4.8 - Special Cases: p = 2

4.8 - Special Cases: p = 2

To further understand the shape of the multivariate normal distribution, let's return to the special case where we have p = 2 variables.

If \(ρ = 0\), there is zero correlation, and the eigenvalues turn out to be equal to the variances of the two variables. So, for example, the first eigenvalue would be equal to \(\sigma^2_1\) and the second eigenvalue would be equal to \(\sigma^2_2\)  as shown below:

\(\lambda_1 = \sigma^2_1\) and \(\lambda_2 = \sigma^2_2\)

the corresponding eigenvectors will have elements 1 and 0 for the first eigenvalue and 0 and 1 for the second eigenvalue.

\(\mathbf{e}_1 = \left(\begin{array}{c} 1\\ 0\end{array}\right)\), \(\mathbf{e}_2 = \left(\begin{array}{c} 0\\ 1\end{array}\right)\)

So, the axis of the ellipse, in this case, are parallel to the coordinate axis.

If there is zero correlation, and the variances are equal so that \(\sigma^2_1\) = \(\sigma^2_2\), then the eigenvalues will be equal to one another, and instead of an ellipse we will get a circle. In this special case, we have a so-called circular normal distribution.

SAS plot

If the correlation is greater than zero, then the longer axis of the ellipse will have a positive slope.

SAS plot

Conversely, if the correlation is less than zero, then the longer axis of the ellipse will have a negative slope.

As the correlation approaches plus or minus 1, the larger eigenvalue will approach the sum of the two variances, and the smaller eigenvalue will approach zero:

\(\lambda_1 \rightarrow \sigma^2_1 +\sigma^2_2\) and \(\lambda_2 \rightarrow 0\)

So, what is going to happen in this case is that the ellipse becomes more and more elongated as the correlation approaches one.

SAS plot

Using Technology

Using SAS

The SAS program below can be used to plot the 95% confidence ellipse corresponding to any specified variance-covariance matrix.

Download the SAS program here: ellplot.sas

Using Minitab

 The bivaraite confidence interval for this example cannot be generated using Minitab.


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