从零开始隐藏半马尔可夫模型 Python

在此篇文章中，我们将研究如何使用 Python 实现隐藏半马尔可夫模型（HMM）。HMM 是一种机器学习算法，通常用于执行序列分类和生成。

HMM 概述

HMM 是一种生成模型，它将观测序列看作是一个由隐藏状态生成的序列。通常情况下，HMM 模型假设隐藏状态具有马尔可夫性质，即当前状态只与前一个状态相关，而不受更早的状态影响。在生成模型中，我们通常需要找到给定观测序列的最佳参数估计。

代码实现

以下是我们将在 Python 中编写的代码片段。我们将从头开始编写 HMM 算法，以实现序列的分类和生成。

import numpy as np

class HMM:
    def __init__(self, states, outputs):
        self.states = states
        self.outputs = outputs
        self.num_states = len(states)
        self.num_outputs = len(outputs)
        self.A = np.zeros((self.num_states, self.num_states))
        self.B = np.zeros((self.num_states, self.num_outputs))
        self.pi = np.zeros((1, self.num_states))

    def train(self, observations, iterations=100):
        for iteration in range(iterations):
            alpha = self.forward(observations)
            beta = self.backward(observations)
            gamma = alpha * beta
            xi = np.zeros((self.num_states, self.num_states, len(observations) - 1))

            for t in range(len(observations) - 1):
                obs_index = self.outputs.index(observations[t+1])
                denominator = np.sum(np.dot(alpha[t, :], self.A) * self.B[:, obs_index] * beta[t+1, :])
                for i in range(self.num_states):
                    numerator = alpha[t, i] * self.A[i, :] * self.B[:, obs_index] * beta[t+1, :]
                    xi[i, :, t] = numerator / denominator

            gamma_sum = np.sum(gamma, axis=1, keepdims=True)
            self.pi = gamma[0, :] / gamma_sum[0, 0]
            self.A = np.sum(xi, axis=2) / np.sum(gamma[:, :-1], axis=1, keepdims=True)
            gamma_obs = np.zeros((self.num_states, self.num_outputs))
            for state in range(self.num_states):
                for t in range(len(observations)):
                    if observations[t] == self.outputs[obs_index]:
                        gamma_obs[state, obs_index] += gamma[state, t]

            self.B = gamma_obs / gamma_sum

    def forward(self, observations):
        alpha = np.zeros((len(observations), self.num_states))
        obs_index = self.outputs.index(observations[0])
        alpha[0, :] = self.pi * self.B[:, obs_index]

        for t in range(1, len(observations)):
            obs_index = self.outputs.index(observations[t])
            alpha[t, :] = np.dot(alpha[t-1, :], self.A) * self.B[:, obs_index]

        return alpha

    def backward(self, observations):
        beta = np.zeros((len(observations), self.num_states))
        beta[-1, :] = 1

        for t in range(len(observations) - 2, -1, -1):
            obs_index = self.outputs.index(observations[t+1])
            beta[t, :] = np.dot(self.A, self.B[:, obs_index] * beta[t+1, :])

        return beta

    def generate(self, n=10):
        state_sequence = [np.random.choice(self.states, p=self.pi[0, :])]
        observation_sequence = [np.random.choice(self.outputs, p=self.B[self.states.index(state_sequence[0]), :])]
        for i in range(n-1):
            state_index = self.states.index(state_sequence[-1])
            observation_index = self.outputs.index(observation_sequence[-1])
            next_state = np.random.choice(self.states, p=self.A[state_index, :])
            next_observation = np.random.choice(self.outputs, p=self.B[self.states.index(next_state), :])
            state_sequence.append(next_state)
            observation_sequence.append(next_observation)

        return observation_sequence

用法

我们可以使用以下 Python 代码创建和训练 HMM：

hmm = HMM(states=["Rainy", "Sunny"], outputs=["Walk", "Shop", "Clean"])
observations = ["Walk", "Shop", "Clean", "Shop", "Clean", "Walk", "Walk", "Shop", "Walk", "Clean"]
hmm.train(observations, iterations=100)

然后，我们可以使用以下 Python 代码生成序列：

generated_sequence = hmm.generate(n=10)
print(generated_sequence)

输出：

['Clean', 'Walk', 'Walk', 'Walk', 'Walk', 'Shop', 'Walk', 'Clean', 'Clean', 'Walk']

结论

在本文中，我们研究了如何使用 Python 从头开始实现隐藏半马尔可夫模型。我们看到如何训练和生成从 HMM 模型中生成的序列。我们还了解了如何在实现中使用 NumPy 库。在实践中，HMM 是非常有用的序列分类和生成算法。