[morpheus.git] / pkg / R / optimParams.R

#' 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?
#'     \item weights Weights on moments when minimizing sum of squares
#'   }
#'
#' @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)
	}

	# 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",
    # Weights matrix (generalized least square)
    W = "matrix"
	),

	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)
      W <<- diag(d+d^2+d^3) #initialize at W = Identity
		},

		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)
		{
			"Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"

			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.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)
		}
	)
)

# 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...)

	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) )
}

# 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
}
Commit	Line	Data
	1	#' Optimize parameters
	2	#'
	3	#' Optimize the parameters of a mixture of logistic regressions model, possibly using
	4	#' \code{mu <- computeMu(...)} as a partial starting point.
	5	#'
	6	#' @param K Number of populations.
	7	#' @param link The link type, 'logit' or 'probit'.
	8	#' @param optargs a list with optional arguments:
	9	#' \itemize{
	10	#' \item 'M' : list of moments of order 1,2,3: will be computed if not provided.
	11	#' \item 'X,Y' : input/output, mandatory if moments not given
	12	#' \item 'exact': use exact formulas when available?
	13	#' \item weights Weights on moments when minimizing sum of squares
	14	#' }
	15	#'
	16	#' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
	17	#' outputs the list of optimized parameters
	18	#' \itemize{
	19	#' \item p: proportions, size K
	20	#' \item β: regression matrix, size dxK
	21	#' \item b: intercepts, size K
	22	#' }
	23	#' x0 is a vector containing respectively the K-1 first elements of p, then β by
	24	#' columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
	25	#'
	26	#' @seealso \code{multiRun} to estimate statistics based on β, and
	27	#' \code{generateSampleIO} for I/O random generation.
	28	#'
	29	#' @examples
	30	#' # Optimize parameters from estimated μ
	31	#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
	32	#' μ = computeMu(io$X, io$Y, list(K=2))
	33	#' M <- computeMoments(io$X, io$Y)
	34	#' o <- optimParams(2, "logit", list(M=M))
	35	#' x0 <- c(1/2, as.double(μ), c(0,0))
	36	#' par0 <- o$run(x0)
	37	#' # Compare with another starting point
	38	#' x1 <- c(1/2, 2*as.double(μ), c(0,0))
	39	#' par1 <- o$run(x1)
	40	#' o$f( o$linArgs(par0) )
	41	#' o$f( o$linArgs(par1) )
	42	#' @export
	43	optimParams = function(K, link=c("logit","probit"), optargs=list())
	44	{
	45	# Check arguments
	46	link <- match.arg(link)
	47	if (!is.list(optargs))
	48	stop("optargs: list")
	49	if (!is.numeric(K) \|\| K < 2)
	50	stop("K: integer >= 2")
	51
	52	M <- optargs$M
	53	if (is.null(M))
	54	{
	55	if (is.null(optargs$X) \|\| is.null(optargs$Y))
	56	stop("If moments are not provided, X and Y are required")
	57	M <- computeMoments(optargs$X,optargs$Y)
	58	}
	59
	60	# Build and return optimization algorithm object
	61	methods::new("OptimParams", "li"=link, "M1"=as.double(M[[1]]),
	62	"M2"=as.double(M[[2]]), "M3"=as.double(M[[3]]), "K"=as.integer(K))
	63	}
	64
	65	# Encapsulated optimization for p (proportions), β and b (regression parameters)
	66	#
	67	# @field li Link, 'logit' or 'probit'
	68	# @field M1 Estimated first-order moment
	69	# @field M2 Estimated second-order moment (flattened)
	70	# @field M3 Estimated third-order moment (flattened)
	71	# @field K Number of populations
	72	# @field d Number of dimensions
	73	#
	74	setRefClass(
	75	Class = "OptimParams",
	76
	77	fields = list(
	78	# Inputs
	79	li = "character", #link 'logit' or 'probit'
	80	M1 = "numeric", #order-1 moment (vector size d)
	81	M2 = "numeric", #M2 easier to process as a vector
	82	M3 = "numeric", #M3 easier to process as a vector
	83	# Dimensions
