Abstract

With the package mHMMbayes you can fit multilevel hidden Markov models. The multilevel hidden Markov model (HMM) is a generalization of the well-known hidden Markov model, tailored to accdeommodate (intense) longitudinal data of multiple individuals simultaneously. Using a multilevel framework, we allow for heterogeneity in the model parameters (transition probability matrix and conditional distribution), while estimating one overall HMM. The model has a great potential of application in many fields, such as the social sciences and medicine. The model can be fitted on multivariate categorical, continous, or count data, and include individual level covariates (allowing for e.g., group comparisons on model parameters). Parameters are estimated using Bayesian estimation utilizing the forward-backward recursion within a hybrid Metropolis within Gibbs sampler. The package also includes a function to simulate data and a function to obtain the most likely hidden state sequence for each individual using the Viterbi algorithm.

Hidden Markov models [HMMs; Rabiner
(1989)] are a machine learning method that have been used in many
different scientific fields to describe a sequence of observations for
several decades. For example, translating a fragment of spoken words
into text (i.e., speech recognition, see e.g.
Rabiner 1989; Woodland and Povey 2002), or the identification of
the regions of DNA that encode genes (i.e., gene
tagging, see e.g., Krogh, Mian, and Haussler 1994; Henderson, Salzberg,
and Fasman 1997; Burge and Karlin 1998). The development of this
package is, however, motivated from the area of social sciences. Due to
technological advancements, it becomes increasingly easy to collect long
sequences of data on behavior. That is, we can monitor behavior as it
unfolds in real time. An example here of is the interaction between a
therapist and a patient, where different types of nonverbal
communication are registered every second for a period of 15 minutes.
When applying HMMs to such behavioral data, they can be used to extract
latent behavioral states over time, and model the dynamics of behavior
over time.

A quite recent development in HMMs is the extension to multilevel HMMs
(see e.g., Altman 2007; Shirley et al. 2010;
Rueda, Rueda, and Diaz-Uriarte 2013; Zhang and Berhane 2014;
Haan-Rietdijk et al. 2017). Using the multilevel framework, we
can model several sequences (e.g., sequences of different persons)
simultaneously, while accommodating the heterogeneity between persons.
As a result, we can quantify the amount of variation between persons in
their dynamics of behavior, easily perform group comparisons on the
model parameters, and investigate how model parameters change as a
result of a covariate. For example, are the dynamics between a patient
and a therapist different for patients with a good therapeutic outcome
and patients with a less favorable therapeutic outcome?

With the package `mHMMbayes`

, one can estimate these
multilevel hidden Markov models. This tutorial starts out with a brief
description of the HMM and the multilevel HMM. For a more elaborate and
gentle introduction to HMMs, we refer to Zucchini, MacDonald, and Langrock (2016). Next,
we show how to use the package `mHMMbayes`

through an
extensive example on categorical data, also touching on the issues of
determining the number of hidden states and checking model convergence.
Information on the used estimation methods and algorithms in the package
is given in the vignette Estimation of the
multilevel hidden Markov model.

We illustrate using the package using the embedded categorical
example data `nonverbal`

. The data contains the nonverbal
communication of 10 patient-therapist couples, recorded for 15 minutes
at a frequency of 1 observation per second (= 900 observations per
couple). The following variables are contained in the dataset:

`id`

: id variable of patient - therapist couple to distinguish which observation belongs to which couple.`p_verbalizing`

: verbalizing behavior of the patient, consisting of 1 = not verbalizing, 2 = verbalizing, 3 = back channeling.`p_looking`

: looking behavior of the patient, consisting of 1 = not looking at therapist, 2 = looking at therapist.`t_verbalizing`

: verbalizing behavior of the therapist, consisting of 1 = not verbalizing, 2 = verbalizing, 3 = back channeling.`t_looking`

: looking behavior of the therapist, consisting of 1 = not looking at patient, 2 = looking at patient. The top 6 rows of the dataset are provided below.

When we plot the data of the first 5 minutes (= the first 300 observations) of the first couple, we get the following:

We can, for example, observe that both the patient and the therapist are mainly looking at each other during the observed 5 minutes. During the first minute, the patient is primarily speaking. During the second minute, the therapists starts, after which the patient takes over while the therapist is back channeling.

