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Provides functions for fast matrix inversion, creation of empirical CDF from sample data including handling of asymmetric errors, and fitting to a funtion using chi-squared. The fitting procedure return the full covariance matrix describing the fitted parameters.

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chi2fit lib distributions frechet.ex
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lib/distributions/frechet.ex 

defmodule Distribution.Frechet do



  # Copyright 2019 Pieter Rijken

  #

  # Licensed under the Apache License, Version 2.0 (the "License");

  # you may not use this file except in compliance with the License.

  # You may obtain a copy of the License at

  #

  #     http://www.apache.org/licenses/LICENSE-2.0

  #

  # Unless required by applicable law or agreed to in writing, software

  # distributed under the License is distributed on an "AS IS" BASIS,

  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.

  # See the License for the specific language governing permissions and

  # limitations under the License.



  @moduledoc """

  The Fréchet distribution, also known inverse Weibull distribution.

  """



  defstruct [:pars]

  

  @type t() :: %__MODULE__{

    pars: [number()] | nil

  }



end



defimpl Distribution, for: Distribution.Frechet do

  import Distribution.Frechet

  alias Distribution.Frechet



  import Exboost.Math, only: [tgamma: 1]



  @spec frechet(scale::number(),shape::number()) :: ((...) -> number)

  defp frechet(scale,shape) when is_number(scale) and is_number(shape) do

   fn ->

     u = :rand.uniform()

     scale * :math.pow(-:math.log(u),-1.0/shape)

   end

  end



  @spec frechetCDF(scale :: float,shape :: float) :: (number -> number)

  defp frechetCDF(scale,shape) when scale>0 and shape>0 do

    fn

      x when x==0.0 ->

        0.0

      x ->

        :math.exp(-:math.pow(x/scale,-shape))

    end

  end

  defp frechetCDF(_scale,_shape), do: raise ArithmeticError, "Fréchet is only defined for positive scale and shape"



  def skewness(%Frechet{pars: nil}) do

    fn [_scale,shape] ->

      g1 = tgamma(1.0-1.0/shape)

      g2 = tgamma(1.0-2.0/shape)

      g3 = tgamma(1.0-3.0/shape)

      (g3 - 3*g2*g1 + 2*g1*g1*g1)/:math.pow(g2 - g1*g1,1.5)

    end

  end

  def kurtosis(%Frechet{pars: nil}) do

    fn [_scale,shape] ->

      g1 = tgamma(1.0-1.0/shape)

      g2 = tgamma(1.0-2.0/shape)

      g3 = tgamma(1.0-3.0/shape)

      g4 = tgamma(1.0-4.0/shape)

      -6 + (g4 - 4*g3*g1 + 3*g2*g2)/:math.pow(g2 - g1*g1,2.0)

    end

  end

  def size(%Frechet{}), do: 2

  

  def cdf(%Frechet{pars: nil}), do: fn x, [scale,shape] -> frechetCDF(scale,shape).(x) end



  def pdf(%Frechet{pars: nil}), do: raise Distribution.FunctionNotSupportedError, message: "pdf is not supported for the Frechet distribution"

  def random(%Frechet{pars: [scale,shape]}), do: frechet(scale, shape).()

end