ACEPTACION DE HERENCIA

A theory of HMMs was introduced in the late 1960s through a series of papers published by

Baum and Petrie (1966);Baum and Eagon (1967);Baum et al. (1970). This class of models

has been successfully used for modelling and classifying dynamic behaviours. HMMs may be applied for different types of data: discrete, continuous, univariate, multivariate, mixed and mixture data. Consequently, they have been widely used in many fields, such as; econometrics (Hamilton,1989;Billio et al.,1999); finance (Bhar and Hamori,2006); speech recognition, image analysis, and time series prediction (Derrode et al., 2006;Rabiner,1989); and psychology (Raijmakers and Molenaar, 2004; Visser et al., 2002). We next give some examples of applications for these models.

determining the real behaviour of patients. These challenges arise because of population heterogeneity, cohabitation of different patients and medical diseases, and diagnostic uncertainty. Hence, it is not easy to measure behavioural indicators of those phenomena where we are interested in the behaviour of a particular disease or of a patient. Therefore, HMMs are more appropriate models in these cases due to their flexibility in the presence of unobservable behaviour.

For HMMs applications, Jackson et al. (2003) proposed a Multi-stage Markov Model to describe aortic aneurysm patients. Sometimes the process of checking patients is not without mistakes and misclassification problems. Hence, HMMs were suggested to estimate transmission averages and likelihoods of state misclassification. A generalized regression model was used to model explanatory variables for transitions between statuses and probabilities of misclassification. In order to reduce uncertainty, Jackson et al. (2003) introduced Hazard Models to link transitions with the age variable for detecting whether there is an age effect. The findings proved that HMM models were sensitive to the assumptions of the study and suggested that the older adults are at increased risk of aortic aneurysm compared to younger adults. The Weibull distribution was proposed as an alternative for the exponential distribution in the estimation process of the sojourn states (Jackson et al.,2003).

Visser et al.(2002) used HMMs for psychological studies. The model was used to quantify

knowledge that subjects express in an implicit learning task. The suggested procedure for comparing models with different constrains imposed on their parameters was the Maximum Likelihood method. They introduced a discrete-time HMM model instead of a continuous time HMM model due to the former being more convenient. Simulation experiments were implemented for the evaluation and model selection. Several candidate criteria were introduced for selecting the best model. They suggested, in addition of the standard criteria: AIC and BIC, two criteria which are the Adjusted Akaike Information Criterion (A-AIC) and Adjusted Bayesian Information Criterion (A-BIC). Their results proved that AIC and BIC are inappropriate in evaluating large models. After having chosen a final model, they used a prediction error measure to test the validity of chosen model (Visser et al., 2002). Nevertheless, these criteria; A-AIC and A-BIC will not be considered in this thesis as do not take into account the uncertainty about the model parameters.

Wall and Li(2009) performed a study to describe unobserved behaviour. They referred to two

types of models to describe the effects of some of the unobserved variables on alcoholism addiction: the Multiple Indicator Hidden Markov Model and the Univariate Hidden Markov

Model. They determined two variables; “healthy” and “unhealthy” which were associated with each case (patient) as latent variables. The study used two kinds of data; longitudinal data that were classified depending on type of patient (Alcohol Specific, Alcohol Chronic, Alcohol Acute, and Not Alcohol), and monthly total data. Since the observations are monthly counts of medical visits, they proposed a two-state Poisson hidden Markov model (HMM). The purpose of the study was to investigate whether medical care reduces the probability of entry to unhealthy state that is identified by the medical visits (Wall and Li,2009).

Furthermore, Hidden Markov Models have been used to analyse clustering and longitudinal data in describing some diseases. Scott et al. (2005) introduced a hidden Markov model for investigating the effect of an anti-psychotic drug and clozapine for schizophrenia patients. The univariate analysis for complicated medical diseases is not suitable for revealing detailed characteristics about the disease under study because of the presence of heterogeneity. This heterogeneity can be interpreted as the different features among patients. Hence, a longitudinal multivariate analysis is more suitable to describe the disease. A clustering method has been used as it is suitable for identifying complicated relationships among medical cases. However, there are some obstacles related to the nature of the data, and the procedure followed in estimating the parameters. Scott et al.(2005) offered a HMM to address those issues because of its ability in dealing with temporal data when estimating model parameters and classifying observations. HMMs in turn have some problems regarding time-homogeneity. Sometimes hypotheses of time-homogeneity may not be valid, particularly since there are unequal temporally intervals during treatment process. Therefore, the authors proposed a non-homogeneous HMM for this purpose. The findings of the study suggest that the clozapine is more effective than haloperidol in antipsychotic therapy.

Various studies have considered Hidden Markov Models as an accurate early-warning system.

Rafei et al. (2012) used a Hidden Markov Model to identify the abnormal cases of a

pulmonary disease, rampant tuberculosis, in Iran over the period 2005-2011. The study sample was based on data gathered weekly from sputum smear of patients. The model’s parameters were estimated using maximum likelihood estimation and the Bayesian framework. The data were presented as a weekly bivariate discrete sequence. The usual phase represented what was expected in the diagnosis process of the disease, and the abnormal phase represented what exceeds the normal phase. Since the data were discrete,Rafei et al.(2012) proposed a Poisson mixture model to fit the data, and introduced two methods for estimating the model parameter (λ ); one without seasonal trends, and one with seasonal trends. HMMs were applied for both

methods. The basic idea of the study was based on the abnormal increase in counts of patients that exceeds normal diagnosis phase. The authors relied on multiple regression models proposed bySerfling(1963), derived from Fourier Equations. The two models below represent a function of the model parameters:

λ1t = E (Yt|St= 1) = β0+ β1t+ β2sin  2πt r  + β3cos  2πt r  . λ2t= E (Yt|St = 2) = (β0+ βe) + β1t+ β2sin  2πt r  + β3cos  2πt r  .

The Bayesian Information Criterion (BIC) and adjusted R-squared were proposed by the authors as criteria to select the best model. Finally, Rafei et al. (2012) concluded from their findings that the results using either criterion suggested that the HMM model with seasonal trend was better than the model without seasonal trend.

HMMs have also been used in epidemiology. Cooper and Lipsitch (2004) used HMMs to analyse hospital infection data. They presented a new method for parameter estimation of structured hidden Markov models for hospital infections data compared to an unstructured (standard) HMM and used their model to evaluate their method. They analysed monthly infection counts that followed a Poisson distribution. They found that both methods can offer considerable improvements over currently used approaches when hospital infection spread is important. Compared to the standard hidden Markov model, the new approach is more biologically plausible, and allows key epidemiological parameters to be estimated.