File:Chi squared.gif
Summary
| Description |
English: If you repeat your set of measurements (e.g. measure y as a function of x) several time, you will get different values each time, and thus a different fit to your model each time, and thus a different χ² each time (and we can calculate how χ² is distributed) |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1464276671469694978 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
\[Sigma] = 1.5; npoints = 15; nparameters = 4; \[Chi]list = {};
frames1 = Table[
data = Table[{i, 0.05*i (i - 5) (i - 12) + 1 + RandomVariate[NormalDistribution[0, \[Sigma]]]}, {i, 1, npoints}];
fit = Fit[data, Evaluate@Table[x^j, {j, 0, nparameters - 1}], x ];
\[Chi]2 = Chop@(Total@Table[(Chop@Evaluate[fit /. {x -> j}] - data[[j, 2]])^2, {j, 1, npoints}]/(\[Sigma]^2));
AppendTo[\[Chi]list, \[Chi]2];
Grid[{{
Show[
ListPlot[Around[#, \[Sigma]] & /@ data[[All, 2]] , PlotStyle -> Black, AxesLabel -> {"x", "y"}, LabelStyle -> {Bold, Black}, PlotLabel -> "f(x)=" Evaluate@Sum[Subscript[a, j] x^j, {j, 0, nparameters - 1}] , PlotRange -> {-10, 30}], Plot[fit, {x, 0, npoints}, PlotStyle -> {Red}, PlotRange -> All], ImageSize -> Medium]
,
Plot[PDF[ChiSquareDistribution[npoints - nparameters], x], {x, -0.1, (npoints - nparameters) + 5 Sqrt[2 (npoints - nparameters)]}, PlotRange -> All, PlotStyle -> {Black, Opacity[0]}, Prolog -> {Pink,
Table[Line[{{\[Chi]list[[j]], -5}, {\[Chi]list[[j]], 100}}], {j, 1, Dimensions[\[Chi]list][[1]]}], Red, Line[{{\[Chi]2, -5}, {\[Chi]2, 100}}]}, AxesLabel -> {"\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\)", "P(\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\))"}, LabelStyle -> {Bold, Black}, PlotLabel -> "\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\)=\!\(\*SubscriptBox[\(\[Sum]\), \(j\)]\)\!\(\*FractionBox[SuperscriptBox[\((\*SubscriptBox[\(y\), \(j\)] - fit)\), \(2\)], SuperscriptBox[\(\[Sigma]\), \(2\)]]\)", ImageSize -> Medium]
}}]
, {75}];
frames2 = Table[
Grid[{{
Show[
ListPlot[Around[#, \[Sigma]] & /@ data[[All, 2]] , PlotStyle -> Black, AxesLabel -> {"x", "y"}, LabelStyle -> {Bold, Black}, PlotLabel -> "f(x)=" Evaluate@Sum[Subscript[a, j] x^j, {j, 0, nparameters - 1}] , PlotRange -> {-10, 30}], Plot[fit, {x, 0, npoints}, PlotStyle -> {Red}, PlotRange -> All], ImageSize -> Medium]
,
Plot[PDF[ChiSquareDistribution[npoints - nparameters], x], {x, -0.1, (npoints - nparameters) + 5 Sqrt[2 (npoints - nparameters)]}, PlotRange -> All, PlotStyle -> {Black, Opacity[\[Tau]]}, Filling -> Axis, FillingStyle -> {Opacity[\[Tau]/2]}, Prolog -> {Pink, Table[Line[{{\[Chi]list[[j]], -5}, {\[Chi]list[[j]], 100}}], {j, 1, Dimensions[\[Chi]list][[1]]}]}, AxesLabel -> {"\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\)", "P(\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\))"}, LabelStyle -> {Bold, Black}, PlotLabel -> "\!\(\*SuperscriptBox[\(\[Chi]\), \(2\)]\)=\!\(\*SubscriptBox[\(\[Sum]\), \(j\)])\!\(\*FractionBox[SuperscriptBox[\((\*SubscriptBox[\(y\), \(j\)] - fit)\), \(2\)], SuperscriptBox[\(\[Sigma]\), \(2\)]]\)", ImageSize -> Medium]
}}]
, {\[Tau], 0, 1, 0.1}];
ListAnimate[
Join[frames1, Table[frames1[[-1]], 5], frames2,
Table[frames2[[-1]], 20]]]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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