	84	K = "integer",
	85	d = "integer",
	86	# Weights matrix (generalized least square)
	87	W = "matrix"
	88	),
	89
	90	methods = list(
	91	initialize = function(...)
	92	{
	93	"Check args and initialize K, d"
	94
	95	callSuper(...)
	96	if (!hasArg("li") \|\| !hasArg("M1") \|\| !hasArg("M2") \|\| !hasArg("M3")
	97	\|\| !hasArg("K"))
	98	{
	99	stop("Missing arguments")
	100	}
	101
	102	d <<- length(M1)
	103	W <<- diag(d+d^2+d^3) #initialize at W = Identity
	104	},
	105
	106	expArgs = function(x)
	107	{
	108	"Expand individual arguments from vector x"
	109
	110	list(
	111	# p: dimension K-1, need to be completed
	112	"p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
	113	"β" = matrix(x[K:(K+d*K-1)], ncol=K),
	114	"b" = x[(K+dK):(K+(d+1)K-1)])
	115	},
	116
	117	linArgs = function(o)
	118	{
	119	" Linearize vectors+matrices into a vector x"
	120
	121	c(o$p[1:(K-1)], as.double(o$β), o$b)
	122	},
	123
	124	f = function(x)
	125	{
	126	"Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"
	127
	128	P <- expArgs(x)
	129	p <- P$p
	130	β <- P$β
	131	λ <- sqrt(colSums(β^2))
	132	b <- P$b
	133
	134	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	135	β2 <- apply(β, 2, function(col) col %o% col)
	136	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	137
	138	return(
	139	sum( ( β %% (p .G(li,1,λ,b)) - M1 )^2 ) +
	140	sum( ( β2 %% (p .G(li,2,λ,b)) - M2 )^2 ) +
	141	sum( ( β3 %% (p .G(li,3,λ,b)) - M3 )^2 ) )
	142	},
	143
	144	grad_f = function(x)
	145	{
	146	"Gradient of f, dimension (K-1) + dK + K = (d+2)K - 1"
	147
	148	P <- expArgs(x)
	149	p <- P$p
	150	β <- P$β
	151	λ <- sqrt(colSums(β^2))
	152	μ <- sweep(β, 2, λ, '/')
	153	b <- P$b
	154
	155	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	156	β2 <- apply(β, 2, function(col) col %o% col)
	157	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	158
	159	# Some precomputations
	160	G1 = .G(li,1,λ,b)
	161	G2 = .G(li,2,λ,b)
	162	G3 = .G(li,3,λ,b)
	163	G4 = .G(li,4,λ,b)
	164	G5 = .G(li,5,λ,b)
	165
	166	# (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
	167	F1 = as.double( 2 * ( β %% (p G1) - M1 ) )
	168	F2 = as.double( 2 * ( β2 %% (p G2) - M2 ) )
	169	F3 = as.double( 2 * ( β3 %% (p G3) - M3 ) )
	170
	171	km1 = 1:(K-1)
	172	grad <- #gradient on p
	173	t( sweep(as.matrix(β [,km1]), 2, G1[km1], '') - G1[K] β [,K] ) %*% F1 +
	174	t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '') - G2[K] β2[,K] ) %*% F2 +
	175	t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '') - G3[K] β3[,K] ) %*% F3
	176
	177	grad_β <- matrix(nrow=d, ncol=K)
	178	for (i in 1:d)
	179	{
	180	# i determines the derivated matrix dβ[2,3]
	181
	182	dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
	183	dβ_right <- matrix(0, nrow=d, ncol=K)
	184	block <- i
	185	dβ_right[block,] <- dβ_right[block,] + 1
	186	dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
	187
	188	dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
	189	dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
	190	sweep(dβ_right, 2, β[j,], '*')
	191	}) )
	192	block <- ((i-1)d+1):(id)
	193	dβ2_right[block,] <- dβ2_right[block,] + β
	194	dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
	195
	196	dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
	197	dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
	198	sweep(dβ2_right, 2, β[j,], '*')
	199	}) )
	200	block <- ((i-1)dd+1):(idd)
	201	dβ3_right[block,] <- dβ3_right[block,] + β2
	202	dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
	203
	204	grad_β[i,] <- t(dβ) %% F1 + t(dβ2) %% F2 + t(dβ3) %*% F3
	205	}
	206	grad <- c(grad, as.double(grad_β))
	207
	208	grad = c(grad, #gradient on b
	209	t( sweep(β, 2, p * G2, '') ) %% F1 +
	210	t( sweep(β2, 2, p * G3, '') ) %% F2 +
	211	t( sweep(β3, 2, p * G4, '') ) %% F3 )
	212
	213	grad
	214	},
	215
	216	run = function(x0)
	217	{
	218	"Run optimization from x0 with solver..."
	219
	220	if (!is.list(x0))
	221	stop("x0: list")
	222	if (is.null(x0$β))
	223	stop("At least x0$β must be provided")
