="color: #008000">int</span>((i)<span style="color: #666666">*</span>(i<span style="color: #666666">+1</span>)<span style="color: #666666">/2</span>)
<span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(i<span style="color: #666666">+1</span>):
X[:,q<span style="color: #666666">+</span>k] <span style="color: #666666">=</span> (x<span style="color: #666666">**</span>(i<span style="color: #666666">-</span>k))<span style="color: #666666">*</span>(y<span style="color: #666666">**</span>k)
<span style="color: #008000; font-weight: bold">return</span> X
<span style="color: #408080; font-style: italic"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
n <span style="color: #666666">=</span> <span style="color: #666666">5</span>
N <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
z <span style="color: #666666">=</span> FrankeFunction(x, y)
X <span style="color: #666666">=</span> create_X(x, y, n<span style="color: #666666">=</span>n)
<span style="color: #408080; font-style: italic"># split in training and test data</span>
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X,z,test_size<span style="color: #666666">=0.2</span>)
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #408080; font-style: italic"># The mean squared error and R2 score</span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;MSE before scaling: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(mean_squared_error(clf<span style="color: #666666">.</span>predict(X_test), y_test)))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;R2 score before scaling </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(clf<span style="color: #666666">.</span>score(X_test,y_test)))
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Feature min values after scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Feature max values after scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;MSE after scaling: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(mean_squared_error(clf<span style="color: #666666">.</span>predict(X_test_scaled), y_test)))
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;R2 score for scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec33">The singular value decomposition </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
The examples we have looked at so far are cases where we normally can
invert the matrix \( \boldsymbol{X}^T\boldsymbol{X} \). Using a polynomial expansion as we
did both for the masses and the fitting of the equation of state,
leads to row vectors of the design matrix which are essentially
orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.
<p>
This may
however not the be case in general and a standard matrix inversion
algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.
<p>
There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions.
<p>
This is given by the <b>Singular Value Decomposition</b> algorithm, perhaps
the most powerful linear algebra algorithm. Let us look at a
different example where we may have problems with the standard matrix
inversion algorithm. Thereafter we dive into the math of the SVD.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec34">Linear Regression Problems </h2>
<p>
One of the typical problems we encounter with linear regression, in particular
when the matrix \( \boldsymbol{X} \) (our so-called design matrix) is high-dimensional,
are problems with near singular or singular matrices. The column vectors of \( \boldsymbol{X} \)
may be linearly dependent, normally referred to as super-collinearity.
This means that the matrix may be rank deficient and it is basically impossible to
to model the data using linear regression. As an example, consider the matrix
$$
\begin{align*}
\mathbf{X} & = \left[
\begin{array}{rrr}
1 & -1 & 2
\\
1 & 0 & 1
\\
1 & 2 & -1
\\
1 & 1 & 0
\end{array} \right]
\end{align*}
$$
<p>
The columns of \( \boldsymbol{X} \) are linearly dependent. We see this easily since the
the first column is the row-wise sum of the other two columns. The rank (more correct,
the column rank) of a matrix is the dimension of the space spanned by the
column vectors. Hence, the rank of \( \mathbf{X} \) is equal to the number
of linearly independent columns. In this particular case the matrix has rank 2.
<p>
Super-collinearity of an \( (n \times p) \)-dimensional design matrix \( \mathbf{X} \) implies
that the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this
$$
\begin{align*}
\boldsymbol{X} & = \left[
\begin{array}{rr}
1 & -1
\\
1 & -1
\end{array} \right].
\end{align*}
$$
We see easily that \( \mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0 \). Hence, \( \mathbf{X} \) is singular and its inverse is undefined.
This is equivalent to saying that the matrix \( \boldsymbol{X} \) has at least an eigenvalue which is zero.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec35">Fixing the singularity </h2>
<p>
If our design matrix \( \boldsymbol{X} \) which enters the linear regression problem
$$
\begin{align}
\boldsymbol{\beta} & = (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y},
\label{_auto1}
\end{align}
$$
has linearly dependent column vectors, we will not be able to compute the inverse
of \( \boldsymbol{X}^T\boldsymbol{X} \) and we cannot find the parameters (estimators) \( \beta_i \).
The estimators are only well-defined if \( (\boldsymbol{X}^{T}\boldsymbol{X})^{-1} \) exits.
This is more likely to happen when the matrix \( \boldsymbol{X} \) is high-dimensional. In this case it is likely to encounter a situation where
the regression parameters \( \beta_i \) cannot be estimated.
<p>
A cheap <em>ad hoc</em> approach is simply to add a small diagonal component to the matrix to invert, that is we change
$$
\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I},
$$
where \( \boldsymbol{I} \) is the identity matrix. When we discuss <b>Ridge</b> regression this is actually what we end up evaluating. The parameter \( \lambda \) is called a hyperparameter. More about this later.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec36">Basic math of the SVD </h2>
<p>
From standard linear algebra we know that a square matrix \( \boldsymbol{X} \) can be diagonalized if and only it is
a so-called <a href="https://en.wikipedia.org/wiki/Normal_matrix" target="_blank">normal matrix</a>, that is if \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \)
we have \( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) or if \( \boldsymbol{X}\in {\mathbb{C}}^{n\times n} \) we have \( \boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X} \).
The matrix has then a set of eigenpairs
$$
(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n),
$$
and the eigenvalues are given by the diagonal matrix
$$
\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n).
$$
The matrix \( \boldsymbol{X} \) can be written in terms of an orthogonal/unitary transformation \( \boldsymbol{U} \)
$$
\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
$$
with \( \boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) or \( \boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I} \).
<p>
Not all square matrices are diagonalizable. A matrix like the one discussed above
$$
\boldsymbol{X} = \begin{bmatrix}
1& -1 \\
1& -1\\
\end{bmatrix}
$$
is not diagonalizable, it is a so-called <a href="https://en.wikipedia.org/wiki/Defective_matrix" target="_blank">defective matrix</a>. It is easy to see that the condition
\( \boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X} \) is not fulfilled.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec37">The SVD, a Fantastic Algorithm </h2>
<p>
However, and this is the strength of the SVD algorithm, any general
matrix \( \boldsymbol{X} \) can be decomposed in terms of a diagonal matrix and
two orthogonal/unitary matrices. The <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition" target="_blank">Singular Value Decompostion
(SVD) theorem</a>
states that a general \( m\times n \) matrix \( \boldsymbol{X} \) can be written in
terms of a diagonal matrix \( \boldsymbol{\Sigma} \) of dimensionality \( m\times n \)
and two orthognal matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \), where the first has
dimensionality \( m \times m \) and the last dimensionality \( n\times n \).
We have then
$$
\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T
$$
<p>
As an example, the above defective matrix can be decomposed as
$$
\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
$$
<p>
with eigenvalues \( \sigma_1=2 \) and \( \sigma_2=0 \).
The SVD exits always!
<p>
The SVD
decomposition (singular values) gives eigenvalues
\( \sigma_i\geq\sigma_{i+1} \) for all \( i \) and for dimensions larger than \( i=p \), the
eigenvalues (singular values) are zero.
<p>
In the general case, where our design matrix \( \boldsymbol{X} \) has dimension
\( n\times p \), the matrix is thus decomposed into an \( n\times n \)
orthogonal matrix \( \boldsymbol{U} \), a \( p\times p \) orthogonal matrix \( \boldsymbol{V} \)
and a diagonal matrix \( \boldsymbol{\Sigma} \) with \( r=\mathrm{min}(n,p) \)
singular values \( \sigma_i\geq 0 \) on the main diagonal and zeros filling
the rest of the matrix. There are at most \( p \) singular values
assuming that \( n > p \). In our regression examples for the nuclear
masses and the equation of state this is indeed the case, while for
the Ising model we have \( p > n \). These are often cases that lead to
near singular or singular matrices.
<p>
The columns of \( \boldsymbol{U} \) are called the left singular vectors while the columns of \( \boldsymbol{V} \) are the right singular vectors.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec38">Economy-size SVD </h2>
<p>
If we assume that \( n > p \), then our matrix \( \boldsymbol{U} \) has dimension \( n
\times n \). The last \( n-p \) columns of \( \boldsymbol{U} \) become however
irrelevant in our calculations since they are multiplied with the
zeros in \( \boldsymbol{\Sigma} \).
