07_RegressionModels/01_02_notation/index.html

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        <h1>Some basic notation and background</h1>
        <h2>Regression</h2>
        <p>Brian Caffo, PhD<br/>Johns Hopkins Bloomberg School of Public Health</p>
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  <hgroup>
    <h2>Some basic definitions</h2>
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  <article>
    <ul>
<li>In this module, we&#39;ll cover some basic definitions and notation used throughout the class.</li>
<li>We will try to minimize the amount of mathematics required for this class.</li>
<li>No caclculus is required. </li>
</ul>

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  <hgroup>
    <h2>Notation for data</h2>
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  <article>
    <ul>
<li>We write \(X_1, X_2, \ldots, X_n\) to describe \(n\) data points.</li>
<li>As an example, consider the data set \(\{1, 2, 5\}\) then 

<ul>
<li>\(X_1 = 1\), \(X_2 = 2\), \(X_3 = 5\) and \(n = 3\).</li>
</ul></li>
<li>We often use a different letter than \(X\), such as \(Y_1, \ldots , Y_n\).</li>
<li>We will typically use Greek letters for things we don&#39;t know. 
Such as, \(\mu\) is a mean that we&#39;d like to estimate.</li>
<li>We will use capital letters for conceptual values of the variables and lowercase letters for realized values.

<ul>
<li>So this way we can write \(P(X_i > x)\). </li>
<li>\(X_i\) is a conceptual random variable.</li>
<li>\(x\) is a number that we plug into.</li>
</ul></li>
</ul>

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    <h2>The empirical mean</h2>
  </hgroup>
  <article>
    <ul>
<li>Define the empirical mean as
\[
\bar X = \frac{1}{n}\sum_{i=1}^n X_i. 
\]</li>
<li>Notice if we subtract the mean from data points, we get data that has mean 0. That is, if we define
\[
\tilde X_i = X_i - \bar X.
\]
The the mean of the \(\tilde X_i\) is 0.</li>
<li>This process is called &quot;centering&quot; the random variables.</li>
<li>The mean is a measure of central tendancy of the data.</li>
<li>Recall from the previous lecture that the mean is 
the least squares solution for minimizing
\[
\sum_{i=1}^n (X_i - \mu)^2
\]</li>
</ul>

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  <hgroup>
    <h2>The emprical standard deviation and variance</h2>
  </hgroup>
  <article>
    <ul>
<li>Define the empirical variance as 
\[
S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2 
= \frac{1}{n-1} \left( \sum_{i=1}^n X_i^2 - n \bar X ^ 2 \right)
\]</li>
<li>The empirical standard deviation is defined as
\(S = \sqrt{S^2}\). Notice that the standard deviation has the same units as the data.</li>
<li>The data defined by \(X_i / s\) have empirical standard deviation 1. This is called &quot;scaling&quot; the data.</li>
<li>The empirical standard deviation is a measure of spread.</li>
<li>Sometimes people divide by \(n\) rather than \(n-1\) (the latter
produces an unbiased estimate.)</li>
</ul>

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  <hgroup>
    <h2>Normalization</h2>
  </hgroup>
  <article>
    <ul>
<li>The the data defined by
\[
Z_i = \frac{X_i - \bar X}{s}
\]
have empirical mean zero and empirical standard deviation 1. </li>
<li>The process of centering then scaling the data is called &quot;normalizing&quot; the data. </li>
<li>Normalized data are centered at 0 and have units equal to standard deviations of the original data. </li>
<li>Example, a value of 2 form normalized data means that data point
was two standard deviations larger than the mean.</li>
</ul>

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  <hgroup>
    <h2>The empirical covariance</h2>
  </hgroup>
  <article>
    <ul>
<li>Consider now when we have pairs of data, \((X_i, Y_i)\).</li>
<li>Their empirical covariance is 
\[
Cov(X, Y) = 
\frac{1}{n-1}\sum_{i=1}^n (X_i - \bar X) (Y_i - \bar Y)
= \frac{1}{n-1}\left( \sum_{i=1}^n X_i Y_i - n \bar X \bar Y\right)
\]</li>
<li>Some people prefer to divide by \(n\) rather than \(n-1\) (the latter
produces an unbiased estimate.)</li>
<li>The correlation is defined is
\[
Cor(X, Y) = \frac{Cov(X, Y)}{S_x S_y}
\]
where \(S_x\) and \(S_y\) are the estimates of standard deviations 
for the \(X\) observations and \(Y\) observations, respectively.</li>
</ul>

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  <hgroup>
    <h2>Some facts about correlation</h2>
  </hgroup>
  <article>
    <ul>
<li>\(Cor(X, Y) = Cor(Y, X)\)</li>
<li>\(-1 \leq Cor(X, Y) \leq 1\)</li>
<li>\(Cor(X,Y) = 1\) and \(Cor(X, Y) = -1\) only when the \(X\) or \(Y\) observations fall perfectly on a positive or negative sloped line, respectively.</li>
<li>\(Cor(X, Y)\) measures the strength of the linear relationship between the \(X\) and \(Y\) data, with stronger relationships as \(Cor(X,Y)\) heads towards -1 or 1.</li>
<li>\(Cor(X, Y) = 0\) implies no linear relationship. </li>
</ul>

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