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

#' Wrapper function for OptimParams class
#'
#' @param X Data matrix of covariables
#' @param Y Output as a binary vector
#' @param K Number of populations.
#' @param link The link type, 'logit' or 'probit'.
#' @param M the empirical cross-moments between X and Y (optional)
#' @param nc Number of cores (default: 0 to use all)
#'
#' @return An object 'op' of class OptimParams, initialized so that
#'   \code{op$run(θ0)} outputs the list of optimized parameters
#'   \itemize{
#'     \item p: proportions, size K
#'     \item β: regression matrix, size dxK
#'     \item b: intercepts, size K
#'   }
#'   θ0 is a list containing the initial parameters. Only β is required
#'   (p would be set to (1/K,...,1/K) and b to (0,...0)).
#'
#' @seealso \code{multiRun} to estimate statistics based on β, and
#'   \code{generateSampleIO} for I/O random generation.
#'
#' @examples
#' # Optimize parameters from estimated μ
#' io <- generateSampleIO(100,
#'   1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
#' μ = computeMu(io$X, io$Y, list(K=2))
#' o <- optimParams(io$X, io$Y, 2, "logit")
#' \donttest{
#' θ0 <- list(p=1/2, β=μ, b=c(0,0))
#' par0 <- o$run(θ0)
#' # Compare with another starting point
#' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
#' par1 <- o$run(θ1)
#' # Look at the function values at par0 and par1:
#' o$f( o$linArgs(par0) )
#' o$f( o$linArgs(par1) )}
#'
#' @export
optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL, nc=0)
{
  # 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), "nc"=as.integer(nc))
}

# 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 Mhat Vector of empirical moments
# @field K Number of populations
# @field n Number of sample points
# @field d Number of dimensions
# @field nc Number of cores (OpenMP //)
# @field W Weights matrix (initialized at identity)
#
setRefClass(
  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",
    nc = "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") || !hasArg("nc"))
      {
        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)
    },

    # TODO: relocate computeW in utils.R
    computeW = function(θ)
    {
      "Compute the weights matrix from a parameters list"

      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),
        pnc=as.integer(nc), 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)
    {
      "Compute the vector of theoretical moments (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)
    {
      "Function to minimize: t(hat_Mi - Mi(θ)) . W . (hat_Mi - Mi(θ))"

      L <- expArgs(θ)
      A <- as.matrix(Mhat - Moments(L))
      t(A) %*% W %*% A
    },

    grad_f = function(θ)
    {
      "Gradient of f: vector of size (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 moments vector: matrix of size 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 #degrades performances (TODO: why?)
      }

