-#' Optimize parameters
-#'
-#' Optimize the parameters of a mixture of logistic regressions model, possibly using
-#' \code{mu <- computeMu(...)} as a partial starting point.
+#' Wrapper function for OptimParams class
#'
#' @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?
-#' \item weights Weights on moments when minimizing sum of squares
-#' }
+#' @param X Data matrix of covariables
+#' @param Y Output as a binary vector
#'
#' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
#' outputs the list of optimized parameters
#' \item β: regression matrix, size dxK
#' \item b: intercepts, size K
#' }
-#' x0 is a vector containing respectively the K-1 first elements of p, then β by
-#' columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
+#' θ0 is a vector containing respectively the K-1 first elements of p, then β by
+#' columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
#'
#' @seealso \code{multiRun} to estimate statistics based on β, and
#' \code{generateSampleIO} for I/O random generation.
#' # Optimize parameters from estimated μ
#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
#' μ = computeMu(io$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)
+#' o <- optimParams(io$X, io$Y, 2, "logit")
+#' θ0 <- list(p=1/2, β=μ, b=c(0,0))
+#' par0 <- o$run(θ0)
#' # Compare with another starting point
-#' x1 <- c(1/2, 2*as.double(μ), c(0,0))
-#' par1 <- o$run(x1)
+#' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
+#' par1 <- o$run(θ1)
#' o$f( o$linArgs(par0) )
#' o$f( o$linArgs(par1) )
#' @export
-optimParams = function(K, link=c("logit","probit"), optargs=list())
+optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL)
{
- # Check arguments
- link <- match.arg(link)
- if (!is.list(optargs))
- stop("optargs: list")
- if (!is.numeric(K) || K < 2)
- stop("K: integer >= 2")
-
- M <- optargs$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)
- }
-
- weights <- optargs$weights
- if (is.null(weights))
- weights <- rep(1, K)
-
- # 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]]),
- "weights"=weights, "K"=as.integer(K))
+ # Check arguments
+ if (!is.matrix(X) || any(is.na(X)))
+ stop("X: numeric matrix, no NAs")
+ if (!is.numeric(Y) || any(is.na(Y)) || any(Y!=0 & Y!=1))
+ stop("Y: binary vector with 0 and 1 only")
+ link <- match.arg(link)
+ if (!is.numeric(K) || K!=floor(K) || K < 2)
+ stop("K: integer >= 2")
+
+ if (is.null(M))
+ {
+ # Precompute empirical moments
+ Mtmp <- computeMoments(X, Y)
+ M1 <- as.double(Mtmp[[1]])
+ M2 <- as.double(Mtmp[[2]])
+ M3 <- as.double(Mtmp[[3]])
+ M <- c(M1, M2, M3)
+ }
+ else
+ M <- c(M[[1]], M[[2]], M[[3]])
+
+ # Build and return optimization algorithm object
+ methods::new("OptimParams", "li"=link, "X"=X,
+ "Y"=as.integer(Y), "K"=as.integer(K), "Mhat"=as.double(M))
}
-# 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 weights Vector of moments' weights
-# @field K Number of populations
-# @field d Number of dimensions
-#
+#' Encapsulated optimization for p (proportions), β and b (regression parameters)
+#'
+#' Optimize the parameters of a mixture of logistic regressions model, possibly using
+#' \code{mu <- computeMu(...)} as a partial starting point.
+#'
+#' @field li Link function, 'logit' or 'probit'
+#' @field X Data matrix of covariables
+#' @field Y Output as a binary vector
+#' @field K Number of populations
+#' @field d Number of dimensions
+#' @field W Weights matrix (iteratively refined)
+#'
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$b
-
- # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
- β2 <- apply(β, 2, function(col) col %o% col)
- β3 <- apply(β, 2, function(col) col %o% col %o% col)
-
- return(
- weights[1] * sum( ( β %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
- weights[2] * sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
- weights[3] * sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
- },
-
- grad_f = function(x)
- {
- "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
-
- P <- expArgs(x)
- p <- P$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
- weights[1] * t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
- weights[2] * t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
- weights[3] * 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,] <-
- weights[1] * t(dβ) %*% F1 +
- weights[2] * t(dβ2) %*% F2 +
- weights[3] * t(dβ3) %*% F3
- }
- grad <- c(grad, as.double(grad_β))
-
- grad = c(grad, #gradient on b
- weights[1] * t( sweep(β, 2, p * G2, '*') ) %*% F1 +
- weights[2] * t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
- weights[3] * t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
-
- grad
- },
-
- run = function(x0)
- {
- "Run optimization from x0 with solver..."
