From d06a8231965117f67578d4eb30d6093bf80d5540 Mon Sep 17 00:00:00 2001 From: Jari Kasandiredjo Date: Wed, 7 Feb 2024 17:33:59 +0100 Subject: [PATCH] fix typo --- preliminaries/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/preliminaries/index.md b/preliminaries/index.md index b11f9ce..14f534a 100644 --- a/preliminaries/index.md +++ b/preliminaries/index.md @@ -1857,7 +1857,7 @@ slides: true
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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.

If the covariance is the identity matrix, 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.

If we change the values on the diagonal, 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.

If we change the off-diagonal values to positive values we get correlation. 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.

If the off-diagonal value is negative, we get anti-correlation. A high positive value on one axis most likely corresponds to a high negative value along the other axis.

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.

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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.

If the covariance is the identity matrix, 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.

If we change the values on the diagonal, 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.

If we change the off-diagonal values to positive values we get correlation. 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.

If the off-diagonal value is negative, we get anti-correlation. A high positive value on one axis most likely corresponds to a high negative value along the other axis.

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.

click image for animation