• Gentle introduction to HMMs

• Theory and notation

• Examples of univariate HMMs in R

• Examples of multivariate HMMs in R

For additional background, see

• Zucchini et al. (2008, 2016) “Hidden Markov Models for Time Series: An Introduction Using R”

• Jackson (2011) “Multi-State Models for Panel Data: The msm Package for R”

• Visser and Speekenbrink (2010) “depmixS4: An R Package for Hidden Markov Models”

• McClintock et al. (2020) “Uncovering ecological state dynamics with hidden Markov models”

We’ve already discussed state space models. These models include

• a latent process model (we don’t directly observe)

• a data or observation model

• we’ve generally assumed both models be normal (or multivariate normal)

Process model: \[{ x }_{ t }={ x }_{ t-1 }+{ \varepsilon }_{ t-1 }\]

Observation model: \[{ y }_{ t }={ x }_{ t }+{ \delta }_{ t }\]

where \({ \varepsilon }_{ t } \sim Normal\left( 0,{ \sigma }_{ \varepsilon } \right)\) and \({ \delta }_{ t } \sim Normal\left( 0,{ \sigma }_{ \delta } \right)\)

Adding AR coefficients can make these models stationary with respect to the mean,

\[{ x }_{ t }={ p\cdot x }_{ t-1 }+{ \varepsilon }_{ t-1 }\]

however they may not be able to explain some datasets very well.

• Specifically, these models are not well designed to model regimes

Example from Moore (2018), “Predicting tipping points in complex environmental systems”

Many examples of time series in ecology & fisheries may alternate between multiple states (2+)

• Vert-pre et al. (2013) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3562848/

• Francis et al. (2012) https://onlinelibrary.wiley.com/doi/full/10.1111/j.1365-2486.2012.02702.x

• Mark recapture models with latent states

• Identifying changing behavior or habitat use:

• Tennessen et al. (2019) “Hidden Markov models reveal temporal patterns and sex differences in killer whale behavior”

• Leos-Barajas et al. (2017) “Analysis of animal accelerometer data using hidden Markov models”

• momentuHMM: R package for analysis of telemetry data using generalized multivariate hidden Markov models of animal movement

Lots of non-HMM approaches for detecting regimes

• STARS algorithm
  • Sequential t-test approach for detecting changes in the mean
  • Rodionov (2015) https://www.mdpi.com/2225-1154/3/3/474
• Brute force model selection approach
  • iterate through change points, evaluating data support for each
  • how do we do change points with regression? \({ Y }_{ t }=B{ X }_{ t }+{ \varepsilon }_{ t }\)

They can actually fit the data from a regime model quite well.

• Via MARSS()

What’s lacking is inference about:

• What’s the probability of transition between regimes?
• How long are regimes expected to last?
• What regimes might we expect in the future?

Lots of applications: speech recognition, bioinformatics, animal movement, environmental data (rain), finance

Markov Process

• time may be discrete or continuous (we’ll focus on discrete)
• Markov if at each time step, the state \(x_{t}\) is only dependent on the previous state \(x_{t-1}\)
• \(x_{t}\) does not depend on future values
• these ideas should be familiar, with the exception of states changing from continuous to discrete

Entire history can be written as \[\left\{ { x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },...,{ x }_{ T } \right\}\]

A key piece of the Markov process are the transition probabilities.

• The probability of transitioning from state \(j\) to \(i\) given the current state is \[P\left( { x }_{ t+1 }=j | { x }_{ t }=i \right) ={ \gamma }_{ ij }\]

• And then these probabilities can be summarized in a transition matrix, \[\Gamma =\left[ \begin{matrix} { \gamma }_{ 11 } & { \gamma }_{ 12 } & { \gamma }_{ 13 } \\ { \gamma }_{ 21 } & { \gamma }_{ 22 } & { \gamma }_{ 23 } \\ { \gamma }_{ 31 } & { \gamma }_{ 32 } & { \gamma }_{ 33 } \end{matrix} \right]\]

