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Modeling wind speed distributions with machine learning

Vitaliy Kaurov, WOLFRAM Research
Posted 10 years ago

What is distribution of wind speed magnitudes at a given geographic location? You could approach this manually like the authors of this paper:

Mixture probability distribution functions to model wind speed distributions

where main conclusion was:

Results show that mixture probability functions are better alternatives to conventional Weibull, two-component mixture Weibull, gamma, and lognormal PDFs to describe wind speed characteristics.

Or you can use Machine Learning and new function FindDistribution. Let's first get a sample of data, say for Boston for recent 5 years:

windBOSTON = WeatherData["Boston", "WindSpeed", {{2010}, {2015}, "Day"}];    
DateListPlot[windBOSTON]

enter image description here

Now get the magnitudes and apply FindDistribution

mags = QuantityMagnitude[windBOSTON["Values"]];
dis = FindDistribution[mags]

which gives, guess what, a MixtureDistribution :

MixtureDistribution[{0.711353, 0.288647}, 
{NormalDistribution[12.8117, 4.74919], LogNormalDistribution[3.06178, 0.308954]}]

Visualizing model versus experimental data looks neat!

Show[
 Histogram[mags, Automatic, "ProbabilityDensity", PlotTheme -> "Detailed"],
 Plot[PDF[dis, x], {x, 0, 50}, PlotRange -> All]]

enter image description here

Try playing with other locations and see what distributions you get. Not always we will get a MixtureDistribution, wind data at different locations can be quite different.

POSTED BY: Vitaliy Kaurov
6 Replies
Sort By:

Hi,

me again... In the paper cited in the first post they study 4 sites/stations, I believe, and they use shorter time series than we do here. As mentioned in my previous post, I want to show the analysis for Europe.

citiesEurope = Flatten[CountryData[#, "LargestCities"] & /@ EntityList[EntityClass["Country", "Europe"]]];
dataEurope = 
  ParallelTable[{citiesEurope[[i]], citiesEurope[[i]]["Coordinates"], FindDistribution[Select[QuantityMagnitude[WeatherData[citiesEurope[[i]], 
 "WindSpeed", {{2004, 1, 1}, Date[], "Day"}]["Values"]], NumberQ]]}, {i, 2, Length[citiesEurope]}];

These are the distributions we find:

Tally[Head /@ (DeleteCases[dataEurope[[All, -1]], _FindDistribution])]
(*{{MixtureDistribution, 1251}, {ExtremeValueDistribution, 
  1446}, {FrechetDistribution, 58}, {InverseGaussianDistribution, 
  91}, {LogNormalDistribution, 224}, {GammaDistribution, 
  493}, {ChiSquareDistribution, 142}, {MaxwellDistribution, 
  75}, {WeibullDistribution, 3}, {LogisticDistribution, 6}}*)

The bar chart representation as above can be calculated like so:

BarChart[Apply[Labeled, 
  Reverse[Reverse@SortBy[{{MixtureDistribution, 1251}, {ExtremeValueDistribution, 1446}, {FrechetDistribution, 58}, {InverseGaussianDistribution,91}, {LogNormalDistribution, 224}, {GammaDistribution, 493}, {ChiSquareDistribution, 142}, {MaxwellDistribution, 75}, {WeibullDistribution, 3}, {LogisticDistribution, 6}}, Last], 2], {1}]]

enter image description here

Separating the MixtureDistributions gives:

