Domain Parameter Estimation

Domain -- Model parameter estimation

Recover the parameters of a known nonlinear model from noisy samples, with an interval that means what it says, from clean or contaminated data and from records too short or too uneven to be comfortable.

Notebooks: 15_families, 14_robust_image, 31_reference_datasets.

Routes and baselines

• dtfit: fit(basis="legendre"), fit(basis="block"), fit(basis="auto"), each with robust=True for the reweighted image.
• Baselines: SciPy curve_fit (Levenberg-Marquardt and trust-region NLLS), least_squares with the soft-L1 loss, the historical integral forms from dtfit-legacy, and the certified values of the NIST StRD corpus.

What is measured

Sixteen families from control, spectroscopy, growth and kinetics (damped oscillation, sine, first-order rise, biexponential, Gaussian and Lorentzian peaks, logistic, Gompertz, Weibull, Michaelis-Menten, Hill and others), 40 seeds at each of five noise levels, and dtfit's own 25-scenario accuracy corpus.

	Legendre / NLLS	block / NLLS
median parameter-error ratio, sixteen families, 5 percent noise	1.000	1.025
range of that median over 2 to 30 percent noise	0.997 to 1.000	0.985 to 1.055
median over the 25-scenario corpus	1.000	0.980

The image reaches the pointwise fit's own optimum: the largest relative distance between the two estimates runs 4.0e-06 to 2.6e-02 over the families. The parity is a median. Thirteen of the sixteen families trail a basis by more than a tenth somewhere in the noise grid, none at all five levels, and the list of trailing families changes with the noise draw. basis="auto" returns the Legendre candidate on all sixteen.

Sampling regimes. A concentrated transient reads 1.04 and 1.12 times NLLS (Legendre, block) against seed spreads of 0.08 and 0.29; sparse irregular sampling 1.00 and 1.05; a short record 1.01 and 0.96. None separates the image from NLLS by more than the scatter.

Contamination. The plain fit has no robustness in either basis. At 10 percent scattered outliers at 8 sigma the median parameter error is 5.30 percent for NLLS, 5.11 for plain Legendre and 4.77 for plain block; the reweighted image reads 0.44 and 0.45. On clean data the reweighting costs a few percent at most. Under bursts the Legendre basis is the one to use: robust block grows 2.9 times over burst lengths 1 to 40 where robust Legendre stays flat.

NIST StRD. On the ten nonlinear-regression sets from both NIST starts the Legendre image's error falls toward the certified parameters as the order rises wherever the record holds enough observations. On MGH17 and BoxBOD, where curve_fit diverges from the far start, the default order guards and a fixed order recovers both under 1 percent. The block image does not keep pace on these short non-uniform records: where both build at twice order_for its error is 19.5 times Legendre's or more. The Puromycin replicate design lands both bases within 0.6 percent of NLLS.

Model mismatch. No estimator rescues a structurally wrong model, and the image and NLLS reach the same in-sample optimum on it. The R2 gap flags two of three mismatches; a biexponential fitted as one decay shows only in the residual autocorrelation, at 4.1 times the correct fit's.

Where dtfit loses

• The historical integral forms trail the image on cycles (7.74 times NLLS on the damped oscillation against 1.01), and direct equal areas reads 1.153 times NLLS in the corpus median.
• The block basis on short, non-uniform records (NIST) and under bursts.
• A record that holds fewer observations than the order needs: the Legendre image refuses on four of the ten NIST sets at some order or start and returns no number.

Reading it

The image is NLLS restricted to the span of a basis, and at the default order the restriction costs nothing measurable. What the image adds is not accuracy on a resident record: it is the fixed-size additive statistic (Big data, Archive showcase), the reweighting that lives in the statistic, and the window form (Embedded control).