Find Fit for Non-linear data

Three weeks ago I replied to a question in the Wolfram Community site about finding a function fit to points of non-linear data.

Here is the data:

Non-linear data

I proposed a different way of doing the non-linear fitting using Quantile regression with B-splines. The advantages of the approach are that it does not need guessing of the fit functions and it requires less experimentation.

Load the package QuantileRegression.m:

Needs["QuantileRegression`"]

After several experiments with the number of knots we can find a regression quantile using 3d order B-splines of 18 knots that approximates the data well.

qfunc = Simplify[QuantileRegression[data, 18, {0.5}, InterpolationOrder -> 2][[1]]];

qfGr = Plot[qfunc[x], {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}, PlotPoints -> 700];
Show[{dGr, qfGr}, Frame -> True, ImageSize -> 700]

NonLinearFitWithBSplineQuantileRegression-18RegularKnots

Here is the relative error estimate:

ListPlot[Map[{#[[1]], (qfunc[#1[[1]]] - #1[[2]])/#1[[2]]} &, data],
Filling -> Axis, Frame -> True, ImageSize -> 700]

RelativeErrorNonLinearFitWithBSplineQuantileRegression-18RegularKnots

I would like to point out that if we put large number of knots we can get aliasing errors effect:

qfunc = Simplify[
QuantileRegression[data, 40, {0.5}, InterpolationOrder -> 2][[1]]];

qfGr = Plot[qfunc[x], {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}, PlotPoints -> 700];
Show[{dGr, qfGr}, Frame -> True, ImageSize -> 700]

NonLinearFitWithBSplineQuantileRegression-40RegularKnots

If we select a non-uniform grid of knots according to gradients of the data we get a good approximation with one attempt.


In[107]:= knots =
Join[
Rescale[Range[0, 1, 0.2], {0, 1}, {Min[data[[All, 1]]],
Max[data[[All, 1]]] 3/4}],
Rescale[Range[0, 1, 0.1], {0, 1}, {Max[data[[All, 1]]] 3/4,
Max[data[[All, 1]]]}]
]

Out[107]= {0., 3.75, 7.5, 11.25, 15., 18.75, 18.75, 19.375, 20., 20.625, 21.25, 21.875, 22.5, 23.125, 23.75, 24.375, 25.}

In[108]:= qfunc = QuantileRegression[data, knots, {0.5}][[1]];

The grid lines in the plot below are made with the knots.

qfGr = Plot[qfunc[x], {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}];
Show[{dGr, qfGr}, GridLines -> {knots, None},
GridLinesStyle -> Directive[GrayLevel[0.8], Dashed], Frame -> True,
ImageSize -> 700]

NonLinearFitWithBSplineQuantileRegression-17NonRegularKnots

Here are the relative errors:

ListPlot[Map[{#[[1]], (qfunc[#1[[1]]] - #1[[2]])/#1[[2]]} &, data],
Filling -> Axis, Frame -> True, ImageSize -> 700]

RelativeErrorNonLinearFitWithBSplineQuantileRegression-17NonRegularKnots

Here is the approximation function after using PiecewiseExpand and Simplify:

qfunc = Evaluate[Simplify[PiecewiseExpand[qfunc[[1]]]]] &;
qfunc

NonLinearFitWithBSplineQuantileRegression-17NonRegularKnotsFunction

Follow-up

If it is desired to make the fitting with a certain, known model function then the function QuantileRegressionFit can be used. QuantileRegressionFit is provided by the same package, QuantileRegression.m, used in the blog post.

This older blog post of mine discusses the use of QuantileRegressionFit and points to a PDF with further details (theory and Mathematica commands).

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