File:Fitting and extrapolation.gif
Summary
| Description |
English: A lot of different models can be a good fit for your data. That by itself doesn't mean your model is good.
And extrapolating from your fit is easily a recipe for disaster. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1234528010809810944 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
\[Sigma] = 15;
data = Table[{j, j^2 + RandomVariate[NormalDistribution[0, \[Sigma]]]}, {j, 5, 20}];
dim = Dimensions[data][[1]]
ed = Table[{data[[j, 1]], data[[j, 2]] \[PlusMinus] \[Sigma]}, {j, 1, dim}];
expfit = FindFit[data, a*E^(b x) + c, {a, b, c}, x];
powfit = FindFit[data, a*x^b + c, {a, b, c}, x];
sigmoidalfit = FindFit[data, a*Erf[b*x + c] + a, {{a, 200}, {b, 0.15}, {c, -2}}, x];
p0 = Table[
Show[
Plot[x^2, {x, 0, j}, PlotStyle -> {Thick, Gray, Dashed}], PlotRange -> {{0, 25}, {0, 500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium}, Epilog -> {Text[Style["Ground truth", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 0.1, 25, 0.4}];
p1 = Table[
Show[
Plot[x^2, {x, 0, 100}, PlotStyle -> {Thick, Gray, Dashed}], ListPlot[ed[[1 ;; j]], PlotStyle -> {Thick, Black, PointSize[0.02]}], PlotRange -> {{0, 25}, {0, 500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium}, Epilog -> {Text[Style["Data Points", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 1, 16, 1}];
p2 = Table[
Show[
Plot[x^2, {x, 0, 100}, PlotStyle -> {Thick, Gray, Dashed}], ListPlot[ed, PlotStyle -> {Thick, Black, PointSize[0.02]}], Plot[(a*E^(b x) + c) /. expfit, {x, 0, j}, PlotStyle -> {Thick, Purple}], PlotRange -> {{0, 25}, {0, 500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium}, Epilog -> {Text[Style["Exponential fit", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 0.1, 25, 0.4}];
p3 = Table[
Show[
Plot[x^2, {x, 0, 100}, PlotStyle -> {Thick, Gray, Dashed}], ListPlot[ed, PlotStyle -> {Thick, Black, PointSize[0.02]}], Plot[(a*E^(b x) + c) /. expfit, {x, 0, 100}, PlotStyle -> {Thick, Purple}],
Plot[(a*x^b + c) /. powfit, {x, 0, j}, PlotStyle -> {Thick, Orange}], PlotRange -> {{0, 25}, {0, 500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium},
Epilog -> {Text[Style["Polynomial fit", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 0.1, 25, 0.4}];
p4 = Table[
Show[
Plot[x^2, {x, 0, 100}, PlotStyle -> {Thick, Gray, Dashed}], ListPlot[ed, PlotStyle -> {Thick, Black, PointSize[0.02]}], Plot[(a*E^(b x) + c) /. expfit, {x, 0, 100}, PlotStyle -> {Thick, Purple}], Plot[(a*x^b + c) /. powfit, {x, 0, 100}, PlotStyle -> {Thick, Orange}], Plot[(a*Erf[b*x + c] + a) /. sigmoidalfit, {x, 0, j}, PlotStyle -> {Thick, Cyan}], PlotRange -> {{0, 25}, {0, 500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium}, Epilog -> {Text[Style["Sigmoidal fit", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 0.1, 25, 0.4}];
p5 = Table[
Show[
Plot[x^2, {x, 0, 100}, PlotStyle -> {Thick, Gray, Dashed}], ListPlot[ed, PlotStyle -> {Thick, Black, PointSize[0.02]}], Plot[(a*E^(b x) + c) /. expfit, {x, 0, 100}, PlotStyle -> {Thick, Purple}], Plot[(a*x^b + c) /. powfit, {x, 0, 100}, PlotStyle -> {Thick, Orange}], Plot[(a*Erf[b*x + c] + a) /. sigmoidalfit, {x, 0, 250}, PlotRange -> All, PlotStyle -> {Thick, Cyan}], PlotRange -> {{0, j*25}, {0, j*500}}, AxesOrigin -> {0, 0}, Ticks -> None, AxesLabel -> {"x", "y(x)"}, LabelStyle -> {Black, Bold, Medium}, Epilog -> {Text[Style["Extrapolations", Bold, FontSize -> 14], Scaled[{0.5, 0.9}]]}], {j, 1, 7, 0.1}];
ListAnimate[Join[p0, p1, p2, p3, p4, p5]]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
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