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

#' Wrapper function for OptimParams class
#'
#' @param K Number of populations.
#' @param link The link type, 'logit' or 'probit'.
#' @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$run(x0)}
#'   outputs the list of optimized parameters
#'   \itemize{
#'     \item p: proportions, size K
#'     \item β: regression matrix, size dxK
#'     \item b: intercepts, size K
#'   }
#'   θ0 is a vector containing respectively the K-1 first elements of p, then β by
#'   columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
#'
#' @seealso \code{multiRun} to estimate statistics based on β, and
#'   \code{generateSampleIO} for I/O random generation.
#'
#' @examples
#' # Optimize parameters from estimated μ
#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
#' μ = computeMu(io$X, io$Y, list(K=2))
#' o <- optimParams(io$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
#' 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(X, Y, K, link=c("logit","probit"))
{
	# 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")

	# Build and return optimization algorithm object
	methods::new("OptimParams", "li"=link, "X"=X,
    "Y"=as.integer(Y), "K"=as.integer(K))
}

#' 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 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"))
				stop("Missing arguments")

      # Precompute empirical moments
      M <- computeMoments(X, Y)
      M1 <- as.double(M[[1]])
      M2 <- as.double(M[[2]])
      M3 <- as.double(M[[3]])
      Mhat <<- c(M1, M2, M3)

			n <<- nrow(X)
			d <<- length(M1)
      W <<- diag(d+d^2+d^3) #initialize at W = Identity
		},

		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" = matrix(v[K:(K+d*K-1)], ncol=K),
				"b" = v[(K+d*K):(K+(d+1)*K-1)])
		},

		linArgs = function(L)
		{
			"Linearize vectors+matrices from list L into a vector"

			c(L$p[1:(K-1)], as.double(L$β), L$b)
		},

    computeW = function(θ)
    {
      dim <- d + d^2 + d^3
      W <<- solve( matrix( .C("Compute_Omega",
        X=as.double(X), Y=as.double(Y), M=as.double(Moments(θ)),
        pn=as.integer(n), pd=as.integer(d),
        W=as.double(W), PACKAGE="morpheus")$W, nrow=dim, ncol=dim) )
      NULL #avoid returning W
    },

    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(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"

      L <- expArgs(θ)
			A <- as.matrix(Moments(L) - Mhat)
      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] ))

      # TODO: understand derivatives order and match the one in optim init param
			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)) || ncol(θ0$β) != K)
				stop("θ0$β: matrix, no NA, ncol == K")
      if (is.null(θ0$p))
        θ0$p = rep(1/K, K-1)
      else if (length(θ0$p) != K-1 || sum(θ0$p) > 1)
        stop("θ0$p should contain positive integers and sum to < 1")
      # Next test = heuristic to detect missing b (when matrix is called "beta")
      if (is.null(θ0$b) || all(θ0$b == θ0$β))
        θ0$b = rep(0, K)
      else if (any(is.na(θ0$b)))
        stop("θ0$b cannot have missing values")

			op_res = constrOptim( linArgs(θ0), .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)) )

      # debug:
      computeW(expArgs(op_res$par))
      print(W)
      # We get a first non-trivial estimation of W
      # TODO: loop, this redefine f, so that we can call constrOptim again...
      # Stopping condition? N iterations? Delta <= epsilon ?