      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
# @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...)
  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;
    # 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)
#
# @param x Ratio of polynoms of exponentials, as in .S[[i]]
#
.zin <- function(x)
{
  x[is.nan(x)] <- 0.
  x
}
Commit	Line	Data
	1	#' Wrapper function for OptimParams class
	2	#'
	3	#' @param X Data matrix of covariables
	4	#' @param Y Output as a binary vector
	5	#' @param K Number of populations.
	6	#' @param link The link type, 'logit' or 'probit'.
	7	#' @param M the empirical cross-moments between X and Y (optional)
	8	#' @param nc Number of cores (default: 0 to use all)
	9	#'
	10	#' @return An object 'op' of class OptimParams, initialized so that
	11	#' \code{op$run(θ0)} outputs the list of optimized parameters
	12	#' \itemize{
	13	#' \item p: proportions, size K
	14	#' \item β: regression matrix, size dxK
	15	#' \item b: intercepts, size K
	16	#' }
	17	#' θ0 is a list containing the initial parameters. Only β is required
	18	#' (p would be set to (1/K,...,1/K) and b to (0,...0)).
	19	#'
	20	#' @seealso \code{multiRun} to estimate statistics based on β, and
	21	#' \code{generateSampleIO} for I/O random generation.
	22	#'
	23	#' @examples
	24	#' # Optimize parameters from estimated μ
	25	#' io <- generateSampleIO(100,
	26	#' 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
	27	#' μ = computeMu(io$X, io$Y, list(K=2))
	28	#' o <- optimParams(io$X, io$Y, 2, "logit")
	29	#' \donttest{
	30	#' θ0 <- list(p=1/2, β=μ, b=c(0,0))
	31	#' par0 <- o$run(θ0)
	32	#' # Compare with another starting point
	33	#' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
	34	#' par1 <- o$run(θ1)
	35	#' # Look at the function values at par0 and par1:
	36	#' o$f( o$linArgs(par0) )
	37	#' o$f( o$linArgs(par1) )}
	38	#'
	39	#' @export
	40	optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL, nc=0)
	41	{
	42	# Check arguments
	43	if (!is.matrix(X) \|\| any(is.na(X)))
	44	stop("X: numeric matrix, no NAs")
	45	if (!is.numeric(Y) \|\| any(is.na(Y)) \|\| any(Y!=0 & Y!=1))
	46	stop("Y: binary vector with 0 and 1 only")
	47	link <- match.arg(link)
	48	if (!is.numeric(K) \|\| K!=floor(K) \|\| K < 2)
	49	stop("K: integer >= 2")
	50
	51	if (is.null(M))
	52	{
	53	# Precompute empirical moments
	54	Mtmp <- computeMoments(X, Y)
	55	M1 <- as.double(Mtmp[[1]])
	56	M2 <- as.double(Mtmp[[2]])
	57	M3 <- as.double(Mtmp[[3]])
	58	M <- c(M1, M2, M3)
	59	}
	60	else
	61	M <- c(M[[1]], M[[2]], M[[3]])
	62
	63	# Build and return optimization algorithm object
	64	methods::new("OptimParams", "li"=link, "X"=X,
	65	"Y"=as.integer(Y), "K"=as.integer(K), "Mhat"=as.double(M), "nc"=as.integer(nc))
	66	}
	67
	68	# Encapsulated optimization for p (proportions), β and b (regression parameters)
	69	#
	70	# Optimize the parameters of a mixture of logistic regressions model, possibly using
	71	# \code{mu <- computeMu(...)} as a partial starting point.
	72	#
	73	# @field li Link function, 'logit' or 'probit'
	74	# @field X Data matrix of covariables
	75	# @field Y Output as a binary vector
	76	# @field Mhat Vector of empirical moments
	77	# @field K Number of populations
	78	# @field n Number of sample points
	79	# @field d Number of dimensions
	80	# @field nc Number of cores (OpenMP //)
	81	# @field W Weights matrix (initialized at identity)
	82	#
	83	setRefClass(
	84	Class = "OptimParams",
	85
	86	fields = list(
	87	# Inputs
	88	li = "character", #link function
	89	X = "matrix",
	90	Y = "numeric",
	91	Mhat = "numeric", #vector of empirical moments
	92	# Dimensions
	93	K = "integer",
	94	n = "integer",
	95	d = "integer",
	96	nc = "integer",
	97	# Weights matrix (generalized least square)
	98	W = "matrix"
	99	),
	100
	101	methods = list(
	102	initialize = function(...)
	103	{
	104	"Check args and initialize K, d, W"
	105
	106	callSuper(...)
	107	if (!hasArg("X") \|\| !hasArg("Y") \|\| !hasArg("K")
