## 3 Methods
### Hidden Markov Model Analytic Approach--A Discrete Valued State Process  
The Hidden Markov Model is a probabilistic model proposed by [PERSON] and [PERSON] [59] to describe the statistical properties of random processes with parameters. A Hidden Markov Model (HMM) generates a hidden sequence of states from an observable sequence of observations. Mixture and hidden Markov models are statistical models that are useful when an observed stochastic, dynamic system over time occupies a number of distinct \"regimes\" or unobserved (hidden) states [60].  
HMMs are widely used in a variety of fields, including environmental science, artificial intelligence, biology, and psychology [61]. HMMs have been shown to be useful specifically in weather prediction, robotics, detecting protein homologies and computer vision [60, 62]. Hidden Markov Models (HMMs) are also extensively used in applications, such as natural language processing (NLP) and finance [63]. Likewise, a recent utilization of HMMs has been in earthquake prediction [64] and in computational biology, where an HMM may be used to model the evolution of a protein or DNA sequence, and observations generally are a sequence of amino acids or nucleotides [65]. In speech recognition, for example, an HMM can be used to model the underlying sounds or phonemes that generate the speech signal, and the observations generally are the features extracted from the speech signal [62].  
Hidden Markov Models can be viewed as a generalization of two types of simpler models. Firstly, HMMs are a generalization of simple Markov models, where the generalization involves additional modeling of the error structure. Secondly, Hidden Markov models can be viewed as an extension of mixture models, to model transitions between states over time. Here, the generalization involves additional modeling of the sequential nature of the data. In addition, recent advances in HMM theory [18, 60] allow the easy incorporation of time-dependent covariates into the model. This means that both the transition and the emission probabilities can depend on a set of covariates, so that they can vary over time and thereby the traditional and unrealistic stationarity assumptions need not be made.  
HMM model sequences/series are either derived from continuous or discrete probability distributions. HMMs are related to state space and Gaussian mixture models(GMMs) [66] where HMMs aim to estimate the state(s) which have given rise to the observations. The states are unknown or 'hidden' and HMMs attempt to estimate the states similar to an unsupervised clustering procedure. Hidden Markov Models can classify observable sequences into unobservable state sequences. State transitions refer to the random change in states of the Markov process in discrete time. The Markov model thus follows the concept of a memory-less property, i.e., the transition from one state to another state depends only on the present state [67].  
The HMM algorithm involves defining the state space, observation space, and the parameters of the state transition probabilities and observation likelihoods. The training of the model generally uses the Baum-Welch algorithm or the forward-backward algorithm and decoding the most likely sequence of hidden states is performed by the Viterbi algorithm [68], as well as evaluating the performance of the model. The Baum-Welch algorithm is a version of the Estimation Maximization (EM) algorithm [60].  
To date, HMM modeling has been used to analyze ozone levels by several authors. Two examples of such research, specifically in predicting the maximum 8 h average ozone concentration, are [PERSON] et al. [69] who use a Hidden Markov Model for measurements assumed to come from a Gamma distribution, rather than a Gaussian distribution. The other study is by [PERSON] et al. [70] who use a supervised HMM combined with generalized linear models.