Directional component analysis (DCA)^[1][2][3] is a statistical method used in climate science for identifying representative patterns of variability in space-time data-sets such as historical climate observations,^[1] weather prediction ensembles^[2] or climate ensembles.^[3]

The first DCA pattern is a pattern of weather or climate variability that is both likely to occur (measured using likelihood) and has a large impact (for a specified linear impact function, and given certain mathematical conditions: see below).

The first DCA pattern contrasts with the first PCA pattern, which is likely to occur, but may not have a large impact, and with a pattern derived from the gradient of the impact function, which has a large impact, but may not be likely to occur.

DCA differs from other pattern identification methods used in climate research, such as EOFs,^[4] rotated EOFs^[5] and extended EOFs^[6] in that it takes into account an external vector, the gradient of the impact.

DCA provides a way to reduce large ensembles from weather forecasts^[2] or climate models^[3] to just two patterns. The first pattern is the ensemble mean, and the second pattern is the DCA pattern, which represents variability around the ensemble mean in a way that takes impact into account. DCA contrasts with other methods that have been proposed for the reduction of ensembles^[7][8] in that it takes impact into account in addition to the structure of the ensemble.

Overview

Two Dimensional Example

Source:^[1]

Figure 1: PCA (blue) and DCA (red) 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 total 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 from this bivariate normal, with higher values inside the ellipse
• 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 spatial 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 (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 along or 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, the first PCA pattern is not a good representative example of a pattern with large total rainfall anomaly, while the first DCA pattern is.

In ${\displaystyle n}$ dimensions the ellipse becomes an ellipsoid, the diagonal line becomes an ${\displaystyle n-1}$ dimensional plane, and the PCA and DCA patterns are vectors in ${\displaystyle n}$ dimensions.

Applications

Derivation of the First DCA Pattern

Source:^[1]

Consider a space-time data-set ${\displaystyle X}$, containing individual spatial pattern vectors ${\displaystyle x}$, where the individual patterns are each considered as single samples from a multivariate normal distribution with mean zero and covariance matrix ${\displaystyle C}$.

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

We define a linear impact function of a spatial pattern as ${\displaystyle r^{t}x}$, where ${\displaystyle r}$ 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

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

Differentiating by ${\displaystyle x}$ and setting to zero gives the solution

${\displaystyle x\propto Cr}$

Normalising so that ${\displaystyle x}$ is unit vector gives

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

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.

References