Hidden Markov Models methods for selecting optimal number of states

Question

Package RHmm (R)

I have a vector which I fit into a hmm model in an attempt to select an optimal number of states for a hidden markov model.

x<-c(-0.0961421466,-0.0375458485,0.0681121271,0.0259201028,0.0016780785,0.0311860542,      
0.0067940299,0.0126520055,0.0357599812,0.0007679569,0.0409759326,0.0560839083,-0.0272581160,-0.0439501404,0.0321578353,0.0196158110,-0.0097262133,-0.0226182376,0.0119897380,-0.0099522863,-0.0359443106,-0.0039363349,-0.0476283592,-0.0383203835,-0.0518624079,0.0187455678,0.0950535435,0.0057115192,-0.0307805051,-0.0272725295,-0.0254645538,-0.0102565781,-0.0267986024,-0.0482906267,-0.0256826510,-0.0414746754,-0.0470666997,0.0284912760,0.1021992517,0.0875572274,0.0064152031,0.0200731787,-0.0091688456,-0.0575608699,-0.0442028942,-0.0277449185,-0.0115369429,0.0084710328,0.0745290085,0.0159369842,-0.0784550401,-0.0934970644,-0.0978390888,0.0160188869,0.0275268626,-0.0552651617,0.0033928140,0.0468507896,0.0374087653,0.0521167410,-0.0177752833,-0.0592673076,0.0514406681,0.0847486437,0.0738066194,-0.0098354049,-0.0572274292,0.0478305465,0.0096885221,-0.0445535022,-0.0153455265,-0.0105375508,0.0100704249,-0.0035215994,0.0243363762,0.0504443519,0.0570023276,0.0395103033,-0.0612817210,-0.0557737453,-0.0273657697,-0.0220077940,0.0083501817,0.0275081574,0.0323161331,0.0385741087,0.0175820844-0.0410599399,-0.0071019642,0.0431060115,-0.0107360128,-0.0007280372,0.0360799385,-0.0061620858  0.0164458899 -0.0050461344 -0.0578381588  0.0097198169  0.0027277926 -0.0127642317,
-0.0037062560, -0.0045482803,  0.0367596953, 0.0021176710,-0.0319243533,-0.0194663776,0.00 91915981,0.0061495737,-0.0090424506,0.0127655251,0.0161735008,0.0193814765,-0.0208605478,-0.0598025722,0.0022554035,0.0473633792,0.0247213549,-0.0063206694,-0.0201626938,0.0207952819,0.0379032576,0.0151612333,0.0038692090,0.0111271847,0.0497851603,0.0273431360,-0.0172488883,-0.0038909126,0.0264670631,-0.0065249612,-0.0467169856,-0.0255090099,0.0082489658, 0.0352569415,0.0272149172,0.0074228928,-0.0040191315,-0.0170611558,-0.0309531801,-0.0327952044,-0.0239372287,-0.0212792531,-0.0132712774,0.0086866983,-0.0007553260,0.0107026497,0.0065106253,-0.0321813990,-0.0081734233,0.0296845524,0.0268925281,-0.0025994962,-0.0038915206, -0.0126335449,0.0040244308,0.0227324065,0.0114903822,-0.0031516422,0.0031563335,0.0137143092,0.0026222849,0.0035802606,0.0111382363,-0.0008037881, -0.0282458124, 0.0056121633, 0.0254201390,0.0033781147,-0.0166139097,-0.0124559340,0.0088520417,0.0072600174, -0.0050320069,-0.0114740312,-0.0066160556, -0.0042080799, -0.0205501042,0.0027078715,  0.0122158472,-0.0206261771,-0.0267682015,-0.0107602258,0.0088477499,0.0165057256, 0.0106637013,0.0115216769,0.0278296526,0.0026376283,-0.0231543960,-0.0141964203)

#partitions test/train
nhs <- c(2,3,4) #number of possible states
S<-runif(length (x))<= .66
train<-print(S)

# mean conditional density of log probability of seeing the partial sequence of obs 
for(i in 1:length(nhs)){
pred <- vector("list", length(x))
    for(fold in 1:length(x)){
        fit <- HMMFit(x [which(train==TRUE)],dis="NORMAL",nStates=nhs[i],
        asymptCov=FALSE)
        pred[[fold]] <-  forwardBackward(fit, x[which(train==FALSE)])
   }
error[i] <- pred[[fold]]$LLH
  }
nhs[which.max(error)]    # Optimal number of hidden states (method max log-likehood)

Every time I run the model trying to obtain the best number of states for the hidden markov model, I get a different number of states as I believe the model is trained over randomnly selected new values and also the local min. This does not happen if I just fit the model.

#score proportional to probability that a sequence is generated by a given model
nhs <- c(2,3,4)
for(i in 1:length(nhs)){
    fit <- HMMFit(x, dis="NORMAL", nStates= nhs[i], asymptCov=FALSE)
    VitPath = viterbi(fit, x)
   error[i] <- fit[[3]]
}
error<-c(error) 
error[is.na(error)] <- 10000
nhs[which.min(error)]    # Optimal number of hidden states (method min AIC)

However the results of the two are very different. Which one is better? On one hand I have a model where I can test on new samples. On the other hand, the second provides best fit on seen samples. In the case of the model if I repeat the test given that the training/test set change (random) the resulting number of states changes as sample train/test changes. In this case what method should I use as to be certain that the model provides generalization (the number of states result is best).

What additional methods may I employ as to be able to select an optimal number of states

Many thanks

Answer 1

To determine the optimal number of states in a HMM is indeed an intricate one.

Please have a look at the following paper:

The Number of Regimes Across Asset Returns: Identification and Economic Value by M. Gatumel and F. Ielpo (2011)

From the abstract:

A shared belief in the financial industry is that markets are driven by two types of regimes. Bull markets would be characterized by high returns and low volatility whereas bear markets would display low returns coupled with high volatility. Modeling the dynamics of different asset classes (stocks, bonds, commodities and currencies) with a Markov-Switching model and using a density-based test, we reject the hypothesis that two regimes are enough to capture asset returns' evolutions for many of the investigated assets. Once the accuracy of our test methodology has been assessed through Monte Carlo experiments, our empirical results point out that between two and five regimes are required to capture the features of each asset's distribution. Moreover, we show that only a part of the underlying number of regimes is explained by the distributional characteristics of the returns such as kurtosis. A thorough out-of-sample analysis provides additional evidence that there are more than just bulls and bears in financial markets. Finally, we highlight that taking into account the real number of regimes allows both improved portfolio returns and density forecasts.

On the other hand you have to think about what to do with these states after you identified them. Normally you want to put a trading strategy on top of it. So if you have five states... now what? My motto is to keep it simple, so in most cases I would still use only two states since this translates very intuitively into being long and being flat.

If you chose three states you will often find that one state is a very short crash state which captures all the extreme (left) tails. You cannot really use that because as soon as you identify this crash state in real time it is too late already. You can normally dodge these tails better with a two state model.

But I would love to hear other experiences too!

Answer 2

Besides using the valuable domain knowledge given in other answer(s) to determine the optimal number of states, and furthermore as an alternative to using the AIC or BIC, we can consider Bayesian estimation of the parameters including the optimal number of states:

Scott, S.L. (2002). Bayesian methods for hidden Markov models: Recursive computing in the 21st century. J. Amer. Statist. Assoc. 97, 337–351.

Congdon, P. (2006). Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Computat. Statist. & Data Analysis 50, 346–357.