Choosing an Estimator

MCUP provides three estimators targeting different error scenarios. Use this guide to pick the right one.

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Decision tree

Do your x measurements have significant errors?
│
├─ No  → WeightedRegressor
│
└─ Yes → Do you need the most exact treatment?
         │
         ├─ No  (mild nonlinearity, speed matters) → XYWeightedRegressor
         │
         └─ Yes (large x errors, strong nonlinearity) → DemingRegressor

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WeightedRegressor

Use when: only y has measurement errors; x is known exactly (or its errors are negligible).

Minimises the weighted chi-squared objective:

Σ (y_i - f(x_i, β))² / σ_yi²

This is the standard weighted least squares formulation and is the fastest option.

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XYWeightedRegressor

Use when: both x and y carry measurement errors, and your model is mildly nonlinear.

Combines x and y variances via first-order error propagation:

σ_combined,i² = σ_yi² + (∂f/∂x_i)² · σ_xi²

The gradient ∂f/∂x is computed numerically at each iteration. The fit is iterated (IRLS — Iteratively Reweighted Least Squares) until convergence, typically in ~10 steps.

Tradeoff vs DemingRegressor: faster and simpler, but the first-order propagation becomes inaccurate when x errors are large or the model curves sharply.

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DemingRegressor

Use when: both x and y have errors and you need the most rigorous treatment.

Performs joint optimisation over model parameters and latent true x values η:

min_{β, η}  Σ (x_i - η_i)²/σ_xi²  +  Σ (y_i - f(η_i, β))²/σ_yi²

This is total least squares (Deming regression) generalised to arbitrary nonlinear models.

Tradeoff vs XYWeightedRegressor: more accurate when x errors are large, but the parameter space grows with the number of data points, making each optimisation step slower.

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Analytical vs Monte Carlo solver

Both solvers are available for all three estimators via the method argument.

	method="analytical"	method="mc"
Speed	Fast — one optimisation pass	Slower — thousands of optimisation runs
Covariance	(J^T W J)^{-1} Jacobian approximation	Welford online covariance over MC samples
Best for	Linear or mildly nonlinear models	Strongly nonlinear models
Robustness	May underestimate errors if model is nonlinear	Correct by construction regardless of nonlinearity
Convergence control	Not applicable	rtol / atol stopping criteria available

When to prefer "mc"

• Your model is strongly nonlinear (e.g. exponentials, power laws, oscillatory functions)
• You want confidence intervals that don't rely on the linear approximation
• You need to verify that analytical results are trustworthy for your model

When to prefer "analytical"

• Your model is linear or close to linear
• Speed is important (e.g. fitting many datasets in a loop)
• You're doing a quick exploratory fit and can validate later with MC