[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)
	}

  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))
}

# 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
#
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)
		}
	)
)

# 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
cbd88fe5 BA	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?
98b8a5dd	13	#' \item weights Weights on moments when minimizing sum of squares
cbd88fe5 BA	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
98b8a5dd BA	60	weights <- optargs$weights
	61	if (is.null(weights))
	62	weights <- rep(1, K)
	63
cbd88fe5 BA	64	# Build and return optimization algorithm object
cbd88fe5 BA	65	methods::new("OptimParams", "li"=link, "M1"=as.double(M[[1]]),
98b8a5dd BA	66	"M2"=as.double(M[[2]]), "M3"=as.double(M[[3]]),
98b8a5dd BA	67	"weights"=weights, "K"=as.integer(K))
cbd88fe5 BA	68	}
	69
	70	# Encapsulated optimization for p (proportions), β and b (regression parameters)
	71	#
	72	# @field li Link, 'logit' or 'probit'
	73	# @field M1 Estimated first-order moment
	74	# @field M2 Estimated second-order moment (flattened)
	75	# @field M3 Estimated third-order moment (flattened)
98b8a5dd	76	# @field weights Vector of moments' weights
cbd88fe5 BA	77	# @field K Number of populations
	78	# @field d Number of dimensions
	79	#
	80	setRefClass(
	81	Class = "OptimParams",
	82
	83	fields = list(
	84	# Inputs
	85	li = "character", #link 'logit' or 'probit'
	86	M1 = "numeric", #order-1 moment (vector size d)
	87	M2 = "numeric", #M2 easier to process as a vector
	88	M3 = "numeric", #M3 easier to process as a vector
	89	# Dimensions
	90	K = "integer",
	91	d = "integer"
	92	),
	93
	94	methods = list(
	95	initialize = function(...)
	96	{
	97	"Check args and initialize K, d"
	98
	99	callSuper(...)
	100	if (!hasArg("li") \|\| !hasArg("M1") \|\| !hasArg("M2") \|\| !hasArg("M3")
	101	\|\| !hasArg("K"))
	102	{
	103	stop("Missing arguments")
	104	}
	105
	106	d <<- length(M1)
	107	},
	108
	109	expArgs = function(x)
	110	{
	111	"Expand individual arguments from vector x"
	112
	113	list(
	114	# p: dimension K-1, need to be completed
	115	"p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
	116	"β" = matrix(x[K:(K+d*K-1)], ncol=K),
	117	"b" = x[(K+dK):(K+(d+1)K-1)])
	118	},
	119
	120	linArgs = function(o)
	121	{
	122	" Linearize vectors+matrices into a vector x"
	123
	124	c(o$p[1:(K-1)], as.double(o$β), o$b)
	125	},
	126
	127	f = function(x)
	128	{
	129	"Sum of squares (Mi - hat_Mi)^2 where Mi is obtained from formula"
	130
	131	P <- expArgs(x)
	132	p <- P$p
	133	β <- P$β
	134	λ <- sqrt(colSums(β^2))
	135	b <- P$b
	136
	137	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	138	β2 <- apply(β, 2, function(col) col %o% col)
	139	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	140
141	return(
98b8a5dd BA	142	weights[1] * sum( ( β %% (p .G(li,1,λ,b)) - M1 )^2 ) +
	143	weights[2] * sum( ( β2 %% (p .G(li,2,λ,b)) - M2 )^2 ) +
	144	weights[3] * sum( ( β3 %% (p .G(li,3,λ,b)) - M3 )^2 ) )
cbd88fe5 BA	145	},
	146
	147	grad_f = function(x)
	148	{
	149	"Gradient of f, dimension (K-1) + dK + K = (d+2)K - 1"
	150
	151	P <- expArgs(x)
	152	p <- P$p
	153	β <- P$β
	154	λ <- sqrt(colSums(β^2))
	155	μ <- sweep(β, 2, λ, '/')
	156	b <- P$b
	157
	158	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	159	β2 <- apply(β, 2, function(col) col %o% col)
	160	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	161
	162	# Some precomputations
	163	G1 = .G(li,1,λ,b)
	164	G2 = .G(li,2,λ,b)
	165	G3 = .G(li,3,λ,b)