<p>
The economy-size decomposition removes extra rows or columns of zeros
from the diagonal matrix of singular values, \( \boldsymbol{\Sigma} \), along with the columns
in either \( \boldsymbol{U} \) or \( \boldsymbol{V} \) that multiply those zeros in the expression.
Removing these zeros and columns can improve execution time
and reduce storage requirements without compromising the accuracy of
the decomposition.
<p>
If \( n > p \), we keep only the first \( p \) columns of \( \boldsymbol{U} \) and \( \boldsymbol{\Sigma} \) has dimension \( p\times p \).
If \( p > n \), then only the first \( n \) columns of \( \boldsymbol{V} \) are computed and \( \boldsymbol{\Sigma} \) has dimension \( n\times n \).
The \( n=p \) case is obvious, we retain the full SVD.
In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec39">Codes for the SVD </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #408080; font-style: italic"># SVD inversion</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">SVDinv</span>(A):
<span style="color: #BA2121; font-style: italic">&#39;&#39;&#39; Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).</span>
<span style="color: #BA2121; font-style: italic"> SVD is numerically more stable than the inversion algorithms provided by</span>
<span style="color: #BA2121; font-style: italic"> numpy and scipy.linalg at the cost of being slower.</span>
<span style="color: #BA2121; font-style: italic"> &#39;&#39;&#39;</span>
U, s, VT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(A)
<span style="color: #408080; font-style: italic"># print(&#39;test U&#39;)</span>
<span style="color: #408080; font-style: italic"># print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
<span style="color: #408080; font-style: italic"># print(&#39;test VT&#39;)</span>
<span style="color: #408080; font-style: italic"># print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
<span style="color: #008000">print</span>(U)
<span style="color: #008000">print</span>(s)
<span style="color: #008000">print</span>(VT)
D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(U),<span style="color: #008000">len</span>(VT)))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>,<span style="color: #008000">len</span>(VT)):
D[i,i]<span style="color: #666666">=</span>s[i]
UT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(U); V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(VT); invD <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(D)
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>matmul(V,np<span style="color: #666666">.</span>matmul(invD,UT))
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1.0</span>, <span style="color: #666666">-1.0</span>, <span style="color: #666666">2.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">0.0</span>, <span style="color: #666666">1.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">2.0</span>, <span style="color: #666666">-1.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">1.0</span>, <span style="color: #666666">0.0</span>] ])
<span style="color: #008000">print</span>(X)
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(X) <span style="color: #666666">@</span> X
<span style="color: #008000">print</span>(A)
<span style="color: #408080; font-style: italic"># Brute force inversion of super-collinear matrix</span>
<span style="color: #408080; font-style: italic">#B = np.linalg.inv(A)</span>
<span style="color: #408080; font-style: italic">#print(B)</span>
C <span style="color: #666666">=</span> SVDinv(A)
<span style="color: #008000">print</span>(C)
</pre></div>
<p>
The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first
column is the row-wise sum of the other two columns. The rank of a
matrix (the column rank) is the dimension of space spanned by the
column vectors. The rank of the matrix is the number of linearly
independent columns, in this case just \( 2 \). We see this from the
singular values when running the above code. Running the standard
inversion algorithm for matrix inversion with \( \boldsymbol{X}^T\boldsymbol{X} \) results
in the program terminating due to a singular matrix.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec40">Mathematical Properties </h2>
<p>
There are several interesting mathematical properties which will be
relevant when we are going to discuss the differences between say
ordinary least squares (OLS) and <b>Ridge</b> regression.
<p>
We have from OLS that the parameters of the linear approximation are given by
$$
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
$$
<p>
The matrix to invert can be rewritten in terms of our SVD decomposition as
$$
\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T.
$$
Using the orthogonality properties of \( \boldsymbol{U} \) we have
$$
\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T = \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T,
$$
with \( \boldsymbol{D} \) being a diagonal matrix with values along the diagonal given by the singular values squared.
<p>
This means that
$$
(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D},
$$
that is the eigenvectors of \( (\boldsymbol{X}^T\boldsymbol{X}) \) are given by the columns of the right singular matrix of \( \boldsymbol{X} \) and the eigenvalues are the squared singular values. It is easy to show (show this) that
$$
(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D},
$$
that is, the eigenvectors of \( (\boldsymbol{X}\boldsymbol{X})^T \) are the columns of the left singular matrix and the eigenvalues are the same.
<p>
Going back to our OLS equation we have
$$
\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}.
$$
We will come back to this expression when we discuss Ridge regression.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec41">Ridge and LASSO Regression </h2>
<p>
Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
our optimization problem is
$$
{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
$$
or we can state it as
$$
{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
$$
where we have used the definition of a norm-2 vector, that is
$$
\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
$$
<p>
By minimizing the above equation with respect to the parameters
\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
parameters \( \boldsymbol{\beta} \). We can add a regularization parameter \( \lambda \) by
defining a new cost function to be optimized, that is
$$
{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
$$
<p>
which leads to the Ridge regression minimization problem where we
require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
a finite number larger than zero. By defining
$$
C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
$$
<p>
we have a new optimization equation
$$
{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
$$
which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
<p>
Here we have defined the norm-1 as
$$
\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec42">More on Ridge Regression </h2>
<p>
Using the matrix-vector expression for Ridge regression,
$$
C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta},
$$
<p>
by taking the derivatives with respect to \( \boldsymbol{\beta} \) we obtain then
a slightly modified matrix inversion problem which for finite values
of \( \lambda \) does not suffer from singularity problems. We obtain
$$
\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y},
$$
<p>
with \( \boldsymbol{I} \) being a \( p\times p \) identity matrix with the constraint that
$$
\sum_{i=0}^{p-1} \beta_i^2 \leq t,
$$
<p>
with \( t \) a finite positive number.
<p>
We see that Ridge regression is nothing but the standard
OLS with a modified diagonal term added to \( \boldsymbol{X}^T\boldsymbol{X} \). The
consequences, in particular for our discussion of the bias-variance tradeoff
are rather interesting.
<p>
Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had
$$
(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}.
$$
<p>
We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \( \boldsymbol{U} \) as
$$
\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}
$$
<p>
For Ridge regression this becomes
$$
\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y},
$$
<p>
with the vectors \( \boldsymbol{u}_j \) being the columns of \( \boldsymbol{U} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec43">Interpreting the Ridge results </h2>
<p>
Since \( \lambda \geq 0 \), it means that compared to OLS, we have
$$
\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1.
$$
<p>
Ridge regression finds the coordinates of \( \boldsymbol{y} \) with respect to the
orthonormal basis \( \boldsymbol{U} \), it then shrinks the coordinates by
\( \frac{\sigma_j^2}{\sigma_j^2+\lambda} \). Recall that the SVD has
eigenvalues ordered in a descending way, that is \( \sigma_i \geq
\sigma_{i+1} \).
<p>
For small eigenvalues \( \sigma_i \) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.
Actually, calculating the variance of \( \boldsymbol{X}\boldsymbol{v}_j \) shows that this quantity is equal to \( \sigma_j^2/n \).
With a parameter \( \lambda \) we can thus shrink the role of specific parameters.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec44">More interpretations </h2>
<p>
For the sake of simplicity, let us assume that the design matrix is orthonormal, that is
$$
\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}.
$$
<p>
In this case the standard OLS results in
$$
\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y},
$$
<p>
and
$$
\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}},
$$
<p>
that is the Ridge estimator scales the OLS estimator by the inverse of a factor \( 1+\lambda \), and
the Ridge estimator converges to zero when the hyperparameter goes to
infinity.
<p>
We will come back to more interpreations after we have gone through some of the statistical analysis part.
<p>
For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
<p>
<!-- !split -->
<h2 id="___sec45">A better understanding of regularization </h2>
<p>
The parameter \( \lambda \) that we have introduced in the Ridge (and
Lasso as well) regression is often called a regularization parameter
or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?
