Re-implementation of "A Revealing Introduction to Hidden Markov Models".

HMM.py

'''
Copyright (c) 2018 [email protected]

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

====
Re-implementaion of:

A Revealing Introduction to Hidden Markov Models
https://www.cs.sjsu.edu/~stamp/RUA/HMM.pdf
'''
import numpy as np
import math
from time import time


class HMM(object):
    def __init__(self, n_states: int, n_symbols: int, max_iter: int):
        self.N: int = n_states
        self.M: int = n_symbols

        self.a: np.ndarray = np.zeros(shape=(self.N, self.N))  # transition
        self.a.fill(1/self.N)
        self.b: np.ndarray = np.zeros(shape=(self.N, self.M))  # emission
        self.b.fill(1/self.M)
        self.pi: np.ndarray = np.zeros(shape=(self.N,))  # initial distribution
        self.pi.fill(1/self.N)

        self.max_iter: int = max_iter

    def __str__(self):
        return ('\n{\n a:\n' + str(self.a) +
                '\n b:\n' + str(self.b) +
                '\n c:\n' + str(self.pi) + '\n}')

    def forward(self, sequence: np.ndarray) -> np.ndarray:
        assert len(sequence.shape) == 1, f'Got: {sequence.shape()}.'
        self.c: np.ndarray = np.zeros(shape=(sequence.size,))  # scale factors
        self.c[0] = 0

        alpha: np.ndarray = np.zeros(shape=(sequence.size, self.N))
        for i in range(0, self.N):
            alpha[0, i] = self.pi[i] * self.b[i, sequence[0]]
            self.c[0] = self.c[0] + alpha[0, i]

        self.c[0] = 1 / self.c[0]

        for i in range(0, self.N):
            alpha[0, i] = self.c[0] * alpha[0, i]
        for t in range(1, sequence.size):
            self.c[t] = 0
            for i in range(0, self.N):
                alpha[t, i] = 0
                for j in range(0, self.N):
                    alpha[t, i] = alpha[t, i] + alpha[t-1, j] * self.a[j, i]
                alpha[t, i] = alpha[t, i] * self.b[i, sequence[t]]
                self.c[t] = self.c[t] + alpha[t, i]

            self.c[t] = 1 / self.c[t]
            for i in range(0, self.N):
                alpha[t, i] = self.c[t] * alpha[t, i]
        return alpha

    def backward(self, sequence: np.ndarray) -> np.ndarray:
        T: int = sequence.size
        beta: np.ndarray = np.zeros(shape=(T, self.N))
        for i in range(0, self.N):
            beta[T-1, i] = self.c[T-1]

        t: int = T-2
        while t >= 0:
            for i in range(0, self.N):
                beta[t, i] = 0
                for j in range(0, self.N):
                    beta[t, i] = (
                        beta[t, i] +
                        self.a[i, j] * self.b[j, sequence[t+1]] * beta[t+1, j])
                beta[t, i] = self.c[t] * beta[t, i]
            t = t - 1
        return beta

    def gammas(self, sequence: np.ndarray) -> (np.ndarray, np.ndarray):
        T: int = sequence.size
        gamma: np.ndarray = np.zeros(shape=(T, self.N))
        digamma: np.ndarray = np.zeros(shape=(T, self.N, self.N))
        alpha: np.ndarray = self.forward(sequence)
        beta: np.ndarray = self.backward(sequence)

        for t in range(0, T-1):
            for i in range(0, self.N):
                gamma[t, i] = 0
                for j in range(0, self.N):
                    digamma[t, i, j] = (
                        alpha[t, i] * self.a[i, j] *
                        self.b[j, sequence[t+1]] * beta[t+1, j])
                    gamma[t, i] = gamma[t, i] + digamma[t, i, j]
        for i in range(0, self.N):
            gamma[T-1, i] = alpha[T-1, i]
        return (gamma, digamma)

    def restimate(self, sequence: np.ndarray):
        gamma, digamma = self.gammas(sequence)
        T: int = sequence.size
        for i in range(0, self.N):
            self.pi[i] = gamma[0, i]
        for i in range(0, self.N):
            denom = 0
            for t in range(0, T-1):
                denom = denom + gamma[t, i]
            for j in range(0, self.N):
                numer = 0
                for t in range(0, T-1):
                    numer = numer + digamma[t, i, j]
                self.a[i, j] = numer / denom

        for i in range(0, self.N):
            denom = 0
            for t in range(0, T):
                denom = denom + gamma[t, i]
            for j in range(0, self.M):
                numer = 0
                for t in range(0, T):
                    if sequence[t] == j:
                        numer = numer + gamma[t, i]
                self.b[i, j] = numer / denom

    def train(self, sequence: np.ndarray) -> (
            np.ndarray, np.ndarray, np.ndarray):
        it = 0
        old_prob = - np.inf
        new_prob = - 1000
        while it < self.max_iter and new_prob >= old_prob:
            old_prob = new_prob
            self.restimate(sequence)
            new_prob = self.log_prob(sequence)
            print(f'It: {it}, Prob: {new_prob}')
            it += 1
        return (self.pi, self.a, self.b)

    def log_prob(self, sequence: np.ndarray):
        prob, T = 0, sequence.size
        for i in range(0, T):
            prob = prob + math.log(self.c[i])
        prob = -prob
        return prob


if __name__ == '__main__':
    hmm = HMM(32, 16, max_iter=8)
    np.random.seed(0)
    sequence = np.random.randint(1, 16, size=32)
    start = time()
    model = hmm.train(sequence)
    duration = time() - start
    print(f'Duration: {duration}')