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# stucchio/mle_compute_z.jl

Last active August 29, 2015 14:01
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Maximum likelihood computation of ad probability
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 #Pkg.add("Optim") using Optim function logLikelihood(z::Float64, clicks::Array{Float64,1}, shows::Array{Float64,1}, alpha::Array{Float64,1}) @assert size(clicks) == size(shows) @assert size(shows) == size(alpha) az = z * alpha return sum(clicks .* log(az) .+ (shows .- clicks) .* log(1-az)) end function derivLogLikelihood(z::Float64, clicks::Array{Float64,1}, shows::Array{Float64,1}, alpha::Array{Float64,1}) @assert size(clicks) == size(shows) @assert size(shows) == size(alpha) az = z*alpha return sum((clicks / z) .- alpha .* (shows .- clicks) ./ (1 .- (alpha*z))) end function adQuality(clicks::Array{Int64,1}, shows::Array{Int64,1}, alpha::Array{Float64,1}) return adQuality(map( x -> convert(Float64, x), clicks), map(x -> convert(Float64, x), shows), alpha) end function adQuality(clicks::Array{Float64,1}, shows::Array{Float64,1}, alpha::Array{Float64,1}) function yToZ(y::Float64) return 0.5+atan(y)/pi end function dzdy(y::Float64) return (1.0/pi)/(1+y*y) end function zToY(z::Float64) return tan(pi*z-(pi/2.0)) end function f(y::Array{Float64,1}) #Uses transformed variables return -1*logLikelihood(yToZ(y[1]), clicks, shows, alpha) end function df!(y::Array{Float64,1}, storage::Array{Float64,1}) storage[1] = -1 * derivLogLikelihood(yToZ(y[1]), clicks, shows, alpha) * dzdy(y[1]) end result = optimize(f, df!, [0.0], method = :gradient_descent) return yToZ(result.minimum[1]) end alpha = [1.0, 1.0, 1.0, 1.0] clicks = [25,25,25,25] shows = [100,100,100,100] println("MLE estimated probability: ", string(adQuality(clicks, shows, alpha)))
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 #Pkg.add("Optim") using Optim function logLikelihood(z::Array{Float64,1}, clicks::Array{Int64,2}, shows::Array{Int64,2}, alpha::Array{Float64,1}) M, N = size(clicks) @assert size(z) == (M,) @assert size(shows) == (M,N) @assert size(alpha) == (N,) result = 0.0 for i=1:N for j=1:M result += clicks[j,i] * log(alpha[i]*z[j]) + (shows[j,i]-clicks[j,i])*log(1-alpha[i]*z[j]) end end return result end function betaToAlpha(beta::Array{Float64,1}) result = Array(Float64, size(beta)[1] + 1) result[1] = 1.0 result[2:] = cumprod(beta) return result end function gradBetaLogLikelihood(z::Array{Float64,1}, clicks::Array{Int64,2}, shows::Array{Int64,2}, beta::Array{Float64,1}) alpha = betaToAlpha(beta) M, N = size(clicks) @assert size(z) == (M,) @assert size(alpha) == (N,) result = Array(Float64, N-1) for k=2:N for i=k:N for j=1:M result[k-1] += clicks[j,i] / beta[k-1] - (shows[j,i] - clicks[j,i]) * z[j] * (alpha[i] / beta[k-1]) / (1 - alpha[i]*z[j]) end end end return result end function gradZLogLikelihood(z::Array{Float64,1}, clicks::Array{Int64,2}, shows::Array{Int64,2}, alpha::Array{Float64,1}) M, N = size(clicks) result = Array(Float64, M) for i=1:N result += ( (slice(clicks, :, i) ./ z) .- (((slice(shows, :, i) .- slice(clicks, :, i)) .* alpha[i]) ./ (1 .- alpha[i] * z)) ) end return result end function computeAlphaZ(clicks::Array{Int64,2}, shows::Array{Int64,2}) M, N = size(clicks) function yToZ(y::Float64) return 0.5+atan(y)/pi end function dzdy(y::Float64) return (1.0/pi)/(1+y*y) end function zToY(z::Float64) return tan(pi*z-(pi/2.0)) end # Here the y-variable is an M+N-dimensional Array{Float64,1} the first N represent transformed # alpha, the remainder represent transformed z function f(y::Array{Float64,1}) z = map(yToZ, y[1:M]) beta = map(yToZ, y[M+1:M+N-1]) alpha = betaToAlpha(beta) return -1*logLikelihood(z, clicks, shows, alpha) end function df!(y::Array{Float64,1}, storage::Array{Float64,1}) z = map(yToZ, y[1:M]) beta = map(yToZ, y[M+1:M+N-1]) alpha = betaToAlpha(beta) storage[1:M] = -1*gradZLogLikelihood(z, clicks, shows, alpha) .* map(dzdy, y[1:M]) gb = gradBetaLogLikelihood(z, clicks, shows, beta) storage[M+1:M+N-1] = -1*gradBetaLogLikelihood(z, clicks, shows, beta) .* map( dzdy, y[M+1:M+N-1]) end init = Array(Float64, M+N-1) init[:] = 0.0 result = optimize(f, df!, init, method = :gradient_descent) resultUntransformed = map( yToZ, result.minimum) alpha = betaToAlpha(resultUntransformed[M+1:M+N-1]) z = resultUntransformed[1:M] return (alpha, z) end clicks = [ 25 13 10; 20 9 4; 10 4 1; 12 4 0] shows = [ 100 105 97; 99 103 96; 102 100 101; 103 101 107] println(computeAlphaZ(clicks, shows))
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