To fit a simple 2 state multilevel model with the function
`mHMM`

, one first has to specify some general model
properties and starting values:

```
library(mHMMbayes)
# specifying general model properties:
m <- 2
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)
# specifying starting values
start_TM <- diag(.8, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .2
start_EM <- list(matrix(c(0.05, 0.90, 0.05,
0.90, 0.05, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[1]), # vocalizing patient
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[2]), # looking patient
matrix(c(0.90, 0.05, 0.05,
0.05, 0.90, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[3]), # vocalizing therapist
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[4])) # looking therapist
```

The first line of code loads the `mHMMbayes`

package and
the `nonverbal`

data. Next we specify the general model
properties: the number of states used is set by `m <- 2`

,
the number of dependent variables in the dataset used to infer the
hidden states is specified by `n_dep <- 4`

, and the number
of categorical outcomes for each of the dependent variables is specified
by `q_emiss <- c(3, 2, 3, 2)`

.

The subsequent lines of code specify starting values for both the
transition probability matrix and the emission distribution(s), which
are given to the model in the argument `start_val`

(see next
piece of code). These starting values are used for the first run of the
forward backward algorithm. Although the hidden states cannot be
observed, one often has an idea for probable compositions of the states.
In the example, we expect that there is a state in which the patient
mostly speaks, and the therapist is silent, and a state during which the
patient is silent and the therapists speaks. In addition, we expect that
during both states, the therapist and the patient will be mainly looking
at each other instead of looking away. One usually also has some (vague)
idea on likely and unlikely switches between states, and the size of
self-transition probabilities. In the example, we think a state will
usually last quite some seconds, and thus expect a rather high
self-transition probability. All these ideas can be used to construct a
set of sensible starting values. Using sensible starting values
increases convergence speed, and often prevents a problem called ‘label
switching’. Hence, using random or uniform starting values is not
recommended, and a default option to do so is not included in the
package. Note that it is strongly advised to check model convergence and
label switching. That is, one should check if the algorithm reaches the
same solution when a set of different (but often conceptually similar)
starting values are used, and if label switching is not a problem. See
the section *Checking model convergence and label switching* for
an example. See the vignette Estimation of
the multilevel hidden Markov model) for more information on the
forward backward algorithm and on the problem of label switching.

As the estimation proceeds within a Bayesian context, a (hyper-)prior
distribution has to be defined for the group level parameters, i.e., the
group level emission and transition probabilities. Default,
non-informative priors are used unless specified otherwise by the user.
Below, we present some information on this. For a more elaborate
explanation on the used (hyper-)prior distributions and their
parameters, see the vignette Estimation of
the multilevel hidden Markov model.

First of all, note that the prior distributions on the emission
distribution and transition probabilities are not on the probabilities
directly, but on the intercepts (and regression coefficients given that
covariates are used) of the Multinomial regression model used to
accommodate the multilevel framework of the data. Second, parameters do
not each have their own independent prior distribution. As each row of
the emission distribution matrix and transition probability matrix sum
to one, the individual parameters of these rows are connected. Hence,
each row is seen as a set of parameters which are estimated jointly, and
each set of parameters has a multivariate prior distribution.

The sets of intercepts of the Multinomial regression model have a
multivariate normal distribution. The (hyper-) prior for these
intercepts thus consists of a prior distribution on the vector of means,
and a prior distribution on the covariance matrix. The hyper-prior for
the mean intercepts is a multivariate Normal distribution, with, as
default, a vector of means equal to 0, and a parameter \(K_0\) with which the covariance matrix is
multiplied. Here, \(K_0\) denotes the
number of observations (i.e., the number of hypothetical prior subjects)
on which the prior mean vector of zero’s is based. By default, \(K_0\) is set to 1. The hyper-prior for the
covariance matrix between the set of (state specific) intercepts has an
Inverse Wishart distribution, for which the variance in the default
setting equals 3 + \(m\) - 1 for the
transition probabilities and 3 + \(q\)
- 1 for the emission probabilities, and the covariance equals 0. The
degrees of freedom of the Inverse Wishart distribution is set to 3 +
\(m\) - 1 for the transition
probabilities and 3 + \(q\) - 1 for the
emission probabilities.

