A novel Spatio-temporal principal component analysis based on Geary's contiguity ratio

Multivariate statistics have gained a respectable place in quantitative research, especially in the economic geography, socio-economic development, urban and regional planning and spatio-temporal analysis. The main goal is to reduce multidimensional data to simple, but meaningful representative information. One of the powerful methods in multivariate statistics is Principal Component Analysis (PCA). The aim of this paper is to define a novel Spatio-Temporal Principal Component Analysis (STPCA). It is the first solution that sensibly combines at the same time variability of the values of the observed features, time of observation of the considered features and place of observation. It is therefore a solution for spatio-temporal data and a very valuable tool for practitioners wishing to obtain useful inferences from a PCA. The inclusion of the time and place of observation, in addition to the variability of the values of features, results in more detailed division of the examined objects into homogeneous clusters. Space and time, which interact with each other, are used on equal terms in the construction of the STPCA. The definition of these principal components is based on the product of two factors. The first factor is equal to the variance of the functional principal components, and the second factor is Geary's contiguity ratio C. The proposed new method of Spatio-Temporal Principal Components was used to show the mutual location of 16 Polish regions characterized by 12 socio-economic features observed in the years 2002–2018 in the system of the first two principal components and to identify homogeneous clusters of these regions in the system of all 15 constructed principal components.