<p>
Here we will first look at how to analyze the difference between the
standard OLS equations and the Ridge expressions in terms of a linear
algebra analysis using the SVD algorithm. Thereafter, we will link
(see the material on the bias-variance tradeoff below) these
observation to the statisical analysis of the results. In particular
we consider how the variance of the parameters \( \boldsymbol{\beta} \) is
affected by changing the parameter \( \lambda \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec46">Decomposing the OLS and Ridge expressions </h2>
<p>
We have our design matrix
\( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). With the SVD we decompose it as
$$
\boldsymbol{X} = \boldsymbol{U\Sigma V^T},
$$
<p>
with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in {\mathbb{R}}^{n\times p} \)
and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
<p>
The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec47">Introducing the Covariance and Correlation functions </h2>
<p>
Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about
the definition of the covariance and the correlation function. These are quantities
<p>
Suppose we have defined two vectors
\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
\end{bmatrix},
$$
where for example
$$
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
$$
With this definition and recalling that the variance is defined as
$$
\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
$$
we can rewrite the covariance matrix as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
\end{bmatrix}.
$$
<p>
The covariance takes values between zero and infinity and may thus
lead to problems with loss of numerical precision for particularly
large values. It is common to scale the covariance matrix by
introducing instead the correlation matrix defined via the so-called
correlation function
$$
\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
$$
<p>
The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
\in [-1,1] \). This avoids eventual problems with too large values. We
can then define the correlation matrix for the two vectors \( \boldsymbol{x} \)
and \( \boldsymbol{y} \) as
$$
\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
\end{bmatrix},
$$
<p>
In the above example this is the function we constructed using <b>pandas</b>.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec48">Correlation Function and Design/Feature Matrix </h2>
<p>
In our derivation of the various regression algorithms like <b>Ordinary Least Squares</b> or <b>Ridge regression</b>
we defined the design/feature matrix \( \boldsymbol{X} \) as
$$
\boldsymbol{X}=\begin{bmatrix}
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
\dots & \dots & \dots & \dots \dots & \dots \\
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
\end{bmatrix},
$$
with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the
entries \( n \) being the row elements.
We can rewrite the design/feature matrix in terms of its column vectors as
$$
\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix},
$$
with a given vector
$$
\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
$$
<p>
With these definitions, we can now rewrite our \( 2\times 2 \)
correaltion/covariance matrix in terms of a moe general design/feature
matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \)
covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \)
$$
\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots & \dots & \dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots & \dots & \dots \\
\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\
\end{bmatrix},
$$
and the correlation matrix
$$
\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots & \dots & \dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots & \dots & \dots \\
\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\
\end{bmatrix},
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec49">Covariance Matrix Examples </h2>
<p>
The Numpy function <b>np.cov</b> calculates the covariance elements using
the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
the exact mean values. The following simple function uses the
<b>np.vstack</b> function which takes each vector of dimension \( 1\times n \)
and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
$$
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\
x_1 & y_1 \\
x_2 & y_2\\
\dots & \dots \\
x_{n-2} & y_{n-2}\\
x_{n-1} & y_{n-1} &
\end{bmatrix},
$$
<p>
which in turn is converted into into the \( 2\times 2 \) covariance matrix
\( \boldsymbol{C} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
covariance matrix through the <b>np.linalg.eig()</b> function.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(x))
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(y))
W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y))
C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
<span style="color: #008000">print</span>(C)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec50">Correlation Matrix </h2>
<p>
The previous example can be converted into the correlation matrix by
simply scaling the matrix elements with the variances. We should also
subtract the mean values for each column. This leads to the following
code which sets up the correlations matrix for the previous example in
a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
<span style="color: #408080; font-style: italic"># define two vectors </span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>random(size<span style="color: #666666">=</span>n)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
<span style="color: #408080; font-style: italic">#scaling the x and y vectors </span>
x <span style="color: #666666">=</span> x <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(x)
y <span style="color: #666666">=</span> y <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y)
variance_x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>n
variance_y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(y<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
<span style="color: #008000">print</span>(variance_x)
<span style="color: #008000">print</span>(variance_y)
cov_xy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
cov_xx <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>n
cov_yy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(y<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>n
C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">2</span>,<span style="color: #666666">2</span>))
C[<span style="color: #666666">0</span>,<span style="color: #666666">0</span>]<span style="color: #666666">=</span> cov_xx<span style="color: #666666">/</span>variance_x
C[<span style="color: #666666">1</span>,<span style="color: #666666">1</span>]<span style="color: #666666">=</span> cov_yy<span style="color: #666666">/</span>variance_y
C[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>]<span style="color: #666666">=</span> cov_xy<span style="color: #666666">/</span>np<span style="color: #666666">.</span>sqrt(variance_y<span style="color: #666666">*</span>variance_x)
C[<span style="color: #666666">1</span>,<span style="color: #666666">0</span>]<span style="color: #666666">=</span> C[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>]
<span style="color: #008000">print</span>(C)
</pre></div>
<p>
We see that the matrix elements along the diagonal are one as they
should be and that the matrix is symmetric. Furthermore, diagonalizing
this matrix we easily see that it is a positive definite matrix.
<p>
The above procedure with <b>numpy</b> can be made more compact if we use <b>pandas</b>.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec51">Correlation Matrix with Pandas </h2>
<p>
We whow here how we can set up the correlation matrix using <b>pandas</b>, as done in this simple code
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
x <span style="color: #666666">=</span> x <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(x)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
y <span style="color: #666666">=</span> y <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y)
X <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>vstack((x, y)))<span style="color: #666666">.</span>T
<span style="color: #008000">print</span>(X)
Xpd <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
<span style="color: #008000">print</span>(Xpd)
correlation_matrix <span style="color: #666666">=</span> Xpd<span style="color: #666666">.</span>corr()
<span style="color: #008000">print</span>(correlation_matrix)
</pre></div>
<p>
We expand this model to the Franke function discussed above.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec52">Correlation Matrix with Pandas and the Franke function </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">FrankeFunction</span>(x,y):
term1 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">0.25*</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>))
term2 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>((<span style="color: #666666">9*</span>x<span style="color: #666666">+1</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">/49.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.1*</span>(<span style="color: #666666">9*</span>y<span style="color: #666666">+1</span>))
term3 <span style="color: #666666">=</span> <span style="color: #666666">0.5*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-7</span>)<span style="color: #666666">**2/4.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-3</span>)<span style="color: #666666">**2</span>))
term4 <span style="color: #666666">=</span> <span style="color: #666666">-0.2*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-4</span>)<span style="color: #666666">**2</span> <span style="color: #666666">-</span> (<span style="color: #666666">9*</span>y<span style="color: #666666">-7</span>)<span style="color: #666666">**2</span>)
<span style="color: #008000; font-weight: bold">return</span> term1 <span style="color: #666666">+</span> term2 <span style="color: #666666">+</span> term3 <span style="color: #666666">+</span> term4
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_X</span>(x, y, n ):
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #008000">len</span>(x<span style="color: #666666">.</span>shape) <span style="color: #666666">&gt;</span> <span style="color: #666666">1</span>:
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(x)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(y)
N <span style="color: #666666">=</span> <span style="color: #008000">len</span>(x)
l <span style="color: #666666">=</span> <span style="color: #008000">int</span>((n<span style="color: #666666">+1</span>)<span style="color: #666666">*</span>(n<span style="color: #666666">+2</span>)<span style="color: #666666">/2</span>) <span style="color: #408080; font-style: italic"># Number of elements in beta</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones((N,l))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n<span style="color: #666666">+1</span>):
q <span style="color: #666666">=</span> <span style="color: #008000">int</span>((i)<span style="color: #666666">*</span>(i<span style="color: #666666">+1</span>)<span style="color: #666666">/2</span>)
<span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(i<span style="color: #666666">+1</span>):
X[:,q<span style="color: #666666">+</span>k] <span style="color: #666666">=</span> (x<span style="color: #666666">**</span>(i<span style="color: #666666">-</span>k))<span style="color: #666666">*</span>(y<span style="color: #666666">**</span>k)
<span style="color: #008000; font-weight: bold">return</span> X
<span style="color: #408080; font-style: italic"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
n <span style="color: #666666">=</span> <span style="color: #666666">4</span>
N <span style="color: #666666">=</span> <span style="color: #666666">100</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
z <span style="color: #666666">=</span> FrankeFunction(x, y)
X <span style="color: #666666">=</span> create_X(x, y, n<span style="color: #666666">=</span>n)
Xpd <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
<span style="color: #408080; font-style: italic"># subtract the mean values and set up the covariance matrix</span>
Xpd <span style="color: #666666">=</span> Xpd <span style="color: #666666">-</span> Xpd<span style="color: #666666">.</span>mean()
covariance_matrix <span style="color: #666666">=</span> Xpd<span style="color: #666666">.</span>cov()
<span style="color: #008000">print</span>(covariance_matrix)
</pre></div>
<p>
We note here that the covariance is zero for the first rows and
columns since all matrix elements in the design matrix were set to one
(we are fitting the function in terms of a polynomial of degree \( n \)).
