As we noted in Multivariate Normal Distribution Basic Concepts, the 1 – α hyper-ellipse plays a similar role to the confidence interval in univariate statistics. We now describe some additional characteristics of these hyper-ellipses, and in particular, their relationship with eigenvalues.
Observation: The axes of the ellipse have lengths which are a function of the eigenvalues of the Σ. Also each axis is in the direction of the eigenvector corresponding to one of the eigenvalues.
Figure 1 – Confidence ellipse
The half length of the axis corresponding to eigenvalue λ is given by the formula
where is the critical value for χ2(k).
All these properties extend to the case where k > 2. In this case we speak of hyper-ellipses (or ellipsoids) instead of ellipses.
Observation: The volume of the hyper-ellipse (or area in the case where k = 2) is given by the function
Note too that by Property 1 of Eigenvalues and Eigenvectors, |Σ| = product of the eigenvalues of Σ.
Example 1: Calculate the eigenvalues and eigenvectors of the sample covariance matrix for Example 1 of Descriptive Multivariate Statistics.
Figure 2 – Eigenvalues and eigenvectors for Example 1
Using the supplemental array function eVECTORS(H5:L9) we can generate the output displayed in Figure 2. Alternatively, we can get the same result by using the supplemental Matrix data analysis tool (as described in Matrix Operations and Supplemental Data Analysis Tools), and choosing the Eigenvalue/eigenvector option. The top row of Figure 2 lists the 5 eigenvalues. Below each eigenvalue is a corresponding unit eigenvector.
As noted above, the half-lengths of the axes corresponding to the eigenvalues are
where is the critical value for χ2(5).
Using the sample covariance matrix as an approximation for the population covariance matrix and the eigenvalues obtained in Figure 2, the lengths of these axes are:
Figure 3 – Axes lengths for Example 1
The volume of the 95% confidence ellipse is 24,842,086 calculated as follows:
Figure 4 – Volume of 95% confidence ellipse for Example 1
Note that |S| (cell A12) can also be calculated as the product of the five eigenvalues.