3.4 Measures of Multivariate Scatter

The matrix \(S\) is one important matrix in multivariate generalisation of the univariate notion of variance, measures scatter about the mean. However, sometimes it is convenient to have a single number to measure multivariate scatter. Two common such measures are:

1.: The generalised variance \(|S|\) and
2.: The total variance, \(Tr(S)\).

for both measures, large values indicate a high degree of scatter about \(\overline {X}\) and lower values indicate concentration about \(\overline {X}\). However, each measure reflects different aspects of the availability in the data. Then generalised variance plays an important role in maximum likelihood estimation and the total variance is an important concept in principal component analysis.

Example 3.5. (\(M\) G nanadesikam and Gupta 1970)
An experimental subject spoke to 10 different Ways, 7 times each and 5 speech taken at each entrance. The \(5\times 5\) Covariance matrix for each word the generalised variances were as follows:

2.9	1.3	641.6	26828.8
262404.3		169.2	3106.8
617671.2		6.7	3.4

ordering these generalised variances we find that the first speaker the second has the least variation and the \(8^{\text {th}}\) word has the most variation.

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