CertifyPAR: Adequacy Certification and Coefficient Inference for Piecewise Affine Regression

Precise coefficient estimates in piecewise affine regression do not establish that a common affine description is adequate within each region. We introduce Certify-PAR, a reporting contract that certifies regional affine adequacy before releasing coefficients with valid confidence intervals. Existing learners supply regions fixed independently of the audit responses; coefficients are estimated anew from the au- dit data. The targets are least-squares projection coefficients on the observed audit inputs, defined without assuming an affine mean. In the two-axis construction, two equivalence tests assess non-affinity within prespecified sub-regions (audit cells) and disagreement among their affine projections. Both noise-normalized root-mean-square departures must be certified below their respective prespecified margins. A pooled alternative certifies total departure from a common affine fit. In either construction, a region’s requested coefficients are released together only after certification. Under independent homoskedastic Gaussian errors, certification based on audit residual projections preserves the coefficient-error distribution while reusing audit observations for estimation. For unknown error variance, two pure-error samples, independent of each other and the audit data, provide separate certification and inference scales. Under these assumptions, we establish finite-sample control of the familywise false-certification probability and, separately, simultaneous confidence-interval coverage conditional on every complete report- ing pattern of positive probability. Simulations using synthetic data and real-data covariates assess error control under the assumed noise model and quantify trade- offs among reporting availability, coefficient precision, and measurement cost. GPU-runtime and fluorescence-calibration examples illustrate reporting decisions under benchmark adequacy margins and highlight obstacles to satisfying the noise assumptions required for exact guarantees.