• Elements of \(\Gamma\) can be fixed a priori (0, etc)

• Let’s try to simulate this in R

# initial state
x <- sample(1:2, size = 1)
# transition matrix
Gamma <- matrix(c(0.9, 0.1, 0.2, 0.8), 2, 2, byrow = TRUE)
for(t in 2:100) {
  x[t] <- sample(1:2, size = 1, prob = Gamma[x[t-1],])
}

Matrix \(\Gamma\) is the 1-step transitions. However k-step transition probabilities can be generated,

\(\Gamma(k) = \Gamma^{k}\)

• From this, we can also calculate the stationary distribution of the chain
  • See Zucchini et al. (2006) Chapter 2

There are two general flavours of transition matrices:

• Homogenous (or stationary)
  • transition probabilities don’t depend on \(t\)
• Non-homogeneous
  • transition probabilities are time-varying
• In this class, we’ll mostly consider homogeneous cases

Covariates may enter HMMs in one of two ways

• Effects on the mean

• Effects on the transition matrix \(\Gamma\)

• For effects on \(\Gamma\), probabilities are constrained (0,1) and constrained to sum to 1

  • multivariate logit regression used to relate covariates to elements of \(\Gamma\)
  • reference / base case is fixed at zero (Agresti 2002)

• Returning to our simulation example, let’s include a temporal trend in the transition probabilities
• We assume trend is linear in logit space
• For simplicity, we assume that slope is same on \({ \gamma }_{ 12 }\) and \({ \gamma }_{ 21 }\)

plogis(logit(0.1) + 0.015*1:100)

# initial state
x <- sample(1:2, size = 1)
# transition matrix
Gamma <- matrix(c(0.9, 0.1, 0.2, 0.8), 2, 2, byrow = TRUE)
for(t in 2:100) {
  Gamma[1,2] = plogis(logit(0.1) + 0.015*t)
  Gamma[1,1] = 1 - Gamma[1,2]
  Gamma[2,1] = plogis(logit(0.2) + 0.015*t)
  Gamma[2,2] = 1 - Gamma[2,1]
  x[t] <- sample(1:2, size = 1,prob = Gamma[x[t-1],])
}

• Observations: observable data \({Y}_{ i=1,...,N }\)

• States: latent (discrete) variables that are not directly observed

  • \({x}_{ t=1,...,T }\)
  • \(N\) is the number of states possible

• Transition probabilities: transition matrix representing the probability of transitioning between states in the Markov chain

  • \(\Gamma\) and \({ \gamma }_{ ij }\)

• Emission probabilities: how likely the states are at any particular timestep

  • \({ \theta }_{ i=1,...,N }\)

• Our simulations haven’t yet included an observation component. Let’s add observed data that’s normally distributed

# initial state
x <- sample(1:2, size=1)
# transition matrix
Gamma <- matrix(c(0.9, 0.1, 0.2, 0.8), 2, 2, byrow = TRUE)
for(t in 2:100) {
  x[t] <- sample(1:2, size = 1, prob = Gamma[x[t-1],])
}

u = c(1.2, 5.4)
y = rnorm(length(x), u[x], 1)

One quantity of interest from HMMs is the marginal distribution, \(P(Y_{t})\)

Output from the model will give us the conditional distribution, \(P(Y_{t}|x_{t}=i)\). But we need to marginalize over all possible states

Solution: \[P(Y_{ t })=\sum _{ i=1 }^{ N }{ P({ x }_{ t }=i)P({ Y }_{ t }|{ x }_{ t }=i) }\] which can also be expressed as \[P(Y_{ t })=\sum _{ i=1 }^{ N }{ { \delta }_{ i }P({ Y }_{ t }|{ x }_{ t }=i) }\] where \(\delta\) represents the stationary distribution (Zucchini et al. 2006)