<p>
This means that the variance for these elements will be zero and will
cause problems when we set up the correlation matrix. We can simply
drop these elements and construct a correlation
matrix without these elements.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec53">Rewriting the Covariance and/or Correlation Matrix </h2>
<p>
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
$$
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}].
$$
<p>
To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
$$
\boldsymbol{X}=\begin{bmatrix}
x_{00} & x_{01}\\
x_{10} & x_{11}\\
\end{bmatrix}=\begin{bmatrix}
\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
\end{bmatrix}.
$$
<p>
If we then compute the expectation value
$$
\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix}
x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
\end{bmatrix},
$$
which is just
$$
\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
\end{bmatrix},
$$
where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
<p>
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec54">Linking with SVD </h2>
<p>
See lecture september 11. More text to be added here soon.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec55">Where are we going? </h2>
<p>
Before we proceed, we need to rethink what we have been doing. In our
eager to fit the data, we have omitted several important elements in
our regression analysis. In what follows we will
<ol>
<li> look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff</li>
<li> introduce resampling techniques like cross-validation, bootstrapping and jackknife and more</li>
</ol>
This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec56">Resampling methods </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Resampling methods are an indispensable tool in modern
statistics. They involve repeatedly drawing samples from a training
set and refitting a model of interest on each sample in order to
obtain additional information about the fitted model. For example, in
order to estimate the variability of a linear regression fit, we can
repeatedly draw different samples from the training data, fit a linear
regression to each new sample, and then examine the extent to which
the resulting fits differ. Such an approach may allow us to obtain
information that would not be available from fitting the model only
once using the original training sample.
<p>
Two resampling methods are often used in Machine Learning analyses,
<ol>
<li> The <b>bootstrap method</b></li>
<li> and <b>Cross-Validation</b></li>
</ol>
In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
cross-validation and the bootstrap method.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec57">Resampling approaches can be computationally expensive </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
Resampling approaches can be computationally expensive, because they
involve fitting the same statistical method multiple times using
different subsets of the training data. However, due to recent
advances in computing power, the computational requirements of
resampling methods generally are not prohibitive. In this chapter, we
discuss two of the most commonly used resampling methods,
cross-validation and the bootstrap. Both methods are important tools
in the practical application of many statistical learning
procedures. For example, cross-validation can be used to estimate the
test error associated with a given statistical learning method in
order to evaluate its performance, or to select the appropriate level
of flexibility. The process of evaluating a model&#8217;s performance is
known as model assessment, whereas the process of selecting the proper
level of flexibility for a model is known as model selection. The
bootstrap is widely used.
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec58">Why resampling methods ? </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b>Statistical analysis.</b>
<p>
<ul>
<li> Our simulations can be treated as <em>computer experiments</em>. This is particularly the case for Monte Carlo methods</li>
<li> The results can be analysed with the same statistical tools as we would use analysing experimental data.</li>
<li> As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.</li>
</ul>
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec59">Statistical analysis </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<ul>
<li> As in other experiments, many numerical experiments have two classes of errors:</li>
<ul>
<li> Statistical errors</li>
<li> Systematical errors</li>
</ul>
<li> Statistical errors can be estimated using standard tools from statistics</li>
<li> Systematical errors are method specific and must be treated differently from case to case.</li>
</ul>
</div>
<p>
<!-- !split -->
<h2 id="___sec60">Linking the regression analysis with a statistical interpretation </h2>
<p>
The
advantage of doing linear regression is that we actually end up with
analytical expressions for several statistical quantities.
Standard least squares and Ridge regression allow us to
derive quantities like the variance and other expectation values in a
rather straightforward way.
<p>
It is assumed that \( \varepsilon_i
\sim \mathcal{N}(0, \sigma^2) \) and the \( \varepsilon_{i} \) are
independent, i.e.:
$$
\begin{align*}
\mbox{Cov}(\varepsilon_{i_1},
\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right.
\end{align*}
$$
The randomness of \( \varepsilon_i \) implies that
\( \mathbf{y}_i \) is also a random variable. In particular,
\( \mathbf{y}_i \) is normally distributed, because \( \varepsilon_i \sim
\mathcal{N}(0, \sigma^2) \) and \( \mathbf{X}_{i,\ast} \, \boldsymbol{\beta} \) is a
non-random scalar. To specify the parameters of the distribution of
\( \mathbf{y}_i \) we need to calculate its first two moments.
<p>
Recall that \( \boldsymbol{X} \) is a matrix of dimensionality \( n\times p \). The
notation above \( \mathbf{X}_{i,\ast} \) means that we are looking at the
row number \( i \) and perform a sum over all values \( p \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec61">Assumptions made </h2>
<p>
The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \)
which describe our data
$$
\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
$$
<p>
We approximate this function with our model from the solution of the linear regression equations, that is our
function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with
$$
\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}.
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec62">Expectation value and variance </h2>
<p>
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).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec63">Expectation value and variance for \( \boldsymbol{\beta} \) </h2>
<p>
With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we can evaluate the expectation value
$$
\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}.
$$
This means that the estimator of the regression parameters is unbiased.
<p>
We can also calculate the variance
<p>
The variance of \( \boldsymbol{\beta} \) is
$$
\begin{eqnarray*}
\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \}
\\
& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \}
\\
% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\\
& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
% \\
% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
% \\
% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T
\\
& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T}
\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
\end{eqnarray*}
$$
<p>
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:
\( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{
[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} } \). This may be used to
construct a confidence interval for the estimates.
<p>
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.
<p>
It is rather straightforward to show that
$$
\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}.
$$
We see clearly that
\( \mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}} \) for any \( \lambda > 0 \). We say then that the ridge estimator is biased.
<p>
We can also compute the variance as
$$
\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
$$
and it is easy to see that if the parameter \( \lambda \) goes to infinity then the variance of Ridge parameters \( \boldsymbol{\beta} \) goes to zero.
<p>
With this, we can compute the difference
$$
\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
$$
The difference is non-negative definite since each component of the
matrix product is non-negative definite.
This means the variance we obtain with the standard OLS will always for \( \lambda > 0 \) be larger than the variance of \( \boldsymbol{\beta} \) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec64">Resampling methods </h2>
<p>
With all these analytical equations for both the OLS and Ridge
regression, we will now outline how to assess a given model. This will
lead us to a discussion of the so-called bias-variance tradeoff (see
below) and so-called resampling methods.
<p>
One of the quantities we have discussed as a way to measure errors is
the mean-squared error (MSE), mainly used for fitting of continuous
functions. Another choice is the absolute error.
<p>
In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,
we discuss the
<ol>
<li> prediction error or simply the <b>test error</b> \( \mathrm{Err_{Test}} \), where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the</li>
<li> training error \( \mathrm{Err_{Train}} \), which is the average loss over the training data.</li>
</ol>
As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.
For a certain level of complexity the test error will reach minimum, before starting to increase again. The
training error reaches a saturation.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec65">Resampling methods: Jackknife and Bootstrap </h2>
<p>
Two famous
resampling methods are the <b>independent bootstrap</b> and <b>the jackknife</b>.
<p>
The jackknife is a special case of the independent bootstrap. Still, the jackknife was made
popular prior to the independent bootstrap. And as the popularity of
the independent bootstrap soared, new variants, such as <b>the dependent bootstrap</b>.
