Probabilistic Modeling of Climate Systems through Markov Chains: A Computational Analysis of the Temporal Dynamics of Meteorological States

Abstract

This work presents a computational analysis of the application of discrete Markov chains in the modeling of simplified climate systems, using three fundamental weather states: Sunny, Cloudy, and Rainy. Through the implementation of a specialized Python class, a framework was developed for analyzing stochastic properties, including stationary distribution, mean return times, and probabilistic convergence. The results show that the model reaches a stationary distribution with a 45.65% probability for sunny days, 28.26% for cloudy days, and 26.09% for rainy days. The convergence analysis revealed that the system stabilizes quickly, reaching the stationary regime in approximately 10 time steps. The calculated mean return times indicated periodicities of 2.19 days for Sun, 3.54 days for Cloudy, and 3.83 days for Rain. Monte Carlo simulations validated the theoretical consistency of the model, showing convergence of empirical frequencies to the predicted theoretical values. This study contributes to the understanding of the applicability of Markovian models in computational meteorology and provides a methodological basis for predictive analyses in discrete dynamic systems.