Hidden Markov Model (HMM) Toolbox

The Hidden Markov Model (HMM) Toolbox is a powerful framework designed for processing sequential data, with applications spanning speech recognition, bioinformatics, and financial time series analysis. This toolbox provides comprehensive functions for model training, evaluation, and prediction, featuring optimized implementations of core algorithms through MATLAB's vectorization capabilities for enhanced computational efficiency.

### Core Functionality Model Initialization: Users initialize HMMs by specifying the number of states and observation symbols. The implementation involves defining initial state probabilities (pi), state transition matrices (A), and emission probability matrices (B), typically through randomized initialization or user-defined priors. Model Training: The Baum-Welch algorithm (an Expectation-Maximization variant) performs unsupervised learning by iteratively refining model parameters through forward-backward probability calculations. Key functions handle expectation steps (computing state occupancy probabilities) and maximization steps (updating A and B matrices). Sequence Prediction: The Viterbi algorithm decodes the most probable hidden state sequence using dynamic programming, while forward-backward algorithms compute observation sequence likelihoods through recursive alpha/beta probability calculations. Model Evaluation: Log-likelihood scores quantify model-data alignment, with functions implementing scaling techniques to prevent numerical underflow during probability chain multiplication.

### Implementation Example For weather observation sequences (e.g., "sunny/rainy/cloudy"), the toolbox enables: Training: Historical weather data trains the model to learn transition patterns between hidden states (e.g., seasonal trends). Code structures observation sequences as cell arrays or numeric matrices for batch processing. State Prediction: New observation sequences trigger Viterbi decoding to output optimal state paths (e.g., "sunny→cloudy→rainy"), with functions handling variable-length inputs via zero-padding or sliding windows. Probability Computation: Forward algorithms evaluate sequence probabilities for anomaly detection, incorporating log-space calculations for numerical stability in long sequences.

The toolbox's modular design supports customization for diverse sequential modeling tasks, particularly effective for time-series data with latent state dependencies.