Directional component analysis

Directional component analysis(DCA) is a statistical method used in the atmospheric sciences for extracting representative extreme patterns from space-time data-sets such as historical climate observations or weather and climate prediction ensembles. It calculates spatial patterns that are both extreme in terms in impact, and likely in terms of probability. Extreme, in this context, is defined as meaning large values for an impact metric that is a weighted sum of the elements of the field. The first DCA pattern is then, by definition, the pattern that maximises probability density for a given level of this impact metric (subject to certain statistical assumptions about the nature of the variability of data: see below). The shape of the pattern is independent of the actual value of the metric. For instance, in a rainfall anomaly dataset the first DCA pattern, using uniform weights, is the spatial pattern that has the highest probability density for a given total rainfall anomaly. If the given total rainfall anomaly is a large one, then this pattern combines being extreme in terms of the metric (i.e., representing large amounts of total rainfall) with being likely in terms of the pattern, and so is well suited as a representative extreme pattern. Any modification of the first DCA pattern will lead to a pattern that is either less extreme, or has a lower probability density.

In comparison with Principal component analysis (PCA) the main differences are that

• PCA is a function of just the covariance matrix, and is defined so as to maximise explained variance
• DCA is a function of the covariance matrix and a vector direction (that defines the metric), and is defined so as to maximise probability density for a given value of the impact metric

As a result, for unit vectors:

• PCA always has the higher explained variance, but may have very low values of the impact metric and the variance of the impact metric
• DCA always has a higher value of the impact metric, but may have low explained variance

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.