Combining Multiple Time Series Using Probability

Hidden Markov models (HMMs) are fundamental tools in fields such as bioinformatics, signal processing, and machine learning. In biosurveillance, Le Strat and Carrat (1999) used HMMs to monitor influenza-like illnesses and poliomyelitis using a Gaussian observation model. Rath et al. (2003) improved their model by, among other things, replacing the Gaussian observation model with an exponential distribution. Other applications include Cooper and Lipsitch (2004) modeling hospital infections with HMMs. Madigan (2005) reviewed the literature and discussed issues such as model selection and random observation-time HMM. He also noted the excellent fit between the capabilities of HMMs and the requirements of multivariate time series.

Figure 15.3 shows an HMM. Arrows indicate causality, shaded variables are observed, and the unshaded state variables

figure 15.3 Hidden Markov model structure.

are unobserved. At time t, the underlying disease state is St, which has N possible values. The observation Ot is a vector of K observations, which can assume count values, such as specific over-the-counter (OTC) sales, ER visits, and school absenteeism. The third variable type is for environmental influences, Et, on the observations. The environment variable is a vector of categorical variables, and has no effect on the disease state.

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