			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
4263503b	1	#' Wrapper function for OptimParams class
cbd88fe5 BA	2	#'
	3	#' @param K Number of populations.
	4	#' @param link The link type, 'logit' or 'probit'.
4263503b BA	5	#' @param X Data matrix of covariables
4263503b BA	6	#' @param Y Output as a binary vector
cbd88fe5 BA	7	#'
	8	#' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
	9	#' outputs the list of optimized parameters
	10	#' \itemize{
	11	#' \item p: proportions, size K
	12	#' \item β: regression matrix, size dxK
	13	#' \item b: intercepts, size K
	14	#' }
7737c2fa BA	15	#' θ0 is a vector containing respectively the K-1 first elements of p, then β by
7737c2fa BA	16	#' columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
cbd88fe5 BA	17	#'
	18	#' @seealso \code{multiRun} to estimate statistics based on β, and
	19	#' \code{generateSampleIO} for I/O random generation.
	20	#'
	21	#' @examples
	22	#' # Optimize parameters from estimated μ
	23	#' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
	24	#' μ = computeMu(io$X, io$Y, list(K=2))
4263503b	25	#' o <- optimParams(io$X, io$Y, 2, "logit")
7737c2fa BA	26	#' θ0 <- list(p=1/2, β=μ, b=c(0,0))
7737c2fa BA	27	#' par0 <- o$run(θ0)
cbd88fe5	28	#' # Compare with another starting point
7737c2fa BA	29	#' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
7737c2fa BA	30	#' par1 <- o$run(θ1)
cbd88fe5 BA	31	#' o$f( o$linArgs(par0) )
	32	#' o$f( o$linArgs(par1) )
	33	#' @export
b389a46a	34	optimParams <- function(X, Y, K, link=c("logit","probit"))
cbd88fe5 BA	35	{
cbd88fe5 BA	36	# Check arguments
4263503b BA	37	if (!is.matrix(X) \|\| any(is.na(X)))
4263503b BA	38	stop("X: numeric matrix, no NAs")
0a630686	39	if (!is.numeric(Y) \|\| any(is.na(Y)) \|\| any(Y!=0 & Y!=1))
4263503b	40	stop("Y: binary vector with 0 and 1 only")
cbd88fe5	41	link <- match.arg(link)
4263503b BA	42	if (!is.numeric(K) \|\| K!=floor(K) \|\| K < 2)
4263503b BA	43	stop("K: integer >= 2")
cbd88fe5	44
cbd88fe5	45	# Build and return optimization algorithm object
4263503b BA	46	methods::new("OptimParams", "li"=link, "X"=X,
4263503b BA	47	"Y"=as.integer(Y), "K"=as.integer(K))
cbd88fe5 BA	48	}
cbd88fe5 BA	49
4263503b BA	50	#' Encapsulated optimization for p (proportions), β and b (regression parameters)
	51	#'
	52	#' Optimize the parameters of a mixture of logistic regressions model, possibly using
	53	#' \code{mu <- computeMu(...)} as a partial starting point.
	54	#'
	55	#' @field li Link function, 'logit' or 'probit'
	56	#' @field X Data matrix of covariables
	57	#' @field Y Output as a binary vector
	58	#' @field K Number of populations
	59	#' @field d Number of dimensions
	60	#' @field W Weights matrix (iteratively refined)
	61	#'
cbd88fe5 BA	62	setRefClass(
	63	Class = "OptimParams",
	64
	65	fields = list(
	66	# Inputs
4263503b BA	67	li = "character", #link function
	68	X = "matrix",
	69	Y = "numeric",
7737c2fa	70	Mhat = "numeric", #vector of empirical moments
cbd88fe5 BA	71	# Dimensions
cbd88fe5 BA	72	K = "integer",
4263503b	73	n = "integer",
e92d9d9d BA	74	d = "integer",
	75	# Weights matrix (generalized least square)
	76	W = "matrix"
cbd88fe5 BA	77	),
	78
	79	methods = list(
	80	initialize = function(...)
	81	{
4263503b	82	"Check args and initialize K, d, W"
cbd88fe5	83
4263503b BA	84	callSuper(...)
4263503b BA	85	if (!hasArg("X") \|\| !hasArg("Y") \|\| !hasArg("K") \|\| !hasArg("li"))
cbd88fe5	86	stop("Missing arguments")
cbd88fe5	87
4263503b	88	# Precompute empirical moments
0a630686	89	M <- computeMoments(X, Y)
7737c2fa BA	90	M1 <- as.double(M[[1]])
	91	M2 <- as.double(M[[2]])
	92	M3 <- as.double(M[[3]])
0a630686	93	Mhat <<- c(M1, M2, M3)
4263503b BA	94
4263503b BA	95	n <<- nrow(X)
cbd88fe5	96	d <<- length(M1)
e92d9d9d	97	W <<- diag(d+d^2+d^3) #initialize at W = Identity
cbd88fe5 BA	98	},
cbd88fe5 BA	99
7737c2fa	100	expArgs = function(v)
cbd88fe5	101	{
7737c2fa	102	"Expand individual arguments from vector v into a list"
cbd88fe5 BA	103
	104	list(
