# Breathing From ACC Formula Sheet

Download Markdown Download PDF

## Provenance This breathing-volume model is repository-specific code ported from the existing Unity runtime. It is not presented as a published external method reference. Important status note: - this ACC breathing-volume model has not yet been externally validated - a validation study is actively being worked on ## Data Path - input stream: PMD accelerometer samples from the Polar H10 - preprocessing: exponential moving average on the raw ACC signal - calibration: estimate a center point and a principal breathing axis from a calibration window - projection: map filtered ACC onto the chosen breathing axis - bounds: derive lower and upper projection bounds from calibration quantiles - output: convert the projected signal into a `0..1` breathing-volume estimate In the notation below, `a_raw(t)` is the raw accelerometer vector at time `t`, `a_f(t)` is the filtered vector, `alpha` is the smoothing factor, `c` is the calibration center, `u` is the dominant breathing axis, `p(t)` is the scalar projection onto that axis, and `Q_lower` / `Q_upper` are the calibration quantile operators used to define a practical operating range. The tracker works in accelerometer coordinates internally and only turns the signal into a single breathing scalar after the projection step. ## Core Approximation Filtered ACC sample: ```latex a_f(t) = (1 - \alpha)a_f(t-1) + \alpha a_{raw}(t) ``` This smooths the raw accelerometer stream so the later breathing estimate is less sensitive to frame-level jitter. Calibration center: ```latex c = \frac{1}{N}\sum_{i=1}^{N} a_f(i) ``` This is the average filtered posture during calibration and acts as the reference point for later motion. Principal-axis projection: ```latex p(t) = \langle a_f(t) - c, u \rangle ``` Where `u` is the dominant calibration axis estimated from the calibration-window covariance. This projection turns 3-D chest motion into one signed scalar that should rise and fall mainly along the learned breathing direction. If the optional `XZ` model is available, the tracker also keeps a 2-D projection in the chest plane: ```latex p_{xz}(t) = \langle (x(t), z(t)) - c_{xz}, u_{xz} \rangle ``` This is the same idea as `p(t)`, but constrained to the chest plane for the optional `XZ` model. Lower and upper projection bounds come from calibration quantiles: ```latex b_{min} = Q_{lower}(p), \qquad b_{max} = Q_{upper}(p) ``` These bounds define the expected low and high ends of the breathing excursion without trusting a single noisy minimum or maximum sample. The runtime then applies the configured edge-ease and optional adaptive-bounds updates before mapping the projection to volume: ```latex v(t) = \operatorname{clamp}\left(\frac{p(t) - b_{min}}{b_{max} - b_{min}}, 0, 1\right) ``` This maps the projected motion into a normalized `0..1` breathing volume, where values near `0` sit near the lower calibration bound and values near `1` sit near the upper calibration bound. The final base volume uses either the 3-D axis model or the `XZ` model, depending on the tracker settings. The displayed output may then be inverted if the configured inhale/exhale direction is flipped. ## State Classification The tracker derives `Inhaling`, `Exhaling`, and `Pausing` from the first difference of the output volume: ```latex \Delta v(t) = v(t) - v(t-1) ``` Positive `Delta v` means the estimated volume is rising, negative `Delta v` means it is falling, and values near zero mean the tracker sees little movement in either direction. - `Inhaling` when `Delta v` exceeds the configured positive threshold - `Exhaling` when `Delta v` is below the negative threshold - `Pausing` otherwise ## Alignment Status There is no external paper claim here. The correct interpretation is: - this page documents what the app computes - the model is intended as a practical ACC-only breathing approximation - external validation is still pending ## References - [Breathing Workflow](breathing-workflow.md) - [App Overview](app-overview.md) - [References](references.md)