alprd 2.2.0

Adaptive Local Polynomial Regression Differentiation framework

numdiff-research

Adaptive Local Polynomial Regression Differentiation (ALPRD) — an ongoing, honesty-first research project on numerical differentiation of unevenly sampled, noisy 1-D data.

Status: Phase 4 concluded at V2.2 (estimator frozen at the V2.0 formula; V2.2 added physical-sensor validation, a completed WLS-path oracle inequality, two falsified-and-reverted fixes, a documented scope boundary, and the paper draft) (evidence-based stopping condition: all remaining improvements require external literature, external code/data, or genuinely new mathematics — see the V1.4 log section). Current self-assessed publication readiness: 7.5/10. Claims are strictly separated into Proven / Derived / Tested / Hypothesized, and negative results are kept in the record (V1.0's outlier-robustness claim was retracted after a control experiment; V1.0-V1.2's noise estimator was found miscalibrated by 2.2x and fixed in V1.3; two V1.4 hypotheses were tested, falsified, and recorded).

Layout

docs/RESEARCH_LOG.md      the full record: derivations, assumptions, benchmarks,
                          failure analyses, version history (V1.0, V1.1, ...)
src/methods.py            ALPRD variants + all baseline methods
src/datasets.py           benchmark dataset generators (exact ground-truth
                          derivatives, incl. a Lorenz trajectory via solve_ivp)
src/benchmark.py          the full 35-scenario suite; writes results/*.csv
tests/test_methods.py     correctness + equivalence tests (plain python, no
                          pytest required)
results/                  raw per-run CSVs and scaling measurements — every
                          number cited in the log traces to a file here

Reproduce

pip install numpy scipy pandas
python tests/test_methods.py     # 5 tests, all must pass
python src/benchmark.py          # ~2 min; rewrites results/results_v1_1.csv

Everything is seeded (seed=12345 throughout); re-running the suite must reproduce the CSVs bit-for-bit on the same numpy/scipy versions (numpy 2.5.1 / scipy 1.18.0 were used for the committed results).

Method in one paragraph

At each query point, ALPRD fits a weighted local polynomial (tricube kernel, degree 2) over a k-nearest-neighbor window of the actual sample times — no resampling to a uniform grid — and reads the derivative off the linear coefficient, together with a finite-sample variance estimate. V1.1 selects k per point by minimizing an estimated Bias² + Variance (plug-in rule built from the derived leading-order bias and exact variance functionals). V1.2 makes every fit Tukey-biweight-robust (IRLS), refines the curvature pilot at the selected bandwidth, and vectorizes the whole selector; under 3% gross contamination its error is now ~16x lower than any non-robust method in the suite while remaining near-best on clean and noisy data. Known open problems, failure modes, and the improvement queue are tracked in docs/RESEARCH_LOG.md Steps 8–13 of each version.

Quick Start

import numpy as np
from alprd import robust_differentiation

t = np.sort(np.random.uniform(0, 10, 100))
y = np.sin(t) + np.random.normal(0, 0.1, 100)
derivative = robust_differentiation(t, y)