Principal component model in macroseismicity

Abstract

Seismic processes are complex and diverse, since their formation is caused by complex, diverse geological and geophysical processes occurring in the Earth’s interior, and are characterized by a set of parameters, and the results of observations over them are presented as multidimensional random variables. When studying such multiparametric processes, the question arises: can we discard some of the parameters, or replace them with a smaller number of some functions from them, while preserving all the information? To solve this problem, we use factor analysis, which is based on determining the minimum number of factors that make up the largest share in the data variance. In the study of the complex nature of seismicity, factor analysis helps to understand better the essence of seismic processes, since the interdependence between seismic parameters must be due to the relationships between parameters, the identification of which is the task of factor analysis.

In order for the regression analysis based on the usual least squares method to give the best results, the random error must satisfy the Gauss-Markov conditions: the mathematical expectation of the random error in any observation must be zero, which means it should not have a systematic bias. Usually, if the regression equation includes a free term, then this means that the condition is satisfied automatically, since the role of the constant is to determine any systematic tendency of the explained variable included in the regression equation. Multicollinearity means a high cross-correlation of the explanatory regression variables. The lack of high collinearity of the regressors is one of the conditions for applying the least squares method to estimate the parameters of multidimensional linear regression. To assess the values of the coefficients of the attenuation function, in the presence of multicollinearity, we use regression analysis on the main components, where the strongly correlated regressors are replaced by components F₁, F₂, F₃, F₄, identified by the model of the main components of factor analysis, between which there is no correlation.