# Bayesian Homogeneous Markov Model (NHMM)

### Description

`HMM`

calculates an HMM for multiple sequences of
data. The sequences can actually be short sets of equal
length sequences (subseq). A set of input variables (W) can
be included to influence the mixture proportions of the
emission distributions. The HMM follows the general weather
state formulation of Hughes and Guttorp but in a Bayesian
fashion. All parameters are sampled via Gibbs steps (latent
variables such that no tuning is needed.) The W variable
coefficients are sampled through an ordered Mulinomial
probit (Albert and Chib). The X variable coefficients are
sampled through an unordered Multinomial logit model
Polya-Gamma formulation (Polson, Scott, Windle). The hidden
states are sampled through a blocked Gibbs sampler.

### Usage

1 2 3 4 5 |

### Arguments

`y` |
T by J matrix of data (J=1 is sufficient) - missing data is denoted with NA |

`subseq` |
[optional] if y is actually a set of subsequences then give the length of those sequences (122 for JJAS) (365 is not it!). Default is subseq=T. |

`dirprior` |
[optional] prior for Dirichlet prior on the rows of the transition matrix only for the HMM, must be size KxK. If not supplied a flat prior is used. |

`K` |
number of states (default=2) |

`iters` |
number of iterations to keep after burn in (default=1000) |

`burnin` |
the number of burn in (default=200) |

`emdist` |
emission distribution: "normal", "poisson", "gamma" actual choices are Normal, Poisson, Gamma, Exponential, or finite mixtures of mixtures or zero inflated version of any of these |

`nmix` |
[optional] number of mixture components for emdist, default is one (do not include delta) |

`delta` |
[TRUE/FALSE] TRUE-if we are using a zero inflated distribution (adds a delta function at zero as the first mixture component) |

`W` |
[optional] is an A by T by J array of emission input data (A different inputs), missing values are not allowed, do not include an intercept term here The mixture components of emission depend on W. |

`psipriorm` |
[optional] default=NULL which is reference prior. Or a [K+A by J matrix] for the mean of the Normal prior for the beta coefficients. |

`psipriorp` |
[optional] default=NULL which is reference prior. Or a [K+A by J matrix] for the precision prior(1/sig^2) of the Normal prior for the beta coefficients. |

`priors` |
priors for emission components, each state can have a different prior dimension 5 by nmix by K by J (some of the 5 dimensions are filled with zeros for some distributions) Normal(mu,sig2) or reference prior: mu~Normal(a.mu, 1/b.mu^2), sig2~IG(a.sig,b.sig) -the first 4 rows are used [a.mu, b.mu, a.sig, b.sig] but the 5th row is not used but must be present. -reference prior: priors=NULL Poisson(lambda): lambda~Gamma(a.lam,b.lam) -the first 2 rows are used [a.lam, b.lam] but the 3rd-5th rows are not used but must be present. Gamma(alpha,beta): alpha~Gamma(a.alpha,b.alpha) beta~Gamma(a.beta,b.beta) -if the second row is supplied as NA then the first row supplies fixed first parameters for the Gamma this is the suggested parametrization of the Gamma. Estimating the first parameter of the Gamma distribution switches to an MCMC that is not as stable. -to create an Exponential distribution the first row is set to 1 and the second is set to NA then the third and fourth rows contain the (a.lam, b.lam) priors for the parameter of the Exponential if prior=NULL then these are set to ones. setting these parameters to zeros is not a good idea. -a non-informative prior may not work, use a matrix of ones for a lowly informative prior instead of zeros. -the 5th row is for the tuning parameter for the MCMC (try 0.5) if a two parameter Gamma is desired. A 2 parameter Gamma needs an informative prior and results may be more unstable. -default prior=NULL is the Exponential distribution with lowly informative prior of ones |

`outdir` |
[optional] can output each set of parameters to output files in a directory use this with larger dimension data sets or with large number of iterations the output will be written line by line and not overburden the memory limit outdir needs to end with a slash or double slash depending on OS. |

