Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantile regression over weather time series
Introduction
In this notebook we show two primary use cases of for Quantile Regression (QR):
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Fitting regression quantiles over
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Estimation of conditional density distributions
Quantile regression over heteroscedastic data
Load the paclet
In[4]:=
Needs["AntonAntonov`QuantileRegression`"]
Take temperature data
In[5]:=
tsTemp=WeatherData[{"Atlanta","Georgia"},"Temperature",{{2018,1,1},{2023,1,1},"Day"}]
Out[5]=
TimeSeries
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Plot time series points
In[6]:=
opts={ImageSizeLarge,AspectRatio1/2,PlotTheme"Detailed",JoinedFalse};DateListPlot[tsTemp,opts]
Out[6]=
Data summary
In[7]:=
ResourceFunction["RecordsSummary"][tsTemp["Path"]]
Out[7]=
,
1 column 1 | ||||||||||||
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2 column 2 | ||||||||||||||
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Find regression quantiles for 0.1, 0.5, 0.9
In[8]:=
AbsoluteTimingprobs={0.1,0.5,0.9};qFuncs=
[QuantityMagnitude[tsTemp["Path"]],16,probs];
 | QuantileRegression |
Out[8]=
{1.0097,Null}
Plot time series points with fitted regression quantiles
In[9]:=
DateListPlot[{tsTemp["Path"],Map[Function[{f},{#,f[#]}&/@tsTemp["Times"]],qFuncs]},opts,Joined{False,True,True,True},PlotLegends{"data",Sequence@@probs}]
Out[9]=
Remark: It should be obvious from the plot above that the time series data is heteroscedastic.
Count the number points under each surface
In[10]:=
sepPoints=Association@MapThread[Function[{p,f},pLength[Select[QuantityMagnitude[tsTemp["Path"]],#〚2〛<f[#〚1〛]&]]],{probs,qFuncs}]
Out[10]=
0.1182,0.5915,0.91647
Show the corresponding fractions (should correspond to the probabilities)
In[11]:=
sepFractions=N[sepPoints/Length[tsTemp["Path"]]]
Out[11]=
0.10.0996169,0.50.500821,0.90.901478
Estimation of conditional density distributions
Find regression quantiles for a "comprehensive" set of probabilities
In[12]:=
AbsoluteTimingprobs=Sort[Join[Range[0.1,0.9,0.1],{0.01,0.99}]];qFuncs=
[QuantityMagnitude[tsTemp["Path"]],16,probs];
| QuantileRegression |
Out[12]=
{2.31692,Null}
Plot time series points with fitted regression quantiles
In[13]:=
DateListPlot[{tsTemp["Path"],Map[Function[{f},{#,f[#]}&/@tsTemp["Times"]],qFuncs]},opts,Joined{False,True,True,True},PlotLegends{"data",Sequence@@probs}]
Out[13]=
Reconstruct the CDF distribution at the focus point (January 1st, 2020)
In[14]:=
focusPoint=AbsoluteTime[{2020,1,1}];xs=Through[qFuncs[focusPoint]];cdfPairs=Transpose[{xs,probs}];
Plot the empirical CDF
In[17]:=
ListLinePlotcdfPairs,
Out[17]=
In order to plot the corresponding PDF function define a CDF reconstruction function
In[18]:=
CDFEstimate[t0_]:=CDFEstimate[probs,qFuncs,t0];CDFEstimate[qs_,qFuncs_,t0_]:=Interpolation[Transpose[{Through[qFuncs[t0]],qs}],InterpolationOrder1];
Plot empirical PDF