## Appendix 12.3: Pattern Correlation Methods

The pattern correlation methods discussed in this section are closely related
to optimal detection with one signal pattern. Pattern correlation studies use
either a centred statistic, *R*, which correlates observed and signal anomalies
in space relative to their respective spatial means, or an uncentred statistic,
*C* (Barnett and Schlesinger, 1987), that correlates these fields without
removing the spatial means. It has been argued that the latter is better suited
for detection, because it includes the response in the global mean, while the
former is more appropriate for attribution because it better measures the similarity
between spatial patterns. The similarity between the statistics is emphasised
by the fact that they can be given similar matrix-vector representations. In
the one pattern case, the optimal (regression) estimate of signal amplitude
is given by

The uncentred statistics may be written similarly as

where **I** is the *n**n*
identity matrix. Similarly, the centred statistic can be written (albeit with
an extra term in the denominator) as

where **U** is the *n**n*
matrix with elements *u*_{i,j}=1/*n*. The matrix **U**
removes the spatial means. Note that area, mass or volume weighting, as appropriate,
is easily incorporated into these expressions. The main point is that each statistic
is proportional to the inner product with respect to a matrix "kernel" between
the signal pattern and the observations (Stephenson, 1997). In contrast with
the pattern correlation statistics, the optimal signal amplitude estimate, which
is proportional to a correlation coefficient using the so-called Mahalonobis
kernel (Stephenson, 1997), maximises the signal-to-noise ratio.