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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 wald.ex
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lib/distributions/wald.ex 

defmodule Distribution.Wald 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 """

  Wald or Inverse Gauss distribution.

  """



  defstruct [:pars]

  

  @type t() :: %__MODULE__{

    pars: [number()] | nil,

  }



end



defimpl Distribution, for: Distribution.Wald do

  import Distribution.Wald

  alias Distribution.Wald



  @spec phi(x :: float) :: float

  defp phi(x) do

    d = %Distribution.Normal{pars: [0.0, 1.0]}

    Distribution.cdf(d).(x)

  end



  @spec wald(mu::number(),lambda::number()) :: ((...) -> number)

  defp wald(mu,lambda) when is_number(mu) and is_number(lambda) do

    d = %Distribution.Normal{pars: [0.0,1.0]}

    fn () ->

      w = Distribution.random(d)

      y = w*w

      x = mu + mu*mu*y/2/lambda - mu/2/lambda*:math.sqrt(4*mu*lambda*y + mu*mu*y*y)



      z = :rand.uniform()

      if z <= mu/(mu+x), do: x, else: mu*mu/x

    end

  end

  defp wald([avg: average],lambda), do: wald(average,lambda)



  @spec waldCDF(number,number) :: (number -> number)

  defp waldCDF(mu,_) when mu < 0, do: raise ArithmeticError, "Wald is only defined for positive average"

  defp waldCDF(_,lambda) when lambda < 0, do: raise ArithmeticError, "Wald is only defined for positive shape"

  defp waldCDF(mu,lambda) do

    fn

      x when x == 0 -> 0.0

      x when x < 0 -> 0.0

      x when x > 0 ->

        phi(:math.sqrt(lambda/x) * (x/mu-1.0)) + :math.exp(2.0*lambda/mu) * phi(-:math.sqrt(lambda/x) * (x/mu+1.0))

    end

  end



  def skewness(%Wald{pars: nil}), do: fn [k,lambda] -> 3*:math.sqrt(k/lambda) end

  def kurtosis(%Wald{pars: nil}), do: fn [k,lambda] -> 15*k/lambda end

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

  def cdf(%Wald{pars: nil}), do: fn x,[k,lambda] -> waldCDF(k,lambda).(x) end

  def pdf(%Wald{pars: nil}), do: fn x, [mu,lambda] -> :math.sqrt(lambda/2/:math.pi/x/x/x) * :math.exp( -lambda*(x-mu)*(x-mu)/2/x/mu/mu ) end

  def random(%Wald{pars: [k,lambda]}), do: wald(k,lambda).()

  def random(%Wald{pars: nil}), do: fn [k,lambda] -> wald(k,lambda).() end



end