Hidden Markov model

Overview

A hidden Markov model (HMM) describes a sequence in which each step has a hidden state, the next state depends only on the current one through a transition matrix, and each state produces an observed value through an emission distribution. The states are never seen directly; they are inferred from the observations. In the one paper on this concept so far, the HMM is used after a neural network: the network scores each sample for its class, and the HMM turns those per-sample scores into a consistent sequence of segments 1807.02569 § Methods (Enhancing ECG segmentation with HMMs).

The shared core across uses is the decoding step: given the observations, find the most likely hidden state path. Viterbi decoding does this exactly for a first-order HMM. Whether the state order is fixed and known is the main design question, because it determines which transitions the model should allow.

Variants

Only one variant is covered so far.

CNN scores refined by a six-state HMM

The HMM has one state per segment class of a 12-lead ECG: P wave, PR segment, QRS complex, ST segment, T wave and TP segment. Start and transition probabilities are set from manual labels, and emission probabilities are learned from the CNN output on the training data. A post-processing rule then removes complexes shorter than 10 ms 1807.02569 § Methods (Enhancing ECG segmentation with HMMs). The paper reports intersection-over-union values for each class, with the QRS class highest, but reports no run without the HMM, so its gain from the HMM step is not measured 1807.02569 § Results (Validation of ecgAI Machine Learning-based ECG Segmentation). Card: Patient ECG profiles (1807.02569).

When to use which

There is only one row so far, so the table below states constraints rather than a comparison of variants.

The paper's result tables do not show segmentation with and without the HMM step, so they do not show that the HMM is needed for the reported accuracy 1807.02569 § Results (Validation of ecgAI Machine Learning-based ECG Segmentation). Do not present the HMM as the source of the gain.

Implementation sketch

Our own pseudocode, following the paper's description. The observation at each step is the class the network predicts; the emission table gives the probability of each observed class given each hidden state, learned on training data. The transition and start values are inputs the implementer must supply.

# Viterbi refinement of per-sample network labels
# Input:  observed class per sample (T,), start (S,), transition (S, S), emission (S, C)
# Output: most likely state per sample (T,)
# [arxiv:1807.02569 § Methods] (Enhancing ECG segmentation with HMMs)

def viterbi(obs, start, trans, emit):
    # 1. Work in log space to avoid underflow on long sequences
    log_start, log_trans, log_emit = log(start), log(trans), log(emit)
    # 2. Initialise each state with its start score and the first observation
    score = log_start + log_emit[:, obs[0]]              # (S,)
    back = []
    # 3. Extend the best path one sample at a time, keeping back-pointers
    for t in range(1, len(obs)):
        cand = score[:, None] + log_trans                # (S, S)
        back.append(argmax(cand, axis=0))                # (S,)
        score = max(cand, axis=0) + log_emit[:, obs[t]]  # (S,)
    # 4. Trace back from the best final state
    path = [argmax(score)]
    for ptr in reversed(back):
        path.append(ptr[path[-1]])
    return reversed(path)

A post-processing step that drops very short segments is a separate heuristic. The paper uses a 10 ms threshold and does not state its exact form 1807.02569 § Methods (Enhancing ECG segmentation with HMMs).

Contradictions

None known. Only one paper covers this concept. Within that paper, the released code and the text differ on where the emission probabilities are estimated and on the transition values; see the card's "Paper vs code" section (Patient ECG profiles). This page follows the paper's text.

Papers

• Patient ECG profiles (1807.02569): a CNN with a six-state HMM segments 12-lead ECGs, and the segment timings feed a fixed-length profile for downstream models.