An Algorithm for Optimally Fitting a Wiener Model

Algorithm 4

Nelder-Mead algorithm used. (Taken from [6, 12]).

Choose 𝑁= number of iterations to be run based on starting value of 𝑅² and 𝑖=1.

Generate a simplex around the starting parameters 𝜃̂; denote them (𝜃̃₁,𝜃̃₂,…,𝜃̃_𝑛+1).

While 𝑖≤𝑁

 Reorder the points of the simplex such that 𝐹(𝐱|𝜃̃₁)≤𝐹(𝐱|𝜃̃₂)≤⋯≤𝐹(𝐱|𝜃̃_𝑛+1).

 𝜃=∑^𝑛𝑖=1𝜃̃_𝑖

 𝜃̃^∗=2𝜃−𝜃̃_𝑛+1

 if  𝐹(𝐱∣𝜃̃₁)≤𝐹(𝐱∣𝜃̃^∗)<𝐹(𝐱∣𝜃̃_𝑛)  then

  𝜃̃_𝑛+1=𝜃̃^∗

  𝑖=𝑖+1; next

 else if  𝐹(𝐱∣𝜃̃^∗)<𝐹(𝐱∣𝜃̃₁) then

  𝜃̃^∗∗=3𝜃−2𝜃̃_𝑛+1

  if  𝐹(𝐱∣𝜃̃^∗∗)<𝐹(𝐱∣𝜃̃^∗)  then

   𝜃̃_𝑛+1=𝜃̃^∗∗

  else

   𝜃̃_𝑛+1=𝜃̃^∗

  end if

  𝑖=𝑖+1; next

 else

  if  𝐹(𝐱∣𝜃̃_𝑛)≤𝐹(𝐱∣𝜃̃^∗)<𝐹(𝐱∣𝜃̃_𝑛+1)  then

   𝜃̃^∗∗=32𝜃−12𝜃̃^𝑛+1

   if  𝐹(𝐱∣𝜃̃^∗∗≤𝐹(𝐱∣𝜃̃^∗)  then

    𝜃̃_𝑛+1=𝜃̃^∗∗

    𝑖=𝑖+1; next

   end if

  else

   𝜃̃^∗∗=12𝜃+12𝜃̃^𝑛+1

   if  𝐹(𝐱∣𝜃̃^∗∗<𝐹(𝐱∣𝜃̃_𝑛+1)  then

    𝜃̃_𝑛+1=𝜃̃^∗∗

    𝑖=𝑖+1; next

   end if

  end if

  for  𝑖=2,3,…,𝑛+1  do

   𝜃̃_𝑖=12(𝜃̃¹+𝜃̃_𝑖)

  end for

  𝑖=𝑖+1

  end if

end while