Directional component analysis

Directional component analysis(DCA) is a statistical method used in the atmospheric sciences for extracting patterns of extremes from space-time data-sets. For instance, in a rainfall dataset, it can be used to find the spatial pattern that has both highest probability density for a given total rainfall anomaly, and, conversely, the highest total rainfall anomaly for a given probability density (subject to certain statistical assumptions about the nature of the variability of data: see below). In comparison with PCA the main differences are that PCA is a function of just the covariance matrix, and is defined so as to maximise explained variance, while DCA is a function of the covariance matrix and a vector direction, and is defined so as to maximise probability density for a given value of the total anomaly.

Derivation of the First DCA Pattern

Consider a space-time data-set $X$, containing individual spatial pattern vectors $x$, where the individual patterns are each considered a single realisation of a multivariate normal distribution with mean zero and covariance matrix $C$.

As a function of $x$, the log probability density is proportional to $-x^{t}C^{-1}x$.

Define a linear impact function of a spatial pattern as $r^{t}x$, where $r$ is a vector of spatial weights.

We seek to find the spatial pattern that maximises the probability density for a given value of the linear impact function, which is equivalent to finding the spatial pattern that maximises the log probability density for a given value of the linear impact function.

This is a constrained maximisation problem, and can be solved using the method of Lagrange multipliers.

The Lagrangian function is given by

$L(x,\lambda )=-x^{t}C^{-1}x-\lambda (r^{t}x-1)$

Differentiating by $x$ and setting to zero gives the solution

$x\propto Cr$

Normalising so that $x$ is unit vector gives

$x=Cr/(r^{t}CCr)^{1/2}$

This is the first DCA pattern.

Subsequent patterns can be derived which are orthogonal to the first, to form an orthonormal set and a method for matrix factorisation.