An Alternative to PCA for Estimating Dominant Patterns of Climate Variability and Extremes, with Application to U.S. and China Seasonal Rainfall

Abstract

Floods and droughts are driven, in part, by spatial patterns of extreme rainfall. Heat waves are driven by spatial patterns of extreme temperature. It is therefore of interest to design statistical methodologies that allow the rapid identification of likely patterns of extreme rain or temperature from observed historical data. The standard work-horse for the rapid identification of patterns of climate variability in historical data is Principal Component Analysis (PCA) and its variants. But PCA optimizes for variance not spatial extremes, and so there is no particular reason why the first PCA spatial pattern should identify, or even approximate, the types of patterns that may drive floods, droughts or heatwaves, even if the linear assumptions underlying PCA are correct. We present an alternative pattern identification algorithm that makes the same linear assumptions as PCA, but which can be used to explicitly optimize for spatial extremes. We call the method Directional Component Analysis (DCA), since it involves introducing a preferred direction, or metric, such as “sum of all points in the spatial field”. We compare the first PCA and DCA spatial patterns for U.S. and China winter and summer rainfall anomalies, using the sum metric for the definition of DCA in order to focus on total rainfall anomaly over the domain. In three out of four of the examples the first DCA spatial pattern is more uniform over a wide area than the first PCA spatial pattern and as a result is more obviously relevant to large-scale flooding or drought. Also, in all cases the definitions of PCA and DCA result in the first PCA spatial pattern having the larger explained variance of the two patterns, while the first DCA spatial pattern, when scaled appropriately, has a higher likelihood and greater total rainfall anomaly, and indeed is the pattern with the highest total rainfall anomaly for a given likelihood. The first DCA spatial pattern is arguably the best answer to the question: what single spatial pattern is most likely to drive large total rainfall anomalies in the future? It is also simpler to calculate than PCA. In combination PCA and DCA patterns yield more insight into rainfall variability and extremes than either pattern on its own.

Presenting Climate Projection Ensembles as Mean and Reasonable Worst Case, with Application to EURO-CORDEX Precipitation

“…We now give a definition and brief overview of DCA, based on Jewson (2020) and SJM. For the ensembles we are considering in this article, we define DCA as follows.…”

“…Equations ( 1) and ( 2) are proportional relationships and define the direction of the DCA vector (i.e., the shape of the spatial pattern described by ), but not the magnitude of the vector (i.e., not the amplitude of the spatial pattern decribed by ). To make the definition of DCA unique, Jewson (2020) defines the first DCA pattern as a unit vector. The length of the unit vector DCA pattern can then be scaled to an appropriate value, depending on the application.…”

“…'Reasonable worst' in the phrase reasonable worst-case means that the probability of this pattern should not be too low, so that the pattern is realistic, and that the pattern should have a large positive value of the linear impact function relative to the members of the ensemble. Two of the four methods that SJM compared were based on Directional Component Analysis (DCA, Jewson (2020)), a third method consisted of taking the worst single member from the ensemble and a fourth method consisted of averaging the worst 10% of the members of the ensemble. They compared these four methods and found the two DCA methods to be the most robust.…”

“…We will expand on the ideas in Jewson (2020) and SJM and explore further the idea of defining a reasonable worst-case pattern from an ensemble using DCA. DCA is a method for deriving extreme patterns from space-time datasets such that the patterns have both a large linear impact and a high likelihood (where we use likelihood synonymously with probability density).…”

“…In this situation, a Europe-wide worst-case might consist of greater than the ensemble mean precipitation in northern Europe and less than the ensemble mean precipitation in southern Europe. Second, we present various additional mathematical properties of DCA that were not described in Jewson (2020) or SJM, that help to clarify the interpretation of DCA. Third, we discuss in detail how the linear impact function used in DCA can be considered as a linearisation of a non-linear impact function, and how the method can be generalized to quadratic and cubic approximations.…”