Tutorial on Continuously Observational Hidden Markov Model

Hidden Markov models (HMM) are a fundamental tool in statistical modeling for prediction and recognition tasks. This paper extends the understanding of HMMs by focusing on cases where observations are continuous, such as real numbers or vectors, instead of discrete values. While the states in HMMs remain discrete, incorporating continuous observations significantly enhances the model's capability to solve complex problems. The research presented here delves into HMMs characterized by single probabilistic distributions and explores mixture HMMs, where observations are described by a mixture model of partial probability density functions. The paper provides mathematical proofs and practical techniques crucial for implementing continuously observational HMMs, thereby offering a comprehensive tutorial for researchers seeking to deepen their understanding and application of these models.