Package ‘mhsmm’

August 23, 2023

Type Package
Title Inference for Hidden Markov and Semi-Markov Models
Version 0.4.21
Date 2023-08-21
Author Jared O'Connell <jaredoconnell@gmail.com>, Søren Højsgaard
<sorenh@math.aau.dk>
Maintainer Jared O'Connell <jaredoconnell@gmail.com>
Description Parameter estimation and prediction for hidden Markov and semi-Markov mod-
els for data with multiple observation sequences. Suitable for equidistant time se-
ries data, with multivariate and/or missing data. Allows user defined emission distributions.
License GPL (>= 2)
LazyLoad yes
Depends mvtnorm
Repository CRAN
Encoding UTF-8
NeedsCompilation yes
LazyData true
Imports methods
Date/Publication 2023-08-23 03:10:02 UTC

R topics documented:
addStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
dmvnorm.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
dnorm.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
dpois.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
gammafit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
hmmfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
hmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
hsmmfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1
2 addStates

hsmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
mstep.mvnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
mstep.norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
mstep.pois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
plot.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
plot.hsmm.data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
predict.hmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
predict.hmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
predict.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
predict.hsmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
print.hmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
print.hmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
print.hsmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
reproai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
reprocows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
reproppa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
rmvnorm.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
rnorm.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
rpois.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
sim.mc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
simulate.hmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
simulate.hsmmspec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
smooth.discrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
summary.hmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
summary.hsmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Index 34

addStates Adds a bar representing state sequence.

Description

Add a colour coded horizontal bar representing the state sequence to a plot of (presumably time-
series) data.

Usage

addStates(states, x=NULL,ybot = axTicks(2)[1],

ytop=ybot + (axTicks(2)[2] - axTicks(2)[1])/5,dy = ytop - ybot,
greyscale = FALSE,leg = NA, J = length(unique(states)), time.scale = 1,
shiftx = 0)
dmvnorm.hsmm 3

Arguments
states A vector of integers representing the states traversed
x The time values where the states are observed ((1:length(states)-shiftx)/time.scale
if NULL)
ybot Vertical bottom limit of the bar.
ytop Vertical top limit of the bar.
dy Height of the bar.
greyscale If TRUE produces a bar in greyscale.
leg Array of state names, if present, produces a legend.
J Number of states
time.scale Resolution of the timescale
shiftx Shift the bar forward or backwards horizontal by shiftx distance.

Author(s)
Soren Hojsgaard sorenh@math.aau.dk

See Also
addStates

Examples
plot(rnorm(100),type='l')
addStates(rep(c(1,2),each=50))

plot(seq(0.01,1,.01),rnorm(100),type='l')
addStates(rep(c(1,2),each=50),seq(0.01,1,.01))

dmvnorm.hsmm Emission ensity function for a multivariate normal emission distribu-

tion

Description
Calculates the density of observations x for state j given the parameters in model. This is used for
a multivariate Gaussian emission distribution of a HMM or HSMM and is a suitable prototype for
user’s to make their own custom distributions.

Usage
dmvnorm.hsmm(x, j, model)
4 dnorm.hsmm

Arguments
x Observed value
j State
model A hsmmspec or hmmspec object

Details
This is used by hmm and hsmm to calculate densities for use in the E-step of the EM algorithm. It can
also be used as a template for users wishing to building their own emission distributions

Value
A vector of probability densities.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
mstep.mvnorm, rmvnorm.hsmm

Examples
J<-2
initial <- rep(1/J,J)
P <- matrix(c(.3,.5,.7,.5),nrow=J)
b <- list(mu=list(c(-3,0),c(1,2)),sigma=list(diag(2),matrix(c(4,2,2,3), ncol=2)))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dmvnorm.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rmvnorm.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.mvnorm)

dnorm.hsmm Emission density function for normal emission distribution

Description
Calculates the density of observations x for state j given the parameters in model. This is used for
the Gaussian emission distribution of a HMM or HSMM and is a suitable prototype for user’s to
make their own custom distributions.

Usage
dnorm.hsmm(x, j, model)
dpois.hsmm 5

Arguments
x Observed value
j State
model A hsmmspec or hmmspec object

Details
This is used by hmm and hsmm to calculate densities for use in the E-step of the EM algorithm. It can
also be used as a template for users wishing to building their own emission distributions

Value
A vector of probability densities.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

dpois.hsmm Emission density function for Poisson emission distribution

Description
Calculates the density of observations x for state j given the parameters in model. This is used for
a Poisson emission distribution of a HMM or HSMM and is a suitable prototype for user’s to make
their own custom distributions.

