如何利用Hidden Markov Model从心率数据中无监督学习状态转移概率？

Great question! You’re exactly right to consider Hidden Markov Models (HMMs) here—they’re tailor-made for problems where you have sequential observed data (your heart rate readings) and need to infer both hidden state assignments and the transition probabilities between those states. Let’s break down how to approach this, step by step:

Your problem matches the core use case of HMMs perfectly:

• You have observed data: The heart rate values (50-140) in a sequential list.
• You have hidden states: The 8 unknown states you want to assign the data to.
• You need to learn two key things:
  • How to map heart rate values to hidden states (observation probabilities).
  • How often one hidden state transitions to another (the transition matrix you’re after).

The Baum-Welch algorithm (an implementation of the Expectation-Maximization framework) is designed specifically for this kind of unsupervised learning with HMMs—it will iteratively estimate both the state assignments and the transition probabilities from your raw data.

HMMs can handle both discrete and continuous observations. Since your heart rate is a continuous value range, you have two solid options:

Option A: Gaussian HMM (Recommended) #

Skip manual discretization entirely by using a Gaussian HMM. Each hidden state will be modeled as a Gaussian distribution (mean and variance) of heart rate values. This lets the algorithm learn natural clusters of heart rates without you having to define arbitrary bins.

Option B: Discretize Heart Rates #

If you prefer discrete observations, split the 50-140 range into 8 equal (or data-weighted) intervals. For example:

• State 1: 50–61
• State 2: 62–73
• ...
• State 8: 129–140

Just make sure each interval has enough data points to avoid skewing the model.

The hmmlearn library makes this straightforward. Here’s a quick implementation using a Gaussian HMM:

import numpy as np
from hmmlearn import hmm

# Assume your heart rate data is stored as a list of integers
heart_rate_data = [72, 75, 68, 80, 102, ...]  # Your full dataset

# Reshape data to fit hmmlearn's expected format (n_samples, n_features)
X = np.array(heart_rate_data).reshape(-1, 1)

# Initialize a Gaussian HMM with 8 hidden states
# covariance_type="diag" works well for single-feature data (heart rate)
model = hmm.GaussianHMM(n_components=8, covariance_type="diag", n_iter=1000, random_state=42)

# Train the model using Baum-Welch
model.fit(X)

# Extract your desired outputs
transition_matrix = model.transmat_  # 8x8 matrix of transition probabilities
state_means = model.means_  # Average heart rate for each hidden state
most_likely_states = model.predict(X)  # Assign each heart rate to a state

• Transition Matrix: Each row transition_matrix[i] gives the probability of moving from state i to every other state. You can visualize this with a heatmap to spot frequent transitions (e.g., low-heart-rate states often transitioning to other low states).
• State Means: Check that the 8 state means are logically ordered (e.g., state 1 has the lowest average heart rate, state 8 the highest). If not, you can reorder the states manually or adjust the model’s initialization.
• Model Fit: Use model.score(X) to get a log-likelihood score of how well the model fits your data. You can also try different covariance types (like full) or increase n_iter if the model isn’t converging properly.

If you want a simpler approach (without HMM’s probabilistic modeling), you can use K-Means to cluster your heart rates into 8 states first, then calculate transition probabilities from the sequential state assignments:

from sklearn.cluster import KMeans

# Cluster heart rates into 8 states
kmeans = KMeans(n_clusters=8, random_state=42)
state_assignments = kmeans.fit_predict(X)

# Initialize transition matrix
transition_matrix = np.zeros((8, 8))

# Count state transitions
for i in range(len(state_assignments) - 1):
    current = state_assignments[i]
    next_state = state_assignments[i+1]
    transition_matrix[current][next_state] += 1

# Normalize rows to get probabilities
transition_matrix = transition_matrix / transition_matrix.sum(axis=1, keepdims=True)

This is simpler, but it treats state assignments as hard (each heart rate belongs to exactly one state) and doesn’t account for the uncertainty in transitions that HMM handles naturally. For sequential data like heart rate, HMM will almost always give more meaningful transition probabilities.

内容的提问来源于stack exchange，提问作者Bar Shaul