To specify user specific prior distributions, one uses the input option
`emiss_hyp_prior`

for the emission distribution and
`gamma_hyp_prior`

for the transition probabilities in the
function `mHMM`

. These input arguments take an object from
the class `mHMM_prior_emiss`

and
`mHMM_prior_gamma`

created by the functions
`prior_emiss_cat`

and `prior_gamma`

, respectively.
Both objects are a list, containing the following key elements:

`mu0`

, a lists containing the hypothesized hyper-prior mean values of the intercepts of the Multinomial logit model.`K0`

, the number of hypothetical prior subjects on which the set of hyper-prior mean intercepts specified in`mu0`

are based.`nu`

, degrees of freedom of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.`V`

, the variance-covariance of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.

Note that `K0`

, `nu`

and `V`

are
assumed equal over the states. The mean values of the intercepts (and
regression coefficients of the covariates) denoted by `mu0`

are allowed to vary over the states. All elements in the list either
have the prefix `gamma_`

or `emiss_`

, depending on
which list they belong to. When specifying prior distributions, note
that the first element of each row in the probability domain does not
have an intercept, as it serves as baseline category in the Multinomial
logit regression model. This means, for example, that if we would
specify a model with 3 states, `mu0`

is a vector with 2
elements, `K0`

and `nu`

contain 1 element and
`V`

is a 2 by 2 matrix.

The multilevel HMM is fitted using the function
`mHMM`

:

```
# Run a model without covariate(s) and default priors:
set.seed(14532)
out_2st <- mHMM(s_data = nonverbal,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 1000, burn_in = 200))
```

The call to `mHMM`

specifies the model with several
arguments. The `s_data`

argument specifies the input data
used to infer the hidden states over time. The `gen`

and
`start_val`

argument specify the general model properties and
the starting values, as discussed above. The arguments needed for the
MCMC algorithm are given in `mcmc`

: `J`

specifies
the number of iterations used by the hybrid metropolis within Gibbs
algorithm and `burn_in`

specifies the number of iterations to
discard when obtaining the model parameter summary statistics. The
function `mHMM`

returns an object of class `mHMM`

,
which has `print`

and `summary`

methods to see the
results. The `print`

method provides basic information on the
model fitted. That is, the number of subjects in the dataset analyzed,
the number of iterations and burn-in period, the average log likelihood
over all subjects and model fit indices AIC, the number of states
specified, and the number of dependent variables the states are based
on:

```
out_2st
#> Number of subjects: 10
#>
#> 1000 iterations used in the MCMC algorithm with a burn in of 200
#> Average Log likelihood over all subjects: -1622.789
#> Average AIC over all subjects: 3273.579
#> Average AICc over all subjects: 3274.053
#>
#> Number of states used: 2
#>
#> Number of dependent variables used: 4
#>
#> Type of dependent variable(s): categorical
```

The `summary`

method provides information on the estimated
parameters. That is, the point estimates of the posterior distribution
for the transition probability matrix and the emission distribution of
each of the dependent variables at the group level:

```
summary(out_2st)
#> State transition probability matrix
#> (at the group level):
#>
#> To state 1 To state 2
#> From state 1 0.930 0.070
#> From state 2 0.073 0.927
#>
#>
#> Emission distribution ( categorical ) for each of the dependent variables
#> (at the group level):
#>
#> $p_vocalizing
#> Category 1 Category 2 Category 3
#> State 1 0.010 0.972 0.018
#> State 2 0.812 0.033 0.155
#>
#> $p_looking
#> Category 1 Category 2
#> State 1 0.246 0.754
#> State 2 0.087 0.913
#>
#> $t_vocalizing
#> Category 1 Category 2 Category 3
#> State 1 0.822 0.058 0.120
#> State 2 0.030 0.956 0.015
#>
#> $t_looking
#> Category 1 Category 2
#> State 1 0.045 0.955
#> State 2 0.272 0.729
```

The resulting model indicates 2 well separated states: one in which
the patient is speaking and one in which the therapist is speaking.
Looking behavior is quite similar for both the patient and the therapist
in the 2 states. Information on the estimated parameters can also be
obtained using the function `obtain_gamma`

and
`obtain_emiss`

. These functions allow the user not only to
inspect the estimated parameters at the group level, but for each
subject individually as well, by specifying the input variable
`level = "subject"`