Usage
dpois.hsmm(x, j, model)

Arguments
x Observed value
j State
model A hsmmspec or hmmspec object

Details
This is used by hmm and hsmm to calculate densities for use in the E-step of the EM algorithm. It can
also be used as a template for users wishing to building their own emission distributions

Value
A vector of probability densities.
6 gammafit

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
mstep.pois, rpois.hsmm

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(lambda=c(1,3,6))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dpois.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rpois.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.pois)

gammafit Parameter estimation for the Gamma distribution

Description
Estimates parameters for the Gamma distribution using the Method of Maximum Likelihood, works
with weighted data.

Usage
gammafit(x, wt = NULL)

Arguments
x A vector of observations
wt Optional set of weights

Value
shape The shape parameter
scale The scale parameter (equal to 1/rate)

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Choi, S. and Wette, R. (1969), Maximum likelihood estimation of the parameters of the gamma
distribution and their bias, Technometrics, 11, 683-96-690.
hmmfit 7

Examples
gammafit(rgamma(1000,shape=10,scale=13))

hmmfit fit a hidden Markov model

Description
Estimates parameters of a HMM using the EM algorithm.

Usage
hmmfit(x,start.val,mstep=mstep.norm,lock.transition=FALSE,tol=1e-08,maxit=1000)

Arguments
x A hsmm.data object (see Details)
start.val Starting parameters for the model (see Details)
mstep Re-estimates the parameters of density function on each iteration
lock.transition
If TRUE will not re-estimate the transition matrix
maxit Maximum number of iterations
tol Convergence tolerance

Value
start A vector of the starting probabilities for each state
a The transition matrix of the embedded Markov chain
emission A list of the parameters of the emission distribution

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Jared O’Connell, Soren Hojsgaard (2011). Hidden Semi Markov Models for Multiple Observa-
tion Sequences: The mhsmm Package for R., Journal of Statistical Software, 39(4), 1-22., URL
http://www.jstatsoft.org/v39/i04/.
Rabiner, L. (1989), A tutorial on hidden Markov models and selected applications in speech recog-
nition, Proceedings of the IEEE, 77, 257-286.

See Also
predict.hmm
8 hmmspec

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(mu=c(-3,0,2),sigma=c(2,1,.5))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dnorm.hsmm)
model

train <- simulate(model, nsim=300, seed=1234, rand.emis=rnorm.hsmm)

plot(train,xlim=c(0,100))

init0 <- rep(1/J,J)

P0 <- matrix(1/J,nrow=J,ncol=J)
b0 <- list(mu=c(-3,1,3),sigma=c(1,1,1))
startval <- hmmspec(init=init0, trans=P0,parms.emission=b0,dens.emission=dnorm.hsmm)
h1 = hmmfit(train,startval,mstep=mstep.norm)

plot(h1$loglik,type='b',ylab='Log-likelihood',xlab='Iteration')
summary(h1)

#proportion of incorrect states

mean(train$s!=predict(h1,train)$s)

#simulate a new test set

test <- simulate(model, nsim=c(100,200,300), seed=1234,rand.emis=rnorm.hsmm)
mean(test$s!=predict(h1,test)$s)

hmmspec Specificatin of HMMs

Description
Creates a model specficiation for a hidden Markov model

Usage
hmmspec(init, trans, parms.emission, dens.emission, rand.emission=NULL,mstep=NULL)

Arguments
init Distribution of states at t=1 ie. P(S=s) at t=1
trans The transition matrix of the Markov chain
parms.emission A list containing the parameters of the emission distribution
dens.emission Density function of the emission distribution.
rand.emission The function used to generate observations from the emission distribution
mstep Re-estimates the parameters of density function on each iteration
hsmmfit 9

Value
A hmmspec object

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Jared O’Connell, Soren Hojsgaard (2011). Hidden Semi Markov Models for Multiple Observa-
tion Sequences: The mhsmm Package for R., Journal of Statistical Software, 39(4), 1-22., URL
http://www.jstatsoft.org/v39/i04/.
Rabiner, L. (1989), A tutorial on hidden Markov models and selected applications in speech recog-
nition, Proceedings of the IEEE, 77, 257-286.

See Also
simulate.hmmspec, simulate.hmmspec, hmmfit, predict.hmm

hsmmfit fit a hidden semi-Markov model

Description
Estimates parameters of a HSMM using the EM algorithm.

