+#' Optimize parameters
+#'
+#' Optimize the parameters of a mixture of logistic regressions model, possibly using
+#' \code{mu <- computeMu(...)} as a partial starting point.
+#'
+#' @param K Number of populations.
+#' @param link The link type, 'logit' or 'probit'.
+#' @param optargs a list with optional arguments:
+#' \itemize{
+#' \item 'M' : list of moments of order 1,2,3: will be computed if not provided.
+#' \item 'X,Y' : input/output, mandatory if moments not given
+#' \item 'exact': use exact formulas when available?
+#' }
+#'
+#' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
+#' outputs the list of optimized parameters
+#' \itemize{
+#' \item p: proportions, size K
+#' \item β: regression matrix, size dxK
+#' \item b: intercepts, size K
+#' }
+#' x0 is a vector containing respectively the K-1 first elements of p, then β by
+#' columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
+#'
+#' @seealso \code{multiRun} to estimate statistics based on β, and
+#' \code{generateSampleIO} for I/O random generation.
+#'
+#' @examples
+#' # Optimize parameters from estimated μ
+#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
+#' μ = computeMu(io$X, io$Y, list(K=2))
+#' M <- computeMoments(io$X, io$Y)
+#' o <- optimParams(2, "logit", list(M=M))
+#' x0 <- c(1/2, as.double(μ), c(0,0))
+#' par0 <- o$run(x0)
+#' # Compare with another starting point
+#' x1 <- c(1/2, 2*as.double(μ), c(0,0))
+#' par1 <- o$run(x1)
+#' o$f( o$linArgs(par0) )
+#' o$f( o$linArgs(par1) )
+#' @export
+optimParams = function(K, link=c("logit","probit"), optargs=list())
+{
+ # Check arguments
+ link <- match.arg(link)
+ if (!is.list(optargs))
+ stop("optargs: list")
+ if (!is.numeric(K) || K < 2)
+ stop("K: integer >= 2")
+
+ M <- optargs$M
+ if (is.null(M))
+ {
+ if (is.null(optargs$X) || is.null(optargs$Y))
+ stop("If moments are not provided, X and Y are required")
+ M <- computeMoments(optargs$X,optargs$Y)
+ }
+
+ # TODO: field?!
+ exactComp <<- optargs$exact
+ if (is.null(exactComp))
+ exactComp <<- FALSE
+
+ # Build and return optimization algorithm object
+ methods::new("OptimParams", "li"=link, "M1"=as.double(M[[1]]),
+ "M2"=as.double(M[[2]]), "M3"=as.double(M[[3]]), "K"=as.integer(K))
+}
+
+# Encapsulated optimization for p (proportions), β and b (regression parameters)
+#
+# @field li Link, 'logit' or 'probit'
+# @field M1 Estimated first-order moment
+# @field M2 Estimated second-order moment (flattened)
+# @field M3 Estimated third-order moment (flattened)
+# @field K Number of populations
+# @field d Number of dimensions
+#
+setRefClass(
+ Class = "OptimParams",
+
+ fields = list(
+ # Inputs
+ li = "character", #link 'logit' or 'probit'
+ M1 = "numeric", #order-1 moment (vector size d)
+ M2 = "numeric", #M2 easier to process as a vector
+ M3 = "numeric", #M3 easier to process as a vector
+ # Dimensions
+ K = "integer",
+ d = "integer"
+ ),
+
+ methods = list(
+ initialize = function(...)
+ {
+ "Check args and initialize K, d"
+
+ callSuper(...)
+ if (!hasArg("li") || !hasArg("M1") || !hasArg("M2") || !hasArg("M3")
+ || !hasArg("K"))
+ {
+ stop("Missing arguments")
+ }
+
+ d <<- length(M1)
+ },
+
+ expArgs = function(x)
+ {
+ "Expand individual arguments from vector x"
+
+ list(
+ # p: dimension K-1, need to be completed
+ "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
+ "β" = matrix(x[K:(K+d*K-1)], ncol=K),
+ "b" = x[(K+d*K):(K+(d+1)*K-1)])
+ },
+
+ linArgs = function(o)
+ {
+ " Linearize vectors+matrices into a vector x"
+
+ c(o$p[1:(K-1)], as.double(o$β), o$b)
+ },
+
+ f = function(x)
+ {
+ "Sum of squares (Mi - hat_Mi)^2 where Mi is obtained from formula"
+
+ P <- expArgs(x)
+ p <- P$p
+ β <- P$β
+ λ <- sqrt(colSums(β^2))
+ b <- P$b
+
+ # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
+ β2 <- apply(β, 2, function(col) col %o% col)
+ β3 <- apply(β, 2, function(col) col %o% col %o% col)
+
+ return(
+ sum( ( β %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
+ sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
+ sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
+ },
+
+ grad_f = function(x)
+ {
+ "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
+
+ P <- expArgs(x)
+ p <- P$p
+ β <- P$β
+ λ <- sqrt(colSums(β^2))
+ μ <- sweep(β, 2, λ, '/')
+ b <- P$b
+
+ # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
+ β2 <- apply(β, 2, function(col) col %o% col)
+ β3 <- apply(β, 2, function(col) col %o% col %o% col)
+
+ # Some precomputations
+ G1 = .G(li,1,λ,b)
+ G2 = .G(li,2,λ,b)
+ G3 = .G(li,3,λ,b)
+ G4 = .G(li,4,λ,b)
+ G5 = .G(li,5,λ,b)
+
+ # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
+ F1 = as.double( 2 * ( β %*% (p * G1) - M1 ) )
+ F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
+ F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
