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. Principal Components (PCs) were used in regionalisation and to identify dominant modes, signals, processes or other hydrological properties of the analysed system. The possibility that the PCs of such analysis 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 size and shape of the analysed spatial domain. Domain size means the spatial extent of the analysed data set, domain shape the spatial arrangement of the data sets´ locations. 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, a common feature in hydrological data sets. DD patterns are distinct, with strong gradients and contrasts, and 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. DD patterns calculated for the analysed spatial domain can be used as reference 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. Finally, considering DD is 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.