Usage
hsmmfit(x,model,mstep=NULL,M=NA,maxit=100,
lock.transition=FALSE,lock.d=FALSE,graphical=FALSE)

Arguments
x A hsmm.data object (see Details)
model Starting parameters for the model (see hsmmspec)
mstep Re-estimates the parameters of density function on each iteration
maxit Maximum number of iterations
M Maximum number of time spent in a state (truncates the waiting distribution)
lock.transition
If TRUE will not re-estimate the transition matrix
lock.d If TRUE will not re-estimate the sojourn time density
graphical If TRUE will plot the sojourn densities on each iteration
10 hsmmfit

Value
start A vector of the starting probabilities for each state
a The transition matrix of the embedded Markov chain
emission A list of the parameters of the emission distribution
waiting A list of the parameters of the waiting distribution

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Jared O’Connell, Soren Hojsgaard (2011). Hidden Semi Markov Models for Multiple Observa-
tion Sequences: The mhsmm Package for R., Journal of Statistical Software, 39(4), 1-22., URL
http://www.jstatsoft.org/v39/i04/.
Guedon, Y. (2003), Estimating hidden semi-Markov chains from discrete sequences, Journal of
Computational and Graphical Statistics, Volume 12, Number 3, page 604-639 - 2003

See Also
hsmmspec,simulate.hsmmspec,predict.hsmm

Examples
J <- 3
init <- c(0,0,1)
P <- matrix(c(0,.1,.4,.5,0,.6,.5,.9,0),nrow=J)
B <- list(mu=c(10,15,20),sigma=c(2,1,1.5))
d <- list(lambda=c(10,30,60),shift=c(10,100,30),type='poisson')
model <- hsmmspec(init,P,parms.emission=B,sojourn=d,dens.emission=dnorm.hsmm)
train <- simulate(model,r=rnorm.hsmm,nsim=100,seed=123456)
plot(train,xlim=c(0,400))
start.poisson <- hsmmspec(init=rep(1/J,J),
transition=matrix(c(0,.5,.5,.5,0,.5,.5,.5,0),nrow=J),
parms.emission=list(mu=c(4,12,23),
sigma=c(1,1,1)),
sojourn=list(lambda=c(9,25,40),shift=c(5,95,45),type='poisson'),
dens.emission=dnorm.hsmm)

M=500
# not run (takes some time)
# h.poisson <- hsmmfit(train,start.poisson,mstep=mstep.norm,M=M)
# plot(h.poisson$loglik,type='b',ylab='Log-likelihood',xlab='Iteration') ##has it converged?
# summary(h.poisson)
# predicted <- predict(h.poisson,train)
# table(train$s,predicted$s) ##classification matrix
# mean(predicted$s!=train$s) ##misclassification rate

d <- cbind(dunif(1:M,0,50),dunif(1:M,100,175),dunif(1:M,50,130))
start.np <- hsmmspec(init=rep(1/J,J),
hsmmspec 11

transition=matrix(c(0,.5,.5,.5,0,.5,.5,.5,0),nrow=J),
parms.emission=list(mu=c(4,12,23),
sigma=c(1,1,1)),
sojourn=list(d=d,type='nonparametric'),
dens.emission=dnorm.hsmm)
# not run (takes some time)
# h.np <- hsmmfit(train,start.np,mstep=mstep.norm,M=M,graphical=TRUE)
# mean(predicted$s!=train$s) ##misclassification rate

#J <- 2
#init <- c(1, 0)
#P <- matrix(c(0, 1, 1, 0), nrow = J)
#B <- list(mu = list(c(2, 3), c(3, 4)), sigma = list(matrix(c(4, 2, 2, 3), ncol = 2), diag(2)))
#d <- list(shape = c(10, 25), scale = c(2, 2), type = "gamma")
#model <- hsmmspec(init, P, parms.emis = B, sojourn = d, dens.emis = dmvnorm.hsmm)
#train <- simulate(model, c(1000,100), seed = 123, rand.emis = rmvnorm.hsmm)

#yhat <- predict(model, train)

#mean(predict(model,train)$s==train$s)

hsmmspec Hidden semi-Markov model specification

Description

Creates a model specification of a hidden semi-Markov model.

Usage

hsmmspec(init,transition,parms.emission,sojourn,dens.emission,
rand.emission=NULL,mstep=NULL)

Arguments

init Distribution of states at t=1 ie. P(S=s) at t=1

transition The transition matrix of the embedded Markov chain (diagonal must be 0)
parms.emission A list containing the parameters of the emission distribution
sojourn A list containining the parameters and type of sojourn distribtuion (see Details)
dens.emission Density function of the emission distribution
rand.emission The function used to generate observations from the emission distribution
mstep Re-estimates the parameters of density function on each iteration
12 mstep.mvnorm

Details
The sojourn argument provides a list containing the parameters for the available sojourn distribu-
tions. Available sojourn distributions are shifted Poisson, Gamma and non-parametric.
In the case of the Gamma distribution, sojourn is a list with vectors shape and scale (the Gamma
parameters in dgamma), both of length J. Where J is the number of states. See reprocows for an
example using Gamma sojourn distributions.
In the case of shifted Poisson, sojourn is list with vectors shift and lambda, both of length J. See
hsmmfit for an example using shifted Poisson sojourn distributions.
In the case of non-parametric, sojourn is a list containing a M x J matrix. Where entry (i,j) is
the probability of a sojourn of length i in state j. See hsmmfit for an example using shifted non-
parametric sojourn distributions.