Reverse@SortBy[Tally[If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ DeleteCases[dataEurope[[All, -1]], _FindDistribution]], Last]
(*{{ExtremeValueDistribution, 1446}, {GammaDistribution, 493}, {{NormalDistribution, LogNormalDistribution}, 388}, {{GammaDistribution, LogNormalDistribution}, 322}, {{NormalDistribution, GammaDistribution}, 292}, {LogNormalDistribution, 224}, {ChiSquareDistribution,142}, {InverseGaussianDistribution, 91}, {MaxwellDistribution, 75}, {{GammaDistribution, GammaDistribution}, 59}, {FrechetDistribution, 58}, {{LogNormalDistribution, LogNormalDistribution}, 46}, {{LogisticDistribution, LogNormalDistribution}, 46}, {{NormalDistribution, NormalDistribution}, 37}, {{MaxwellDistribution, GammaDistribution}, 18}, {{MaxwellDistribution, LogNormalDistribution}, 17}, {{LogisticDistribution, GammaDistribution}, 17}, {{LogNormalDistribution, GammaDistribution}, 7}, {LogisticDistribution, 6}, {WeibullDistribution, 3}, {{GammaDistribution, NormalDistribution, GammaDistribution}, 1}, {{GammaDistribution, GammaDistribution, GammaDistribution}, 1}}*)

Here is the BarChart:

BarChart[Apply[Labeled, 
  Reverse[{Rotate[#[[1]], Pi/2], #[[2]]} & /@ Reverse@SortBy[Tally[If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ 
  DeleteCases[dataEurope[[All, -1]], _FindDistribution]], Last],2], {1}]]

enter image description here

As before we can attach values to the different distributions:

rules = MapThread[Rule, {Reverse@SortBy[Tally[If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ 
        DeleteCases[dataEurope[[All, -1]], _FindDistribution]], Last][[All, 1]], Range[Length[Reverse@SortBy[Tally[If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ DeleteCases[dataEurope[[All, -1]], _FindDistribution]], Last]]]}]

This is the corresponding plot:

GeoRegionValuePlot[#[[1]] -> #[[2]] & /@ (Transpose[{Select[dataEurope, ! (Head[#[[3]]] === FindDistribution) &][[All, 1]], If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ DeleteCases[dataEurope[[All, -1]], _FindDistribution]}] /. rules), ColorFunction -> ColorData["Rainbow"]]

enter image description here

Note that there are too many red dots - they should represent the rare distributions and there should be few. This can be fixed by setting the PlotRange like so:

GeoRegionValuePlot[#[[1]] -> #[[2]] & /@ (Transpose[{Select[dataEurope, ! (Head[#[[3]]] === FindDistribution) &][[All, 1]], If[Head[#] === MixtureDistribution, Head /@ #[[2]], Head[#]] & /@ DeleteCases[dataEurope[[All, -1]], _FindDistribution]}] /. rules), ColorFunction -> ColorData["Rainbow"], PlotRange -> {-0.5, 24}]

enter image description here

This is obviously still very naïve, but it appears that the "distributions are not randomly distributed".

Cheers,

M.
POSTED BY: Marco Thiel

This is amazing idea, @Marco, it makes more sense now. Thanks for sharing! I find it curious, that the Weibull distribution, a popular model for wind, is quite rare and never enters MixtureDistribution. Perhaps because it works better for hourly/ten-minute wind speeds sampling, or at least this is what I understood.

POSTED BY: Vitaliy Kaurov

Nice work! I still think that there might be a seasonal dependence as well in the distributions.
POSTED BY: Kay Herbert

Interesting, but I can't duplicate your answer on my version 10.4:

In[59]:= mags = QuantityMagnitude[windBOSTON["Values"]];
dis = FindDistribution[mags, 2]

Out[60]= {ExtremeValueDistribution[12.6967, 5.66123], 
 MixtureDistribution[{0.680313, 
   0.319687}, {NormalDistribution[13.118, 4.54959], 
   GammaDistribution[7.80677, 2.78237]}]}

or

In[65]:= mags = QuantityMagnitude[windBOSTON["Values"]];
dis = FindDistribution[mags]

Out[66]= ExtremeValueDistribution[12.6967, 5.66123]

on the same data.

I actually was wondering whether the mixture distribution is a consequence of different weather patterns. Like here in Boston we typically have either warm weather coming out the S to SW or the jet stream dipping down from the W to NW. If so, then one distribution would be more prevalent in the winter and another in the summer.
POSTED BY: Kay Herbert
POSTED BY: Vitaliy Kaurov
POSTED BY: Marco Thiel
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