# Hidden Markov model constructors

### Description

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.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
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)
hmmTNorm(mean, sd, lower, upper)
hmmMETNorm(mean, sd, lower, upper, sderr, meanerr=0)
hmmMEUnif(lower, upper, sderr, meanerr=0)
hmmNBinom(disp, prob)
hmmBeta(shape1,shape2)
hmmT(mean,scale,df)
``` |

### Arguments

`prob` |
( |

`basecat` |
( |

`x` |
( |

`mean` |
( |

`sd` |
( |

`meanlog` |
( |

`sdlog` |
( |

`rate` |
( |

`shape` |
( |

`shape1,shape2` |
First and second parameters of a beta
distribution (see |

`scale` |
( |

`df` |
Degrees of freedom of the Student t distribution. |

`size` |
Order of a Binomial distribution (see |

`disp` |
Dispersion parameter of a negative binomial distribution,
also called |

`lower` |
( |

`upper` |
( |

`sderr` |
( |

`meanerr` |
( |

### 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.

`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 |

`prob` | (multinomial logistic regression) |

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`

).

### Value

Each function returns an object of class `hmodel`

, 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.

### Author(s)

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

### 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).

### See Also

`msm`