-
- if (!is.list(x0))
- stop("x0: list")
- if (is.null(x0$β))
- stop("At least x0$β must be provided")
- if (!is.matrix(x0$β) || any(is.na(x0$β)) || ncol(x0$β) != K)
- stop("x0$β: matrix, no NA, ncol == K")
- if (is.null(x0$p))
- x0$p = rep(1/K, K-1)
- else if (length(x0$p) != K-1 || sum(x0$p) > 1)
- stop("x0$p should contain positive integers and sum to < 1")
- # Next test = heuristic to detect missing b (when matrix is called "beta")
- if (is.null(x0$b) || all(x0$b == x0$β))
- x0$b = rep(0, K)
- else if (any(is.na(x0$b)))
- stop("x0$b cannot have missing values")
-
- op_res = constrOptim( linArgs(x0), .self$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)
- }
- )
+ Class = "OptimParams",
+
+ fields = list(
+ # Inputs
+ li = "character", #link function
+ X = "matrix",
+ Y = "numeric",
+ Mhat = "numeric", #vector of empirical moments
+ # Dimensions
+ K = "integer",
+ n = "integer",
+ d = "integer",
+ # Weights matrix (generalized least square)
+ W = "matrix"
+ ),
+
+ methods = list(
+ initialize = function(...)
+ {
+ "Check args and initialize K, d, W"
+
+ callSuper(...)
+ if (!hasArg("X") || !hasArg("Y") || !hasArg("K")
+ || !hasArg("li") || !hasArg("Mhat"))
+ {
+ stop("Missing arguments")
+ }
+
+ n <<- nrow(X)
+ d <<- ncol(X)
+ # W will be initialized when calling run()
+ },
+
+ expArgs = function(v)
+ {
+ "Expand individual arguments from vector v into a list"
+
+ list(
+ # p: dimension K-1, need to be completed
+ "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
+ "β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
+ "b" = v[(K+d*K):(K+(d+1)*K-1)])
+ },
+
+ linArgs = function(L)
+ {
+ "Linearize vectors+matrices from list L into a vector"
+
+ # β linearized row by row, to match derivatives order
+ c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
+ },
+
+ computeW = function(θ)
+ {
+ require(MASS)
+ dd <- d + d^2 + d^3
+ M <- Moments(θ)
+ Omega <- matrix( .C("Compute_Omega",
+ X=as.double(X), Y=as.integer(Y), M=as.double(M),
+ pn=as.integer(n), pd=as.integer(d),
+ W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
+ MASS::ginv(Omega)
+ },
+
+ Moments = function(θ)
+ {
+ "Vector of moments, of size d+d^2+d^3"
+
+ p <- θ$p
+ β <- θ$β
+ λ <- sqrt(colSums(β^2))
+ b <- θ$b
+
+ # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
+ β2 <- apply(β, 2, function(col) col %o% col)
+ β3 <- apply(β, 2, function(col) col %o% col %o% col)
+
+ c(
+ β %*% (p * .G(li,1,λ,b)),
+ β2 %*% (p * .G(li,2,λ,b)),
+ β3 %*% (p * .G(li,3,λ,b)))
+ },
+
+ f = function(θ)
+ {
+ "Product t(hat_Mi - Mi) W (hat_Mi - Mi) with Mi(theta)"
+
+ L <- expArgs(θ)
+ A <- as.matrix(Mhat - Moments(L))
+ t(A) %*% W %*% A
+ },
+
+ grad_f = function(θ)
+ {
+ "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
+
+ L <- expArgs(θ)
+ -2 * t(grad_M(L)) %*% W %*% as.matrix(Mhat - Moments(L))
+ },
+
+ grad_M = function(θ)
+ {
+ "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
+
+ p <- θ$p
+ β <- θ$β
+ λ <- sqrt(colSums(β^2))
+ μ <- sweep(β, 2, λ, '/')
+ b <- θ$b
+
+ res <- matrix(nrow=nrow(W), ncol=0)
+
+ # 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)
