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 *χ*^{2}(*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 *χ*^{2}(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.

Is this sentence “The axes of the ellipse have lengths which are a function of the eigenvalues” also true for the ellipsoid? Thank you

Yes.

Thank you Charles for these information. In this case what will be the equation of the hyper-ellipses if I wanted to code it in VBA or Fortran?

To code the equation in Fortran or VBA requires the following:

1. Calculate the eigenvalues. This is the hard part, although a description of how to do this is described elsewhere on the Real Statistics website

2. Use the formula under Figure 1 of the referenced webpage for the length of the axes of the hyper-ellipse (based on the eigenvalues)

3. Plus these values into the equation for a hyper-ellipse. The equation for a hyper-ellipse (the equation for an ellipse can be found in any elementary calculus book; the equation for a hyper-ellipse i ssimply the generalization of this equation to more than 2 dimensions).

Charles

Thank you Charles for these explanations.

Hi Charles,

Have you a reference for the function that compute the volume of the hyper-ellipse?

Thank you.

Hi Manu,

See the following webpage:

https://onlinecourses.science.psu.edu/stat505/node/36

Charles