Shared-parameter experiments¶

Suppose two experiments measure the same first-order decay rate but begin with different concentrations. Joint fitting uses one rate degree of freedom and one amplitude per experiment, even when their time grids differ.

Fit once, project carefully¶

import numpy as np
from curvefit_stats import GlobalFitGroup, generate_error_bands

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

rng = np.random.default_rng(52)
time_a = np.linspace(0, 4, 35)
time_b = np.linspace(0.1, 6, 52)
datasets = {
    "control": {"x": time_a, "y": decay(time_a, 3.0, 0.7) + rng.normal(0, 0.08, 35), "sigma": 0.08},
    "treated": {"x": time_b, "y": decay(time_b, 4.5, 0.7) + rng.normal(0, 0.1, 52), "sigma": 0.1},
}
group = GlobalFitGroup(decay, datasets, shared_parameters={"rate"})
result = group.fit(
    {"rate": 0.5, "control.amplitude": 2.5, "treated.amplitude": 4.0},
    bounds={"rate": (0.01, 3), "control.amplitude": (0, np.inf), "treated.amplitude": (0, np.inf)},
    absolute_sigma=True,
)
assert result.success and result.diagnostics.identifiable
assert result.parameter_names == ("rate", "control.amplitude", "treated.amplitude")
assert result.pcov.shape == (3, 3)

indices = group.parameter_indices["control"]
local_parameters = result.popt[indices]
local_covariance = result.pcov[np.ix_(indices, indices)]
grid = np.linspace(0, 5, 80)
lower, upper = generate_error_bands(
    decay, grid, local_parameters, local_covariance, reference="normal",
)
assert lower.shape == upper.shape == grid.shape
print(result.diagnostics)

The result includes per-dataset parameters, predictions, residuals, explicit global/local mappings and solver diagnostics. Covariance is calculated from the joint Jacobian, never by concatenating independent dataset covariances. Mapping aliases can tie local arguments to a shared variable; tied derivative contributions add into its one global column.

Known absolute measurement errors justify normal-reference bands here. For an interior relative-weight fit, pass the global dof explicitly with no local y; per-dataset N-p does not describe joint parameter uncertainty. There is one common unknown variance multiplier, not separate unknown dataset scales.

For prediction intervals, supply the future observation variance from the same global noise model. A per-dataset audit is not global AIC/BIC. Rank failure or optimizer failure invalidates covariance; active bounds leave NaN active rows and a conditional free block. Check these before extracting local bands.

See joint-fit integration and GlobalFitGroup for the exact mapping, solver and result contracts.