Linear and Nonlinear Fitting

Fitting linear and nonlinear models to experimental data with uncertainty.

Conceptual context

In experimental physics, a cloud of points is not yet a law. Every datum combines system response, instrumental resolution, fluctuations, and possible bias. Fitting a model means proposing a relation between $x$ and $y$, estimating its parameters, and checking whether disagreement between data and prediction is compatible with $\sigma_{i}$. The task is not to select the prettiest curve, but the simplest explanation that retains physical meaning, respects units, and leaves no pattern in $r_i$. A linear model describes constant sensitivity, whereas a nonlinear model allows the response rate to change. Model choice therefore requires parameters, residuals, experimental domain, and measurement quality to be read together.

🟢 Essential level

A fit turns a collection of measurements into an interpretable relation. First, plot $y$ against $x$ and decide whether the trend looks straight, curved, or divided into regimes. In a linear model, $m$ states how much the response changes per unit input, while $b$ describes the extrapolated starting level. In a power law, $A$ sets the scale and $n$ controls how growth accelerates or slows. Selection must not rely only on $R2$. Inspect $r_i$ as well: positive and negative values scattered without a pattern support the model, whereas curves, fans, or bands suggest missing physics or an incorrect uncertainty description. $chi2nu$ helps decide whether deviations have the size expected from $\sigma_{i}$.

A good choice also remains stable when one ordinary point is removed and gives parameters with sensible units. A good fit should also remain reasonable when a single ordinary point is removed: if the result changes drastically, either the model or the datum itself must be reviewed before accepting the obtained parameters.

🔵 Formal level

Basic formulas

The linear model represents a response with constant rate and a possible offset:

Linear model

Ordinary least squares estimates $m$ by minimizing the sum of squared vertical deviations:

Least-squares slope

When the response follows scaling behavior, the power-law model introduces $A$ and $n$:

Power-law model

Real student questions

Is the model with the largest $R2$ always best? No. Extra flexibility can raise it while leaving structured residuals or meaningless parameters.

Why inspect $r_i$ if the curve already looks good? The eye is dominated by the global scale. Residuals amplify small systematic deviations.

Does $chi2nu$ near one prove the model is correct? It only indicates compatibility with stated uncertainties under specific assumptions.

When should I use a nonlinear model? When physical reasoning, persistent curvature, or changing sensitivity cannot be captured linearly.

Can I linearize a power law with logarithms? Yes, if the transformation is valid and uncertainties are handled properly, because transformation changes the error structure.

What should I do with an outlier? Investigate it. Automatic deletion can hide physics or manufacture a clean fit.

Final synthesis

Choosing between linear and nonlinear fitting requires more than comparing curves. Units, parameters, $r_i$, $\sigma_{i}$, global metrics, and mechanism must agree. The best model is the simplest one that explains the observed dependence without $r_i$ structure, preserves physical meaning, and states its validity limits honestly.