An Algorithm for Optimally Fitting a Wiener Model

Algorithm 5

BFGS algorithm used.

Let 𝑛= number of iterations to be run, 𝜎=1×10⁻¹², and 𝐵⁻¹=𝐼.

Choose 𝛿 based on current value of 𝑅² and subset of parameters to be fit.

for  𝑖=1 to 𝑛  do

 if  𝑖>1  then

  𝑡=𝐽′𝐽ℎ+(𝐽−𝐽^∗)′𝑓

  𝐵⁻¹=𝐵⁻¹+ℎ′𝑡+𝑡′𝐵⁻¹𝑡𝑡′ℎℎ′𝑡ℎℎ′−𝐵⁻¹𝑡ℎ′+ℎ𝑡′𝐵⁻¹ℎ′𝑡

 end if

 𝐹^∗=𝐽′𝐟

 ℎ=−𝐵⁻¹𝐹^∗

 𝑑=√ℎ′ℎ

 if  𝑑>𝛿  then

  ℎ=𝛿𝑑²ℎ

  𝑑=𝛿

 end if

 𝜃̂=𝜃̂+ℎ

 𝜌=𝐹(𝐱∣𝜃̂−ℎ)−𝐹(𝐱∣𝜃̂)−ℎ′𝐹^∗−(1/2)‖𝐽ℎ‖²

 if  𝜌<.25  then

  𝛿=𝛿2

 else

  if  𝜌>.75  then

   𝛿=max(𝛿,3𝑑)

  end if

 end if

  𝐽^∗=𝐽

end for