+
+ km1 = 1:(K-1)
+ grad <- #gradient on p
+ t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
+ t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
+ t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
+
+ grad_β <- matrix(nrow=d, ncol=K)
+ for (i in 1:d)
+ {
+ # i determines the derivated matrix dβ[2,3]
+
+ dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
+ dβ_right <- matrix(0, nrow=d, ncol=K)
+ block <- i
+ dβ_right[block,] <- dβ_right[block,] + 1
+ dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
+
+ dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
+ dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
+ sweep(dβ_right, 2, β[j,], '*')
+ }) )
+ block <- ((i-1)*d+1):(i*d)
+ dβ2_right[block,] <- dβ2_right[block,] + β
+ dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
+
+ dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
+ dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
+ sweep(dβ2_right, 2, β[j,], '*')
+ }) )
+ block <- ((i-1)*d*d+1):(i*d*d)
+ dβ3_right[block,] <- dβ3_right[block,] + β2
+ dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
+
+ grad_β[i,] <- t(dβ) %*% F1 + t(dβ2) %*% F2 + t(dβ3) %*% F3
+ }
+ grad <- c(grad, as.double(grad_β))
+
+ grad = c(grad, #gradient on b
+ t( sweep(β, 2, p * G2, '*') ) %*% F1 +
+ t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
+ t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
+
+ grad
+ },
+
+ run = function(x0)
+ {
+ "Run optimization from x0 with solver..."
+
+ if (!is.numeric(x0) || any(is.na(x0)) || length(x0) != (d+2)*K-1
+ || any(x0[1:(K-1)] < 0) || sum(x0[1:(K-1)]) > 1)
+ {
+ stop("x0: numeric vector, no NA, length (d+2)*K-1, sum(x0[1:(K-1) >= 0]) <= 1")
+ }
+
+ op_res = constrOptim( x0, .self$f, .self$grad_f,
+ ui=cbind(
+ rbind( rep(-1,K-1), diag(K-1) ),
+ matrix(0, nrow=K, ncol=(d+1)*K) ),
+ ci=c(-1,rep(0,K-1)) )
+
+ expArgs(op_res$par)
+ }
+ )
+)
+
+# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
+# = E[g^{(order)}(z)] with z~N(b,diag(λ))
+#
+# @param link Link, 'logit' or 'probit'
+# @param order Order of derivative
+# @param λ Norm of columns of β
+# @param b Intercept
+#
+.G <- function(link, order, λ, b)
+{
+ # NOTE: weird "integral divergent" error on inputs:
+ # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
+ # Switch to pracma package for that (but it seems slow...)
+
+ if (exactComp && link == "probit")
+ {
+ # Use exact computations
+ sapply( seq_along(λ), function(k) {
+ .exactProbitIntegral(order, λ[k], b[k])
+ })
+ }
+
+ else
+ {
+ # Numerical integration
+ sapply( seq_along(λ), function(k) {
+ res <- NULL
+ tryCatch({
+ # Fast code, may fail:
+ res <- stats::integrate(
+ function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
+ lower=-Inf, upper=Inf )$value
+ }, error = function(e) {
+ # Robust slow code, no fails observed:
+ sink("/dev/null") #pracma package has some useless printed outputs...
+ res <- pracma::integral(
+ function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
+ xmin=-Inf, xmax=Inf, method="Kronrod")
+ sink()
+ })
+ res
+ })
+ }
+}
+
+# TODO: check these computations (wrong atm)
+.exactProbitIntegral <- function(order, λ, b)
+{
+ c1 = (1/sqrt(2*pi)) * exp( -.5 * b/((λ^2+1)^2) )
+ if (order == 1)
+ return (c1)
+ c2 = b - λ^2 / (λ^2+1)
+ if (order == 2)
+ return (c1 * c2)
+ if (order == 3)
+ return (c1 * (λ^2 - 1 + c2^2))
+ if (order == 4)
+ return ( (c1*c2/((λ^2+1)^2)) * (-λ^4*((b+1)^2+1) -
+ 2*λ^3 + λ^2*(2-2*b*(b-1)) + 6*λ + 3 - b^2) )
+ if (order == 5) #only remaining case...
+ return ( c1 * (3*λ^4+c2^4+6*c1^2*(λ^2-1) - 6*λ^2 + 6) )
+}
+
+# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
+#
+.deriv <- list(
+ "probit"=list(
+ # 'probit' derivatives list;
+ # TODO: exact values for the integral E[g^(k)(λz+b)]
+ function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
+ function(x) exp(-x^2/2)/(sqrt(2*pi)) * -x, #g''
+ function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1), #g^(3)
+ function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x), #g^(4)
+ function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
+ ),
+ "logit"=list(
+ # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
+ # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
+ function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
+ function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
+ function(x) {e=exp(x); .zin(e*( e^2 - 4*e + 1) /(e+1)^4)}, #g^(3)
+ function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
+ function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)} #g^(5)
+ )
+)
+
+# Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
+#
+# @param x Ratio of polynoms of exponentials, as in .S[[i]]
+#
+.zin <- function(x)
+{
+ x[is.nan(x)] <- 0.
+ x
+}