Gaussian Process State Space Model

In this article, we provide a comprehensive exploration of Gaussian Process State Space Models (GP-SSMs). Gaussian Processes (GPs) represent probability distributions commonly used for modeling continuous random variables. They serve as powerful tools for estimating posterior distributions of unknown functions and predicting future observations. In state space modeling, we apply Gaussian Processes to represent system states and use observational data to update posterior state distributions. This approach finds widespread applications across various fields including finance, biology, engineering, and social sciences. Before delving into GP-SSMs, let's first examine fundamental Gaussian Process theory and its practical implementations. From a computational perspective, GP implementation typically involves kernel function selection (e.g., RBF or Matern kernels) and optimization of hyperparameters through maximum likelihood estimation. Key functions in GP libraries often include covariance matrix computation and Cholesky decomposition for efficient inference.