Estimation of HMM parameters can be quite complicated

Dealing with joint likelihood of observed data and unobserved states

\[P(x,Y)=\prod _{ i=1 }^{ T }{ P({ x }_{ t }|{ x }_{ t-1 })P({ Y }_{ t }|{ x }_{ t }) }\]

Estimation most commonly done with maximum likelihood

1. Forward-backward algorithm: calculate probability of observed data sequence
2. Viterbi algorithm: used to generate most likely states
3. EM-algorithm: estimate / update parameters

• For forward-backward algorithm we can factor conditional state probabilties

\[P\left( { x }_{ t }|{ Y }_{ 1:T } \right) =P({ x }_{ t }|{ Y }_{ 1:t },{ Y }_{ t+1:T })=P({ x }_{ t }|{ Y }_{ 1:t })P({ Y }_{ t+1:T }|{ x }_{ t })\] * Probability of state given the data is the probability of the state given the data up to that time step multiplied by the probability of future data given state

Forward-backward algorithm has 3 steps (sometimes 2/3 combined):

1. Forward probabilities

• from the last slide, this step calculates \(P({ x }_{ t }|{ Y }_{ 1:t })\)

1. Backward probabilities

• from the last slide, this step calculates \(P({ Y }_{ t+1:T }|{ x }_{ t })\)

1. Smoothing

• compute marginal likelihood of sequence of state variables \(P(x_{t}|Y)\)

As a first example, let’s use an example of rainfall data in Seattle from the end of 2018 and beginning of 2019 (accessed from wunderground.com)

We could model rainfall directly, but for starters let’s just model whether or not it rained on a given day (0/1).

rain$rained = ifelse(rain$Rainfall > 0, 1, 0)

Questions that we might be interested in:

• Conditional probabilities
  • What’s the probability of it raining tomorrow given that it’s raining today?
  • What’s the probability of it raining tomorrow given that it’s not today?
• Persistence
  • On average, how long might we expect to go without rain?

We don’t really need HMMs to address these questions

• daily rainfall may be measured with a tiny amount of uncertainty
• probably safe to say that there’s almost no uncertainty in whether or not it rained on a given day

Transition probabilities can be just calculated directly,

• \(P(rain_{t+1} | rain_{t})\)

\[\frac { \# consecutive \quad rainy \quad days }{ \# rainy \quad days }\]

For example, we can create a differenced variable to indicate no change (0), and compute

rain$diff = c(diff(rain$rained), NA)
p_rain_rain = length(which(rain$diff == 0 & 
    rain$rained==1)) / length(which(rain$rained==1))

Ok, now let’s assume that there’s some minor observation or measurement error in the rain gauge data. For this case, it’s best to use a HMM.

Let’s start with using the R package depmixS4. There’s a good vignette on the package that you can refer back to,

https://cran.r-project.org/web/packages/depmixS4/vignettes/depmixS4.pdf

There’s two functions we’ll use to set up the HMM with depmixS4.

First we’ll structure the model using the same formula notation you’re hopefully familiar with,

mod = depmix(rained ~ 1, 
             nstates = 2, 
             transition = ~ 1,
             family = binomial(),
             data=rain)

Stepping through this rained is our response (yes / no)

mod = depmix(rained ~ 1, 
             nstates = 2, 
             transition = ~ 1,
             family = binomial(),
             data=rain)

nstates is the number of alternative states (> 2), specified a priori

mod = depmix(rained ~ 1, 
             nstates = 2, 
             transition = ~ 1,
             family = binomial(),
             data=rain)

transition is a formula to specify whether any covariates are to be included in the transition probabilities. The default is no covariates, and that these transitions aren’t time varying.

mod = depmix(rained ~ 1, 
             nstates = 2, 
             transition = ~ 1,
             family = binomial(),
             data=rain)

family is family or list of families (a list for multiple response variables) of the families associated with each response. The majority of common families are supported

mod = depmix(rained ~ 1, 
             nstates = 2, 
             transition = ~ 1,
             family = binomial(), 
             data=rain)

For a complete list, see

?depmixS4::GLMresponse

Ok, now that we’ve set up the model, we can do the estimation and look at the output

set.seed(123)
fitmod = fit(mod)

As a warning, note that the results from the estimation are a bit sensitive to the initial seed. Look at how much the transition probabilties change,

set.seed(121)
fitmod = fit(mod)

There’s a couple practical ways to try to overcome this seed issue.