	108	\|\| !hasArg("li") \|\| !hasArg("Mhat") \|\| !hasArg("nc"))
	109	{
	110	stop("Missing arguments")
	111	}
	112
	113	n <<- nrow(X)
	114	d <<- ncol(X)
	115	# W will be initialized when calling run()
	116	},
	117
	118	expArgs = function(v)
	119	{
	120	"Expand individual arguments from vector v into a list"
	121
	122	list(
	123	# p: dimension K-1, need to be completed
	124	"p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
	125	"β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
	126	"b" = v[(K+dK):(K+(d+1)K-1)])
	127	},
	128
	129	linArgs = function(L)
	130	{
	131	"Linearize vectors+matrices from list L into a vector"
	132
	133	# β linearized row by row, to match derivatives order
	134	c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
	135	},
	136
	137	# TODO: relocate computeW in utils.R
	138	computeW = function(θ)
	139	{
	140	"Compute the weights matrix from a parameters list"
	141
	142	require(MASS)
	143	dd <- d + d^2 + d^3
	144	M <- Moments(θ)
	145	Omega <- matrix( .C("Compute_Omega",
	146	X=as.double(X), Y=as.integer(Y), M=as.double(M),
	147	pnc=as.integer(nc), pn=as.integer(n), pd=as.integer(d),
	148	W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
	149	MASS::ginv(Omega)
	150	},
	151
	152	Moments = function(θ)
	153	{
	154	"Compute the vector of theoretical moments (size d+d^2+d^3)"
	155
	156	p <- θ$p
	157	β <- θ$β
	158	λ <- sqrt(colSums(β^2))
	159	b <- θ$b
	160
	161	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	162	β2 <- apply(β, 2, function(col) col %o% col)
	163	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	164
	165	c(
	166	β %% (p .G(li,1,λ,b)),
	167	β2 %% (p .G(li,2,λ,b)),
	168	β3 %% (p .G(li,3,λ,b)))
	169	},
	170
	171	f = function(θ)
	172	{
	173	"Function to minimize: t(hat_Mi - Mi(θ)) . W . (hat_Mi - Mi(θ))"
	174
	175	L <- expArgs(θ)
	176	A <- as.matrix(Mhat - Moments(L))
	177	t(A) %% W %% A
	178	},
	179
	180	grad_f = function(θ)
	181	{
	182	"Gradient of f: vector of size (K-1) + dK + K = (d+2)K - 1"
	183
	184	L <- expArgs(θ)
	185	-2 * t(grad_M(L)) %% W %% as.matrix(Mhat - Moments(L))
	186	},
	187
	188	grad_M = function(θ)
	189	{
	190	"Gradient of the moments vector: matrix of size d+d^2+d^3 x K-1+K+d*K"
	191
	192	p <- θ$p
	193	β <- θ$β
	194	λ <- sqrt(colSums(β^2))
	195	μ <- sweep(β, 2, λ, '/')
	196	b <- θ$b
	197
	198	res <- matrix(nrow=nrow(W), ncol=0)
	199
	200	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	201	β2 <- apply(β, 2, function(col) col %o% col)
	202	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	203
	204	# Some precomputations
	205	G1 = .G(li,1,λ,b)
	206	G2 = .G(li,2,λ,b)
	207	G3 = .G(li,3,λ,b)
	208	G4 = .G(li,4,λ,b)
	209	G5 = .G(li,5,λ,b)
	210
	211	# Gradient on p: K-1 columns, dim rows
	212	km1 = 1:(K-1)
	213	res <- cbind(res, rbind(
	214	sweep(as.matrix(β [,km1]), 2, G1[km1], '') - G1[K] β [,K],
	215	sweep(as.matrix(β2[,km1]), 2, G2[km1], '') - G2[K] β2[,K],
	216	sweep(as.matrix(β3[,km1]), 2, G3[km1], '') - G3[K] β3[,K] ))
	217
	218	for (i in 1:d)
	219	{
	220	# i determines the derivated matrix dβ[2,3]
	221
	222	dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
	223	dβ_right <- matrix(0, nrow=d, ncol=K)
	224	block <- i
	225	dβ_right[block,] <- dβ_right[block,] + 1
	226	dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
	227
	228	dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
	229	dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
	230	sweep(dβ_right, 2, β[j,], '*')
	231	}) )
	232	block <- ((i-1)d+1):(id)
	233	dβ2_right[block,] <- dβ2_right[block,] + β
	234	dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
	235
	236	dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
	237	dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
	238	sweep(dβ2_right, 2, β[j,], '*')
	239	}) )
	240	block <- ((i-1)dd+1):(idd)
	241	dβ3_right[block,] <- dβ3_right[block,] + β2
	242	dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
	243
	244	res <- cbind(res, rbind(dβ, dβ2, dβ3))
	245	}
	246
