dsp {itdr}

Distance Between Two Subspaces.

Description

The “dsp()” function calculates the distance between two subspaces, which are spanned by the columns of two matrices.

Usage

dsp(A, B)

Arguments

A

A matrix with dimension p-by-d.

B

A matrix with dimension p-by-d.

Details

Let A and B be two full rank matrices of size p \times q. Suppose \mathcal{S}(\textbf{A}) and \mathcal{S}(\textbf{B}) are the column subspaces of matrices A and B, respectively. And, let \lambda_i 's with 1 \geq \lambda_1^2 \geq \lambda_2^2 \geq,\cdots,\lambda_p^2\geq 0, be the eigenvalues of the matrix \textbf{B}^T\textbf{A}\textbf{A}^T\textbf{B}.

1.Trace correlation, (Hotelling, 1936):

\gamma=\sqrt{\frac{1}{p}\sum_{i=1}^{p}\lambda_i^2}

2.Vector correlation, (Hooper, 1959):

\theta=\sqrt{\prod_{i=1}^{p}\lambda_i^2}

Value

Outputs are the following scale values.

r

One mines the trace correlation. That is, r=1-\gamma

q

One mines the vector correlation. That is, q=1-\theta

References

Hooper J. (1959). Simultaneous Equations and Canonical Correlation Theory. Econometrica 27, 245-256.

Hotelling H. (1936). Relations Between Two Sets of Variates. Biometrika 28, 321-377.