Confidence Hyper-ellipse and Eigenvalues

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.

Confidence elleipse

Figure 1 – Confidence ellipse

The half length of the axis corresponding to eigenvalue λ is given by the formula

Half axis confidence ellipse

where \chi^2_{crit} 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

volume of hyper-ellipse

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.

Eigenvalue eigenvectors covariance Excel

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

Half axis confidence ellipse

where \chi^2_{crit} 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:

Axes lengths hyper-ellipse

Figure 3 – Axes lengths for Example 1

The volume of the 95% confidence ellipse is 24,842,086 calculated as follows:

Confidence ellipse volume

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.

7 Responses to Confidence Hyper-ellipse and Eigenvalues

  1. Manu says:

    Hi Charles,
    Have you a reference for the function that compute the volume of the hyper-ellipse?
    Thank you.

  2. Mattucci says:

    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?

    • Charles says:

      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).


  3. Bandar says:

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

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