Kullback-Leibler Divergence of Mixture Autoregressive Random Processes via Extreme-Value-Distributions (EVDs) Noise with Application of the Processes to Climate Change

Abstract

This paper designs inter-switch autoregressive random processes in a mixture manner with Extreme-Value-Distributions (EVDs) random noises to give EVDs-MAR model. The EVDs-MAR model comprises of Fréchet, Gumbel, and Weibull distributional error terms to form FMA, GMA, and WMA models with their embedded inter-switching transitional weights (wk) , distributional parameters, and autoregressive coefficients . The Kullback-Leibler divergence was used to measure the proximity (D) between finite/ delimited mixture density  and infinite mixture density of the EVDs-MAR model with Expectation-Maximization (EM) algorithm adopted as the parameter estimation technique for the extreme mixture model. The FMA, GMA, and WMA models were subjected to monthly temperature in Celsius (oC) from 1900 to 2020 and annual rainfall in Millimeter (mm) from 1960 to 2020 datasets in Nigeria context.