Value
An object of class hsmmspec

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Jared O’Connell, Soren Hojsgaard (2011). Hidden Semi Markov Models for Multiple Observa-
tion Sequences: The mhsmm Package for R., Journal of Statistical Software, 39(4), 1-22., URL
http://www.jstatsoft.org/v39/i04/.
Guedon, Y. (2003), Estimating hidden semi-Markov chains from discrete sequences, Journal of
Computational and Graphical Statistics, Volume 12, Number 3, page 604-639 - 2003

See Also
hsmmfit ,simulate.hsmmspec, predict.hsmm

mstep.mvnorm Performs re-estimation (the M-step) for a multivariate normal emis-

sion distribution

Description
Re-estimates the parameters of a multivariate normal emission distribution as part of the EM al-
gorithm for HMMs and HSMMs. This is called by the hmm and hsmm functions. It is a suitable
prototype function for users wishing to design their own emission distributions.

Usage
mstep.mvnorm(x, wt)
mstep.norm 13

Arguments
x A vector of observed values
wt A T x J matrix of weights. Column entries are the weights for respective states.

Details
Users may write functions that take the same arguments and return the same values for their own
custom emission distributions.

Value
Returns the emission slot of a hmmspec or hsmmspec object

mu A list of length J contain the mean vectors

sigma A list of length J containing the covariance matrices

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
dmvnorm.hsmm, rmvnorm.hsmm

Examples
J<-2
initial <- rep(1/J,J)
P <- matrix(c(.3,.5,.7,.5),nrow=J)
b <- list(mu=list(c(-3,0),c(1,2)),sigma=list(diag(2),matrix(c(4,2,2,3), ncol=2)))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dmvnorm.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rmvnorm.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.mvnorm)

mstep.norm Performs re-estimation (the M-step) for a normal emission distribution

Description
Re-estimates the parameters of a normal emission distribution as part of the EM algorithm for
HMMs and HSMMs. This is called by the hmm and hsmm functions. It is a suitable prototype
function for users wishing to design their own emission distributions.

Usage
mstep.norm(x, wt)
14 mstep.pois

Arguments
x A vector of observed values
wt A T x J matrix of weights. Column entries are the weights for respective states.

Details
Users may write functions that take the same arguments and return the same values for their own
custom emission distributions.

Value
Returns the emission slot of a hmmspec or hsmmspec object
mu Vector of length J contain the means
sigma Vector of length J containing the variances

Author(s)
Jared O’Connell jaredoconnell@gmail.com

mstep.pois Performs re-estimation (the M-step) for a Poisson emission distribu-

tion

Description
Re-estimates the parameters of a Poisson emission distribution as part of the EM algorithm for
HMMs and HSMMs. This is called by the hmm and hsmm functions. It is a suitable prototype
function for users wishing to design their own emission distributions.

Usage
mstep.pois(x, wt)

Arguments
x A vector of observed values
wt A T x J matrix of weights. Column entries are the weights for respective states.

Details
Users may write functions that take the same arguments and return the same values for their own
custom emission distributions.

Value
Returns the emission slot of a hmmspec or hsmmspec object
lambda Vector of length J containing the Poisson parameters for each state j
plot.hsmm 15

Author(s)

Jared O’Connell jaredoconnell@gmail.com

See Also

rpois.hsmm, dpois.hsmm

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(lambda=c(1,3,6))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dpois.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rpois.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.pois)

plot.hsmm Plot function for hsmms

Description

Displays the densities for the sojourn distributions of each state.

Usage

## S3 method for class 'hsmm'

plot(x, ...)

Arguments

x A hsmm object
... Arguments passed to plot

Author(s)

Jared O’Connell jaredoconnell@gmail.com

16 predict.hmm

plot.hsmm.data Plot function for hsmm data

Description
Produces a plot of the observed sequences, and displays a coloured bar signifying the hidden states
(if available)

Usage
## S3 method for class 'hsmm.data'
plot(x, ...)

Arguments
x A hsmm.data object
... Arguments passed to plot.ts

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
addStates

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(mu=c(-3,0,2),sigma=c(2,1,.5))
model <- hmmspec(init=initial, trans=P, parms.emission=b, dens.emission=dnorm.hsmm)

train <- simulate(model, nsim=300, seed=1234, rand.emis=rnorm.hsmm)

plot(train,xlim=c(0,100))

predict.hmm Prediction function for hmm

Description
Predicts the underlying state sequence for an observed sequence newdata given a hmm model
predict.hmm 17

Usage
## S3 method for class 'hmm'
predict(object, newdata,method = "viterbi", ...)