<p>
The Jackknife and independent bootstrap work for
independent, identically distributed random variables.
If these conditions are not
satisfied, the methods will fail. Yet, it should be said that if the data are
independent, identically distributed, and we only want to estimate the
variance of \( \overline{X} \) (which often is the case), then there is no
need for bootstrapping.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec66">Resampling methods: Jackknife </h2>
<p>
The Jackknife works by making many replicas of the estimator \( \widehat{\theta} \).
The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \( \boldsymbol{x} = (x_1,x_2,\cdots,X_n) \).
Let \( \boldsymbol{x}_i \) denote the vector
$$
\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n),
$$
<p>
which equals the vector \( \boldsymbol{x} \) with the exception that observation
number \( i \) is left out. Using this notation, define
\( \widehat{\theta}_i \) to be the estimator
\( \widehat{\theta} \) computed using \( \vec{X}_i \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec67">Jackknife code example </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">jackknife</span>(data, stat):
n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data);t <span style="color: #666666">=</span> zeros(n); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
<span style="color: #408080; font-style: italic">## &#39;jackknifing&#39; by leaving out an observation for each i </span>
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
t[i] <span style="color: #666666">=</span> stat(delete(data,i) )
<span style="color: #408080; font-style: italic"># analysis </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Jackknife Statistics :&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;original bias std. error&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (stat(data),(n<span style="color: #666666">-1</span>)<span style="color: #666666">*</span>mean(t)<span style="color: #666666">/</span>n, (n<span style="color: #666666">*</span>var(t))<span style="color: #666666">**.5</span>))
<span style="color: #008000; font-weight: bold">return</span> t
<span style="color: #408080; font-style: italic"># Returns mean of data samples </span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
<span style="color: #008000; font-weight: bold">return</span> mean(data)
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
<span style="color: #408080; font-style: italic"># jackknife returns the data sample </span>
t <span style="color: #666666">=</span> jackknife(x, stat)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec68">Resampling methods: Bootstrap </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Bootstrapping is a nonparametric approach to statistical inference
that substitutes computation for more traditional distributional
assumptions and asymptotic results. Bootstrapping offers a number of
advantages:
<ol>
<li> The bootstrap is quite general, although there are some cases in which it fails.</li>
<li> Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.</li>
<li> It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.</li>
<li> It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).</li>
</ol>
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec69">Resampling methods: Bootstrap background </h2>
<p>
Since \( \widehat{\theta} = \widehat{\theta}(\boldsymbol{X}) \) is a function of random variables,
\( \widehat{\theta} \) itself must be a random variable. Thus it has
a pdf, call this function \( p(\boldsymbol{t}) \). The aim of the bootstrap is to
estimate \( p(\boldsymbol{t}) \) by the relative frequency of
\( \widehat{\theta} \). You can think of this as using a histogram
in the place of \( p(\boldsymbol{t}) \). If the relative frequency closely
resembles \( p(\vec{t}) \), then using numerics, it is straight forward to
estimate all the interesting parameters of \( p(\boldsymbol{t}) \) using point
estimators.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec70">Resampling methods: More Bootstrap background </h2>
<p>
In the case that \( \widehat{\theta} \) has
more than one component, and the components are independent, we use the
same estimator on each component separately. If the probability
density function of \( X_i \), \( p(x) \), had been known, then it would have
been straight forward to do this by:
<ol>
<li> Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).</li>
<li> Then using these numbers, we could compute a replica of \( \widehat{\theta} \) called \( \widehat{\theta}^* \).</li>
</ol>
By repeated use of (1) and (2), many
estimates of \( \widehat{\theta} \) could have been obtained. The
idea is to use the relative frequency of \( \widehat{\theta}^* \)
(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec71">Resampling methods: Bootstrap approach </h2>
<p>
But
unless there is enough information available about the process that
generated \( X_1,X_2,\cdots,X_n \), \( p(x) \) is in general
unknown. Therefore, <a href="https://projecteuclid.org/euclid.aos/1176344552" target="_blank">Efron in 1979</a> asked the
question: What if we replace \( p(x) \) by the relative frequency
of the observation \( X_i \); if we draw observations in accordance with
the relative frequency of the observations, will we obtain the same
result in some asymptotic sense? The answer is yes.
<p>
Instead of generating the histogram for the relative
frequency of the observation \( X_i \), just draw the values
\( (X_1^*,X_2^*,\cdots,X_n^*) \) with replacement from the vector
\( \boldsymbol{X} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec72">Resampling methods: Bootstrap steps </h2>
<p>
The independent bootstrap works like this:
<ol>
<li> Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).</li>
<li> Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).</li>
<li> Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\theta}^* \) by evaluating \( \widehat \theta \) under the observations \( \boldsymbol{x}^* \).</li>
<li> Repeat this process \( k \) times.</li>
</ol>
When you are done, you can draw a histogram of the relative frequency
of \( \widehat \theta^* \). This is your estimate of the probability
distribution \( p(t) \). Using this probability distribution you can
estimate any statistics thereof. In principle you never draw the
histogram of the relative frequency of \( \widehat{\theta}^* \). Instead
you use the estimators corresponding to the statistic of interest. For
example, if you are interested in estimating the variance of \( \widehat
\theta \), apply the etsimator \( \widehat \sigma^2 \) to the values
\( \widehat \theta ^* \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec73">Code example for the Bootstrap method </h2>
<p>
The following code starts with a Gaussian distribution with mean value
\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
used in the bootstrap analysis. The bootstrap analysis returns a data
set after a given number of bootstrap operations (as many as we have
data points). This data set consists of estimated mean values for each
bootstrap operation. The histogram generated by the bootstrap method
shows that the distribution for these mean values is also a Gaussian,
centered around the mean value \( \mu=100 \) but with standard deviation
\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
this case the same as the number of original data points). The value
of the standard deviation is what we expect from the central limit
theorem.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
<span style="color: #008000; font-weight: bold">return</span> mean(data)
<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
<span style="color: #408080; font-style: italic"># analysis </span>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Bootstrap Statistics :&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;original bias std. error&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
<span style="color: #008000; font-weight: bold">return</span> t
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;red&#39;</span>, alpha<span style="color: #666666">=0.75</span>)
<span style="color: #408080; font-style: italic"># add a &#39;best fit&#39; line </span>
y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">&#39;r--&#39;</span>, linewidth<span style="color: #666666">=1</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Smarts&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Probability&#39;</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split -->
<h2 id="___sec74">Various steps in cross-validation </h2>
<p>
When the repetitive splitting of the data set is done randomly,
samples may accidently end up in a fast majority of the splits in
either training or test set. Such samples may have an unbalanced
influence on either model building or prediction evaluation. To avoid
this \( k \)-fold cross-validation structures the data splitting. The
samples are divided into \( k \) more or less equally sized exhaustive and
mutually exclusive subsets. In turn (at each split) one of these
subsets plays the role of the test set while the union of the
remaining subsets constitutes the training set. Such a splitting
warrants a balanced representation of each sample in both training and
test set over the splits. Still the division into the \( k \) subsets
involves a degree of randomness. This may be fully excluded when
choosing \( k=n \). This particular case is referred to as leave-one-out
cross-validation (LOOCV).