	105	# p: dimension K-1, need to be completed
7737c2fa BA	106	"p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
	107	"β" = matrix(v[K:(K+d*K-1)], ncol=K),
	108	"b" = v[(K+dK):(K+(d+1)K-1)])
cbd88fe5 BA	109	},
cbd88fe5 BA	110
7737c2fa	111	linArgs = function(L)
cbd88fe5	112	{
7737c2fa	113	"Linearize vectors+matrices from list L into a vector"
cbd88fe5	114
7737c2fa	115	c(L$p[1:(K-1)], as.double(L$β), L$b)
cbd88fe5 BA	116	},
cbd88fe5 BA	117
7737c2fa	118	computeW = function(θ)
4263503b BA	119	{
4263503b BA	120	dim <- d + d^2 + d^3
7737c2fa	121	W <<- solve( matrix( .C("Compute_Omega",
b389a46a	122	X=as.double(X), Y=as.double(Y), M=as.double(Moments(θ)),
7737c2fa BA	123	pn=as.integer(n), pd=as.integer(d),
	124	W=as.double(W), PACKAGE="morpheus")$W, nrow=dim, ncol=dim) )
	125	NULL #avoid returning W
4263503b BA	126	},
4263503b BA	127
b389a46a	128	Moments = function(θ)
4263503b	129	{
7737c2fa	130	"Vector of moments, of size d+d^2+d^3"
cbd88fe5	131
7737c2fa BA	132	p <- θ$p
7737c2fa BA	133	β <- θ$β
cbd88fe5	134	λ <- sqrt(colSums(β^2))
7737c2fa	135	b <- θ$b
cbd88fe5 BA	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
0a630686	141	c(
7737c2fa BA	142	β %% (p .G(li,1,λ,b)),
7737c2fa BA	143	β2 %% (p .G(li,2,λ,b)),
0a630686	144	β3 %% (p .G(li,3,λ,b)))
7737c2fa BA	145	},
	146
	147	f = function(θ)
	148	{
	149	"Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"
	150
0a630686 BA	151	L <- expArgs(θ)
0a630686 BA	152	A <- as.matrix(Moments(L) - Mhat)
4263503b BA	153	t(A) %% W %% A
4263503b BA	154	},
cbd88fe5	155
7737c2fa	156	grad_f = function(θ)
cbd88fe5 BA	157	{
	158	"Gradient of f, dimension (K-1) + dK + K = (d+2)K - 1"
	159
0a630686 BA	160	L <- expArgs(θ)
0a630686 BA	161	-2 * t(grad_M(L)) %% W %% as.matrix((Mhat - Moments(L)))
b389a46a	162	},
4263503b	163
7737c2fa	164	grad_M = function(θ)
4263503b	165	{
7737c2fa	166	"Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
4263503b	167
0a630686 BA	168	p <- θ$p
0a630686 BA	169	β <- θ$β
cbd88fe5 BA	170	λ <- sqrt(colSums(β^2))
cbd88fe5 BA	171	μ <- sweep(β, 2, λ, '/')
0a630686	172	b <- θ$b
7737c2fa BA	173
7737c2fa BA	174	res <- matrix(nrow=nrow(W), ncol=0)
cbd88fe5 BA	175
	176	# Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
	177	β2 <- apply(β, 2, function(col) col %o% col)
	178	β3 <- apply(β, 2, function(col) col %o% col %o% col)
	179
	180	# Some precomputations
	181	G1 = .G(li,1,λ,b)
	182	G2 = .G(li,2,λ,b)
	183	G3 = .G(li,3,λ,b)
	184	G4 = .G(li,4,λ,b)
	185	G5 = .G(li,5,λ,b)
	186
7737c2fa	187	# Gradient on p: K-1 columns, dim rows
cbd88fe5	188	km1 = 1:(K-1)
0a630686	189
7737c2fa	190	res <- cbind(res, rbind(
0a630686 BA	191	sweep(as.matrix(β [,km1]), 2, G1[km1], '') - G1[K] β [,K],
	192	sweep(as.matrix(β2[,km1]), 2, G2[km1], '') - G2[K] β2[,K],
	193	sweep(as.matrix(β3[,km1]), 2, G3[km1], '') - G3[K] β3[,K] ))
cbd88fe5	194
d08fef42	195	# TODO: understand derivatives order and match the one in optim init param
cbd88fe5 BA	196	for (i in 1:d)
	197	{
	198	# i determines the derivated matrix dβ[2,3]
	199
	200	dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
	201	dβ_right <- matrix(0, nrow=d, ncol=K)
	202	block <- i
	203	dβ_right[block,] <- dβ_right[block,] + 1
	204	dβ <- dβ_left + sweep(dβ_right, 2, p * G1, '*')
	205
	206	dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
	207	dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
	208	sweep(dβ_right, 2, β[j,], '*')
	209	}) )
	210	block <- ((i-1)d+1):(id)
	211	dβ2_right[block,] <- dβ2_right[block,] + β
	212	dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
	213
	214	dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
	215	dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
	216	sweep(dβ2_right, 2, β[j,], '*')
	217	}) )
	218	block <- ((i-1)dd+1):(idd)
	219	dβ3_right[block,] <- dβ3_right[block,] + β2
	220	dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
	221
0a630686	222	res <- cbind(res, rbind(dβ, dβ2, dβ3))
cbd88fe5	223	}
cbd88fe5	224
7737c2fa BA	225	# Gradient on b