+
+ # Gradient on p: K-1 columns, dim rows
+ km1 = 1:(K-1)
+ res <- cbind(res, rbind(
+ sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
+ sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
+ sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,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, '*')
+
+ res <- cbind(res, rbind(dβ, dβ2, dβ3))
+ }
+
+ # Gradient on b
+ res <- cbind(res, rbind(
+ sweep(β, 2, p * G2, '*'),
+ sweep(β2, 2, p * G3, '*'),
+ sweep(β3, 2, p * G4, '*') ))
+
+ res
+ },
+
+ run = function(θ0)
+ {
+ "Run optimization from θ0 with solver..."
+
+ if (!is.list(θ0))
+ stop("θ0: list")
+ if (is.null(θ0$β))
+ stop("At least θ0$β must be provided")
+ if (!is.matrix(θ0$β) || any(is.na(θ0$β))
+ || nrow(θ0$β) != d || ncol(θ0$β) != K)
+ {
+ stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
+ }
+ if (is.null(θ0$p))
+ θ0$p = rep(1/K, K-1)
+ else if (!is.numeric(θ0$p) || length(θ0$p) != K-1
+ || any(is.na(θ0$p)) || sum(θ0$p) > 1)
+ {
+ stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
+ }
+ if (is.null(θ0$b))
+ θ0$b = rep(0, K)
+ else if (!is.numeric(θ0$b) || length(θ0$b) != K || any(is.na(θ0$b)))
+ stop("θ0$b: length K, no NA")
+
+ # (Re)Set W to identity, to allow several run from the same object
+ W <<- diag(d+d^2+d^3)
+
+ loopMax <- 2 #TODO: loopMax = 3 ? Seems not improving...
+ x_init <- linArgs(θ0)
+ for (loop in 1:loopMax)
+ {
+ op_res = constrOptim( x_init, .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)) )
+ if (loop < loopMax) #avoid computing an extra W
+ W <<- computeW(expArgs(op_res$par))
+ x_init <- op_res$par
+ #print(op_res$value) #debug
+ #print(expArgs(op_res$par)) #debug
+ }
+
+ expArgs(op_res$par)
+ }
+ )
)
# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
# = E[g^{(order)}(z)] with z~N(b,diag(λ))
+# by numerically evaluating the integral.
#
# @param link Link, 'logit' or 'probit'
# @param order Order of derivative
#
.G <- function(link, order, λ, b)
{
- # NOTE: weird "integral divergent" error on inputs:
- # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
- # Switch to pracma package for that (but it seems slow...)
-
- exactComp <- FALSE #TODO: global, or argument...
-
- if (exactComp && link == "probit")
- {
- # Use exact computations
- sapply( seq_along(λ), function(k) {
- .exactProbitIntegral(order, λ[k], b[k])
- })
- }
-
- else
- {
- # Numerical integration
- sapply( seq_along(λ), function(k) {
- res <- NULL
- tryCatch({
- # Fast code, may fail:
- res <- stats::integrate(
- function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
- lower=-Inf, upper=Inf )$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) )
+ # 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...)
+ 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
+ })
}
# 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)
- )
+ "probit"=list(
+ # 'probit' derivatives list;
+ # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
+ 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)
#
.zin <- function(x)
{
- x[is.nan(x)] <- 0.
- x
+ x[is.nan(x)] <- 0.
+ x
}
projects / morpheus.git / blobdiff