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Fréchet Distance (FD)

Key: fd · Family: distribution · Better: lower ↓ · Per-lead: yes

FD asks: do simple statistics of the signals (amplitude, spread, shape and frequency content) follow the same joint distribution in the real and synthetic sets? It uses the same Fréchet formula as FED, but on 11 handcrafted features instead of deep embeddings, so it needs no neural network or GPU.

Try it

Each dot is one ECG lead described by two of its features (z-scored). FD fits a Gaussian to each cloud and measures the distance between the two Gaussians.

Interactive demo loading… (needs JavaScript)

What to notice:

  • The mean term grows when the synthetic features are systematically off, for example amplitudes that are too low.
  • The covariance term grows when the synthetic set is too uniform (too little variety), too varied, or when features co-vary the wrong way (wrong correlation).
  • With few samples, even identical distributions score above 0. Compare against the real-vs-real noise floor.

How it works

For each lead separately:

  1. Compute 11 features per recording: mean, standard deviation, skewness, kurtosis, RMS, energy, maximum |x|, zero-crossing rate, lag-1 autocorrelation, spectral centroid and spectral bandwidth.
  2. Z-score the features so that each has a comparable scale.
  3. Fit a Gaussian (mean vector and covariance matrix) to the real features and another to the synthetic features.
  4. Compute the Fréchet distance between them.

The overall score is the mean over leads.

The math

\[ \mathrm{FD} = \lVert \mu_r - \mu_s \rVert_2^2 + \operatorname{Tr}\!\left(\Sigma_r + \Sigma_s - 2\,(\Sigma_r\Sigma_s)^{1/2}\right) \]

with \(\mu, \Sigma\) the mean and covariance of the 11-dimensional feature vectors, and \(\varepsilon = 10^{-6}\) added to the covariance diagonals. For two 2-D Gaussians (as in the demo) the trace of the matrix square root has a closed form, \(\operatorname{Tr}\sqrt{AB} = \sqrt{\operatorname{Tr}(AB) + 2\sqrt{\det(AB)}}\).

Reading the score

  • 0 means identical feature means and covariances. There is no upper bound.
  • FD only sees what the 11 features capture. Two sets with similar global statistics but different beat morphology can score well. Pair it with a morphology metric such as SQI, or with FED.
  • Like every Fréchet distance it is biased upward for small samples. Keep the number of recordings fixed across models (n_subsample, default 2000).
  • Per-lead scores show which leads the generator struggles with.

Use it with ECGEN-Eval

ecgen-eval eval --real real.npy --synthetic synth.npy -m fd --output report.html

from ecgen_eval import load_ecg_npy, AVAILABLE_METRICS

real = load_ecg_npy("real.npy", fs=500)
synth = load_ecg_npy("synth.npy", fs=500)

result = AVAILABLE_METRICS["fd"]().compute(real.data, synth.data, lead_names=real.lead_names)
print(result.score)            # mean over leads
print(result.per_lead_scores)  # {"I": ..., "II": ..., ...}

Parameters (n_subsample, eps) are in the Metric Reference.

Learn more

  • Heusel et al., NeurIPS 2017 (FID); applied to ECG with handcrafted features in Thambawita et al., Scientific Reports 2021.
  • FED replaces the handcrafted features with foundation-model embeddings.