	166	G4 = .G(li,4,λ,b)
	167	G5 = .G(li,5,λ,b)
	168
	169	# (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
	170	F1 = as.double( 2 * ( β %% (p G1) - M1 ) )
	171	F2 = as.double( 2 * ( β2 %% (p G2) - M2 ) )
	172	F3 = as.double( 2 * ( β3 %% (p G3) - M3 ) )
	173
	174	km1 = 1:(K-1)
	175	grad <- #gradient on p
98b8a5dd BA	176	weights[1] * t( sweep(as.matrix(β [,km1]), 2, G1[km1], '') - G1[K] β [,K] ) %*% F1 +
	177	weights[2] * t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '') - G2[K] β2[,K] ) %*% F2 +
	178	weights[3] * t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '') - G3[K] β3[,K] ) %*% F3
cbd88fe5 BA	179
	180	grad_β <- matrix(nrow=d, ncol=K)
	181	for (i in 1:d)
	182	{
	183	# i determines the derivated matrix dβ[2,3]
	184
	185	dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
	186	dβ_right <- matrix(0, nrow=d, ncol=K)
	187	block <- i
	188	dβ_right[block,] <- dβ_right[block,] + 1
	189	dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
	190
	191	dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
	192	dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
	193	sweep(dβ_right, 2, β[j,], '*')
	194	}) )
	195	block <- ((i-1)d+1):(id)
	196	dβ2_right[block,] <- dβ2_right[block,] + β
	197	dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
	198
	199	dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
	200	dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
	201	sweep(dβ2_right, 2, β[j,], '*')
	202	}) )
	203	block <- ((i-1)dd+1):(idd)
	204	dβ3_right[block,] <- dβ3_right[block,] + β2
	205	dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
	206
98b8a5dd BA	207	grad_β[i,] <-
	208	weights[1] * t(dβ) %*% F1 +
	209	weights[2] * t(dβ2) %*% F2 +
	210	weights[3] * t(dβ3) %*% F3
cbd88fe5 BA	211	}
	212	grad <- c(grad, as.double(grad_β))
	213
	214	grad = c(grad, #gradient on b
98b8a5dd BA	215	weights[1] * t( sweep(β, 2, p * G2, '') ) %% F1 +
	216	weights[2] * t( sweep(β2, 2, p * G3, '') ) %% F2 +
	217	weights[3] * t( sweep(β3, 2, p * G4, '') ) %% F3 )
cbd88fe5 BA	218
	219	grad
	220	},
	221
	222	run = function(x0)
	223	{
	224	"Run optimization from x0 with solver..."
	225
d294ece1 BA	226	if (!is.list(x0))
	227	stop("x0: list")
	228	if (is.null(x0$β))
	229	stop("At least x0$β must be provided")
	230	if (!is.matrix(x0$β) \|\| any(is.na(x0$β)) \|\| ncol(x0$β) != K)
	231	stop("x0$β: matrix, no NA, ncol == K")
	232	if (is.null(x0$p))
	233	x0$p = rep(1/K, K-1)
	234	else if (length(x0$p) != K-1 \|\| sum(x0$p) > 1)
	235	stop("x0$p should contain positive integers and sum to < 1")
	236	# Next test = heuristic to detect missing b (when matrix is called "beta")
	237	if (is.null(x0$b) \|\| all(x0$b == x0$β))
	238	x0$b = rep(0, K)
	239	else if (any(is.na(x0$b)))
	240	stop("x0$b cannot have missing values")
	241
	242	op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
cbd88fe5 BA	243	ui=cbind(
	244	rbind( rep(-1,K-1), diag(K-1) ),
	245	matrix(0, nrow=K, ncol=(d+1)*K) ),
	246	ci=c(-1,rep(0,K-1)) )
	247
	248	expArgs(op_res$par)
	249	}
	250	)
	251	)
	252
	253	# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
	254	# = E[g^{(order)}(z)] with z~N(b,diag(λ))
	255	#
	256	# @param link Link, 'logit' or 'probit'
	257	# @param order Order of derivative
	258	# @param λ Norm of columns of β
	259	# @param b Intercept
	260	#
	261	.G <- function(link, order, λ, b)
	262	{
	263	# NOTE: weird "integral divergent" error on inputs:
	264	# link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
	265	# Switch to pracma package for that (but it seems slow...)
