Straight Line Fitting Algorithm

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Straight line fitting estimates the two parameters of a linear model from a set of measured points, in closed form, along with their variances, covariance, and a goodness-of-fit probability. It is the algorithm behind StraightLineFitter in the com.irurueta.numerical.fitting package, and is described in Press et al., 2007, Section 15.2 (page 781).

Problem statement

Given data points with known measurement errors on (the are assumed known exactly), the model is:

and, as with every fitter in this package, and are chosen to minimize the chi-square merit function:

This is the simplest possible case of general linear least-squares fitting — only two basis functions, and — solved with dedicated closed-form formulas rather than the general normal-equations matrix machinery, making it considerably cheaper for this one common case.

Closed-form solution

Setting and gives two linear equations in and . Defining the weighted sums:

the minimization condition becomes:

whose solution, with , is:

Uncertainties, covariance, and correlation

Propagating the measurement errors through and (using ) gives their variances and covariance:

and the correlation coefficient between the errors in and :

Goodness of fit

The probability that a fit this poor (or worse) could occur by chance, under the assumption of normally-distributed errors with degrees of freedom, is:

where is the chi-square cumulative distribution function with degrees of freedom (computed via the incomplete gamma function). As a rule of thumb: indicates a believable fit, may still be acceptable if the errors are non-normal or somewhat underestimated, and calls the model — or the assumed measurement errors — into question.

A numerically stable reformulation

The closed-form expressions above are susceptible to roundoff error. StraightLineFitter instead uses the equivalent, numerically stable formulation from Press et al., 2007, built around a per-point normalized abscissa :

This is algebraically identical to the naive formulas above but avoids the cancellation that makes unreliable in floating-point arithmetic.

Algorithm

flowchart TD A["Data (x_i, y_i, sigma_i)"] --> B["Accumulate weighted sums S, S_x, S_y"] B --> C["Compute t_i = (x_i - S_x/S) / sigma_i and S_tt = sum(t_i^2)"] C --> D["b = sum(t_i * y_i / sigma_i) / S_tt"] D --> E["a = (S_y - S_x * b) / S"] E --> F["sigma_a^2, sigma_b^2, Cov(a,b) from S, S_x, S_tt"] F --> G["chi-square = sum(((y_i - a - b*x_i) / sigma_i)^2)"] G --> H["Q = goodness-of-fit probability (chi-square distribution, N-2 dof)"]

Unlike the iterative Levenberg-Marquardt fitter, there is nothing to configure and no risk of non-convergence: the fit is exact given the formulas above.

When measurement errors are unknown

If the are not known, StraightLineFitter sets for every point, solves for and using the unweighted sums, and then estimates a common per-point error after the fact from the residuals:

rescaling and by this factor. Since the data itself was used to estimate the error bars, this procedure assumes the line is a good fit and provides no independent goodness-of-fit check — getQ() stays at its default value of in this case.

Library implementation

StraightLineFitter extends Fitter directly rather than SingleDimensionFitter — the model is fixed to , so unlike the other fitters in the package it needs no evaluator listener, just the raw x/y (and optionally sig) arrays:

  • StraightLineFitter(double[] x, double[] y) — standard deviations unknown; estimated from the residuals as described above.

  • StraightLineFitter(double[] x, double[] y, double[] sig) — standard deviations known.

After fit() completes:

  • getA() / getB() — the fitted parameters and .

  • getSigA() / getSigB() — their estimated standard deviations and .

  • getChi2() — the final value.

  • getQ() — the goodness-of-fit probability (only meaningful when sig was provided).

  • getSigdat() — the estimated common per-point error , set only when sig was not provided.

Example

final var fitter = new StraightLineFitter(x, y, sigma);
fitter.fit();

final var a = fitter.getA();       // intercept
final var b = fitter.getB();       // slope
final var chiSq = fitter.getChi2(); // goodness-of-fit statistic
final var q = fitter.getQ();        // goodness-of-fit probability