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# bdeonovic/example.jl

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Conditional Maximum Likelihood for Rasch Model
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 using Optim using DelimitedFiles using ForwardDiff #data X = readdlm("lsat.txt"); n, m = size(X) r = vec(mapslices(sum, X, dims=2)) s = vec(mapslices(sum, X, dims=1)) f = [sum(r.==j) for j in 1:m] #auxiliary functions needed function esf_sum!(S::AbstractArray{T,1}, x::AbstractArray{T,1}) where T <: Real n = length(x) fill!(S,zero(T)) S[1] = one(T) @inbounds for col in 1:n for r in 1:col row = col - r + 1 S[row+1] = S[row+1] + x[col] * S[row] end end end function esf_ext!(S::AbstractArray{T,1}, H::AbstractArray{T,3}, x::AbstractArray{T,1}) where T <: Real n = length(x) esf_sum!(S, x) H[:,:,1] .= zero(T) H[:,:,2] .= one(T) @inbounds for i in 3:n+1 for j in 1:n H[j,j,i] = S[i-1] - x[j] * H[j,j,i-1] for k in j+1:n H[k,j,i] = S[i-1] - ((x[j]+x[k])*H[k,j,i-1] + x[j]*x[k]*H[k,j,i-2]) H[j,k,i] = H[k,j,i] end end end end ##Newton Raphson esf_input = ones(Float64, m) beta = zeros(Float64, m) last_beta = fill(NaN, m) S = zeros(Float64, m+1) H = zeros(Float64, m, m, m+1) function calculate_common!(x, last_x) if x != last_x copyto!(last_x, x) beta .= x esf_input .= exp.(-beta) esf_ext!(S, H, esf_input) end end function logLC(beta) calculate_common!(beta, last_beta) return -s'log.(esf_input) + f'log.(S[2:end]) end function g!(storage, beta) calculate_common!(beta, last_beta) for j in 1:m storage[j] = s[j] for l in 1:m storage[j] -= esf_input[j] * f[l] * (H[j,j,l+1] / S[l+1]) end end end function h!(storage, beta) calculate_common!(beta, last_beta) for j in 1:m for k in 1:m storage[k,j] = 0.0 for l in 1:m if j == k storage[j,j] += f[l] * (esf_input[j]*H[j,j,l+1] / S[l+1]) * (1 - esf_input[j]*H[j,j,l+1] / S[l+1]) elseif k > j storage[k,j] += esf_input[j] * esf_input[k] * f[l] * ((H[k,j,l] / S[l+1]) - (H[j,j,l+1] * H[k,k,l+1]) / S[l+1] ^ 2) else #k < j storage[k,j] += esf_input[j] * esf_input[k] * f[l] * ((H[j,k,l] / S[l+1]) - (H[j,j,l+1] * H[k,k,l+1]) / S[l+1] ^ 2) end end end end end con_c!(c, x) = (c[1] = sum(x); c) function con_jacobian!(J, x) J[1,:] .= ones(length(x)) end function con_h!(h, x, λ) for i in 1:size(h)[1] for j in 1:size(h)[2] h[i,j] += (i == j) ? λ[1] : 0.0 end end end lx = Float64[]; ux = Float64[] lc = [0.0]; uc = [0.0] df = TwiceDifferentiable(logLC, g!, h!, zeros(Float64, m)) dfc = TwiceDifferentiableConstraints(con_c!, con_jacobian!, con_h!, lx, ux, lc, uc) res = optimize(df, dfc, zeros(Float64,m), IPNewton()) #LBFGS with autodiff struct DiffCache{T<:AbstractArray, S<:AbstractArray} du::T dual_du::S end function DiffCache(T, size, ::Type{Val{chunk_size}}) where chunk_size DiffCache(zeros(T, size...), zeros(ForwardDiff.Dual{nothing,T,chunk_size}, size...)) end DiffCache(u::AbstractArray) = DiffCache(eltype(u), size(u), Val{ForwardDiff.pickchunksize(length(u))}) get_cache(dc::DiffCache, ::Type{T}) where T<: ForwardDiff.Dual = dc.dual_du get_cache(dc::DiffCache, T) = dc.du beta_0 = zeros(Float64, m-1) #cache for ESF values z_cache1 = DiffCache(Float64, (m+1,), Val{ForwardDiff.pickchunksize(m-1)}) z_cache2 = DiffCache(Float64, (m,), Val{ForwardDiff.pickchunksize(m-1)}) function logLC(beta) esf_input = get_cache(z_cache2, eltype(beta)) esf_input_typed = reinterpret(eltype(beta[1]), esf_input) esf_input_typed[1] = exp(sum(beta)) esf_input_typed[2:end] .= exp.(-beta) #get correctly typed cache value and fix tag esf = get_cache(z_cache1, eltype(beta)) esf_typed = reinterpret(eltype(beta[1]), esf) #in-place ESF esf_sum!(esf_typed, esf_input_typed) return -s'log.(esf_input_typed) + f'log.(esf_typed[2:end]) end df = OnceDifferentiable(logLC, beta_0; autodiff = :forward) res = Optim.optimize(df, beta_0)
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