HMM is useful for temporal pattern recognition of time series data and is primarily used for speech recognition, context prediction, protein and DNA structure analysis, and asset price prediction. HMM is based on the Markov chain, which is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. The HMM assumes that states are hidden and unobservable. Therefore, this model is a process of inferring states stochastically using observations. In other words, the HMM is a dual stochastic process that models the stochastic process in an unobservable state through another stochastic process of observations. The HMM incorporates two sets of states, that is, hidden states and observation, and it incorporates three probability sets: π vector, state transition matrix, and observation probability matrix. The model has two parameters: transition probabilities and emission probabilities. There are several methods of estimating the parameters of the HMM given observations. In the beginning, forward-backward retention and marginal smoothing probabilities were used to estimate the HMM parameters. Later, extensive research using the HMM was conducted after Baum proposed the Baum-Welch algorithm and established a basic theory. Then, Viterbi developed an algorithm that is a dynamic planning algorithm to identify the sequence of hidden states. In addition, a method of optimizing the parameters of the Markov model to match the observed signal patterns has been developed. As a result, the HMM has been employed in a wide range of applications, such as the development of systems for advanced speech recognition and human posture recognition.

In the standard type of HMM, the state space of the hidden variables is discrete, while the observations themselves can be either discrete or continuous. The discrete HMM presents the relationship between the state and variables in a feature emission matrix of emission probabilities, while the continuous HMM describes it using a probability density function. Figure 1 shows the structure of states in the HMM. The variable π represents the initial probability of states, and there are a set of observations (On) that correspond to a set of hidden states (Sn). The state transition matrix consists of a square matrix that exhibits the probabilities of different states going from one to another, and the observation probability matrix presents the observation probabilities from every state.

 Figure 1. Structure of states in the Hidden Markov Model (HMM).