<p>
<!-- !split -->
<h2 id="___sec75">How to set up the cross-validation for Ridge and/or Lasso </h2>
<ul>
<li> Define a range of interest for the penalty parameter.</li>
<li> Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.</li>
<li> Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as</li>
</ul>
$$
\begin{align*}
\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T}
\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
\end{align*}
$$
<ul>
<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
</ul>
$$
\begin{align*}
\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
\end{align*}
$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec76">Cross-validation in brief </h2>
<p>
For the various values of \( k \)
<ol>
<li> shuffle the dataset randomly.</li>
<li> Split the dataset into \( k \) groups.</li>
<li> For each unique group:
<ol type="a"></li>
<li> Decide which group to use as set for test data</li>
<li> Take the remaining groups as a training data set</li>
<li> Fit a model on the training set and evaluate it on the test set</li>
<li> Retain the evaluation score and discard the model</li>
</ol>
<li> Summarize the model using the sample of model evaluation scores</li>
</ol>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec77">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
<p>
The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
<span style="color: #408080; font-style: italic"># Generate the data.</span>
nsamples <span style="color: #666666">=</span> <span style="color: #666666">100</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
y <span style="color: #666666">=</span> <span style="color: #666666">3*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
<span style="color: #408080; font-style: italic">## Cross-validation on Ridge regression using KFold only</span>
<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">6</span>)
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
<span style="color: #408080; font-style: italic"># Perform the cross-validation to estimate MSE</span>
scores_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((nlambdas, k))
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
<span style="color: #008000; font-weight: bold">for</span> train_inds, test_inds <span style="color: #AA22FF; font-weight: bold">in</span> kfold<span style="color: #666666">.</span>split(x):
xtrain <span style="color: #666666">=</span> x[train_inds]
ytrain <span style="color: #666666">=</span> y[train_inds]
xtest <span style="color: #666666">=</span> x[test_inds]
ytest <span style="color: #666666">=</span> y[test_inds]
Xtrain <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtrain[:, np<span style="color: #666666">.</span>newaxis])
ridge<span style="color: #666666">.</span>fit(Xtrain, ytrain[:, np<span style="color: #666666">.</span>newaxis])
Xtest <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtest[:, np<span style="color: #666666">.</span>newaxis])
ypred <span style="color: #666666">=</span> ridge<span style="color: #666666">.</span>predict(Xtest)
scores_KFold[i,j] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum((ypred <span style="color: #666666">-</span> ytest[:, np<span style="color: #666666">.</span>newaxis])<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>np<span style="color: #666666">.</span>size(ypred)
j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
estimated_mse_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(scores_KFold, axis <span style="color: #666666">=</span> <span style="color: #666666">1</span>)
<span style="color: #408080; font-style: italic">## Cross-validation using cross_val_score from sklearn along with KFold</span>
<span style="color: #408080; font-style: italic"># kfold is an instance initialized above as:</span>
<span style="color: #408080; font-style: italic"># kfold = KFold(n_splits = k)</span>
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
X <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(x[:, np<span style="color: #666666">.</span>newaxis])
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, X, y[:, np<span style="color: #666666">.</span>newaxis], scoring<span style="color: #666666">=</span><span style="color: #BA2121">&#39;neg_mean_squared_error&#39;</span>, cv<span style="color: #666666">=</span>kfold)
<span style="color: #408080; font-style: italic"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
<span style="color: #408080; font-style: italic"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
<span style="color: #408080; font-style: italic">## Plot and compare the slightly different ways to perform cross-validation</span>
plt<span style="color: #666666">.</span>figure()
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;cross_val_score&#39;</span>)
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_KFold, <span style="color: #BA2121">&#39;r--&#39;</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;KFold&#39;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;log10(lambda)&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;mse&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec78">The bias-variance tradeoff </h2>
<p>
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
<p>
Let us assume that the true data is generated from a noisy model
$$
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
$$
<p>
where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
<p>
In our derivation of the ordinary least squares method we defined then
an approximation to the function \( f \) in terms of the parameters
\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model,
that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
<p>
Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
$$
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
$$
<p>
We can rewrite this as
$$
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
$$
<p>
The three terms represent the square of the bias of the learning
method, which can be thought of as the error caused by the simplifying
assumptions built into the method. The second term represents the
variance of the chosen model and finally the last terms is variance of
the error \( \boldsymbol{\epsilon} \).
<p>
To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \).
We use a more compact notation in terms of the expectation value
$$
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right],
$$
and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get
$$
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right],
$$
which, using the abovementioned expectation values can be rewritten as
$$
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2,
$$
that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec79">Example code for Bias-Variance tradeoff </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
n <span style="color: #666666">=</span> <span style="color: #666666">500</span>
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
degree <span style="color: #666666">=</span> <span style="color: #666666">18</span> <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
<span style="color: #408080; font-style: italic"># Make data set.</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-1</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
<span style="color: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</span>
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
<span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> &gt;= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;f(x)&#39;</span>)
plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Data points&#39;</span>)
plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Pred&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec80">Understanding what happens </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
n <span style="color: #666666">=</span> <span style="color: #666666">40</span>
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
<span style="color: #408080; font-style: italic"># Make data set.</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
polydegree[degree] <span style="color: #666666">=</span> degree
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Polynomial degree:&#39;</span>, degree)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Error:&#39;</span>, error[degree])
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Bias^2:&#39;</span>, bias[degree])
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Var:&#39;</span>, variance[degree])
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> &gt;= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Error&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;bias&#39;</span>)
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Variance&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split -->
<h2 id="___sec81">Summing up </h2>
<p>
The bias-variance tradeoff summarizes the fundamental tension in
machine learning, particularly supervised learning, between the
complexity of a model and the amount of training data needed to train
it. Since data is often limited, in practice it is often useful to
use a less-complex model with higher bias, that is a model whose asymptotic
performance is worse than another model because it is easier to
train and less sensitive to sampling noise arising from having a
finite-sized training dataset (smaller variance).
<p>
The above equations tell us that in
order to minimize the expected test error, we need to select a
statistical learning method that simultaneously achieves low variance
and low bias. Note that variance is inherently a nonnegative quantity,
and squared bias is also nonnegative. Hence, we see that the expected
test MSE can never lie below \( Var(\epsilon) \), the irreducible error.
<p>
What do we mean by the variance and bias of a statistical learning
method? The variance refers to the amount by which our model would change if we
estimated it using a different training data set. Since the training
data are used to fit the statistical learning method, different
training data sets will result in a different estimate. But ideally the
estimate for our model should not vary too much between training
sets. However, if a method has high variance then small changes in
the training data can result in large changes in the model. In general, more
flexible statistical methods have higher variance.
<p>
You may also find this recent <a href="https://www.pnas.org/content/116/32/15849" target="_blank">article</a> of interest.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec82">Another Example from Scikit-Learn's Repository </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #BA2121; font-style: italic">============================</span>
<span style="color: #BA2121; font-style: italic">Underfitting vs. Overfitting</span>
<span style="color: #BA2121; font-style: italic">============================</span>
<span style="color: #BA2121; font-style: italic">This example demonstrates the problems of underfitting and overfitting and</span>
<span style="color: #BA2121; font-style: italic">how we can use linear regression with polynomial features to approximate</span>
<span style="color: #BA2121; font-style: italic">nonlinear functions. The plot shows the function that we want to approximate,</span>
<span style="color: #BA2121; font-style: italic">which is a part of the cosine function. In addition, the samples from the</span>
<span style="color: #BA2121; font-style: italic">real function and the approximations of different models are displayed. The</span>
<span style="color: #BA2121; font-style: italic">models have polynomial features of different degrees. We can see that a</span>
<span style="color: #BA2121; font-style: italic">linear function (polynomial with degree 1) is not sufficient to fit the</span>
<span style="color: #BA2121; font-style: italic">training samples. This is called **underfitting**. A polynomial of degree 4</span>
<span style="color: #BA2121; font-style: italic">approximates the true function almost perfectly. However, for higher degrees</span>
<span style="color: #BA2121; font-style: italic">the model will **overfit** the training data, i.e. it learns the noise of the</span>
<span style="color: #BA2121; font-style: italic">training data.</span>
<span style="color: #BA2121; font-style: italic">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
<span style="color: #BA2121; font-style: italic">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
<span style="color: #BA2121; font-style: italic">set, the higher, the less likely the model generalizes correctly from the</span>
<span style="color: #BA2121; font-style: italic">training data.</span>
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #008000">print</span>(<span style="color: #19177C">__doc__</span>)
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">true_fun</span>(X):
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>pi <span style="color: #666666">*</span> X)
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
n_samples <span style="color: #666666">=</span> <span style="color: #666666">30</span>
degrees <span style="color: #666666">=</span> [<span style="color: #666666">1</span>, <span style="color: #666666">4</span>, <span style="color: #666666">15</span>]
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples))
y <span style="color: #666666">=</span> true_fun(X) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_samples) <span style="color: #666666">*</span> <span style="color: #666666">0.1</span>
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">14</span>, <span style="color: #666666">5</span>))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(degrees)):
ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(degrees), i <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
