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 density. 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). 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 impact metric), and is defined so as to maximise probability density for a given value of the impact metric

As a result, for unit vectors:

• The first PCA pattern always has the higher explained variance, but may have a low value of the impact metric (e.g., the total rainfall anomaly)
• The first DCA pattern always has a higher value of the impact metric, but may have a low value of the explained variance

Example

Figure 1: PCA and DCA vectors in a two dimensional example.

Figure 1 gives an example, which can be understood as follows.

• The two axes represent anomalies of annual mean rainfall at two locations, with the highest rainfall anomaly values towards the top right corner of the diagram
• The joint variability of the rainfall anomalies at the two locations is assumed to follow a bivariate normal distribution
• The ellipse shows a single contour of probability density, with higher values inside the ellipses
• The red dot at the centre of the ellipse shows zero rainfall anomalies at both locations
• The blue parallel-line arrow shows the principal axis of the ellipse, which is also the first PCA pattern vector
• In this case, the PCA pattern is scaled so that it touches the ellipse
• The diagonal straight line shows a line of constant positive total rainfall anomaly, assumed to be at some fairly extreme level
• The red dotted-line arrow shows the first DCA pattern, which points towards the point at which the diagonal line is tangent to the ellipse
• In this case, the DCA pattern is scaled so that it touches the ellipse

From this diagram, the DCA pattern can be seen to possess the following properties:

• Of all the points on the diagonal line, it is the one with the highest probability density
• Of all the points on the ellipse, it is the one with the highest total rainfall anomaly
• It has the same probability density as the PCA pattern, but represents higher total rainfall than the PCA pattern (i.e., points further towards the top right hand corner of the diagram)
• Any change of the DCA pattern will reduce either the probability density (if it moves out of the ellipse) or reduce the total rainfall anomaly (if it moves into the ellipse)

In this case the total rainfall anomaly of the PCA pattern is quite small, because of anticorrelations between the rainfall anomalies at the two locations. As a result, PCA is not a good representative example of a pattern with large total rainfall anomaly.

In $n$ dimensions the ellipse becomes an ellipsoid, the diagonal line becomes an $n-1$ dimensional plane, and the PCA and DCA patterns are still vectors.

Application to Climate Variability

DCA has been applied to data-sets of historical climate variability in order to understand the most likely patterns of rainfall extremes in the US and China .

Application to Weather Ensembles

DCA has been applied to medium-range weather forecast ensembles in order to identify the most likely patterns of extreme temperatures.

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 as single samples from of a multivariate normal distribution with mean zero and covariance matrix $C$.

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

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

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

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.