The counts in the observation vectors are driven by more than the disease state. Additional observed information, such as day of week, weather, promotions (in the case of retail data), vacations (for school absenteeism), and other external factors are expected to influence the observed counts. For example, the HMM could expect, with high probability, to observe zero counts for school absenteeism on public holidays, and a 20% increase in OTC sales at stores with promotional activity.
The following simulation illustrates the benefits of fusion of multiple data streams. For details of such an HMM approach, see the references above. The data in the first three panels of Figure 15.4 were generated from a vastly oversimplified HMM where there are three possible disease states: "none," "influenza," and "allergy." The simulated data streams are ED visits, school absenteeism, and OTC antihistamine sales. In this example, the ED visits have a strong weekend effect, while weekend data are missing for school absenteeism. "Influenza" is assumed to double the rates of all three data streams. "Allergy" also doubles OTC antihistamines. The population is in state "allergy" during time steps 10-20, and in "influenza" for time steps 30 to 40. A three-state HMM was applied to the data and the inference results are shown in the bottom panel of Figure 15.4. Inference involves computing the disease state probabilities (which are not in the observed data) from the counts and environmental factors (which are in the observed data). Note that the HMM is not told when allergy is present.
In Figure 15.4, note that the model was successful in explaining the jump in OTC sales and missing absenteeism data did not noticeably impair inference. Individually, these data streams would be inconclusive, or worse, misleading, but the proper statistical model effectively fuses them into a robust and highly informative resource. This example is very simplified, but illustrates several of the components of a realistic multivariate system.
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