`ymiss` |
[optional-TRUE/FALSE] if outdir is specified then draws for any missing data points will be saved to ymiss-J*.txt. There will be one data point for each sequence of length Tmiss (the amount missing from that sequence). Each row of the output file will be an iteration of the algorithm after burnin is removed. |

`yrep` |
[optional] number (ie. 100,200,or 500) of output replicate data sets to print to outdir. The replicates will be the same dimension as y and start after burn in period. Default is zero. These replicate data sets are generated from the same input variables. - must be shorter than iters |

`ypred` |
number of predictive sets (ie. 0,100, 200, 500) - must be shorter than iters Predicted chains: will produce ypred set of predictive [pT by J] to print to outdir. yrep uses the same inputs to make replications, this uses new inputs to make predictions over a different time span (pT) Also outputs a set of predictive z values or length (pT by ypred) |

`pT` |
the length of the new sequence |

`Wp` |
predictive set of Ws of length pT [A by pT by J] -missing valus are not allowed -ensure that the Wp inputs are in the same order as W |

`yhold` |
[optioinal] a sequence of y observed values [pT by J], held out data that is of length ypred that is used to compute the predictive log score (PLS) which is a metric like BIC (ie. hold out last 10 5-fold CV). missing data values are filled in with mean PLS value. PLS is the average PLS across sequences. |

### Value

my.hmm object

### 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 44 45 46 47 48 49 50 51 52 53 54 55 | ```
## Gamma or Exponential
### because we do not supply "priors" as an input it fits an Exponetial distribution
## Not run:
data(NHMMdata)
attach(NHMMdata)
my.hmm1=HMM(y=ygamma, K=3, iters=100, burnin=10, emdist="gamma",
nmix=3, delta=TRUE)
OBIC(my.hmm1)
zz=Oz(my.hmm1) #compare with the truth zgamma
qq=OQQ(my.hmm1)
pp=OWcoef(my.hmm1,FALSE)
tt=Oemparams(my.hmm1,FALSE)
## Normal
my.hmm2=HMM(y=ynormal, subseq=100, K=3, iters=100, burnin=10,
emdist="normal", nmix=2, delta=FALSE)
OBIC(my.hmm2)
## Poisson
my.hmm3=HMM(y=ypoisson, K=3, iters=100, burnin=10, emdist="poisson",
nmix=2, delta=FALSE)
OBIC(my.hmm3)
## Predictive estimation - make 15 predictive data sets (new X) and 20 replicate data sets (same X)
filelocation="C:\\Users\\iamrandom\\Desktop\\here\\"
my.hmm5=HMM(y=ygamma, W=tW, K=3, iters=100, burnin=10,
emdist="gamma", nmix=3, delta=TRUE,
outdir=filelocation, pT=200, yrep=20, Wp=Wp1, ypred=15)
OBIC(my.hmm5)
pp=OWcoef(my.hmm5,filelocation)
## Gamma with fixed first variables nmix=2
nmix=2; K=3; J=dim(ygamma)[2]
prior1=array(1,dim=c(5,nmix,K,J)); prior1[1,1,,]=1; prior1[1,2,,]=2; prior1[2,,,]=NA
my.hmm6=HMM(y=ygamma, priors=prior1, K=3, iters=100, burnin=10,
emdist="gamma", nmix=2, delta=TRUE)
OBIC(my.hmm6)
Oemparams(my.hmm6)
### my.nhmm7 (K=3) (yhold is the last 10% of the data)
filelocation="C:\\Users\\iamrandom\\Desktop\\here\\"
my.hmm7=HMM(y=ygamma[1:1800,], W=array(tW[,1:1800,],
dim=c(1,1800,15)), K=3, iters=50, burnin=10,
emdist="gamma", nmix=3, delta=TRUE, outdir=filelocation,
ymiss=TRUE, yrep=10, pT=200,
Wp=array(tW[,1801:2000,],dim=c(1,200,15)), ypred=10,
yhold=ygamma[1801:2000,])
OBIC(my.hmm7)
# run it with K=3 and then K=1 and compare using both BIC and PLS
## End(Not run)
``` |

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