	247	# Gradient on b
	248	res <- cbind(res, rbind(
	249	sweep(β, 2, p * G2, '*'),
	250	sweep(β2, 2, p * G3, '*'),
	251	sweep(β3, 2, p * G4, '*') ))
	252
	253	res
	254	},
	255
	256	run = function(θ0)
	257	{
	258	"Run optimization from θ0 with solver..."
	259
	260	if (!is.list(θ0))
	261	stop("θ0: list")
	262	if (is.null(θ0$β))
	263	stop("At least θ0$β must be provided")
	264	if (!is.matrix(θ0$β) \|\| any(is.na(θ0$β))
	265	\|\| nrow(θ0$β) != d \|\| ncol(θ0$β) != K)
	266	{
	267	stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
	268	}
	269	if (is.null(θ0$p))
	270	θ0$p = rep(1/K, K-1)
	271	else if (!is.numeric(θ0$p) \|\| length(θ0$p) != K-1
	272	\|\| any(is.na(θ0$p)) \|\| sum(θ0$p) > 1)
	273	{
	274	stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
	275	}
	276	if (is.null(θ0$b))
	277	θ0$b = rep(0, K)
	278	else if (!is.numeric(θ0$b) \|\| length(θ0$b) != K \|\| any(is.na(θ0$b)))
	279	stop("θ0$b: length K, no NA")
	280
	281	# (Re)Set W to identity, to allow several run from the same object
	282	W <<- diag(d+d^2+d^3)
	283
	284	loopMax <- 2 #TODO: loopMax = 3 ? Seems not improving...
	285	x_init <- linArgs(θ0)
	286	for (loop in 1:loopMax)
	287	{
	288	op_res = constrOptim( x_init, .self$f, .self$grad_f,
	289	ui=cbind(
	290	rbind( rep(-1,K-1), diag(K-1) ),
	291	matrix(0, nrow=K, ncol=(d+1)*K) ),
	292	ci=c(-1,rep(0,K-1)) )
	293	if (loop < loopMax) #avoid computing an extra W
	294	W <<- computeW(expArgs(op_res$par))
	295	#x_init <- op_res$par #degrades performances (TODO: why?)
	296	}
	297
	298	expArgs(op_res$par)
	299	}
	300	)
	301	)
	302
	303	# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
	304	# = E[g^{(order)}(z)] with z~N(b,diag(λ))
	305	# by numerically evaluating the integral.
	306	#
	307	# @param link Link, 'logit' or 'probit'
	308	# @param order Order of derivative
	309	# @param λ Norm of columns of β
	310	# @param b Intercept
	311	#
	312	.G <- function(link, order, λ, b)
	313	{
	314	# NOTE: weird "integral divergent" error on inputs:
	315	# link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
	316	# Switch to pracma package for that (but it seems slow...)
	317	sapply( seq_along(λ), function(k) {
	318	res <- NULL
	319	tryCatch({
	320	# Fast code, may fail:
	321	res <- stats::integrate(
	322	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	323	lower=-Inf, upper=Inf )$value
	324	}, error = function(e) {
	325	# Robust slow code, no fails observed:
	326	sink("/dev/null") #pracma package has some useless printed outputs...
	327	res <- pracma::integral(
	328	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	329	xmin=-Inf, xmax=Inf, method="Kronrod")
	330	sink()
	331	})
	332	res
	333	})
	334	}
	335
	336	# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
	337	#
	338	.deriv <- list(
	339	"probit"=list(
	340	# 'probit' derivatives list;
	341	# NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
	342	function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
	343	function(x) exp(-x^2/2)/(sqrt(2pi)) -x, #g''
	344	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^2 - 1), #g^(3)
	345	function(x) exp(-x^2/2)/(sqrt(2pi)) (-x^3 + 3*x), #g^(4)
	346	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^4 - 6*x^2 + 3) #g^(5)
	347	),
	348	"logit"=list(
	349	# Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
	350	# @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
	351	function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
	352	function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
	353	function(x) {e=exp(x); .zin(e( e^2 - 4e + 1) /(e+1)^4)}, #g^(3)
	354	function(x) {e=exp(x); .zin(e(-e^3 + 11e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
	355	function(x) {e=exp(x); .zin(e( e^4 - 26e^3 + 66e^2 - 26e + 1)/(e+1)^6)} #g^(5)
	356	)
	357	)
	358
	359	# Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
	360	#
	361	# @param x Ratio of polynoms of exponentials, as in .S[[i]]
	362	#
	363	.zin <- function(x)
	364	{
	365	x[is.nan(x)] <- 0.
	366	x
	367	}