K₁K₂T₁T₂ Estimation via Hierarchical Bayesian NLSS

Estimating K₁, K₂, T₁, T₂ Using Hierarchical Bayesian NLSS Gabriel Della Mattia · AGMT2 · Argentina · 2026 Abstract The Banister IR-Model describes athletic performance as the superposition of two exponential responses to training load, parameterized by {K₁, K₂, T₁, T₂}. Classical estimation (Skiba & Clarke NLLS) requires many phase-distributed power tests and is numerically unstable with sparse data. TrainingPeaks avoids this problem by fixing K₁=K₂=1 for all athletes — an underestimation by factors of ×2.9 and ×4.1 relative to AGMT2 cohort values, producing systematic taper prediction errors of ±5–8 days. We propose NLSS (Nonlinear Least Squares with Shrinkage): a hierarchical Bayesian extension that regularizes parameter estimation via a Mahalanobis penalty derived from a 30-athlete AGMT2 cohort prior, and recalibrates weekly on a 90-day sliding window. The method requires only daily TSS and occasional power tests — no additional sensors beyond a standard power meter. With 3 distributed tests, NLSS achieves taper prediction errors of ±1–2 days. The complete algorithm, analytic gradient, and a reference Python implementation are provided. Conclusions TrainingPeaks addresses the challenge of estimating 4 nonlinear parameters from sparse data by eliminating individualization entirely — fixing K₁=K₂=1 for all athletes regardless of physiology. Classical NLLS (Skiba & Clarke) is the correct methodological approach, but its practical reach is limited by the test density it requires: most athletes accumulate 4–8 tests per year, insufficient for stable parameter identification without additional constraints. The hierarchical Bayesian approach proposed here resolves both failure modes simultaneously. The Mahalanobis prior, calibrated from the AGMT2 30-athlete cohort, anchors the estimator in physiologically plausible space from day one — correcting the K₁ and K₂ underestimation that makes the TrainingPeaks PMC systematically wrong. The 90-day sliding window with adaptive λ(n) allows progressive individualization as tests accumulate, transitioning smoothly from population-level estimates to athlete-specific parameters without requiring a full season of data. The practical implications are concrete: a 3-test protocol distributed across Base, Build, and post-taper phases is sufficient to achieve taper prediction accuracy (±1–2 days) that classical NLLS only reaches with 10+ tests, and that TrainingPeaks never achieves for athletes whose T₁ deviates from the population mean. Four open limitations are identified and formalized as mathematical extensions: sub-priors by athlete profile (requires N ≥ 100 per group), intensity-domain load decomposition (expands parameter space to 12), Huber loss for outlier-robust estimation (implementable immediately, highest priority), and Ledoit-Wolf shrinkage for the covariance matrix (already estimated in the current AGMT2 implementation). Of these, the Huber-δ modification to the data loss term represents the highest-priority improvement for the next release: it requires minimal changes to the gradient equations while providing the largest gain in robustness to field conditions. All parameters, equations, and the reference implementation are fully specified in this document and are sufficient for independent replication.