# hmmMV: Multivariate hidden Markov models In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

## Description

Constructor for a a multivariate hidden Markov model (HMM) where each of the `n` variables observed at the same time has a (potentially different) standard univariate distribution conditionally on the underlying state. The `n` outcomes are independent conditionally on the hidden state.

If a particular state in a HMM has such an outcome distribution, then a call to `hmmMV` is supplied as the corresponding element of the `hmodel` argument to `msm`. See Example 2 below.

A multivariate HMM where multiple outcomes at the same time are generated from the same distribution is specified in the same way as the corresponding univariate model, so that `hmmMV` is not required. The outcome data are simply supplied as a matrix instead of a vector. See Example 1 below.

The outcome data for such models are supplied as a matrix, with number of columns equal to the maximum number of arguments supplied to the `hmmMV` calls for each state. If some but not all of the variables are missing (`NA`) at a particular time, then the observed data at that time still contribute to the likelihood. The missing data are assumed to be missing at random. The Viterbi algorithm may be used to predict the missing values given the fitted model and the observed data.

Typically the outcome model for each state will be from the same family or set of families, but with different parameters. Theoretically, different numbers of distributions may be supplied for different states. If a particular state has fewer outcomes than the maximum, then the data for that state are taken from the first columns of the response data matrix. However this is not likely to be a useful model, since the number of observations will probably give information about the underlying state, violating the missing at random assumption.

Models with outcomes that are dependent conditionally on the hidden state (e.g. correlated multivariate normal observations) are not currently supported.

## Usage

 `1` ```hmmMV(...) ```

## Arguments

 `...` The number of arguments supplied should equal the maximum number of observations made at one time. Each argument represents the univariate distribution of that outcome conditionally on the hidden state, and should be the result of calling a univariate hidden Markov model constructor (see `hmm-dists`).

## Value

A list of objects, each of class `hmmdist` as returned by the univariate HMM constructors documented in `hmm-dists`. The whole list has class `hmmMVdist`, which inherits from `hmmdist`.

## Author(s)

C. H. Jackson [email protected]

## References

Jackson, C. H., Su, L., Gladman, D. D. and Farewell, V. T. (2015) On modelling minimal disease activity. Arthritis Care and Research (early view).

`hmm-dists`,`msm`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43``` ```## Simulate data from a Markov model nsubj <- 30; nobspt <- 5 sim.df <- data.frame(subject = rep(1:nsubj, each=nobspt), time = seq(0, 20, length=nobspt)) set.seed(1) two.q <- rbind(c(-0.1, 0.1), c(0, 0)) dat <- simmulti.msm(sim.df[,1:2], qmatrix=two.q, drop.absorb=FALSE) ### EXAMPLE 1 ## Generate two observations at each time from the same outcome ## distribution: ## Bin(40, 0.1) for state 1, Bin(40, 0.5) for state 2 dat\$obs1[dat\$state==1] <- rbinom(sum(dat\$state==1), 40, 0.1) dat\$obs2[dat\$state==1] <- rbinom(sum(dat\$state==1), 40, 0.1) dat\$obs1[dat\$state==2] <- rbinom(sum(dat\$state==2), 40, 0.5) dat\$obs2[dat\$state==2] <- rbinom(sum(dat\$state==2), 40, 0.5) dat\$obs <- cbind(obs1 = dat\$obs1, obs2 = dat\$obs2) ## Fitted model should approximately recover true parameters msm(obs ~ time, subject=subject, data=dat, qmatrix=two.q, hmodel = list(hmmBinom(size=40, prob=0.2), hmmBinom(size=40, prob=0.2))) ### EXAMPLE 2 ## Generate two observations at each time from different ## outcome distributions: ## Bin(40, 0.1) and Bin(40, 0.2) for state 1, dat\$obs1 <- dat\$obs2 <- NA dat\$obs1[dat\$state==1] <- rbinom(sum(dat\$state==1), 40, 0.1) dat\$obs2[dat\$state==1] <- rbinom(sum(dat\$state==1), 40, 0.2) ## Bin(40, 0.5) and Bin(40, 0.6) for state 2 dat\$obs1[dat\$state==2] <- rbinom(sum(dat\$state==2), 40, 0.6) dat\$obs2[dat\$state==2] <- rbinom(sum(dat\$state==2), 40, 0.5) dat\$obs <- cbind(obs1 = dat\$obs1, obs2 = dat\$obs2) ## Fitted model should approximately recover true parameters msm(obs ~ time, subject=subject, data=dat, qmatrix=two.q, hmodel = list(hmmMV(hmmBinom(size=40, prob=0.3), hmmBinom(size=40, prob=0.3)), hmmMV(hmmBinom(size=40, prob=0.3), hmmBinom(size=40, prob=0.3))), control=list(maxit=10000)) ```

### Example output

```Call:
msm(formula = obs ~ time, subject = subject, data = dat, qmatrix = two.q,     hmodel = list(hmmBinom(size = 40, prob = 0.2), hmmBinom(size = 40,         prob = 0.2)))

Optimisation probably not converged to the maximum likelihood.
optim() reported convergence but estimated Hessian not positive-definite.

-2 * log-likelihood:  3706.02
[Note, to obtain old print format, use "printold.msm"]
Warning message:
In msm(obs ~ time, subject = subject, data = dat, qmatrix = two.q,  :
Optimisation has probably not converged to the maximum likelihood - Hessian is not positive definite.

Call:
msm(formula = obs ~ time, subject = subject, data = dat, qmatrix = two.q,     hmodel = list(hmmMV(hmmBinom(size = 40, prob = 0.3), hmmBinom(size = 40,         prob = 0.3)), hmmMV(hmmBinom(size = 40, prob = 0.3),         hmmBinom(size = 40, prob = 0.3))), control = list(maxit = 10000))

Maximum likelihood estimates

Transition intensities
Baseline
State 1 - State 1 -0.09458 (-0.13940,-0.06416)
State 1 - State 2  0.09458 ( 0.06416, 0.13940)

Hidden Markov model, 2 states
State 1
Outcome 1 - binomial distribution
Estimate       LCL       UCL
size 40.0000000        NA        NA
prob  0.1054793 0.0948457 0.1171507

Outcome 2 - binomial distribution
Estimate       LCL       UCL
size 40.0000000        NA        NA
prob  0.1958908 0.1818942 0.2106871

State 2
Outcome 1 - binomial distribution
Estimate       LCL      UCL
size 40.0000000        NA       NA
prob  0.5954564 0.5780108 0.612664

Outcome 2 - binomial distribution
Estimate       LCL      UCL
size 40.0000000        NA       NA
prob  0.5110363 0.4933762 0.528669

-2 * log-likelihood:  1523.652
[Note, to obtain old print format, use "printold.msm"]
```

msm documentation built on Feb. 3, 2018, 1:04 a.m.