Variograma marcado na análise de processos pontuais espaçotemporais com marcas

Abstract

Spatial statistics has been widely studied and applied to several phenomena, such as seismic events, forest fires, infection by a specific disease, location of plants of a specific species in a forest, crimes in a city, among others. These types of phenomena can be characterized as realizations of point processes, in which each observed occurrence can be represented as a point in two-dimensional space. If d = 1 there is a one-dimensional point process or a temporal point process (TPP), if d = 2 there is a spatial point process (SPP), and if there is both information about space and time, which has is a space-time point process (STPP). In many cases, in addition to the location and timing of the events, there is also additional information called a mark, which is a specific attribute of the observed event, such as the height of each tree. In this context, the point process is said to be marked, and a tool that provides valuable information about the correlation structure of the marks is the marked variogram, which measures the degree of correlation between them. In addition, the marked variogram can also be used as a tool for the analysis of an unmarked STPP, transforming it into two marked point processes, treating the times as marks of the locations and after, the locations as marks of the times. The marked variogram, however, has not yet been explored as a tool for the analysis of data from a spacetime marked point process (STMPP). Given this context, the main objective of this work was to present a spatio-temporal version of the marked variogram, for joint analysis of the spatial and temporal components of data from a STMPP, to verify whether this approach presents gains in relation to an analysis considering these components separately. The proposed method was used in simulated data under the null model of random labeling (independent and identically distributed marks (iid) and independent of points), to verify the compatibility between the theoretical and the calculated result, and in real data of heat points obtained by remote sensing by satellites. The heat point mark is Radiative Fire Power, which is a measure of the radiant energy of the fire during the burning process, and is associated with the amount of biomass burned in a fire, for example. The data were obtained from the National Institute for Space Research - INPE and were for the years 2018 and 2019, for the municipality of Itapuã do Oeste - RO, in the Amazon region. It was found that the graphic structure of the proposed marked variogram is consistent with the theoretical result for the null model of random labeling, and that in the analysis of the data of heat points, the result showed gains in relation to the marginal analysis of the spatial and temporal components , since it was possible to identify a correlation structure for the variability of the marks, in several space-time scales that were not observed when considering the processes separately.