Technical note: An illustrative introduction to the domain dependence of spatial principal component patterns

Abstract

Principal Component Analysis (PCA) of synchronous time series of one variable, e.g. water level or discharge, measured at multiple locations, has been applied in a wide spectrum of hydrological analyses. The possibility that the Principal Components (PCs) can exhibit domain dependence (DD) found only little recognition in the hydrological PCA literature so far. DD describes the situation in which the spatial PC patterns are mainly determined by the spatial extent of the analysed data set (domain size) and the spatial arrangement of the data set’s locations (domain shape). Thus, instead of the hydrological functioning of the analysed system, the spatial PC patterns rather reflect the functioning of the PCA within the context of the data set’s spatial domain. The effect is caused by homogeneous spatial autocorrelation in the analysed series. DD patterns are distinct, with strong gradients and contrasts. We show that it can come together with substantial accumulation of variance in the leading PCs. In addition, DD can cause effectively degenerate multiplets, i.e. PCs which are not well separable. All these features are highly suggestive and easily lead to wrong hydrological interpretations. Consequently, DD should be considered for any application in which the PCs are used to draw conclusions about spatially distinct properties of the analysed system. For most practical applications checking the first few leading PC patterns should be sufficient. Visual comparison of the spatial PC patterns from subdomains with markedly different shapes and/or sizes can serve as quick qualitative check. Reference patterns can be used to test whether spatial PC patterns differ significantly from pure DD patterns. We present two methods, one stochastic, one analytic, to calculate DD reference patterns for defined spatial correlation properties and arbitrary spatial domains. With a series of synthetic examples, we explore the DD effect with respect to (a) domain shape, (b) domain size and spatial correlation length and (c) effectively degenerate multiplets. Particular focus is given to the effect of DD on the explained variance of the PCs and the contrasts of their spatial patterns. An application example with a precipitation raster data set is presented and different options to detect and diminish DD are discussed. Accompanying this technical note, R-scripts to (i) demonstrate and explore the DD effect, and (ii) perform the presented DD reference methods are provided.