Arguments
object An object of class hmm
newdata A vector or list of observations
method Prediction method (see details)
... further arguments passed to or from other methods.

Details
If method="viterbi", this technique applies the Viterbi algorithm for HMMs, producing the most
likely sequence of states given the observed data. If method="smoothed", then the individually
most likely (or smoothed) state sequence is produced, along with a matrix with the respective prob-
abilities for each state.

Value
Returns a hsmm.data object, suitable for plotting.

newdata A vector or list of observations

s A vector containing the reconstructed state sequence
N The lengths of each sequence
p A matrix where the rows represent time steps and the columns are the probability
for the respective state (only produced when method="smoothed")

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Rabiner, L. (1989), A tutorial on hidden Markov models and selected applications in speech recog-
nition, Proceedings of the IEEE, 77, 257-286.

See Also
hmmfit,hmmspec

Examples
##See examples in 'hmmfit'
18 predict.hmmspec

predict.hmmspec Prediction function for hmmspec

Description
Predicts the underlying state sequence for an observed sequence newdata given a hmmspec model

Usage
## S3 method for class 'hmmspec'
predict(object, newdata,method = "viterbi", ...)

Arguments
object An object of class hmm
newdata A vector or list of observations
method Prediction method (see details)
... further arguments passed to or from other methods.

Details
If method="viterbi", this technique applies the Viterbi algorithm for HMMs, producing the most
likely sequence of states given the observed data. If method="smoothed", then the individually
most likely (or smoothed) state sequence is produced, along with a matrix with the respective prob-
abilities for each state. This function differs from predict.hmm in that it takes the output from
hmmspec ie. this is useful when users already know their parameters and wish to make predictions.

Value
Returns a hsmm.data object, suitable for plotting.

newdata A vector or list of observations

s A vector containing the reconstructed state sequence
N The lengths of each sequence
p A matrix where the rows represent time steps and the columns are the probability
for the respective state (only produced when method="smoothed")

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Rabiner, L. (1989), A tutorial on hidden Markov models and selected applications in speech recog-
nition, Proceedings of the IEEE, 77, 257-286.
predict.hsmm 19

See Also
hmmspec

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(mu=c(-3,0,2),sigma=c(2,1,.5))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dnorm.hsmm)
train <- simulate(model, nsim=300, seed=1234, rand.emis=rnorm.hsmm)
mean(predict(model,train)$s!=train$s) #error rate when true model is known

predict.hsmm Prediction for hsmms

Description
Predicts the underlying state sequence for an observed sequence newdata given a hsmm model

Usage
## S3 method for class 'hsmm'
predict(object, newdata, method = "viterbi", ...)

Arguments
object An object of type hsmm
newdata A vector or dataframe of observations
method Prediction method (see details)
... further arguments passed to or from other methods.

Details
If method="viterbi", this technique applies the Viterbi algorithm for HSMMs, producing the
most likely sequence of states given the observed data. If method="smoothed", then the individu-
ally most likely (or smoothed) state sequence is produced, along with a matrix with the respective
probabilities for each state.

Value
Returns a hsmm.data object, suitable for plotting.
newdata A vector or list of observations
s A vector containing the reconstructed state sequence
N The lengths of each sequence
p A matrix where the rows represent time steps and the columns are the probability
for the respective state (only produced when method="smoothed")
20 predict.hsmmspec

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Guedon, Y. (2003), Estimating hidden semi-Markov chains from discrete sequences, Journal of
Computational and Graphical Statistics, Volume 12, Number 3, page 604-639 - 2003

See Also
hsmmfit,predict.hsmmspec

Examples
##See 'hsmmfit' for examples

predict.hsmmspec Prediction for hsmmspec

Description
Predicts the underlying state sequence for an observed sequence newdata given a hsmm model

Usage
## S3 method for class 'hsmmspec'
predict(object, newdata, method = "viterbi",M=NA, ...)

Arguments
object An object of type hsmmspec
newdata A vector or dataframe of observations
method Prediction method (see details)
M Maximum number of time spent in a state (truncates the waiting distribution)
... further arguments passed to or from other methods.