	266
cf673dee BA	267	exactComp <- FALSE #TODO: global, or argument...
cf673dee BA	268
cbd88fe5 BA	269	if (exactComp && link == "probit")
	270	{
	271	# Use exact computations
	272	sapply( seq_along(λ), function(k) {
	273	.exactProbitIntegral(order, λ[k], b[k])
	274	})
	275	}
	276
	277	else
	278	{
	279	# Numerical integration
	280	sapply( seq_along(λ), function(k) {
	281	res <- NULL
	282	tryCatch({
	283	# Fast code, may fail:
	284	res <- stats::integrate(
	285	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	286	lower=-Inf, upper=Inf )$value
	287	}, error = function(e) {
	288	# Robust slow code, no fails observed:
	289	sink("/dev/null") #pracma package has some useless printed outputs...
	290	res <- pracma::integral(
	291	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	292	xmin=-Inf, xmax=Inf, method="Kronrod")
	293	sink()
	294	})
	295	res
	296	})
	297	}
	298	}
	299
	300	# TODO: check these computations (wrong atm)
	301	.exactProbitIntegral <- function(order, λ, b)
	302	{
	303	c1 = (1/sqrt(2pi)) exp( -.5 * b/((λ^2+1)^2) )
	304	if (order == 1)
	305	return (c1)
	306	c2 = b - λ^2 / (λ^2+1)
	307	if (order == 2)
	308	return (c1 * c2)
	309	if (order == 3)
	310	return (c1 * (λ^2 - 1 + c2^2))
	311	if (order == 4)
	312	return ( (c1c2/((λ^2+1)^2)) (-λ^4*((b+1)^2+1) -
	313	2λ^3 + λ^2(2-2b(b-1)) + 6*λ + 3 - b^2) )
	314	if (order == 5) #only remaining case...
	315	return ( c1 * (3λ^4+c2^4+6c1^2(λ^2-1) - 6λ^2 + 6) )
	316	}
	317
	318	# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
	319	#
	320	.deriv <- list(
	321	"probit"=list(
	322	# 'probit' derivatives list;
	323	# TODO: exact values for the integral E[g^(k)(λz+b)]
	324	function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
	325	function(x) exp(-x^2/2)/(sqrt(2pi)) -x, #g''
	326	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^2 - 1), #g^(3)
	327	function(x) exp(-x^2/2)/(sqrt(2pi)) (-x^3 + 3*x), #g^(4)
	328	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^4 - 6*x^2 + 3) #g^(5)
	329	),
	330	"logit"=list(
	331	# Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
	332	# @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
333	function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
334	function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
335	function(x) {e=exp(x); .zin(e( e^2 - 4e + 1) /(e+1)^4)}, #g^(3)
336	function(x) {e=exp(x); .zin(e(-e^3 + 11e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
337	function(x) {e=exp(x); .zin(e( e^4 - 26e^3 + 66e^2 - 26e + 1)/(e+1)^6)} #g^(5)
338	)
339	)
340
341	# Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
342	#
343	# @param x Ratio of polynoms of exponentials, as in .S[[i]]
344	#
345	.zin <- function(x)
346	{
347	x[is.nan(x)] <- 0.
348	x
349	}