:

```
# When not specified, level defaults to "group"
gamma_pop <- obtain_gamma(out_2st)
gamma_pop
#> To state 1 To state 2
#> From state 1 0.930 0.070
#> From state 2 0.073 0.927
# To obtain the subject specific parameter estimates:
gamma_subj <- obtain_gamma(out_2st, level = "subject")
gamma_subj
#> $`Subject 1`
#> To state 1 To state 2
#> From state 1 0.945 0.055
#> From state 2 0.047 0.953
#>
#> $`Subject 2`
#> To state 1 To state 2
#> From state 1 0.937 0.063
#> From state 2 0.061 0.939
#>
#> $`Subject 3`
#> To state 1 To state 2
#> From state 1 0.968 0.032
#> From state 2 0.050 0.950
#>
#> $`Subject 4`
#> To state 1 To state 2
#> From state 1 0.940 0.060
#> From state 2 0.047 0.953
#>
#> $`Subject 5`
#> To state 1 To state 2
#> From state 1 0.942 0.058
#> From state 2 0.056 0.944
#>
#> $`Subject 6`
#> To state 1 To state 2
#> From state 1 0.939 0.061
#> From state 2 0.084 0.916
#>
#> $`Subject 7`
#> To state 1 To state 2
#> From state 1 0.930 0.070
#> From state 2 0.043 0.957
#>
#> $`Subject 8`
#> To state 1 To state 2
#> From state 1 0.933 0.067
#> From state 2 0.074 0.926
#>
#> $`Subject 9`
#> To state 1 To state 2
#> From state 1 0.948 0.052
#> From state 2 0.057 0.943
#>
#> $`Subject 10`
#> To state 1 To state 2
#> From state 1 0.960 0.040
#> From state 2 0.065 0.935
```

An additional option that the functions `obtain_gamma`

and
`obtain_emiss`

offer is changing the burn-in period used for
obtaining the summary statistics, using the input variable
`burn_in`

.

The package includes several plot functions to display the fitted
model and its parameters. First, one can plot the posterior densities of
a fitted model, for both the transition probability matrix gamma and for
the emission distribution probabilities. The posterior densities are
plotted for the group level and the subject level simultaneously. For
example, for the emission distribution for the variable
`p_vocalizing`

:

```
library(RColorBrewer)
Voc_col <- c(brewer.pal(3,"PuBuGn")[c(1,3,2)])
Voc_lab <- c("Not Speaking", "Speaking", "Back channeling")
plot(out_2st, component = "emiss", dep = 1, col = Voc_col,
dep_lab = c("Patient vocalizing"), cat_lab = Voc_lab)
```

Here, `component`

specifies whether we want to visualize the
posterior densities for the transition probability matrix gamma
(`component = "gamma"`

) or for the emission distribution
probabilities (`component = "emiss"`

), when using
`component = "emiss"`

the input variable `dep`

specifies which dependent variable we want to inspect (as the variable
`p_vocolizing`

is the first variable in the set, we set
`dep = 1`

), `col`

specifies the colors to be used
when plotting the lines, `dep_lab`

denotes the label of the
dependent variable we are plotting, and `cat_lab`

denotes the
labels of the categorical outcomes in the dependent variable. In the
plot, the solid line visualizes the posterior density at the group
level, while each of the dotted lines visualizes the posterior density
of one subject.

Second, one can plot the transition probabilities obtained with the
function `obtain_gamma`

with a riverplot:

```
# Transition probabilities at the group level and for subject number 1, respectively:
plot(gamma_pop, col = rep(rev(brewer.pal(3,"PiYG"))[-2], each = m))
plot(gamma_subj, subj_nr = 1, col = rep(rev(brewer.pal(3,"PiYG"))[-2], each = m))
```

Note that graphically displaying the transition probabilities becomes
more informative as the number of states increase.

Given a well-fitting HMM, it may be of interest to determine the
actual *sequence*, or order of succession, of hidden states that
has most likely given rise to the sequence of outcomes as observed in a
subject. One can either use local decoding, in which the probabilities
of the hidden state sequence are obtained simultaneously with the model
parameters estimates, or the well-known Viterbi algorithm (Viterbi 1967; Forney Jr 1973). In local
decoding, the most likely state is determined separately at each time
point \(t\), in contrast to the Viterbi
algorithm in which one determines the joint probability of the complete
sequence of observations \(O_{1:T}\)
and the complete sequence of hidden states \(S_{1:T}\).

