Implications of space-time orientation for Principal Components Analysis of Earth observation image time series

Abstract

A time series of geographic images can be viewed from two perspectives: as a set of images, each image representing a slice of time, or as a grid of temporal profiles (one at each pixel location). In the context of Principal Components Analysis (PCA), these different orientations are known as T-mode and S-mode analysis respectively. In the sparse literature on these modes it is recognized that they produce different results, but the reasons have not been fully explored. In this paper we investigate the interactions between space-time orientation and standardization and centering in PCA. Standardization refers to the eigenanalysis of the inter-variable correlation matrix rather than the variance-covariance matrix while centering refers to the subtraction of the mean in the development of either matrix. Using time series of monthly anomalies in lower tropospheric temperature from the Microwave Sounding Unit (MSU) as well as in CO₂ in the middle troposphere from the Atmospheric Infrared Sounder (AIRS), we show that with T-mode PCA, standardization has the effect of giving equal weight to each time step while centering has the effect of detrending over time. In contrast, with S-mode PCA, standardization has the effect of giving equal weight to each location in space while centering detrends over space. Further, in the formation of components, S-mode PCA preferences patterns that are prevalent over space while T-mode PCA preferences patterns that are prevalent over time. The two orientations thus provide complementary insights into the nature of variability within the series.