fix typo

preliminaries/index.md

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             <img src="00.Preliminaries.key-stage-0138anim0.svg" data-images="00.Preliminaries.key-stage-0138anim0.svg,00.Preliminaries.key-stage-0138anim1.png,00.Preliminaries.key-stage-0138anim2.png,00.Preliminaries.key-stage-0138anim3.png" class="slide-image" />
 
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-            <p    >Here is an illustration of the way the covariance matrix affects the data we get from a multivariate normal distribution. The mean is at (0, 0) in all four examples. <br></p><p    >If the covariance is<span> the identity matrix</span>, we get the standard normal distribution. This is called a spherical distribution, because the variance along all axes is the same, and there is no correlation between axes, giving the data roughly spherical shape. <br></p><aside    >More precisely the lines of equal probability density are circles in 2D and spherical surfaces in higher dimensions.<br></aside><p    >If we <span class="blue">change the values on the diagonal</span>, we stretch this sphere into an ellipse, but only along the axes. There is still no correlation: knowing the value along one axis tells us nothing about the value along the others. <br></p><p    >If <span class="red">we change the off-diagonal values to positive values</span> we get <strong>correlation</strong>. In this case having a high value along one axis makes it more likely that the value along the other axis is also high. Note that the coviarance matrix needs to be symmetric, so the value on one side of the diagonal must be the same as the value on the other side.<br></p><p    >If <span>the off-diagonal value is negative</span>, we get <strong>anti-correlation</strong>. A high positive value on one axis most likely corresponds to a high negative value along the other axis.<br></p><p    >If we have more than 2 dimensions, say n, then there are (n^2 - n)/2 possible pairs of axes between which we can define a correlation. any of these could be positive, negative or 0. This corresponds exactly to the number of values above the diagonal in an n x n matrix.</p><p    ></p>
+            <p    >Here is an illustration of the way the covariance matrix affects the data we get from a multivariate normal distribution. The mean is at (0, 0) in all four examples. <br></p><p    >If the covariance is<span> the identity matrix</span>, we get the standard normal distribution. This is called a spherical distribution, because the variance along all axes is the same, and there is no correlation between axes, giving the data roughly spherical shape. <br></p><aside    >More precisely the lines of equal probability density are circles in 2D and spherical surfaces in higher dimensions.<br></aside><p    >If we <span class="blue">change the values on the diagonal</span>, we stretch this sphere into an ellipse, but only along the axes. There is still no correlation: knowing the value along one axis tells us nothing about the value along the others. <br></p><p    >If <span class="red">we change the off-diagonal values to positive values</span> we get <strong>correlation</strong>. In this case having a high value along one axis makes it more likely that the value along the other axis is also high. Note that the covariance matrix needs to be symmetric, so the value on one side of the diagonal must be the same as the value on the other side.<br></p><p    >If <span>the off-diagonal value is negative</span>, we get <strong>anti-correlation</strong>. A high positive value on one axis most likely corresponds to a high negative value along the other axis.<br></p><p    >If we have more than 2 dimensions, say n, then there are (n^2 - n)/2 possible pairs of axes between which we can define a correlation. any of these could be positive, negative or 0. This corresponds exactly to the number of values above the diagonal in an n x n matrix.</p><p    ></p>
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             <span class="hint">click image for animation</span>
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