2 Hidden Markov models with arbitrary state dwell-time distributions 1
2.5 Application to Old Faithful eruptions
Nonetheless, due to the long tail of the fitted distribution the mean dwell-time in state 1 is quite large, namely 69.7. The deviation from the geometric case is evident. In state 2 approximately 82% of the dwell times are of length one, on average, and the mean dwell-time is 4.2. The state process mostly switches back and forth between the two states, and occasionally remains in one state for a relatively long period. A con-ventional HMM, with geometric state dwell-time distributions, can not accommodate both of these attributes. ZC
2.5 Application to Old Faithful eruptions
In this section we investigate whether HSMMs can improve the fit to the Old Faith-ful series that was already analysed in Section 1.4. We start again by looking at the dichotomized, binary series of long and short inter-arrival times (Section 2.5.1). Sub-sequently, in Section 2.5.2, the nondichotomized time series is analysed.
2.5.1 Modelling the binary series via HSMMs
Consider again the series of Old Faithful’s eruption inter-arrival times that have been discretized into either “short” or “long” (cf. Section 1.4.1). Up to now the most suitable models for this series have been identified to be a second-order Markov chain, a two-state second-order Bernoulli HMM and a three-two-state Bernoulli HMM. The latter model yielded the highest likelihood of all considered models, but it was surprising to find that two of its states are almost equivalent at the observation level. Consider again the parameter estimates for the three-state Bernoulli HMM, given by equations (1.6) and (1.7). As the Bernoulli parameters of states 2 and 3 are approximately equal, the set of states {2,3} can be regarded as a kind of state aggregate. The time the Markov chain spends in this state aggregate is not geometrically distributed. The main reason for the improved likelihood, compared to the two-state Bernoulli HMM, may thus well be the departure from the assumption of geometric dwell-time distributions. This motivates the use of HSMMs in this particular application.
In Section 1.4.1 we have seen that the dwell-time in the state associated with short inter-arrival times almost surely is of length one (across all models). The dwell-time of this state is thus not worth being modelled by a non-geometric distribution. We consider the following two-state Bernoulli hybrid HMM/HSMM: ZC
• given either of the two states, the observations are Bernoulli distributed,
• state 1 involves a geometric dwell-time distribution and
2 Hidden Markov models with arbitrary state dwell-time distributions
• the dwell-time in state 2 is negative binomially distributed.
The maximum of the log likelihood for this model is −1380.00, the AIC and BIC are 2770.00 and 2803.30, respectively. In terms of these criteria the model performs better than any of the models that were considered in Section 1.4.1 (cf. Table 1.1). The estimated Bernoulli parameter vector is
πb = (0.59,1.00).
Dwell-times in state 1 of the fitted hybrid HMM/HSMM are of length one (almost surely). The parameter estimates of the negative binomial dwell-time distribution in state 2 (cf. (2.6)) are
ˆk= 0.261 and ˆπ= 0.063.
2 4 6 8 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
dwell−time in state 2
probability
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Figure 2.7: Fitted dwell-time distributions for the state associated with long eruption inter-arrival times, for the two-state HMM (circles) and for the two-state hybrid HMM/HSMM (bars).
Figure 2.7 compares the dwell-time distribution in state 2 of the fitted hybrid HMM/
HSMM with that in state 2 of the two-state Bernoulli HMM from Section 1.4.1. One should be aware that the states are not synchronized across the models; those labelled
“state 1” do not lead to the same distribution at the observation level. For the displayed distribution associated with the hybrid HMM/HSMM the deviation from the geometric case is evident. Allowing for more flexibility in the state dwell-time distributions leads to an improved fit in this example.
2.5 Application to Old Faithful eruptions
2.5.2 Modelling the series of inter-arrival times via HSMMs
Now consider again the original, nondichotomized series of eruption inter-arrival times (cf. Section 1.4.2). One of the most promising models that was fitted to this series is the four-state HMM with gamma state-dependent distributions (cf. Table 1.3). We found that a further increase in the number of states improved the likelihood significantly, even though two of the states in the fitted five-state model were hardly distinguishable at the observation level. Intuitively the two components with mean approximately 93 would not be regarded as two separate states (cf. Figure 1.5). These two states resemble once more a state aggregate, at least in an approximate sense. In view of the likelihood gap between the four-state and the five-state model it seems worthwhile to investigate whether the fit of the four-state model can be improved by employing non-geometric, more flexible dwell-time distributions.
Consider again the four-state GHMM that was fitted in Section 1.4.2 (the parameter estimates are given in Appendix A1). The estimated state-dependent means of this model for states 1, 2, 3, and 4 are 65.35, 86.16, 92.99 and 99.94, respectively. The estimated parameters of the (geometric) state dwell-time distributions, ˆπi for statei, are ˆπ1 = 1, ˆπ2 = 1, ˆπ3 = 0.43 and ˆπ4 = 0.98. As sojourn times in states 1, 2 and 4 are infrequently longer than 1, it does not seem worthwhile to replace the dwell-time distributions of these states by non-geometric ones. When moving from the four-state to the five-state model, it is essentially the remaining state 3 that is split up into two states, which are almost equivalent at the observation level. We thus consider the following four-state gamma hybrid HMM/HSMM:
• given any of the four states, the observations are gamma distributed,
• states 1, 2 and 4 have geometric dwell-time distributions and
• the dwell-time in state 3 is assumed to have an unstructured start of order 4 (cf.
Example 4).
(The states are ordered in terms of increasing means of the state-dependent distri-butions.) The reason for using a distribution with unstructured start, rather than the negative binomial distribution, is the small essential range of the dwell-time distribution in the given application; not many additional parameters are needed to obtain sufficient flexibility. The maximum of the log likelihood of the suggested model is −19834.14, which yields AIC = 39716.28 and BIC = 39876.12. The additional flexibility in the dwell-time of state 3 only led to a minor improvement of the fit. This is confirmed by Figure 2.8, which displays the dwell-time distributions in state 3, for the conventional
2 Hidden Markov models with arbitrary state dwell-time distributions
2 4 6 8 10
0.0 0.1 0.2 0.3 0.4
dwell−time in state 3
probability
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Figure 2.8: Fitted dwell-time distributions for the state associated with the second-largest mean of the state-dependent distribution: for the four-state gamma HMM (circles) and for the four-state gamma hybrid HMM/HSMM (bars).
HMM and for the hybrid HSMM/HMM, respectively. The discrepancy between the two distributions is small.
Bearing in mind the findings from Section 1.4.2, it can be concluded that the length of the memory of the state process, rather than a possible semi-Markovian behaviour, is responsible for the likelihood gap between the four-state and the five-state GHMM. ZC
2.6 Concluding remarks
In this chapter we considered a class of HMMs that capture the ‘semi’-property of HSMMs, i.e. that can represent any given state dwell-time distribution, either exact or approximately, where the approximation can be made arbitrarily accurate. The motivation for doing so was to take advantage of the well-established methodology that is available for HMMs. Key advantages of using HMMs with arbitrary dwell-time distributions (rather than HSMMs) include the ease with which it is possible to incorporate covariate information in different ways, and that fitting stationary models is straightforward. The applications given to rainfall occurrences, daily returns and geyser eruption inter-arrival times illustrate the feasibility of the proposed method.