Hidden Markov Model (HMM)

This repository contains a from-scratch Hidden Markov Model implementation utilizing the Forward-Backward algorithm and Expectation-Maximization for probabilities optimization. Please note that this code is not yet optimized for large sequences. More specifically, with a large sequence, expect to encounter problems with computational underflow. This will be resolved in the next release.

Methodology

Hidden Markov models are used to ferret out the underlying, or hidden, sequence of states that generates a set of observations. In his now canonical toy example, Jason Eisner uses a series of daily ice cream consumption (1, 2, 3) to understand Baltimore's weather for a given summer (Hot/Cold days). These are arrived at using transmission probabilities (i.e. the likelihood of moving from one state to another) and emission probabilities (i.e. the likelihood of seeing a particular observation given an underlying state).

This implementation adopts his approach into a system that can take:

• An initial transmission matrix
• An initial emission matrix
• A set of observations

You can see an example input by using the main() function call on the hmm.py file.

HMM models calculate first the probability of a given sequence and its individual observations for possible hidden state sequences, then re-calculate the matrices above given those probabilities. By iterating back and forth (what's called an expectation-maximization process), the model arrives at a local optimum for the tranmission and emission probabilities. It's a pretty good outcome for what might otherwise be a very hefty computationally difficult problem.

This model implements the forward-backward algorithm recursively for probability calculation within the broader expectation-maximization pattern.

Use of Current Release

This system can currently:

• Train an HMM model on a set of observations, given a number of hidden states N
• Determine the likelihood of a new set of observations given the training observations and the learned hidden state probabilities

To Come

I'm a full time student and this is a side project. It's still in progress. Things to come:

• Further methodology & how-to documentation
• Full Testing suite
• Computational underflow mitigation
• Viterbi decoding for understanding the most likely sequence of hidden states

Sample Usage

emission = np.array([[0.7, 0], [0.2, 0.3], [0.1, 0.7]])
transmission = np.array([ [0, 0, 0, 0], [0.5, 0.8, 0.2, 0], [0.5, 0.1, 0.7, 0], [0, 0.1, 0.1, 0]])
observations = ['2','3','3','2','3','2','3','2','2','3','1','3','3','1','1',
        '1','2','1','1','1','3','1','2','1','1','1','2','3','3','2',
        '3','2','2']
model = HMM(transmission, emission)
model.train(observations)
new_seq = ['1', '2', '3']
likelihood = model.likelihood(new_seq)