MATLAB Implementation of Probabilistic Principal Component Analysis (PPCA)

Probabilistic Principal Component Analysis (PPCA) is a classical dimensionality reduction technique that extends traditional PCA through a probabilistic model, enabling better handling of noise and missing values in data. Implementing PPCA in MATLAB typically involves the following key steps: First, initialize model parameters including the dimensionality of latent variables, mean vectors, and noise variance. PPCA assumes data is generated from latent variables through linear transformation with added Gaussian noise. The latent variable dimensionality determines the number of features after dimensionality reduction and usually needs to be predefined during implementation using parameters like 'latent_dim' or through cross-validation. Second, iteratively optimize model parameters using the Expectation-Maximization (EM) algorithm. In the E-step, compute the posterior distribution of latent variables, which involves calculating conditional expectations using matrix operations like 'inv(sigma)*W' where W is the transformation matrix. The M-step updates the linear transformation matrix and noise parameters through maximum likelihood estimation, typically implemented using covariance matrix calculations and eigenvalue decomposition. This iterative process continues until model parameters converge (measured by tolerance thresholds) or reach the preset maximum number of iterations. The implementation often includes convergence checks using norms of parameter differences between iterations. Finally, PPCA results can be used for data dimensionality reduction or new sample generation. For dimensionality reduction, data is mapped to low-dimensional space through the posterior mean of latent variables using the transformation 'W'*(x - mu). For generating new samples, one can sample from the prior distribution of latent variables and obtain high-dimensional data through linear transformation using the learned parameters. The MATLAB program typically includes preprocessing steps like data standardization using 'zscore' or manual normalization to ensure different features have the same scale, preventing numerical computation issues. PPCA implementation not only applies to complete datasets but can also be extended to handle partially missing values through modifications in the E-step calculations, making it more flexible for practical applications. Key functions involved may include 'pca', 'eig', matrix inversion operations, and custom EM algorithm loops.