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Use Hoeffding's D to Quantify and Test Non-monotonic Dependence

Use Hoeffding's D to Quantify and Test Non-monotonic Dependence

Unlike SpearmanRho, KendallTau, and Pearson's Correlation, HoeffdingD can be used to detect a wide variety of dependence structures beyond monotonic association.

    
Wolfram Language code: v1 = {3, -4, 1, 4, 22, 17, -2, 2, 13, -11}; v2 = {-20, -24, 0, 4, 24, 36, -12, -12, 56, -14};
Wolfram Language code: HoeffdingD[v1, v2]
Test whether Hoeffding's is significantly different from zero:
Wolfram Language code: HoeffdingDTest[v1, v2, "TestDataTable"]
A collection of datasets with different dependency structures:
Wolfram Language code: uni = RandomVariate[UniformDistribution[{-2Pi, 2Pi}], 1500]; SetAttributes[f, Listable]; f[x_, σ_] := { RandomVariate[BinormalDistribution[0]], RandomVariate[BinormalDistribution[.6]], {x, -Sqrt[Abs[x]] + RandomVariate[NormalDistribution[0, σ]]}, {x, FresnelC[# / 10]&[(x + RandomVariate[NormalDistribution[0, σ]])^2]}, RandomVariate[BinormalDistribution[-.8]], RandomChoice[{RandomVariate[BinormalDistribution[-.8]], RandomVariate[BinormalDistribution[.8]]}], RandomChoice[{RandomVariate[NormalDistribution[1, .25], 2], RandomVariate[NormalDistribution[0, .5], 2]}], {x, -Sinc[x] + RandomVariate[NormalDistribution[0, σ]]}, {Cos[x], Sin[x] + RandomVariate[NormalDistribution[0, σ]]}} data = f[uni, .2];
Wolfram Language code: GraphicsGrid[Partition[Table[ListPlot[data[[All, i]], Frame -> True, Axes -> None, AspectRatio -> 1, PlotStyle -> Directive[PointSize[Tiny]], FrameTicks -> None, Background -> White], {i, 9}], 3], ImageSize -> 350]
The Hoeffding statistics and test conclusions for the data grid:
Wolfram Language code: Partition[Table[Column@HoeffdingDTest[Sequence@@Transpose@data[[All, i]], {"TestStatistic", "ShortTestConclusion"}], {i, 9}], 3]//TableForm