How Curve Fitting Works

This guide explains the mathematical foundation of nonlinear least squares curve fitting - what it means to “fit” a model to data and how the algorithm finds optimal parameters.

The Fitting Problem

Given: - Data points: (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ) - A model function: f(x; θ) with parameters θ = (θ₁, θ₂, …, θₘ)

Find the parameters θ that minimize the sum of squared residuals:

\[S(θ) = \sum_{i=1}^{n} [y_i - f(x_i; θ)]^2\]

This is called the least squares objective because we’re minimizing the sum of squared differences (residuals) between data and model.

Why Squared Residuals?

  1. Mathematical convenience: The squared function is smooth and differentiable everywhere, making optimization easier.

  2. Statistical interpretation: If errors are normally distributed, minimizing squared residuals gives the maximum likelihood estimate.

  3. Equal treatment: Positive and negative residuals contribute equally.

Linear vs Nonlinear Least Squares

Linear least squares: The model is linear in parameters:

\[y = θ_1 x + θ_2\]

This has a direct algebraic solution (normal equations).

Nonlinear least squares: The model is nonlinear in parameters:

\[y = θ_1 e^{-θ_2 x}\]

No direct solution exists - we need iterative optimization.

The Optimization Process

NLSQ uses the Trust Region Reflective (TRF) algorithm:

  1. Initialize: Start with initial guess θ₀

  2. Evaluate: Compute residuals r = y - f(x; θ)

  3. Linearize: Compute Jacobian J = ∂f/∂θ (automatic with JAX)

  4. Solve subproblem: Find step direction δ within a “trust region”

  5. Update: θ ← θ + δ

  6. Check convergence: Stop if change is small enough

  7. Repeat: Go to step 2

θ₀ → [compute residuals] → [compute Jacobian] → [solve for δ]
 ↑                                                    ↓
 └──────────────── [update θ ← θ + δ] ←──────────────┘

The Jacobian Matrix

The Jacobian J is an n × m matrix of partial derivatives:

\[J_{ij} = \frac{\partial f(x_i; θ)}{\partial θ_j}\]

It tells us how sensitive the model output is to each parameter.

In NLSQ, the Jacobian is computed automatically using JAX’s automatic differentiation - no manual derivatives needed!

# You just write the model
def model(x, a, b):
    return a * jnp.exp(-b * x)


# NLSQ automatically computes ∂f/∂a and ∂f/∂b

Trust Region Method

Instead of taking the full Newton step, TRF restricts the step to a “trust region” - a ball of radius Δ around the current point.

This ensures stability:

  • Step too large? Reduce trust region

  • Good step? Expand trust region

  • Near bounds? Use reflected steps

See Trust Region Reflective Algorithm for more details.

Convergence Criteria

Optimization stops when any of these are satisfied:

  1. Gradient tolerance (gtol): Gradient is nearly zero

    \[\|J^T r\|_\infty < \text{gtol}\]

  2. Function tolerance (ftol): Cost isn’t decreasing

    \[\frac{|S_{k+1} - S_k|}{S_k} < \text{ftol}\]

  3. Step tolerance (xtol): Parameters aren’t changing

    \[\frac{\|δ\|}{\|θ\|} < \text{xtol}\]

  4. Maximum iterations: Safety limit reached

Parameter Uncertainties

After finding optimal θ*, we estimate uncertainties from the covariance matrix:

\[\text{cov}(θ) \approx s^2 (J^T J)^{-1}\]

where s² is the residual variance:

\[s^2 = \frac{S(θ^*)}{n - m}\]

The standard error of each parameter is:

\[σ_{θ_i} = \sqrt{\text{cov}(θ)_{ii}}\]

Goodness of Fit

R-squared (coefficient of determination):

\[R^2 = 1 - \frac{S(θ^*)}{S_{\text{tot}}}\]

where Sₜₒₜ = Σ(yᵢ - ȳ)² is the total variance.

  • R² = 1: Perfect fit

  • R² = 0: Model no better than mean

  • R² < 0: Model worse than mean (rare, indicates problems)

Reduced chi-squared:

\[χ^2_ν = \frac{S(θ^*)}{n - m}\]

Should be approximately 1 for a good fit with correctly estimated errors.

When Fitting Fails

Common issues:

  1. Local minimum: Found a suboptimal solution

    • Use better initial guesses

    • Try global optimization (preset=’global’)

  2. Ill-conditioned Jacobian: Parameters are poorly determined

    • Simplify model

    • Fix some parameters

  3. Divergence: Cost keeps increasing

    • Check data quality

    • Adjust bounds

  4. Oscillation: Parameters alternate without converging

    • Check for parameter correlations

    • Reparameterize model

See How to Debug Bad Fits for troubleshooting.

Summary

Concept

Meaning

Residuals

Difference between data and model

Least squares

Minimize sum of squared residuals

Jacobian

How model changes with parameters

Trust region

Safe step size limit

Covariance

Parameter uncertainties

R²

Fraction of variance explained

See Also