7737c2fa BA	226	res <- cbind(res, rbind(
0a630686 BA	227	sweep(β, 2, p * G2, '*'),
	228	sweep(β2, 2, p * G3, '*'),
	229	sweep(β3, 2, p * G4, '*') ))
cbd88fe5	230
7737c2fa	231	res
cbd88fe5 BA	232	},
cbd88fe5 BA	233
7737c2fa	234	run = function(θ0)
cbd88fe5	235	{
7737c2fa BA	236	"Run optimization from θ0 with solver..."
	237
	238	if (!is.list(θ0))
	239	stop("θ0: list")
	240	if (is.null(θ0$β))
	241	stop("At least θ0$β must be provided")
	242	if (!is.matrix(θ0$β) \|\| any(is.na(θ0$β)) \|\| ncol(θ0$β) != K)
	243	stop("θ0$β: matrix, no NA, ncol == K")
	244	if (is.null(θ0$p))
	245	θ0$p = rep(1/K, K-1)
	246	else if (length(θ0$p) != K-1 \|\| sum(θ0$p) > 1)
	247	stop("θ0$p should contain positive integers and sum to < 1")
d294ece1	248	# Next test = heuristic to detect missing b (when matrix is called "beta")
7737c2fa BA	249	if (is.null(θ0$b) \|\| all(θ0$b == θ0$β))
	250	θ0$b = rep(0, K)
	251	else if (any(is.na(θ0$b)))
	252	stop("θ0$b cannot have missing values")
d294ece1	253
7737c2fa	254	op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
cbd88fe5 BA	255	ui=cbind(
	256	rbind( rep(-1,K-1), diag(K-1) ),
	257	matrix(0, nrow=K, ncol=(d+1)*K) ),
	258	ci=c(-1,rep(0,K-1)) )
	259
7737c2fa	260	# debug:
0a630686 BA	261	computeW(expArgs(op_res$par))
0a630686 BA	262	print(W)
7737c2fa	263	# We get a first non-trivial estimation of W
4263503b	264	# TODO: loop, this redefine f, so that we can call constrOptim again...
b389a46a	265	# Stopping condition? N iterations? Delta <= epsilon ?
4263503b	266
cbd88fe5 BA	267	expArgs(op_res$par)
	268	}
	269	)
	270	)
	271
	272	# Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
	273	# = E[g^{(order)}(z)] with z~N(b,diag(λ))
4263503b	274	# by numerically evaluating the integral.
cbd88fe5 BA	275	#
	276	# @param link Link, 'logit' or 'probit'
	277	# @param order Order of derivative
	278	# @param λ Norm of columns of β
	279	# @param b Intercept
	280	#
	281	.G <- function(link, order, λ, b)
	282	{
	283	# NOTE: weird "integral divergent" error on inputs:
	284	# link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
	285	# Switch to pracma package for that (but it seems slow...)
4263503b BA	286	sapply( seq_along(λ), function(k) {
	287	res <- NULL
	288	tryCatch({
	289	# Fast code, may fail:
	290	res <- stats::integrate(
	291	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	292	lower=-Inf, upper=Inf )$value
	293	}, error = function(e) {
	294	# Robust slow code, no fails observed:
	295	sink("/dev/null") #pracma package has some useless printed outputs...
	296	res <- pracma::integral(
	297	function(z) .deriv[[link]][[order]](λ[k]z+b[k]) exp(-z^2/2) / sqrt(2*pi),
	298	xmin=-Inf, xmax=Inf, method="Kronrod")
	299	sink()
	300	})
	301	res
	302	})
cbd88fe5 BA	303	}
	304
	305	# Derivatives list: g^(k)(x) for links 'logit' and 'probit'
	306	#
	307	.deriv <- list(
	308	"probit"=list(
	309	# 'probit' derivatives list;
4263503b	310	# NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
cbd88fe5 BA	311	function(x) exp(-x^2/2)/(sqrt(2*pi)), #g'
	312	function(x) exp(-x^2/2)/(sqrt(2pi)) -x, #g''
	313	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^2 - 1), #g^(3)
	314	function(x) exp(-x^2/2)/(sqrt(2pi)) (-x^3 + 3*x), #g^(4)
	315	function(x) exp(-x^2/2)/(sqrt(2pi)) ( x^4 - 6*x^2 + 3) #g^(5)
	316	),
	317	"logit"=list(
	318	# Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
	319	# @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
	320	function(x) {e=exp(x); .zin(e /(e+1)^2)}, #g'
	321	function(x) {e=exp(x); .zin(e*(-e + 1) /(e+1)^3)}, #g''
	322	function(x) {e=exp(x); .zin(e( e^2 - 4e + 1) /(e+1)^4)}, #g^(3)
	323	function(x) {e=exp(x); .zin(e(-e^3 + 11e^2 - 11*e + 1) /(e+1)^5)}, #g^(4)
	324	function(x) {e=exp(x); .zin(e( e^4 - 26e^3 + 66e^2 - 26e + 1)/(e+1)^6)} #g^(5)
	325	)
	326	)
	327
	328	# Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
	329	#
	330	# @param x Ratio of polynoms of exponentials, as in .S[[i]]
	331	#
	332	.zin <- function(x)
	333	{
	334	x[is.nan(x)] <- 0.
	335	x
	336	}