Fitting a mixture of independent Poisson distributions

This is an example from Zucchini & MacDonald’s book on Hidden Markov Models for Time Series (exercise 1.3).

The data is annual counts of earthquakes of magnitude 7 or greater, which exhibits both overdispersion for a Poisson (where the mean should equal the variance) as well as serial dependence.

The aim is to fit a mixture of m independent Poisson distributions to this data, using the non-linear minimizer nlm to minimize the negative log-likelihood of the Poisson mixture.

Sounds easy right? I have not done much optimisation stuff so it took me longer than it probably should have. There’s little better for the ego then getting stuck on things by page 10.

Constrained to unconstrained 

There are two sets of parameters, the lambda parameters to the Poisson distributions and the delta mixing distribution, the latter giving the proportions of each distribution to mix.

The Poisson parameters must be greater than zero and there are m of them. The mixing distribution values must all sum to one, and the first mixing value is implied by the m-1 subsequent values.

These are the constraints (lambdas greater than zero, deltas sum to one), and there are 2m – 1 values (m lambdas, m-1 deltas). We need to transform these to unconstrained values for use with nlm (becoming eta and tau respectively).

The formulas for transformation are reproduced from the book here

These transformed values are combined in to one parameter vector and passed to nlm. We also pass a function that calculates the negative log likelihood.

In the function, we first convert the parameters from the unconstrained “working” form to the constrained “natural” form using the inverse transform, and then use these values to do the likelihood calculation.

You can see the code here.

Outro

Now that I look over the code and write this out it does seem all fairly straightforward and relatively trivial, but it was new to me and took a while, so I figured other people might find the example useful as well.

The code has three examples for the cases m = 2, 3, and 4. Obviously feel free to mess around with the initial parameter estimates to see what comes up. 

It does seem to match the fitted models from the book, which I have copied here:

For all the details, check out the book, chapter 1!

HMM example with depmixS4

On a scale of one to straight up voodoo, Hidden Markov Models (HMMs) are definitely up there for me.

They have all sorts of applications, and as the name suggests, they can be very useful when you wish to use a Markovian approach to represent some stochastic process.

In loose terms this just means we wish to represent our process as some set of states and probabilistic transitions between them.

For example if I am in state 1, there may be a 85% chance of staying in state 1, and a 15% chance of moving to state 2.

To complete this simple two state model, we would also have to define the transitions for state 2, namely what is the probability we will stay in state 2 if we are already in state 2, and what is the probability we will transition from state 2 to state 1.

But what if we don’t know this information? What if we just have a set of realizations from some process and a vague idea of the number of states that might be behind it?

Enter HMMs. Among other things, they can be used to determine the state transition probabilities that underlie some process, as well as the distributional parameters for each state.

That is, it can work out all the details that are "hidden" from an observer of a process. All we really need to do is specify how many states we think there are.

Example

To test this out, and to get more familiar with the depmixS4 package, I made a small test program. It creates an observation series with:

a) a known number of states
b) known state transition matrices
c) known underlying distribution parameters

The aim is to use depmixS4 to estimate all this information, which should help get a grip on how to use the package, and also let us see if HMMs are actually any good.

We have two states, lets call them S1 and S2. They were setup with the following properties

Both states provide realizations from the normal distribution, in this case S1 has mean -1, standard deviation 1, and S2 has mean 1, standard deviation 1. We can also see the transition matrix between S1 and S2 in the last two columns.

A trajectory was sampled, and depmixS4 used to fit a HMM. It provided the following estimates

You can see it did a pretty good job of recovering the true values.

In the following plot, you can see the sample trajectory, the estimated state from the HMM and the actual state used to generate the sample. This example worked very well, it’s not always the case that things turn out so nicely.

Voodoo I tell you!

Outro

There is no guarantee that a fitted HMM will be of any use, and even with this simple example, the state estimates can be wildly inaccurate. You can try this for yourself by using different seed values.

This example is based on one from the book Hidden Markov Models and Dynamical Systems, which I found to be an excellent resource on the topic. It is clearly written, covers the basic theory and some actual applications, along with some very illustrative examples. Source code is provided in python.

I've read (or at least tried to read) pretty much every book on HMMs I could find, and found this one to be the most useful if you are new to HMMs and are interested in applications.

You may also find these other posts about HMMs useful as well:

Fun With R and HMMs
Getting Started with Hidden Markov Models in R
Hidden Markov Models Examples in R, part 3/4

There is also the classic paper by Rabiner.

Hopefully this has shed a bit of light on HMMs and the depmixS4 package. Code is up here.