PolarH10Windows-first telemetry toolkit
PolarH10 Developer Reference

Breathing Dynamics Formula Sheet

Use this sheet when you want the derived breath interval and amplitude series, plus ACW50, PSD slope, LZC, sample entropy, and multiscale entropy written out directly.

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Data Path

  • input waveform: calibrated VolumeBase01 from the Breathing tracker
  • event extraction: alternating peaks and troughs from that calibrated waveform
  • derived series:
    • interval series: spacing between same-polarity extrema
    • amplitude series: absolute peak-trough excursion

This is the main adaptation relative to the cited paper: the paper uses a respiration-belt signal, while this app uses the repo's calibrated ACC breathing waveform.

Unless a section says otherwise, the formulas below are applied separately to the derived breath-interval series and the derived breath-amplitude series. In the notation below, t_j is the time of extremum j, x_j is its amplitude, s_i is a generic element of whichever derived series is currently being analyzed, mu is the mean of that series, and rho(ell) is autocorrelation at lag ell.

Derived Breath Series

For accepted extrema times t_k and values x_k:

intervalj=tj−tj−1\text{interval}_j = t_j - t_{j-1}

Where the two extrema belong to the same polarity family, for example peak to peak or trough to trough.

This turns the waveform into a timing series, so it describes how long one breath cycle segment takes from one like-extremum to the next.

amplitudej=∣xj−xj−1∣\text{amplitude}_j = |x_j - x_{j-1}|

Where the two extrema are alternating peak and trough partners in a completed breath excursion.

This turns the waveform into an excursion-depth series, so it describes how large each completed breath movement is.

Basic Statistics

For either derived series s_1 ... s_n:

μ=1n∑i=1nsi\mu = \frac{1}{n}\sum_{i=1}^{n}s_i

mu is the typical interval or excursion size over the current window.

SD=1n−1∑i=1n(si−μ)2SD = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(s_i - \mu)^2}

SD measures absolute variability around that typical value.

CV=SD∣μ∣CV = \frac{SD}{|\mu|}

CV rescales variability by the mean so two sessions with different average breath size or timing can still be compared on a relative basis.

ACW50

The autocorrelation window is the first lag where the normalized autocorrelation falls below 0.5:

ACW50=min⁡{ℓ≥1:ρ(ℓ)<0.5}ACW50 = \min \left\{ \ell \ge 1 : \rho(\ell) < 0.5 \right\}

Large ACW50 values mean the series stays self-similar across more lags; small values mean the pattern decorrelates quickly.

PSD Slope

The implementation now follows the cited NeuroKit2 path more closely:

  • linearly detrend the series
  • standardize it with population SD
  • compute a one-sided FFT power spectrum
  • keep the lower-frequency quartile of the spectrum
  • fit the slope in log-log space

Formula sketch:

PSD slope=slope⁡(log⁡10(f),log⁡10(PSD(f)))\text{PSD slope} = \operatorname{slope}\left( \log_{10}(f), \log_{10}(PSD(f)) \right)

Here f is frequency and PSD(f) is the one-sided power spectral density of the detrended, standardized series. The fitted slope summarizes how strongly slower fluctuations dominate over faster ones in the low-frequency region.

Lempel-Ziv Complexity

The series is binarized around its mean:

bi={1si≥μ0si<μb_i = \begin{cases} 1 & s_i \ge \mu \\ 0 & s_i < \mu \end{cases}

This binary step asks only whether each sample is above or below the series mean, turning the waveform-derived series into a symbolic pattern string.

The Lempel-Ziv counter then scans the binary string left-to-right and adds a new phrase whenever it encounters a substring not yet present in the discovered pattern set. The app publishes a normalized form:

LZCnorm=c(n)n/log⁡2(n)LZC_{norm} = \frac{c(n)}{n / \log_2(n)}

Higher normalized LZC means new binary patterns keep appearing; lower values mean the sequence is more repetitive.

Sample Entropy

Default settings:

  • m = 2
  • delay = 1
  • r = 0.2 * SD

Here m is embedding dimension, delay is the lag between points in the embedded vectors, r is the matching tolerance, and N is the series length.

Formula sketch:

SampEn(m,r,N)=−ln⁡(AB)SampEn(m, r, N) = -\ln\left(\frac{A}{B}\right)

In plain language, sample entropy asks how often similar short patterns remain similar when extended by one more sample.

Where:

  • B counts matching embedded vectors of length m
  • A counts matching embedded vectors of length m + 1
  • matching uses the Chebyshev-distance tolerance r

If the conditional probability is undefined, the app now suppresses the headline instead of inventing a finite fallback value.

Multiscale Entropy

Default settings:

  • m = 3
  • delay = 1
  • r = 0.2 * SD of the original series
  • scales 1..5
  • published summary: area under the entropy-by-scale curve divided by the count of usable scales

For each scale tau, the series is coarse-grained by averaging non-overlapping blocks:

yj(τ)=1τ∑i=(j−1)τ+1jτsiy_j^{(\tau)} = \frac{1}{\tau}\sum_{i=(j-1)\tau + 1}^{j\tau} s_i

This creates a slower, coarser version of the same series at each scale tau.

Sample entropy is then computed on each coarse-grained series with the same base tolerance. The reported MSE summary is:

MSEAUC=trapz⁡(SampEnτ)KMSE_{AUC} = \frac{\operatorname{trapz}(SampEn_{\tau})}{K}

Where K is the number of usable finite scales kept for the summary.

This summary asks whether irregularity persists across multiple time scales rather than only at the native sample-to-sample resolution.

Alignment Status

This implementation is aligned to the cited method family for:

  • interval and amplitude series extraction from extrema
  • mean, SD, CV, ACW50, PSD slope, LZC, SampEn, and MSE
  • NeuroKit2-style SampEn and MSE defaults
  • NeuroKit2-style handling of undefined entropy values
  • NeuroKit2-style PSD slope preprocessing and low-frequency fit region

The remaining explicit adaptation is the waveform source:

  • the paper uses respiration-belt input
  • this repo uses the calibrated ACC breathing waveform from PolarBreathingTracker

References