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<title>Z-Score</title>
<!-- 2018-09-17 Mon 17:03 -->
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<div id="content">
<h1 class="title">Z-Score</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#sec-1">Intro</a></li>
<li><a href="#sec-2">Setup</a></li>
<li><a href="#sec-3">Standardization</a>
<ul>
<li><a href="#sec-3-1">Mean</a></li>
<li><a href="#sec-3-2">Variance</a></li>
<li><a href="#sec-3-3">Standard Deviation</a></li>
<li><a href="#sec-3-4">Standardizing</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-sec-1" class="outline-2">
<h2 id="sec-1">Intro</h2>
<div class="outline-text-2" id="text-1">
<p>
The <code>z score</code> is a method for evaluating how correlated 2 sets of values are. It tries to answer the question "If your <code>x</code> increases or decreases, how does <code>y</code> respond?". It does this by trying to eliminate any offset and scaling and then comparing the values to each other. Then finally through a "least squares fit" it attempts to give an quantatative answer to the question.
</p>
</div>
</div>
<div id="outline-container-sec-2" class="outline-2">
<h2 id="sec-2">Setup</h2>
<div class="outline-text-2" id="text-2">
<p>
To start we import ELisp linear algebra function we've developed
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(load-file <span style="color: #ff1f8b;">"matrix.el"</span>)
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-3" class="outline-2">
<h2 id="sec-3">Standardization</h2>
<div class="outline-text-2" id="text-3">
<p>
The first step is gettimg each set of values into a standard form with mean zero and a standard deviation of one
</p>
</div>
<div id="outline-container-sec-3-1" class="outline-3">
<h3 id="sec-3-1">Mean</h3>
<div class="outline-text-3" id="text-3-1">
<p>
Add up all the elements of the data and divide by the number of values. This is really easy to do with matrices
</p>
\begin{equation}
\begin{bmatrix}
1/n
\end{bmatrix}
\begin{bmatrix}
x_1 & x_{2} & x_{3} & x_{4} && ..\\
\end{bmatrix}
\begin{bmatrix}
1\\
1\\
1\\
1\\
..\\
\end{bmatrix}
=
\begin{bmatrix}
1/n
\end{bmatrix}
\begin{bmatrix}
x_1 + x_{2} + x_{3} + x_{4} + ..\\
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-emacs-lisp"> (<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">column-of-ones</span> (size)
<span style="color: #cc0000;">"Make a column vector full of ones"</span>
(matrix-from-data-list size 1 (make-list size 1)))
(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">mean</span> (row-vector)
<span style="color: #cc0000;">"Get the mean of the elements of data"</span>
(<span style="color: #00af00;">let*</span> ((size (matrix-columns row-vector))
(sum (matrix-get-value
(matrix-product row-vector (col-of-ones size))
0
0)))
(/ sum size)))
<span style="color: #b2b2b2; font-style: italic;">;; </span><span style="color: #b2b2b2; font-style: italic;">(mean (matrix-from-data-list 1 4 (list 1.0 2.0 3.0 4.0)))</span>
<span style="color: #b2b2b2; font-style: italic;">;; </span><span style="color: #b2b2b2; font-style: italic;">=> 2.5</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-3-2" class="outline-3">
<h3 id="sec-3-2">Variance</h3>
<div class="outline-text-3" id="text-3-2">
<p>
is the average of how far points are from the mean
</p>
\begin{equation}
\begin{bmatrix}
1/n
\end{bmatrix}
\begin{bmatrix}
x_1-\mu & x_{2}-\mu & x_{3}-\mu & x_{4}-\mu && ..\\
\end{bmatrix}
\begin{bmatrix}
x_1-\mu\\
x_{2}-\mu\\
x_{3}-\mu\\
x_{4}-\mu\\
..\\
\end{bmatrix}
=
\begin{bmatrix}
1/n
\end{bmatrix}
\begin{bmatrix}
(x_1-\mu)^2 + (x_{2}-\mu)^2 + (x_{3}-\mu)^2 + (x_{4}-\mu)^2 + ..\\
\end{bmatrix}
\end{equation}
<p>
(the numbers are squared.. <a href="https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia">which is the numerically easy version of taking the absolute values</a>)
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">row-of-values</span> (value size)
<span style="color: #cc0000;">"Make a row with a VALUE repeated SIZE number of times"</span>
(matrix-from-data-list 1 size (make-list size value)))
(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">variance</span> (row-vector)
<span style="color: #cc0000;">"Get the variance of the elements of data"</span>
(<span style="color: #00af00;">let*</span> ((row-mean (mean row-vector))
(row-of-deviations (matrix-subtract
row-vector
(row-of-values row-mean (matrix-columns row-vector))))
(sum-of-squares (matrix-get-value
(matrix-product row-of-deviations (matrix-transpose row-of-deviations))
0
0)))
(/ sum-of-squares row-mean)))
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-3-3" class="outline-3">
<h3 id="sec-3-3">Standard Deviation</h3>
<div class="outline-text-3" id="text-3-3">
<div class="org-src-container">
<pre class="src src-emacs-lisp">(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">standard-deviation</span> (row-vector)
<span style="color: #cc0000;">"Get the standard deviation of the elements of data"</span>
(sqrt (variance row-vector)))
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-3-4" class="outline-3">
<h3 id="sec-3-4">Standardizing</h3>
<div class="outline-text-3" id="text-3-4">
<p>
is taking each point, subtracting out the mean (so they all lie around zero) and dividing by the standard deviation (so the "spread" is "standard)
</p>
<div class="org-src-container">
<pre class="src src-emacs-lisp">(<span style="color: #00af00;">defun</span> <span style="color: #ef2929;">standardize</span> (row-vector)
<span style="color: #cc0000;">"See text.."</span>
(<span style="color: #00af00;">let*</span> ((row-mean (mean row-vector))
(row-standard-deviation (standard-deviation row-vector))
(row-of-deviations (matrix-subtract
row-vector
(row-of-values row-mean (matrix-columns row-vector)))))
(matrix-scalar-product
row-of-deviations
(/ 1 row-standard-deviation))))
</pre>
</div>
<p>
The resulting values are the <b>z-scores</b>
</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: George Kontsevich</p>
<p class="date">Created: 2018-09-17 Mon 17:03</p>
<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 25.2.2 (<a href="http://orgmode.org">Org</a> mode 8.2.10)</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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