Forecasting VO₂ as a stochastic dynamical system.
AeroGlyphics does not treat your VO₂ log as an ordinary regression problem. It treats the underlying VO₂ trajectory as a latent Markov process observed with measurement error, and estimates its drift the way modern epidemiology and pharmacokinetics estimate the parameters of a differential-equation model — with a functional integral.
The engine: Sparse Laplace Approximation (SLAM)
The forecast is a reduced-order application of the Sparse Laplace Approximation Method introduced by John Tillinghast (American University, 2015). SLAM computes high-dimensional functional integrals — sums over all possible latent trajectories — by fitting a multivariate Gaussian around the critical path (the maximum-likelihood trajectory), then correcting with higher-order derivative tensors. Its key insight is that for a Markovian dynamical system those derivative tensors are block-tridiagonal, so higher-order corrections cost O(N·d⁴) instead of the naive O(N⁴·d⁴). That is what makes it faster than STAN / HMC on the same problem while returning near-identical posteriors.
The variance-stabilizing transform
Raw VO₂ readings have heteroscedastic Markov variance — big values move more than small values in the same interval. A naïve Laplace fit pulls the critical path toward zero, exactly the pathology Tillinghast identifies (SLAM slides §28–§30). We adopt his fix: fit in the transformed variable v = log(VO₂), which is variance-stabilizing for the log-linear growth regime aerobic capacity follows over months. Projections invert exp(·) so the ±1σ band is lognormal in ml·kg⁻¹·min⁻¹, never crossing zero.
The critical path
Inside log-space we fit an exponentially-weighted linear drift with a 30-day half-life kernel. Recent readings dominate; older readings decay gracefully instead of being hard-clipped. The Laplace posterior around that critical path is a Gaussian in v, whose standard deviation is the weighted RMSE — the single-latent-variable case of SLAM's block Hessian.
Why not a plain regression?
Three reasons. (1) A regression assumes the observation model is the whole model; SLAM separates the latent stochastic process from measurement error, so a noisy smartwatch reading does not distort the estimated drift. (2) SLAM's variance- stabilizing step removes a real bias that log-free fits inherit. (3) SLAM is the same machinery that scales to multi-compartment physiology (SIR, pharmacokinetics), so the roadmap — adding HRV, sleep, and training-load compartments — does not require a different math stack.
Attribution
SLAM is the intellectual property of John Tillinghast, Adjunct Professor, Dept. of Mathematics and Statistics, American University. The canonical reference is arXiv:1504.06352. AeroGlyphics implements a reduced-order specialization of the method for single-latent-variable VO₂ trajectories; the multi- variable block-tridiagonal formulation and the higher-order cumulant correction (Shun & McCullagh, 1995) belong to Tillinghast.