In the package, local decoding can be achieved by saving the sampled
hidden state sequence at each iteration of the Gibbs sampler, by setting
the input variable `return_path = TRUE`

for the function
`mHMM`

. This will result in very large output files, however.
Global decoding can be performed by using the function
`vit_mHMM`

:

```
state_seq <- vit_mHMM(out_2st, s_data = nonverbal)
#> Please note that the output format is changed from wide to long format to facilitate aditionally returning the state probabilities, see the section 'Value' in the help file for more information.
head(state_seq)
#> subj state
#> 1 1 1
#> 2 1 1
#> 3 1 1
#> 4 1 1
#> 5 1 1
#> 6 1 1
```

The function returns the hidden state sequence for each subject in a matrix, where each row represents a point in time and each column represents a subject. We can inspect the obtained hidden state sequence by for example plotting it together with the observed data. Below, the first 5 minutes of the first couple is plotted again, with the addition of the estimated state sequence:

When using Bayesian estimation procedures, it is strongly advised to check model convergence and label switching. That is, one should check if the algorithm reaches the same solution when a set of different (but often conceptually similar) starting values are used, and if label switching is not a problem. With label switching, the label (i.e., which state represents what) ordering of the states switches over the iterations of the estimation algorithm. For example, what started out as state 1, now becomes state 2. One can check model convergence and label switching visually by inspecting the trace plots of parameters of a set of identical models that used varying starting values. Trace plots are plots of the sampled parameter values over the iterations. First, we fit the model with 2 states again, but with different starting values:

```
# specifying general model properties
m <-2
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)
# specifying different starting values
start_TM <- diag(.8, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .2
start_EM_b <- list(matrix(c(0.2, 0.6, 0.2,
0.6, 0.2, 0.2), byrow = TRUE,
nrow = m, ncol = q_emiss[1]), # vocalizing patient
matrix(c(0.4, 0.6,
0.4, 0.6), byrow = TRUE, nrow = m,
ncol = q_emiss[2]), # looking patient
matrix(c(0.6, 0.2, 0.2,
0.2, 0.6, 0.2), byrow = TRUE,
nrow = m, ncol = q_emiss[3]), # vocalizing therapist
matrix(c(0.4, 0.6,
0.4, 0.6), byrow = TRUE, nrow = m,
ncol = q_emiss[4])) # looking therapist
# Run a model identical to out_2st, but with different starting values:
set.seed(9843)
out_2st_b <- mHMM(s_data = nonverbal,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 1000, burn_in = 200))
```

The group level parameter estimates of the emission probabilities and
the transition probability matrix at each iteration of the estimation
algorithm are stored in the objects `emiss_prob_bar`

and
`gamma_prob_bar`

, respectively. The subject level parameter
estimates are stored in the object `PD_subj`

, where PD is an
abbreviation for posterior density. If we, for example, want to inspect
the trace plots for the emission probabilities for looking behavior of
the patient at the group level, we use the following code:

```
par(mfrow = c(m,q_emiss[2]))
for(i in 1:m){
for(q in 1:q_emiss[2]){
plot(x = 1:1000, y = out_2st$emiss_prob_bar[[2]][,(i-1) * q_emiss[2] + q],
ylim = c(0,1.4), yaxt = 'n', type = "l", ylab = "Transition probability",
xlab = "Iteration", main = paste("Patient", Look_lab[q], "in state", i), col = "#8da0cb")
axis(2, at = seq(0,1, .2), las = 2)
lines(x = 1:1000, y = out_2st_b$emiss_prob_bar[[2]][,(i-1) * q_emiss[2] + q], col = "#e78ac3")
legend("topright", col = c("#8da0cb", "#e78ac3"), lwd = 2,
legend = c("Starting value set 1", "Starting value set 2"), bty = "n")
}
}
```

It can be observed that the parameter estimates converge to the same parameter space, and that the chains mix well. Also, there is no evidence of label switching.

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Note that the likelihood ratio test, commonly used to compare nested models, cannot be used in case of the HMM (i.e., the difference in the log-likelihoods between models is not \(\chi^2\) distributed Rydén (2008)).↩︎

We note, however, that the AIC approximates the posterior distribution of the parameters by a Gaussian distribution, which might not be appropriate for models including parameters on the boundary of the parameter space (e.g., close to 0 or 1 in case of probability estimates), or for small data sets, as exemplified by Scott (2002). Model selection is therefore not a straightforward procedure in the context of HMM, and the choices remain subjective.↩︎