Details
If method="viterbi", this technique applies the Viterbi algorithm for HSMMs, producing the
most likely sequence of states given the observed data. If method="smoothed", then the individu-
ally most likely (or smoothed) state sequence is produced, along with a matrix with the respective
probabilities for each state. This method is different to predict.hsmm in that it takes the output from
hsmmspec as input ie. it is useful for people who already know their model parameters.
print.hmm 21

Value
Returns a hsmm.data object, suitable for plotting.

newdata A vector or list of observations

s A vector containing the reconstructed state sequence
N The lengths of each sequence
p A matrix where the rows represent time steps and the columns are the probability
for the respective state (only produced when method="smoothed")

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Guedon, Y. (2003), Estimating hidden semi-Markov chains from discrete sequences, Journal of
Computational and Graphical Statistics, Volume 12, Number 3, page 604-639 - 2003

See Also
hsmmspec, predict.hsmm

Examples
J <- 3
init <- c(0,0,1)
P <- matrix(c(0,.1,.4,.5,0,.6,.5,.9,0),nrow=J)
B <- list(mu=c(10,15,20),sigma=c(2,1,1.5))
d <- list(lambda=c(10,30,60),shift=c(10,100,30),type='poisson')
model <- hsmmspec(init,P,parms.emission=B,sojourn=d,dens.emission=dnorm.hsmm)
train <- simulate(model,r=rnorm.hsmm,nsim=100,seed=123456)
mean(predict(model,train,M=500)$s!=train$s) #error rate when true model is known

print.hmm Print method for hmm objects

Description
Prints the slots of a hmm object

Usage
## S3 method for class 'hmm'
print(x, ...)
22 print.hsmmspec

Arguments
x An object of type hmm
... further arguments passed to or from other methods.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

print.hmmspec Print function for hmmspec

Description
Prints the parameters contained in the object

Usage
## S3 method for class 'hmmspec'
print(x, ...)

Arguments
x An object of type hmmspec
... further arguments passed to or from other methods.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

print.hsmmspec Print function for hsmmspec

Description
Prints the parameters contained in the object

Usage
## S3 method for class 'hsmmspec'
print(x, ...)

Arguments
x An object of type hsmmspec
... further arguments passed to or from other methods.
reproai 23

Author(s)
Jared O’Connell jaredoconnell@gmail.com

reproai Artificial insemination times for seven cows

Description
This is an auxilliary data set to the cows data set containing times of artificial insemination for
respective cows. Only the day of insemination was recorded so time of day is always midday.

Usage
reproai

Format
reproai is a dataframe with 12 rows and id being the cow’s id and days.from.calving recording
the number of days from calving when insemination occurred.

Source
Danish Cattle Research Centre

References
Peters, A. and Ball, P. (1995), "Reproduction in Cattle," 2nd ed.

reprocows Reproductive data from seven dairy cows

Description
This data set contains hourly observations on progesterone and an activity index at hourly intervals
since calving on seven dairy cows.

Usage
reprocows

Format
reprocows is a data frame containing 13040 rows. id is the cow ID, progesterone is a measure-
ment of the hormone in ng/L taken from a milk sample, activity is a relative measure of activity
calculated from a pedometer.
There are a large number of missing values as progesterone is measured only at milking time (and at
a farm manager’s discretion). Missing values in activity occur due to hardware problems can occur
with pedometers.
24 reproppa

Source
Danish Cattle Research Centre

References
Peters, A. and Ball, P. (1995), "Reproduction in Cattle," 2nd ed.

Examples
data(reprocows)
data(reproai)
data(reproppa)
tm = 1600

J <- 3
init <- c(1,0,0)
trans <- matrix(c(0,0,0,1,0,1,0,1,0),nrow=J)
emis <- list(mu=c(0,2.5,0),sigma=c(1,1,1))

N <- as.numeric(table(reprocows$id))
train <- list(x=reprocows$activity,N=N)
class(train) <- "hsmm.data"
tmp <- gammafit(reproppa * 24)
M <- max(N)

d <- cbind(dgamma(1:M,shape=tmp$shape,scale=tmp$scale),
# ppa sojourn directly estimated from ppa data set
dunif(1:M,4,30),
# oestrus between 4 and 30 hours
dunif(1:M,15*24,40*24))
#not-oestrus between 15 and 40 days

startval <- hsmmspec(init,trans,parms.emission=emis,list(d=d,type='gamma'),

dens.emission=dnorm.hsmm)
#not run (takes some time)
#h.activity <- hsmmfit(train,startval,mstep=mstep.norm,maxit=10,M=M,lock.transition=TRUE)

reproppa Observed lengths of post-partum anoestrus for 73 dairy cows

Description
This data set contains the observed length of post-partum anoestrus (in days) for 73 dairy cattle.

Usage
reproppa
rmvnorm.hsmm 25

Format
reproppa a vector containing 73 integers.

Source
Danish Cattle Research Centre

References
Peters, A. and Ball, P. (1995), "Reproduction in Cattle," 2nd ed.

rmvnorm.hsmm Random number generation from a multivariate normal distributed

emission distribution

Description
This generates values from a multivariate normal distributed emission state j given parameters in
model.

