# linearfitxy

linearfitxy(X, Y; σX=0, σY=0, r=0, ci=95)

keywords: GMT, Julia, liner fit, statistical plots

Performs 1D linear fitting of experimental data with uncertainties in X and Y:

• Linear fit: Y = a + b*X [1]

• Errors: X ± σX; Y ± σY [2]

• Errors' correlation: r = = cov(σX, σY) / (σX * σY) [3]

# Arguments:

• X and Y are input data vectors with length ≥ 3

• Optional standard deviation errors σX and σY are vectors or scalars

• Optional r is the correlation between the σX and σY errors. r can be a vector or scalar

• ci is the confidence interval for the statistics. By default it's 95% but any integer number > 0 < 100 will do.

σX and σY errors (error ellipses) with bivariate Gaussian distribution assumed. If no errors, or if only σX or σY are provided, then the results are equivalent to those from the LsqFit.jl package.

Based on York et al. (2004) with extensions (confidence intervals, diluted corr. coeff.).

The results are added as new columns of a GMTdataset structure when they are vectors (σX σY r) and stored as attributes when they are scalars (a, b, σa, σb, σa95, σb95, ρ and S):

• The intercept a, the slope b and their uncertainties σa and σb

• σa95 and σb95: 95%-confidence interval using two-tailed t-Student distribution, e.g.: b ± σb95 = b ± t(0.975,N-2)*σb

• Goodness of fit S (reduced Χ² test): quantity with Χ² N-2 degrees of freedom S ~ 1: fit consistent with errors, S > 1: poor fit, S >> 1: errors underestimated, S < 1: overfitting or errors overestimated

• Pearson's correlation coefficient ρ that accounts for data errors

For more information and references see the LinearFitXYerrors.jl package, from which this function is derived.

## Examples

D = linearfitxy(X, Y)    # no errors in X and Y, no plot displayed

D = linearfitxy(X, Y; σX, σY) # XY errors not correlated (r=0);

D = linearfitxy([91., 104, 107, 107, 106, 100, 92, 92, 105, 108], [9.8, 7.4, 7.9, 8.3, 8.3, 9.0, 9.7, 8.8, 7.6, 6.9]);

D = linearfitxy([0.0, 0.9, 1.8, 2.6, 3.3, 4.4, 5.2, 6.1, 6.5, 7.4], [5.9, 5.4, 4.4, 4.6, 3.5, 3.7, 2.8, 2.8, 2.4, 1.5], sx=1 ./ sqrt.([1000., 1000, 500, 800, 200, 80,  60, 20, 1.8, 1]), sy=1 ./ sqrt.([1., 1.8, 4, 8, 20, 20, 70, 70, 100, 500]));

D = linearfitxy([0.037 0.0080; 0.035 0.0084; 0.032 0.0100; 0.040 0.0085; 0.013 0.0270; 0.038 0.0071; 0.042 0.0043; 0.030 0.0160], sx=0.03, sy=0.1, r=0.7071);