Hidden Markov model constructors

These functions are used to specify the distribution of the response conditionally on the underlying state in a hidden Markov model. A list of these function calls, with one component for each state, should be used for the hmodel argument to msm. The initial values for the parameters of the distribution should be given as arguments. Note the initial values should be supplied as literal values - supplying them as variables is currently not supported.

Usage

hmmCat(prob, basecat)

hmmIdent(x)

hmmUnif(lower, upper)

hmmNorm(mean, sd)

hmmLNorm(meanlog, sdlog)

hmmExp(rate)

hmmGamma(shape, rate)

hmmWeibull(shape, scale)

hmmPois(rate)

hmmBinom(size, prob)

hmmBetaBinom(size, meanp, sdp)

hmmNBinom(disp, prob)

hmmBeta(shape1, shape2)

hmmTNorm(mean, sd, lower = -Inf, upper = Inf)

hmmMETNorm(mean, sd, lower, upper, sderr, meanerr = 0)

hmmMEUnif(lower, upper, sderr, meanerr = 0)

hmmT(mean, scale, df)

Arguments

prob: (hmmCat) Vector of probabilities of observing category 1, 2, ...{}, length(prob) respectively. Or the probability governing a binomial or negative binomial distribution.
basecat: (hmmCat) Category which is considered to be the "baseline", so that during estimation, the probabilities are parameterised as probabilities relative to this baseline category. By default, the category with the greatest probability is used as the baseline.
x: (hmmIdent) Code in the data which denotes the exactly-observed state.
lower: (hmmUnif,hmmTNorm,hmmMEUnif) Lower limit for an Uniform or truncated Normal distribution.
upper: (hmmUnif,hmmTNorm,hmmMEUnif) Upper limit for an Uniform or truncated Normal distribution.
mean: (hmmNorm,hmmLNorm,hmmTNorm) Mean defining a Normal, or truncated Normal distribution.
sd: (hmmNorm,hmmLNorm,hmmTNorm) Standard deviation defining a Normal, or truncated Normal distribution.
meanlog: (hmmNorm,hmmLNorm,hmmTNorm) Mean on the log scale, for a log Normal distribution.
sdlog: (hmmNorm,hmmLNorm,hmmTNorm) Standard deviation on the log scale, for a log Normal distribution.
rate: (hmmPois,hmmExp,hmmGamma) Rate of a Poisson, Exponential or Gamma distribution (see dpois, dexp, dgamma).
shape: (hmmPois,hmmExp,hmmGamma) Shape parameter of a Gamma or Weibull distribution (see dgamma, dweibull).
scale: (hmmGamma) Scale parameter of a Gamma distribution (see dgamma), or unstandardised Student t distribution.
size: Order of a Binomial distribution (see dbinom).
meanp: Mean outcome probability in a beta-binomial distribution
sdp: Standard deviation describing the overdispersion of the outcome probability in a beta-binomial distribution
disp: Dispersion parameter of a negative binomial distribution, also called size or order. (see dnbinom).
shape1, shape2: First and second parameters of a beta distribution (see dbeta).
sderr: (hmmMETNorm,hmmUnif) Standard deviation of the Normal measurement error distribution.
meanerr: (hmmMETNorm,hmmUnif) Additional shift in the measurement error, fixed to 0 by default. This may be modelled in terms of covariates.
df: Degrees of freedom of the Student t distribution.

Value

Each function returns an object of class hmmdist, which is a list containing information about the model. The only component which may be useful to end users is r, a function of one argument n which returns a random sample of size n from the given distribution.

Details

hmmCat represents a categorical response distribution on the set 1, 2, ...{}, length(prob). The Markov model with misclassification is an example of this type of model. The categories in this case are (some subset of) the underlying states.

The hmmIdent distribution is used for underlying states which are observed exactly without error. For hidden Markov models with multiple outcomes, (see hmmMV), the outcome in the data which takes the special hmmIdent value must be the first of the multiple outcomes.

hmmUnif, hmmNorm, hmmLNorm, hmmExp, hmmGamma, hmmWeibull, hmmPois, hmmBinom, hmmTNorm, hmmNBinom and hmmBeta represent Uniform, Normal, log-Normal, exponential, Gamma, Weibull, Poisson, Binomial, truncated Normal, negative binomial and beta distributions, respectively, with parameterisations the same as the default parameterisations in the corresponding base R distribution functions.

hmmT is the Student t distribution with general mean \(\mu\), scale \(\sigma\) and degrees of freedom df. The variance is \(\sigma^2 df/(df + 2)\). Note the t distribution in base R dt is a standardised one with mean 0 and scale 1. These allow any positive (integer or non-integer) df. By default, all three parameters, including df, are estimated when fitting a hidden Markov model, but in practice, df might need to be fixed for identifiability - this can be done using the fixedpars argument to msm.

The hmmMETNorm and hmmMEUnif distributions are truncated Normal and Uniform distributions, but with additional Normal measurement error on the response. These are generalisations of the distributions proposed by Satten and Longini (1996) for modelling the progression of CD4 cell counts in monitoring HIV disease. See medists for density, distribution, quantile and random generation functions for these distributions. See also tnorm for density, distribution, quantile and random generation functions for the truncated Normal distribution.

See the PDF manual msm-manual.pdf in the doc subdirectory for algebraic definitions of all these distributions. New hidden Markov model response distributions can be added to msm by following the instructions in Section 2.17.1.

Parameters which can be modelled in terms of covariates, on the scale of a link function, are as follows.

PARAMETER NAME	LINK FUNCTION
`mean`	identity
`meanlog`	identity
`rate`	log
`scale`	log
`meanerr`	identity
`meanp`	logit
`prob`	logit or multinomial logit

Parameters basecat, lower, upper, size, meanerr are fixed at their initial values. All other parameters are estimated while fitting the hidden Markov model, unless the appropriate fixedpars argument is supplied to msm.

For categorical response distributions (hmmCat) the outcome probabilities initialized to zero are fixed at zero, and the probability corresponding to basecat is fixed to one minus the sum of the remaining probabilities. These remaining probabilities are estimated, and can be modelled in terms of covariates via multinomial logistic regression (relative to basecat).

References

Satten, G.A. and Longini, I.M. Markov chains with measurement error: estimating the 'true' course of a marker of the progression of human immunodeficiency virus disease (with discussion) Applied Statistics 45(3): 275-309 (1996).

Jackson, C.H. and Sharples, L.D. Hidden Markov models for the onset and progresison of bronchiolitis obliterans syndrome in lung transplant recipients Statistics in Medicine, 21(1): 113–128 (2002).

Jackson, C.H., Sharples, L.D., Thompson, S.G. and Duffy, S.W. and Couto, E. Multi-state Markov models for disease progression with classification error. The Statistician, 52(2): 193–209 (2003).

Author

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk