Markov Chain Simulation

A Markov Chain is a mathematical model used to describe stochastic state transition processes in systems, characterized by its "memoryless" property where the next state depends solely on the current state. Implementing Markov Chain simulations in MATLAB typically involves three key steps:

Defining the State Space The first step requires specifying discrete system states (e.g., "sunny/rainy" for weather modeling). These states form the foundational framework of the Markov chain, commonly represented using numerical indices or categorical labels. In MATLAB implementation, this can be structured as a cell array of state names or a numerical vector for computational efficiency.

Constructing the Transition Probability Matrix This square matrix defines transition probabilities where each element P(i,j) represents the probability of moving from state i to state j. The matrix must satisfy the condition that each row sums to 1, obtainable through historical statistical data or theoretical assumptions. MATLAB's matrix operations facilitate efficient probability matrix manipulation and validation using functions like sum() for row-sum verification.

State Evolution Simulation The simulation employs random number generators to drive state transitions: Initialize starting state using randi() or predefined values For each step, apply the roulette wheel selection algorithm based on the current state's transition probabilities using rand() for random sampling Iterate the process until reaching predetermined simulation steps Key implementation involves cumulative probability calculations with cumsum() and logical indexing for state transitions.

MATLAB excels in handling large-scale state spaces through built-in matrix operations and probability functions like rand(). Typical applications include weather prediction, financial market analysis, and text generation, where adjusting probability matrices simulates different system dynamics. Advanced implementations can incorporate visualization modules using plot() or animatedline() to graphically display state evolution trajectories over time.