plt<span style="color: #666666">.</span>setp(ax, xticks<span style="color: #666666">=</span>(), yticks<span style="color: #666666">=</span>())
polynomial_features <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=</span>degrees[i],
include_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
linear_regression <span style="color: #666666">=</span> LinearRegression()
pipeline <span style="color: #666666">=</span> Pipeline([(<span style="color: #BA2121">&quot;polynomial_features&quot;</span>, polynomial_features),
(<span style="color: #BA2121">&quot;linear_regression&quot;</span>, linear_regression)])
pipeline<span style="color: #666666">.</span>fit(X[:, np<span style="color: #666666">.</span>newaxis], y)
<span style="color: #408080; font-style: italic"># Evaluate the models using crossvalidation</span>
scores <span style="color: #666666">=</span> cross_val_score(pipeline, X[:, np<span style="color: #666666">.</span>newaxis], y,
scoring<span style="color: #666666">=</span><span style="color: #BA2121">&quot;neg_mean_squared_error&quot;</span>, cv<span style="color: #666666">=10</span>)
X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">100</span>)
plt<span style="color: #666666">.</span>plot(X_test, pipeline<span style="color: #666666">.</span>predict(X_test[:, np<span style="color: #666666">.</span>newaxis]), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Model&quot;</span>)
plt<span style="color: #666666">.</span>plot(X_test, true_fun(X_test), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;True function&quot;</span>)
plt<span style="color: #666666">.</span>scatter(X, y, edgecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;b&#39;</span>, s<span style="color: #666666">=20</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Samples&quot;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;x&quot;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;y&quot;</span>)
plt<span style="color: #666666">.</span>xlim((<span style="color: #666666">0</span>, <span style="color: #666666">1</span>))
plt<span style="color: #666666">.</span>ylim((<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>))
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">&quot;best&quot;</span>)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Degree </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">MSE = </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">(+/- </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">)&quot;</span><span style="color: #666666">.</span>format(
degrees[i], <span style="color: #666666">-</span>scores<span style="color: #666666">.</span>mean(), scores<span style="color: #666666">.</span>std()))
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec83">More examples on bootstrap and cross-validation and errors </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">30</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),Maxpolydegree))
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
testerror <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
trainingerror <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
polynomial <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
trials <span style="color: #666666">=</span> <span style="color: #666666">100</span>
<span style="color: #008000; font-weight: bold">for</span> polydegree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>, Maxpolydegree):
polynomial[polydegree] <span style="color: #666666">=</span> polydegree
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(polydegree):
X[:,degree] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(degree<span style="color: #666666">/3.0</span>)
<span style="color: #408080; font-style: italic"># loop over trials in order to estimate the expectation value of the MSE</span>
testerror[polydegree] <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
trainingerror[polydegree] <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
<span style="color: #008000; font-weight: bold">for</span> samples <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(trials):
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, Energies, test_size<span style="color: #666666">=0.2</span>)
model <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)<span style="color: #666666">.</span>fit(x_train, y_train)
ypred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(x_train)
ytilde <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(x_test)
testerror[polydegree] <span style="color: #666666">+=</span> mean_squared_error(y_test, ytilde)
trainingerror[polydegree] <span style="color: #666666">+=</span> mean_squared_error(y_train, ypred)
testerror[polydegree] <span style="color: #666666">/=</span> trials
trainingerror[polydegree] <span style="color: #666666">/=</span> trials
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Degree of polynomial: </span><span style="color: #BB6688; font-weight: bold">%3d</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span> polynomial[polydegree])
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Mean squared error on training data: </span><span style="color: #BB6688; font-weight: bold">%.8f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> trainingerror[polydegree])
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Mean squared error on test data: </span><span style="color: #BB6688; font-weight: bold">%.8f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> testerror[polydegree])
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(trainingerror), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Training Error&#39;</span>)
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(testerror), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Test Error&#39;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Polynomial degree&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;log10[MSE]&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split -->
<h2 id="___sec84">The same example but now with cross-validation </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">30</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),Maxpolydegree))
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
polynomial <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
k <span style="color: #666666">=5</span>
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
<span style="color: #008000; font-weight: bold">for</span> polydegree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>, Maxpolydegree):
polynomial[polydegree] <span style="color: #666666">=</span> polydegree
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(polydegree):
X[:,degree] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(degree<span style="color: #666666">/3.0</span>)
OLS <span style="color: #666666">=</span> LinearRegression()
<span style="color: #408080; font-style: italic"># loop over trials in order to estimate the expectation value of the MSE</span>
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(OLS, X, Energies, scoring<span style="color: #666666">=</span><span style="color: #BA2121">&#39;neg_mean_squared_error&#39;</span>, cv<span style="color: #666666">=</span>kfold)
<span style="color: #408080; font-style: italic">#[:, np.newaxis]</span>
estimated_mse_sklearn[polydegree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(estimated_mse_sklearn), label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Test Error&#39;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Polynomial degree&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;log10[MSE]&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec85">Cross-validation with Ridge </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
<span style="color: #408080; font-style: italic"># Generate the data.</span>
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">10</span>)
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, x, y, scoring<span style="color: #666666">=</span><span style="color: #BA2121">&#39;neg_mean_squared_error&#39;</span>, cv<span style="color: #666666">=</span>kfold)
estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
plt<span style="color: #666666">.</span>figure()
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;cross_val_score&#39;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;log10(lambda)&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;MSE&#39;</span>)
plt<span style="color: #666666">.</span>legend()
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec86">The Ising model </h2>
<p>
The one-dimensional Ising model with nearest neighbor interaction, no
external field and a constant coupling constant \( J \) is given by
$$
\begin{align}
H = -J \sum_{k}^L s_k s_{k + 1},
\label{_auto2}
\end{align}
$$
<p>
where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
in the system is determined by \( L \). For the one-dimensional system
there is no phase transition.
<p>
We will look at a system of \( L = 40 \) spins with a coupling constant of
\( J = 1 \). To get enough training data we will generate 10000 states
with their respective energies.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
</pre></div>
<p>
Here we use ordinary least squares
regression to predict the energy for the nearest neighbor
one-dimensional Ising model on a ring, i.e., the endpoints wrap
around. We will use linear regression to fit a value for
the coupling constant to achieve this.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec87">Reformulating the problem to suit regression </h2>
<p>
A more general form for the one-dimensional Ising model is
$$
\begin{align}
H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
\label{_auto3}
\end{align}
$$
<p>
Here we allow for interactions beyond the nearest neighbors and a state dependent
coupling constant. This latter expression can be formulated as
a matrix-product
$$
\begin{align}
\boldsymbol{H} = \boldsymbol{X} J,
\label{_auto4}
\end{align}
$$
<p>
where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
elements \( -J_{jk} \). This form of writing the energy fits perfectly
with the form utilized in linear regression, that is
$$
\begin{align}
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
\label{_auto5}
\end{align}
$$
<p>
We split the data in training and test data as discussed in the previous example
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
y <span style="color: #666666">=</span> energies
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec88">Linear regression </h2>
<p>
In the ordinary least squares method we choose the cost function
$$
\begin{align}
C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
\label{_auto6}
\end{align}
$$
<p>
We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
This yields the expression for \( \boldsymbol{\beta} \) to be
$$
\boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
$$
<p>
which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
intercept, i.e., a constant term, we must make sure that the
first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
axis<span style="color: #666666">=1</span>
)
X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
axis<span style="color: #666666">=1</span>
)
</pre></div>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_inv</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
<span style="color: #008000; font-weight: bold">return</span> scl<span style="color: #666666">.</span>inv(x<span style="color: #666666">.</span>T <span style="color: #666666">@</span> x) <span style="color: #666666">@</span> (x<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
beta <span style="color: #666666">=</span> ols_inv(X_train_own, y_train)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec89">Singular Value decomposition </h2>
<p>
Doing the inversion directly turns out to be a bad idea since the matrix
\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
value decomposition</b>. Using the definition of the Moore-Penrose
pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
$$
\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
$$
<p>
where the pseudoinverse of \( \boldsymbol{X} \) is given by
$$
\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
$$
<p>
Using singular value decomposition we can decompose the matrix \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
\( \omega \) to
$$
\begin{align}
\boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
\label{_auto7}
\end{align}
$$
<p>
Note that solving this equation by actually doing the pseudoinverse
(which is what we will do) is not a good idea as this operation scales
as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
general matrix. Instead, doing \( QR \)-factorization and solving the
linear system as an equation would reduce this down to
\( \mathcal{O}(n^2) \) operations.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-&gt;</span> np<span style="color: #666666">.</span>ndarray:
u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
<span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T <span style="color: #666666">@</span> scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) <span style="color: #666666">@</span> u<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
</pre></div>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
</pre></div>
<p>
When extracting the \( J \)-matrix we need to make sure that we remove the intercept, as is done here
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
</pre></div>
<p>
A way of looking at the coefficients in \( J \) is to plot the matrices as images.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;OLS&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
It is interesting to note that OLS
considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
valid matrix elements for \( J \).