Usage
rmvnorm.hsmm(j, model)

Arguments
j An integer representing the state
model A hmmspec or hsmmspec object

Details
This is essentially a wrapper for rnorm. Users may build functions with the same arguments and
return values so they can use their own custom emission distributions.

Value
A single value from the emission distribution.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
dmvnorm.hsmm, mstep.mvnorm
26 rnorm.hsmm

Examples
J<-2
initial <- rep(1/J,J)
P <- matrix(c(.3,.5,.7,.5),nrow=J)
b <- list(mu=list(c(-3,0),c(1,2)),sigma=list(diag(2),matrix(c(4,2,2,3), ncol=2)))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dmvnorm.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rmvnorm.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.mvnorm)

rnorm.hsmm Random number generation from a normally distributed emission dis-

tribution

Description

This generates values from a normally distributed emission state j given parameters in model.

Usage

rnorm.hsmm(j, model)

Arguments

j An integer representing the state

model A hmmspec or hsmmspec object

Details

This is essentially a wrapper for rnorm. Users may build functions with the same arguments and
return values so they can use their own custom emission distributions.

Value

A single value from the emission distribution.

Author(s)

Jared O’Connell jaredoconnell@gmail.com

rpois.hsmm 27

rpois.hsmm Random number generation from a Poisson distributed emission dis-

tribution

Description
This generates values from a Poisson distributed emission state j given parameters in model.

Usage
rpois.hsmm(j, model)

Arguments
j An integer representing the state
model A hmmspec or hsmmspec object

Details
This is essentially a wrapper for rpois. Users may build functions with the same arguments and
return values so they can use their own custom emission distributions.

Value
A single value from the emission distribution.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

See Also
mstep.pois, dpois.hsmm

Examples
J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(lambda=c(1,3,6))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dpois.hsmm)
model
train <- simulate(model, nsim=300, seed=1234, rand.emis=rpois.hsmm)
plot(train,xlim=c(0,100))
h1 = hmmfit(train,model,mstep=mstep.pois)
28 simulate.hmmspec

sim.mc Markov chain simulation

Description
Simulates a Markov chain

Usage
sim.mc(init, transition, N)

Arguments
init The distribution of states at the first time step
transition The transition probability matrix of the Markov chain
N The number of observations to simulate

Value
A vector of integers representing the state sequence.

Author(s)
Jared O’Connell jaredoconnell@gmail.com

Examples
p <- matrix(c(.1,.3,.6,rep(1/3,3),0,.5,.5),ncol=3,byrow=TRUE)
init <- rep(1/3,3)
sim.mc(init,p,10)

simulate.hmmspec Simulation of hidden Markov models

Description
Simulates data from a hidden Markov model

Usage
## S3 method for class 'hmmspec'
simulate(object, nsim, seed = NULL, rand.emission=NULL,...)
simulate.hmmspec 29

Arguments
object A hmmspec object
nsim An integer or vector of integers (for multiple sequences) specifying the length
of the sequence(s)
seed seed for the random number generator
rand.emission The function used to generate observations from the emission distribution
... further arguments passed to or from other methods.

Details
If nsim is a single integer then a HMM of that length is produced. If nsim is a vector of integers,
then length(nsim) sequences are generated with respective lengths.

Value
An object of class hmmdata

x A vector of length sum(N) - the sequence(s) of observed values

s A vector of length sum(N) - the sequence(s) of hidden states
N A vector of the length of each observation sequence (used to segment x and s)

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Rabiner, L. (1989), A tutorial on hidden Markov models and selected applications in speech recog-
nition, Proceedings of the IEEE, 77, 257-286.

See Also
hmmspec, link{predict.hmm}

Examples

J<-3
initial <- rep(1/J,J)
P <- matrix(c(.8,.5,.1,0.05,.2,.5,.15,.3,.4),nrow=J)
b <- list(mu=c(-3,0,2),sigma=c(2,1,.5))
model <- hmmspec(init=initial, trans=P, parms.emission=b,dens.emission=dnorm.hsmm)
train <- simulate(model, nsim=100, seed=1234, rand.emis=rnorm.hsmm)
plot(train)
30 simulate.hsmmspec

simulate.hsmmspec Simulation for HSMMs

Description
Simulates values for a specified hidden semi-Markov model

Usage
## S3 method for class 'hsmmspec'
simulate(object, nsim, seed = NULL,rand.emission=NULL,...)

Arguments
object A hsmmspec object
nsim An integer or vector of integers (for multiple sequences) specifying the number
of states to generate per sequence
seed seed for the random number generator
rand.emission The function used to generate observations from the emission distribution
... further arguments passed to or from other methods.

