Breathing Rate Estimation

Quick Links

Requested Links

• Final 15s Output CSV
• Computation Script (compute_breathing_rate_final.py)

Additional Helpful Links

• Appendix Notebook (EDA & Figures)
• Mathematical Notes Notebook (math_notes.ipynb)

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Output Requirements

• CSV with two columns:
  • time: HH:MM:SS, localized to PDT
  • bpm: breaths per minute, rounded to 1 decimal place
• Standalone Python script:
  • Recomputes the CSV from raw JSON logs
  • Clear, commented code explaining the pipeline

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Methodology

1. Reconstruct RR interval stream

   • Remove implausible intervals (0.3–2.0 s).
   • Resample to a uniform 4 Hz grid to enable FFT on evenly spaced data.

2. Bandpass filtering (0.1–0.5 Hz)

   • Isolate the physiological breathing band (6–30 bpm).
   • Suppress low-frequency drift and high-frequency noise.

3. Windowed FFT with peak refinement

   • Split the signal into 15 s windows.
   • Apply a Hann window; compute FFT.
   • Identify the dominant peak within 0.1–0.5 Hz.
   • Refine peak frequency via parabolic interpolation.
   • Convert to breaths per minute (bpm).

4. Validity checks and output

   • Require sufficient data coverage and peak prominence.
   • Mark invalid windows as NaN.
   • Export to CSV.

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Why This Approach?

• Physiological grounding: respiratory sinus arrhythmia links breathing to HRV, so the IBI signal carries respiratory-rate content (see Applied/Phys references below).
• FFT fit: after resampling, respiration appears as a narrowband component. Hann windowing reduces leakage; parabolic interpolation gives sub-bin accuracy with short windows.
• Practical tradeoffs: 15 s windows balance temporal resolution and spectral stability. The pipeline is fast, transparent, and reproducible.

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Mathematical Notes

The mathematical derivations are described verbally here, with full LaTeX formulas and worked examples available in math_notes.ipynb.

• FFT Power Spectrum Normalization: why window normalization is required to get consistent power scaling (see notebook §1).

• Frequency Resolution and Zero-Padding: how FFT bin width depends on window length, and why zero-padding improves interpolation (see notebook §2).

• Parabolic Peak Interpolation: technique for refining peak location between bins to achieve sub-bin accuracy (see notebook §3).

• BPM Conversion: how the refined frequency is converted into breaths per minute (see notebook §4).

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Exploratory Data Analysis (EDA)

The appendix notebook (notebooks/appendix_breathing_rate_eda.ipynb) includes:

• Histograms with Gaussian and KDE overlays
• Full-session time series with rolling median
• Missingness analysis (run-lengths and strip plots)
• Outlier detection with the 3xIQR rule
• Summaries of quality and key statistics

Example Figures

Below are representative outputs from the EDA process:

Histogram of 15s Breathing Rates (with KDE overlay)

Full-Session Breathing Rate Time Series

Heatmap of Valid Data Coverage

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Conclusions

• Produces reproducible breathing-rate estimates at 15 s resolution.
• FFT-based estimation is physiologically grounded and mathematically justified.
• Deliverables are transparent, auditable, and easy to re-run.

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Limitations & Future Work

• Motion artifacts: RR intervals can be corrupted by movement; accelerometer-informed cleaning could help.
• Transients: rapid changes (sighs, speech) break stationarity; wavelets or EMD may improve tracking.
• Validation: concurrent reference respiration (capnography, belts) would strengthen accuracy claims.
• Adaptive parameters: band limits and prominence thresholds could be tuned per subject or state.
• Overlapping windows: 50% overlap may increase temporal resolution while keeping spectral reliability.

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Bibliography

Applied / Physiological References

Ref	Citation	Notes
[1]	Task Force of the European Society of Cardiology. Heart rate variability: Standards of measurement, physiological interpretation and clinical use. Circulation, 1996.	HRV frequency bands; HF band aligns with respiration.
[2]	Schäfer, A., & Kratky, K. W. Estimation of breathing rate from respiratory sinus arrhythmia: Comparison of various methods. Ann. Biomed. Eng., 2008.	Compares respiration estimators from HRV, including FFT.
[3]	Bailón, R., et al. Analysis of respiratory frequency from heart rate variability. IEEE TBME, 2006.	HRV-derived respiration validated vs reference signals.

Mathematical / DSP References

Ref	Citation	Notes
[4]	Harris, F. J. On the use of windows for harmonic analysis with the DFT. Proc. IEEE, 1978.	Window design, leakage, practical guidance.
[5]	Quinn, B. G. Estimating frequency by interpolation using Fourier coefficients. IEEE TSP, 1994.	Theory of sub-bin interpolation.
[6]	Oppenheim, A. V., & Schafer, R. W. Discrete-Time Signal Processing. 3rd ed., 2009.	Core DSP: FFT assumptions, windowing, spectral estimation.
[7]	Stoica, P., & Moses, R. Spectral Analysis of Signals. 2005.	Spectral estimation, modified periodograms, PSD theory.
[8]	Lyons, R. G. Understanding Digital Signal Processing. 3rd ed., 2010.	Practical treatment of FFT, windows, interpolation.
[9]	Rife, D. C., & Vincent, G. A. Use of the DFT in the measurement of frequencies and levels of tones. Bell Syst. Tech. J., 1970.	Early, practical frequency estimation from DFT bins.
[10]	Jacobsen, E., & Kootsookos, P. Fast, accurate frequency estimators. IEEE Signal Processing Magazine, 2007.	Survey of parabolic and related estimators.
[11]	Welch, P. The use of FFT for the estimation of power spectra. IEEE Trans. Audio Electroacoust., 1967.	Periodograms and PSD estimation.