linearfitxy(X, Y; σX=0, σY=0, r=0, isplot=false, ratio=1)

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

$σX$ 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.).

Examples:

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

st = linearfitxy(X, Y; σX, σY, isplot=true) # XY errors not correlated (r=0); plot ratio=1

linearfitxy(X, Y; σX,σY,r=0,isplot=true,ratio=:auto) # XY errors not correlated (r=0); plot

The results are in the fields of the returned st::stfitxy structure:

• 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
• Optional display of fit results with error ellipses and confidence intervals

The default argument isplot=false can be turned on to plot the results. Currently Plots.jl's gr() is used