Errors in both axes¶

An uncertain sampling time changes the statistical model: vertical residual least squares alone treats the observed predictor as exact. EIV fits latent adjusted coordinates with independent uncertainties on both axes.

where \(\delta\) is adjusted minus observed x and \(\varepsilon\) is fitted minus observed y. The predictor term vanishes in exact-x mode (sx=None).

Fit and inspect the backend result¶

import warnings
import numpy as np
from scipy.optimize import curve_fit
from curvefit_stats import eiv_curve_fit, generate_error_bands

def decay(time, amplitude, rate):
    return amplitude * np.exp(-rate * time)

rng = np.random.default_rng(82)
true_time = np.linspace(0, 4, 70)
sx, sy = 0.15, 0.08
observed_time = true_time + rng.normal(0, sx, true_time.size)
concentration = decay(true_time, 3.0, 0.7) + rng.normal(0, sy, true_time.size)
ordinary, _ = curve_fit(decay, observed_time, concentration, [2.5, 0.5], sigma=sy)
with warnings.catch_warnings():
    warnings.filterwarnings("default", message="scipy.odr is deprecated", category=FutureWarning)
    popt, pcov, diagnostic = eiv_curve_fit(
        decay, observed_time, concentration, [2.5, 0.5],
        sx=sx, sy=sy, absolute_sigma=True, full_output=True,
    )
assert diagnostic.success and diagnostic.identifiable
assert pcov.shape == (2, 2)
print("Vertical LS:", ordinary, "EIV:", popt)
print(diagnostic.stopreason)

# Coordinates here are exact evaluation points, not noisy future measurements.
grid = np.linspace(0, 4, 80)
lower, upper = generate_error_bands(decay, grid, popt, pcov, reference="normal")
assert np.isfinite(lower).all() and np.isfinite(upper).all()

This single experiment illustrates the distinction; it does not prove that EIV universally outperforms ordinary least squares. The internal adapter accepts ordinary func(x, *parameters) models and handles ODR's parameter-first convention.

Covariance and failure semantics¶

absolute_sigma=True returns ODR's unscaled cov_beta. Relative errors return cov_beta * res_var, with identified res_var=Q_ODR/(N-p) and one common unknown multiplier for both axis variances. Delta bands at exact coordinates are a local approximation; future uncertain x requires an additional observation model.

By default, non-convergence raises RuntimeError. With full_output=True, the function can return current estimates with a failure diagnostic and NaN covariance, plus an explicit warning. Never proceed to uncertainty propagation without checking convergence and covariance validity.

Response-only audit_fit, profile likelihood, bootstrap, bounded covariance and Student-t summaries are not ODR methods. Feeding them ODR parameters does not change their likelihood or calibration. Read EIV integration and eiv_curve_fit for the full contract.