	224	if (!is.matrix(x0$β) \|\| any(is.na(x0$β)) \|\| ncol(x0$β) != K)
	225	stop("x0$β: matrix, no NA, ncol == K")
	226	if (is.null(x0$p))
	227	x0$p = rep(1/K, K-1)
	228	else if (length(x0$p) != K-1 \|\| sum(x0$p) > 1)
	229	stop("x0$p should contain positive integers and sum to < 1")
	230	# Next test = heuristic to detect missing b (when matrix is called "beta")
	231	if (is.null(x0$b) \|\| all(x0$b == x0$β))
	232	x0$b = rep(0, K)
	233	else if (any(is.na(x0$b)))
	234	stop("x0$b cannot have missing values")
	235
	236	op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
	237	ui=cbind(
	238	rbind( rep(-1,K-1), diag(K-1) ),
	239	matrix(0, nrow=K, ncol=(d+1)*K) ),
	240	ci=c(-1,rep(0,K-1)) )
	241
	242	expArgs(op_res$par)
	243	}
	244	)
	245	)
	246
	247	# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
	248	# = E[g^{(order)}(z)] with z~N(b,diag(λ))
	249	#
	250	# @param link Link, 'logit' or 'probit'
	251	# @param order Order of derivative
	252	# @param λ Norm of columns of β
	253	# @param b Intercept
	254	#
	255	.G <- function(link, order, λ, b)
	256	{
	257	# NOTE: weird "integral divergent" error on inputs:
	258	# link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
	259	# Switch to pracma package for that (but it seems slow...)
	260
	261	exactComp <- FALSE #TODO: global, or argument...
	262
	263	if (exactComp && link == "probit")
	264	{
	265	# Use exact computations
	266	sapply( seq_along(λ), function(k) {
	267	.exactProbitIntegral(order, λ[k], b[k])
	268	})
	269	}
	270
	271	else
	272	{
	273	# Numerical integration
	274	sapply( seq_along(λ), function(k) {
	275	res <- NULL
	276	tryCatch({
	277	# Fast code, may fail:
	278	res <- stats::integrate(
	279	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	280	lower=-Inf, upper=Inf )$value
	281	}, error = function(e) {
	282	# Robust slow code, no fails observed:
	283	sink("/dev/null") #pracma package has some useless printed outputs...
	284	res <- pracma::integral(
	285	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	286	xmin=-Inf, xmax=Inf, method="Kronrod")
	287	sink()
	288	})
	289	res
	290	})
	291	}
	292	}
	293
	294	# TODO: check these computations (wrong atm)
	295	.exactProbitIntegral <- function(order, λ, b)
	296	{
	297	c1 = (1/sqrt(2pi)) exp( -.5 * b/((λ^2+1)^2) )
	298	if (order == 1)
	299	return (c1)
	300	c2 = b - λ^2 / (λ^2+1)
	301	if (order == 2)
	302	return (c1 * c2)
	303	if (order == 3)
	304	return (c1 * (λ^2 - 1 + c2^2))
	305	if (order == 4)
	306	return ( (c1c2/((λ^2+1)^2)) (-λ^4*((b+1)^2+1) -
	307	2λ^3 + λ^2(2-2b(b-1)) + 6*λ + 3 - b^2) )
	308	if (order == 5) #only remaining case...
	309	return ( c1 * (3λ^4+c2^4+6c1^2(λ^2-1) - 6λ^2 + 6) )
	310	}
	311
	312	# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
	313	#
	314	.deriv <- list(
	315	"probit"=list(
	316	# 'probit' derivatives list;
	317	# TODO: exact values for the integral E[g^(k)(λz+b)]
	318	function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
	319	function(x) exp(-x^2/2)/(sqrt(2pi)) -x, #g''
	320	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^2 - 1), #g^(3)
	321	function(x) exp(-x^2/2)/(sqrt(2pi)) (-x^3 + 3*x), #g^(4)
	322	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^4 - 6*x^2 + 3) #g^(5)
	323	),
	324	"logit"=list(
	325	# Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
	326	# @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
	327	function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
	328	function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
	329	function(x) {e=exp(x); .zin(e( e^2 - 4e + 1) /(e+1)^4)}, #g^(3)
	330	function(x) {e=exp(x); .zin(e(-e^3 + 11e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
	331	function(x) {e=exp(x); .zin(e( e^4 - 26e^3 + 66e^2 - 26e + 1)/(e+1)^6)} #g^(5)
	332	)
	333	)
	334
	335	# Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
	336	#
	337	# @param x Ratio of polynoms of exponentials, as in .S[[i]]
	338	#
	339	.zin <- function(x)
	340	{
	341	x[is.nan(x)] <- 0.
	342	x
	343	}