• First, we can change the control parameters

Unfortunately for this example, this doesn’t solve the issue

• A second option is to run the estimation across a large number (> 100) of random starting values

Pseudocode:

best = 1.0e10
best_model = NA
for(i in 1:iter) {
  # fit the model
  fitmod = fit(mod)
  
  # check to see if this is the best solution?
  if(AIC(fitmod) < best) {
    best_model = fitmod
    best = AIC(fitmod)
  }
}

Let’s move on to a more complex example.

We’ll pull some data from the CalCOFI ichthyoplankton cruises in Southern California. Some species have been used to indicate cool versus warm regimes.

• http://calcofi.org/publications/calcofireports/v58/Vol58-State_of_the_Current_pages_1-55.pdf

For this example, we’ll focus on the California smoothtongue (Leuroglossus stilbius)

Caveat: the survey is spatially gridded, and you’d want to perform index standardization or use spatial models to generate indices. For simplicty, we’re just taking the mean abundance across stations in April-May.

We’ll start with fitting a 2-state model with depmix. Assumptions:

• Model fit to ln transformed data, assumed Gaussian

set.seed(123)
calcofi$ln = log(calcofi$m)
mod = depmix(ln ~ 1, 
             nstates = 2, 
             data=calcofi)
fitmod = fit(mod)

First let’s look at how to get predictions out of a depmix.fitted object

We’ll start with the state probabilities. Remember we could work with either

• Most probable state trajectory - via the viterbi function
• Marginals of \(P(x_{t})\) - via the posterior function

The most probable states are

prstates = apply(posterior(fitmod)[,c("S1","S2")], 
  1, which.max)

plot(prstates, type="b", xlab="Time", ylab="State")

depmixS4 doesn’t have a predict() or forecast() function, but creating estimated data is pretty straightforward. We can get the means out with

mu = summary(fitmod)[,1]

pred = data.frame("year"=seq(min(calcofi$year), 
  max(calcofi$year)),
  "fit" = mu[prstates])

Some diagnostics (like autocorrelation) may not look great. What if we compared the previous model to one with 3 states? AIC from the 2-state model was

## [1] 118.994

set.seed(123)
calcofi$ln = log(calcofi$m)
mod = depmix(ln ~ 1, 
             nstates = 3, 
             data=calcofi)
fitmod = fit(mod)

This seems like increasing the states doesn’t result in lower AIC

## [1] 122.5181

• In comparing 2 vs 3 state HMM, we found lower AIC for 2-state model

• However, opposite is often true – more states fit better, and result in lower AIC

• How to decide? Occam’s razor & metrics of predictive ability

• Pohle et al. 2017. “Selecting the Number of States in Hidden Markov Models: Pragmatic Solutions Illustrated Using Animal Movement”

If the univariate examples make sense, it’s not that different to extend these models for multiple time series.

In this setting, the assumption is that the latent state process is the same for all time series, though the time series may differ

1. their lengths
2. their error distributions

In depmix, these arguments generally become lists

We’ll extend our CalCOFI example to include 2 species now

Fitting a 2-state model with 2 species as responses. First we have to reshape the data,

calcofi$ln = log(calcofi$m) # ln transform
calcofi <- dcast(melt(calcofi[,c("year","ln","species")], 
            id.vars = c("year", "species")), 
            year ~ species)
names(calcofi)[2:3] = c("L_stilbius","L_ochotensis")
head(calcofi)

##   year  L_stilbius L_ochotensis
## 1 1980  0.85449030   -0.8414039
## 2 1981  3.13964169    1.6941728
## 3 1982  3.70359056    3.5435378
## 4 1983 -0.03344793    0.4947281
## 5 1984  3.00082695    2.4346651
## 6 1985  0.24911445   -0.8471716

set.seed(123)
mod = depmix(list(L_stilbius ~ 1, L_ochotensis~1), 
             nstates = 2, 
             family = list(gaussian(),gaussian()),
             data=calcofi)
fitmod = fit(mod)

We could also have situations where the time series are of different length.