In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
this problem can be removed, partly and only with Lasso regression.
<p>
In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec90">The one-dimensional Ising model </h2>
<p>
Let us bring back the Ising model again, but now with an additional
focus on Ridge and Lasso regression as well. We repeat some of the
basic parts of the Ising model and the setup of the training and test
data. The one-dimensional Ising model with nearest neighbor
interaction, no external field and a constant coupling constant \( J \) is
given by
$$
\begin{align}
H = -J \sum_{k}^L s_k s_{k + 1},
\label{_auto8}
\end{align}
$$
where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
<p>
We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&#39;seismic&#39;</span>)
L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
</pre></div>
<p>
A more general form for the one-dimensional Ising model is
$$
\begin{align}
H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
\label{_auto9}
\end{align}
$$
<p>
Here we allow for interactions beyond the nearest neighbors and a more
adaptive coupling matrix. This latter expression can be formulated as
a matrix-product on the form
$$
\begin{align}
H = X J,
\label{_auto10}
\end{align}
$$
<p>
where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
elements \( -J_{jk} \). This form of writing the energy fits perfectly
with the form utilized in linear regression, viz.
$$
\begin{align}
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
\label{_auto11}
\end{align}
$$
We organize the data as we did above
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
y <span style="color: #666666">=</span> energies
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.96</span>)
X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
axis<span style="color: #666666">=1</span>
)
X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
axis<span style="color: #666666">=1</span>
)
</pre></div>
<p>
We will do all fitting with <b>Scikit-Learn</b>,
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X_train, y_train)
</pre></div>
<p>
When extracting the \( J \)-matrix we make sure to remove the intercept
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>J_sk <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
</pre></div>
<p>
And then we plot the results
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_sk, <span style="color: #666666">**</span>cmap_args)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;LinearRegression from Scikit-learn&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
The results perfectly with our previous discussion where we used our own code.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec91">Ridge regression </h2>
<p>
Having explored the ordinary least squares we move on to ridge
regression. In ridge regression we include a <b>regularizer</b>. This
involves a new cost function which leads to a new estimate for the
weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
cost function is given by
$$
\begin{align}
C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
\label{_auto12}
\end{align}
$$
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X_train, y_train)
J_ridge_sk <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_ridge_sk, <span style="color: #666666">**</span>cmap_args)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Ridge from Scikit-learn&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec92">LASSO regression </h2>
<p>
In the <b>Least Absolute Shrinkage and Selection Operator</b> (LASSO)-method we get a third cost function.
$$
\begin{align}
C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
\label{_auto13}
\end{align}
$$
<p>
Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from <b>Scikit-Learn</b>.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>clf_lasso <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Lasso(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X_train, y_train)
J_lasso_sk <span style="color: #666666">=</span> clf_lasso<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_lasso_sk, <span style="color: #666666">**</span>cmap_args)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Lasso from Scikit-learn&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
It is quite striking how LASSO breaks the symmetry of the coupling
constant as opposed to ridge and OLS. We get a sparse solution with
\( J_{j, j + 1} = -1 \).
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec93">Performance as function of the regularization parameter </h2>
<p>
We see how the different models perform for a different set of values for \( \lambda \).
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">5</span>, <span style="color: #666666">10</span>)
train_errors <span style="color: #666666">=</span> {
<span style="color: #BA2121">&quot;ols_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
<span style="color: #BA2121">&quot;ridge_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
<span style="color: #BA2121">&quot;lasso_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size)
}
test_errors <span style="color: #666666">=</span> {
<span style="color: #BA2121">&quot;ols_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
<span style="color: #BA2121">&quot;ridge_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size),
<span style="color: #BA2121">&quot;lasso_sk&quot;</span>: np<span style="color: #666666">.</span>zeros(lambdas<span style="color: #666666">.</span>size)
}
plot_counter <span style="color: #666666">=</span> <span style="color: #666666">1</span>
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">32</span>, <span style="color: #666666">54</span>))
<span style="color: #008000; font-weight: bold">for</span> i, _lambda <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(tqdm<span style="color: #666666">.</span>tqdm(lambdas)):
<span style="color: #008000; font-weight: bold">for</span> key, method <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">zip</span>(
[<span style="color: #BA2121">&quot;ols_sk&quot;</span>, <span style="color: #BA2121">&quot;ridge_sk&quot;</span>, <span style="color: #BA2121">&quot;lasso_sk&quot;</span>],
[skl<span style="color: #666666">.</span>LinearRegression(), skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda), skl<span style="color: #666666">.</span>Lasso(alpha<span style="color: #666666">=</span>_lambda)]
):
method <span style="color: #666666">=</span> method<span style="color: #666666">.</span>fit(X_train, y_train)
train_errors[key][i] <span style="color: #666666">=</span> method<span style="color: #666666">.</span>score(X_train, y_train)
test_errors[key][i] <span style="color: #666666">=</span> method<span style="color: #666666">.</span>score(X_test, y_test)
omega <span style="color: #666666">=</span> method<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">10</span>, <span style="color: #666666">5</span>, plot_counter)
plt<span style="color: #666666">.</span>imshow(omega, <span style="color: #666666">**</span>cmap_args)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&quot;</span><span style="color: #BB6688; font-weight: bold">%s</span><span style="color: #BA2121">, $\lambda = </span><span style="color: #BB6688; font-weight: bold">%.4f</span><span style="color: #BA2121">$&quot;</span> <span style="color: #666666">%</span> (key, _lambda))
plot_counter <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
We see that LASSO reaches a good solution for low
values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
much. Ridge is more stable over a larger range of values for
\( \lambda \), but eventually also fades away.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec94">Finding the optimal value of \( \lambda \) </h2>
<p>
To determine which value of \( \lambda \) is best we plot the accuracy of
the models when predicting the training and the testing set. We expect
the accuracy of the training set to be quite good, but if the accuracy
of the testing set is much lower this tells us that we might be
subject to an overfit model. The ideal scenario is an accuracy on the
testing set that is close to the accuracy of the training set.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
colors <span style="color: #666666">=</span> {
<span style="color: #BA2121">&quot;ols_sk&quot;</span>: <span style="color: #BA2121">&quot;r&quot;</span>,
<span style="color: #BA2121">&quot;ridge_sk&quot;</span>: <span style="color: #BA2121">&quot;y&quot;</span>,
<span style="color: #BA2121">&quot;lasso_sk&quot;</span>: <span style="color: #BA2121">&quot;c&quot;</span>
}
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> train_errors:
plt<span style="color: #666666">.</span>semilogx(
lambdas,
train_errors[key],
colors[key],
label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Train </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(key),
linewidth<span style="color: #666666">=4.0</span>
)
<span style="color: #008000; font-weight: bold">for</span> key <span style="color: #AA22FF; font-weight: bold">in</span> test_errors:
plt<span style="color: #666666">.</span>semilogx(
lambdas,
test_errors[key],
colors[key] <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;--&quot;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;Test </span><span style="color: #BB6688; font-weight: bold">{0}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(key),
linewidth<span style="color: #666666">=4.0</span>
)
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">&quot;best&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&quot;$\lambda$&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&quot;$R^2$&quot;</span>, fontsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>tick_params(labelsize<span style="color: #666666">=18</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
achieves a very good accuracy on the test set. This by far surpasses the
other models for all values of \( \lambda \).
<p>
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