How to calculate dispersion for a collection of vectors

Below is the calculation made in bedpostx to calculate the dyads dispersion.

If \(\mathbf{x_1},...,\mathbf{x_n}\) denote the sample dyads, then we can construct the average dyadic tensor \(\mathbf{M}\) (average of the rank-1 tensors implied by the dyads) as follows:

\[\mathbf{M} = \frac{1}{n}\sum_{i=1}^n \mathbf{x_i}\mathbf{x_i}^T.\]

The dispersion \(\rho\) is defined as one minus the largest eigenvalue of \(\mathbf{M}\):

\[\rho = 1-\lambda_1.\]

To understand why this is a measure of dispersion, consider dot products between sample dyads and the largest eigenvector \(\mathbf{v_1}\). Since we have \(\mathbf{Mv_1}=\lambda_1\mathbf{v_1}\), then:

\[ \begin{array}{rcl} \rho & = & 1-\mathbf{v_1}^TM\mathbf{v_1} \\ & = & 1-\mathbf{v_1}^T\frac{1}{n}\sum_{i=1}^n \mathbf{x_i}\mathbf{x_i}^T\mathbf{v_1} \\ & = & 1-\frac{1}{n}\sum_{i=1}^n <\mathbf{x_i}, \mathbf{v_1}>^2 \end{array} \]

Thus, the dispersion is related to the average square dot product between the samples and their average orientation (similar to a variance term).