Details
If nsim is a single integer then a HSMM of that length is produced. If nsim is a vector of integers,
then length(nsim) sequences are generated with respective lengths. Note thate length is the num-
ber of states visited, each state will have a sojourn time typically >1 so the vector will be longer
than nsim

Value
An object of class hmmdata

x A vector of length sum(N) - the sequence(s) of observed values

s A vector of length sum(N) - the sequence(s) of hidden states
N A vector of the length of each observation sequence (used to segment x and s)

Author(s)
Jared O’Connell jaredoconnell@gmail.com

References
Guedon, Y. (2003), Estimating hidden semi-Markov chains from discrete sequences, Journal of
Computational and Graphical Statistics, Volume 12, Number 3, page 604-639 - 2003
smooth.discrete 31

See Also
hsmmfit, hsmmspec, predict.hsmm

Examples
J <- 3
init <- c(0,0,1)
P <- matrix(c(0,.1,.4,.5,0,.6,.5,.9,0),nrow=J)
B <- list(mu=c(10,15,20),sigma=c(2,1,1.5))
d <- list(lambda=c(10,30,60),shift=c(10,100,30),type='poisson')
model <- hsmmspec(init,P,parms.emission=B,sojourn=d,dens.emission=dnorm.hsmm)
train <- simulate(model,rand.emis=rnorm.hsmm,nsim=100,seed=123456)
plot(train,xlim=c(0,400))

smooth.discrete Smoothing a discrete time series.

Description
The smooth.discrete() function provides a simple smoothing of a time series of discrete values
measured at equidistant times. Under the hood of smooth.discrete() is a hidden Markov model.

Usage
smooth.discrete(y, init = NULL, trans = NULL, parms.emission = 0.5,
method = "viterbi", details = 0, ...)

Arguments
y A numeric vector
init Initial distribution (by default derived from data; see the vignette for details)
trans Transition matrix (by default derived from data; see the vignette for details)
parms.emission Matrix describing the conditional probabilities of the observed states given the
latent states. (See the vignette for details).
method Either "viterbi" or "smoothed". The viterbi method gives the jointly most likely
sequence; the smoothed method gives the sequence of individually most likely
states.
details Controlling the amount of information printed.
... Further arguments passed on to the "hmmfit" function.

Details
The parameters are estimated using the Baum-Welch algorithm (a special case of the EM-algorithm).
32 summary.hmm

Value
A list with the following components:

s The "smoothed" states

model The underlying hmm (hidden Markov model) object
data The data
initial The initial parameters

Author(s)
S<c3><b8>ren H<c3><b8>jsgaard <sorenh at math.aau.dk>

See Also
hmmspec, hmmfit

Examples
## Please see the vignette

summary.hmm Summary method for hmm objects

Description
Prints the estimated parameters of a hmm object

Usage
## S3 method for class 'hmm'
summary(object, ...)

Arguments
object A hmm object
... further arguments passed to or from other methods.

Value
An object of class ’summary.hmm’

Author(s)
Jared O’Connell jaredoconnell@gmail.com
summary.hsmm 33

summary.hsmm Summary function for hsmm

Description
Returns a summary object for a hsmm object

Usage
## S3 method for class 'hsmm'
summary(object, ...)

Arguments
object An object of type hsmm
... further arguments passed to or from other methods.

Author(s)
Jared O’Connell jaredoconnell@gmail.com
Index

∗ datasets reprocows, 23
reproai, 23 reproppa, 24
reprocows, 23 rmvnorm.hsmm, 4, 13, 25
reproppa, 24 rnorm.hsmm, 26
∗ models rpois.hsmm, 6, 15, 27
smooth.discrete, 31
sim.mc, 28
addStates, 2, 16 simulate.hmmspec, 9, 28
simulate.hsmmspec, 10, 12, 30
createTransition (smooth.discrete), 31 smooth.discrete, 31
summary.hmm, 32
dmvnorm.hsmm, 3, 13, 25 summary.hsmm, 33
dnorm.hsmm, 4 summary.smoothDiscrete
dpois.hsmm, 5, 15, 27 (smooth.discrete), 31
gammafit, 6

hmmfit, 7, 9, 32
hmmspec, 8, 29, 32
hsmmfit, 9, 12, 20, 31
hsmmspec, 10, 11, 21, 31

mstep.mvnorm, 4, 12, 25
mstep.norm, 13
mstep.pois, 6, 14, 27

plot.hsmm, 15
plot.hsmm.data, 16
predict.hmm, 7, 9, 16
predict.hmmspec, 18
predict.hsmm, 10, 12, 19, 21, 31
predict.hsmmspec, 20, 20
predict.smoothDiscrete
(smooth.discrete), 31
print.hmm, 21
print.hmmspec, 22
print.hsmmspec, 22
print.smoothDiscrete (smooth.discrete),
31

reproai, 23

34