Introduction to Hidden Markov Models (HMMs) and their Applications

Hidden Markov Models (HMMs) are a class of statistical models used for modeling temporal sequences. They are widely used in various fields such as speech recognition, bioinformatics, and finance to name a few. In this blog post, we will delve into the world of HMMs, specifically focusing on training multiple observations with the MHSM&M package in R.

What are Hidden Markov Models (HMMs)?

A Hidden Markov Model (HMM) is a mathematical model that describes a system where the underlying mechanisms are not directly observable. Instead, the system emits observable signals, which can be used to infer the underlying state of the system. The HMM consists of three main components:

1. States: These represent the different states or classes that the system can be in.
2. Observations: These represent the measurable outputs of the system.
3. Transition probabilities: These describe the probability of transitioning from one state to another.

What is the MHSM&M Package?

The MHSM&M package in R provides a suite of tools for training Hidden Markov Models (HMMs) and their variants, such as Semi-Markov Hidden-Markov Models (MHSM&M). The package includes functions for estimating model parameters from observations, fitting models to data, and performing inference.

Training Multiple Observations with MHSM&M

The problem of initializing the model is critical when training multiple observations with HMMs. In this section, we will explore how to overcome this challenge using the MHSM&M package in R.

Initializing the Model

In the example provided, the initialization of the model is set as follows:

J <- 3
initial <- rep(1/J, J)
P <- matrix(1/J, nrow = J, ncol = J)
b <- list(lambda=c(1, 3, 6))
model <- hmmspec(init=initial, trans=P, parms.emission=b, dens.emission=dpois.hsmm)

In this code snippet, we define the number of states J, which is set to 3 in this case. We then create an initial state distribution initial and a transition probability matrix P. The emission distribution parameters are set using the b list, where each element corresponds to a state.

However, when training multiple observations with HMMs, we don’t know the initial values of the emission distribution parameters. To overcome this challenge, we can use the following approach:

train <- list(x = data.df$sequences, N = N)
class(train) <- "hsmm.data"

In this code snippet, we define a training object train containing two components: x, which represents the array of observations sequences, and N, which represents the count of observations for each sequence. We then set the class of train to hsmm.data.

Passing Observations to h1 = hmmfit()

Once we have defined the training object train, we can use it to fit a model using the hmmfit() function from the MHSM&M package.

list_of_observations <- simulate(model, N, rand.emis = rpois.hsmm)
h1 <- hmmfit(list_of_observations, model, mstep=mstep.pois)

In this code snippet, we define a list of observations list_of_observations using the simulate() function. We then fit a model to these observations using the hmmfit() function.

Conclusion

Training multiple observations with HMMs and the MHSM&M package in R requires careful consideration of initialization. By understanding how to initialize the model and pass observations to hmmfit(), we can overcome this challenge and fit accurate models to our data. In the next section, we will explore some common pitfalls when working with HMMs.

Common Pitfalls When Working with HMMs

When working with HMMs, there are several pitfalls to watch out for. Here are a few:

Overfitting

HMMs can suffer from overfitting, particularly when the number of states is large compared to the number of observations. To avoid this, it’s essential to carefully select the number of states and regularize the model using techniques such as early stopping.

Underfitting

On the other hand, HMMs can also underfit if the number of states is too small. This can result in poor performance on unseen data. To avoid this, it’s essential to choose a suitable number of states based on domain knowledge or by cross-validation.

Regularization Techniques for HMMs

Regularization techniques are used to prevent overfitting and underfitting in HMMs. Here are some common regularization techniques:

Early Stopping

Early stopping is a technique where the model is stopped before it has a chance to overfit the training data. This can be achieved by monitoring the performance of the model on a validation set and stopping when the performance starts to degrade.

L1 Regularization

L1 regularization adds a penalty term to the loss function that discourages large coefficients. This can help prevent overfitting by reducing the impact of high-coefficient features.

Conclusion

In this blog post, we explored how to train multiple observations with HMMs and the MHSM&M package in R. We discussed the importance of initialization and provided an example code snippet for passing observations to hmmfit(). Additionally, we touched on some common pitfalls when working with HMMs, including overfitting and underfitting.

References

• Bishop, C. M. (2006). Pattern recognition and machine learning. Springer.
• Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for regression. In D. M. Harman et al. (Eds.), Advances in neural information processing systems 19 (pp. 577-584).

Note: The references provided are a selection of resources that may be helpful when working with HMMs and the MHSM&M package.

Last modified on 2023-08-10