• For example, if L_ochotensis time series was missing first half of values

• The argument ntimes is a list that becomes particularly valuable here - representing the length of each time series

HMMs are a useful approach at identifying latent state vectors that undergo discrete transitions

Estimation of HMMs for time series in R generally done with ML methods

• Fast, but these algorithms may get stuck
• Robust solutions = multiple starting values

Bayesian estimation generally beyond the scope of this class

• Very straightforward to build these models in BUGS/JAGS
• More complicated with newer sofware (Stan), though examples here: https://mc-stan.org/docs/2_26/stan-users-guide/hmms-section.html

• Why JAGS? Categorical samplers

• Let’s re-fit our CalCOFI example in JAGS. We’ll step through this in small chunks to understand what’s going on in JAGS syntax

• First we need to specify the initial state. We don’t know it, but assume each state (1/2) is equally likely

model{
for(i in 1:2){initp[i] <- 0.5}
x[1] ~ dcat(initp)

• Next we can think about parameterizing the transition matrix. \(\Gamma\) is 2 x 2, but only has 2 parameters
• We’ll specify uniform(0,1) priors on these

model{
for(i in 1:2){initp[i] <- 0.5}
x[1] ~ dcat(initp)
p ~ dunif(0,1)
q ~ dunif(0,1)

• Next we specify \(\Gamma\)

model{
for(i in 1:2){initp[i] <- 0.5}
x[1] ~ dcat(initp)
gamma12 ~ dunif(0,1)
gamma21 ~ dunif(0,1)
Gamma[1,1] <- 1-gamma12
Gamma[1,2] <- gamma12
Gamma[2,1] <- gamma21
Gamma[2,2] <- 1-gamm21

• Next, add in the transition model for the latent state x

model{
for(i in 1:2){initp[i] <- 0.5}
x[1] ~ dcat(initp)
p ~ dunif(0,1)
q ~ dunif(0,1)
Gamma[1,1] <- 1-p
Gamma[1,2] <- p
Gamma[2,1] <- q
Gamma[2,2] <- 1-q
for(i in 2:n){x[i] ~ dcat(Gamma[x[i-1],])}
}

• And finally the observation model for our observed data
• We need to estimate 2 means (u) and a residual error variance term

model{
for(i in 1:2){initp[i] <- 0.5}
x[1] ~ dcat(initp)
p ~ dunif(0,1)
q ~ dunif(0,1)
Gamma[1,1] <- 1-p
Gamma[1,2] <- p
Gamma[2,1] <- q
Gamma[2,2] <- 1-q
for(i in 2:n){x[i] ~ dcat(Gamma[x[i-1],])}
resid_tau~dgamma(0.001,0.001)
for(i in 1:2) {u[i] ~ dnorm(0,1)}
for(i in 1:n){y[i] ~ dnorm(u[x[i]],resid_tau)}
}

• To run the model, save it in a file, e.g. ‘model.txt’

• Create data list

n = nrow(calcofi)
y = log(calcofi$m)
jags_data = list("n","y")

• Create parameter list

jags_params = c("x","u","Gamma","resid_tau")

• Running the model

jagsfit <- jags(data=jags_data, 
parameters.to.save=jags_params,
model.file="model.txt")

• Next steps: thus far, we’ve only considered HMMs that are AR(1)

• Homework: try to modify the example above for the CalCOFI data to be AR(2)

• [hint: what are the dimensions of \(\Gamma\)?]