Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantile regression 3D examples
In this notebook we develop in detail Quantile Regression (QR) examples over 3D data. The QR examples demonstrate using:
1
.2
. with custom made basis functions
Remark: In order to get a better idea of the how the data looks like, use the 3D graphics interactivity feature rotation.
Preparation
Load the Quantile regression paclet
In[18]:=
Needs["AntonAntonov`QuantileRegression`"]
Set random seed
In[19]:=
SeedRandom[9903];
Define plot function
In[20]:=
QRPlot[data_List,funcs_Association,opts___]:=Show[ListPointPlot3D[data,FilterRules[{opts},{{BoxRatios{1,1,1/1.5}}}],{BoxRatios{1,1,1/1.5}},PlotStyleRed,PlotRangeAll],Plot3D[Evaluate@Through[Values[funcs][x,y]],{x,Min[data〚All,1〛],Max[data〚All,1〛]},{y,Min[data〚All,2〛],Max[data〚All,2〛]},opts,PlotRangeAll,PlotStyle{Opacity[0.7]},MeshTrue,PlotTheme"LightMesh",PerformanceGoal"Quality",PlotLegendsKeys[funcs]],ImageSizeMedium];
Define pure functions converter
In[21]:=
ToPureFunctions[funcs_,vars_List:{x,y}]:=Map[With[{f=#},f&]&,ReplaceAll[funcs,Thread[varsMap[Slot,Range[Length[vars]]]]]];
Noisy arrow wave
Generate random, "noisy wave" data
In[22]:=
{b,c}={-6,6};data=RandomReal[{b,c},{1200,2}];data=Map[Append[#,Sqrt[#〚1〛-b]+Sin[#〚1〛+Abs[#〚2〛]]+RandomVariate[NormalDistribution[0,0.3]]]&,data];Dimensions[data]
Out[25]=
{1200,3}
3D list plot of the data
In[26]:=
ListPointPlot3D[data,PlotRangeAll,PlotLabel"Noisy wave data",BoxRatios{1,1,2/3},ImageSizeMedium]
Out[26]=
Data summary
In[27]:=
ResourceFunction["RecordsSummary"][data]
Out[27]=
,
,
1 column 1 | ||||||||||||
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2 column 2 | ||||||||||||
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3 column 3 | ||||||||||||
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Compute Quantile regression for probability 0.5 using a grid of NURBS basis functions
7×4
In[28]:=
AbsoluteTimingfuncs=
[data,{7,4},0.5];
 | QuantileRegression |
Out[28]=
{0.354104,Null}
Plot the obtained function and data
In[29]:=
QRPlot[data,0.5First[funcs]]
Out[29]=
 |
|
Quantile regression for probabilities 0.1 and 0.9
In[30]:=
probs={0.1,0.9};AbsoluteTimingfuncs=AssociationThreadprobs,
[data,{7,4},probs];
 | QuantileRegression |
Out[31]=
{0.60699,Null}
3D list plot of the data
In[32]:=
QRPlot[data,funcs]
Out[32]=
Count the number of points under each surface
In[33]:=
sepPoints=Map[Function[{f},Length@Select[data,#〚-1〛<First[Apply[f,Most[#]]]&]],funcs]
Out[33]=
0.1119,0.91080
Show the corresponding fractions (should correspond to the probabilities)
In[34]:=
sepFractions=N[sepPoints/Length[data]]
Out[34]=
0.10.0991667,0.90.9
Square sombrero
Generate random, "square sombrero" data
3D list plot of the data
Data summary
Quantile regression for probability 0.9
Plot functions and data
Pyramid basis
In this section we demonstrate the use of custom made basis functions.
Define a pyramid basis function
Here is a plot of a basis function
Make a basis for the sombrero data
Find regression quantiles for probability 0.2
Plot data and regression quantiles together
Partial sombrero
Generate random, "partial sombrero" data
3D list plot of the data
Regression quantiles
Plot data and regression quantiles together
Create a region object for the convex hull of the data points
The plot data and regression quantiles over the region above
Count the number of points under each surface
Show the corresponding fractions (should correspond to the probabilities)