pkg/R/optimParams.R

   1 #' Wrapper function for OptimParams class
   2 #'
   3 #' @param K Number of populations.
   4 #' @param link The link type, 'logit' or 'probit'.
   5 #' @param X Data matrix of covariables
   6 #' @param Y Output as a binary vector
   7 #'
   8 #' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
   9 #'   outputs the list of optimized parameters
  10 #'   \itemize{
  11 #'     \item p: proportions, size K
  12 #'     \item β: regression matrix, size dxK
  13 #'     \item b: intercepts, size K
  14 #'   }
  15 #'   θ0 is a vector containing respectively the K-1 first elements of p, then β by
  16 #'   columns, and finally b: \code{θ0 = c(p[1:(K-1)],as.double(β),b)}.
  17 #'
  18 #' @seealso \code{multiRun} to estimate statistics based on β, and
  19 #'   \code{generateSampleIO} for I/O random generation.
  20 #'
  21 #' @examples
  22 #' # Optimize parameters from estimated μ
  23 #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
  24 #' μ = computeMu(io$X, io$Y, list(K=2))
  25 #' o <- optimParams(io$X, io$Y, 2, "logit")
  26 #' θ0 <- list(p=1/2, β=μ, b=c(0,0))
  27 #' par0 <- o$run(θ0)
  28 #' # Compare with another starting point
  29 #' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
  30 #' par1 <- o$run(θ1)
  31 #' o$f( o$linArgs(par0) )
  32 #' o$f( o$linArgs(par1) )
  33 #' @export
  34 optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL)
  35 {
  36   # Check arguments
  37   if (!is.matrix(X) || any(is.na(X)))
  38     stop("X: numeric matrix, no NAs")
  39   if (!is.numeric(Y) || any(is.na(Y)) || any(Y!=0 & Y!=1))
  40     stop("Y: binary vector with 0 and 1 only")
  41   link <- match.arg(link)
  42   if (!is.numeric(K) || K!=floor(K) || K < 2)
  43     stop("K: integer >= 2")
  44
  45   if (is.null(M))
  46   {
  47     # Precompute empirical moments
  48     Mtmp <- computeMoments(X, Y)
  49     M1 <- as.double(Mtmp[[1]])
  50     M2 <- as.double(Mtmp[[2]])
  51     M3 <- as.double(Mtmp[[3]])
  52     M <- c(M1, M2, M3)
  53   }
  54
  55   # Build and return optimization algorithm object
  56   methods::new("OptimParams", "li"=link, "X"=X,
  57     "Y"=as.integer(Y), "K"=as.integer(K), "Mhat"=as.double(M))
  58 }
  59
  60 #' Encapsulated optimization for p (proportions), β and b (regression parameters)
  61 #'
  62 #' Optimize the parameters of a mixture of logistic regressions model, possibly using
  63 #' \code{mu <- computeMu(...)} as a partial starting point.
  64 #'
  65 #' @field li Link function, 'logit' or 'probit'
  66 #' @field X Data matrix of covariables
  67 #' @field Y Output as a binary vector
  68 #' @field K Number of populations
  69 #' @field d Number of dimensions
  70 #' @field W Weights matrix (iteratively refined)
  71 #'
  72 setRefClass(
  73   Class = "OptimParams",
  74
  75   fields = list(
  76     # Inputs
  77     li = "character", #link function
  78     X = "matrix",
  79     Y = "numeric",
  80     Mhat = "numeric", #vector of empirical moments
  81     # Dimensions
  82     K = "integer",
  83     n = "integer",
  84     d = "integer",
  85     # Weights matrix (generalized least square)
  86     W = "matrix"
  87   ),
  88
  89   methods = list(
  90     initialize = function(...)
  91     {
  92       "Check args and initialize K, d, W"
  93
  94       callSuper(...)
  95       if (!hasArg("X") || !hasArg("Y") || !hasArg("K")
  96         || !hasArg("li") || !hasArg("Mhat"))
  97       {
  98         stop("Missing arguments")
  99       }
 100
 101       n <<- nrow(X)
 102       d <<- ncol(X)
 103       W <<- diag(d+d^2+d^3) #initialize at W = Identity
 104     },
 105
 106     expArgs = function(v)
 107     {
 108       "Expand individual arguments from vector v into a list"
 109
 110       list(
 111         # p: dimension K-1, need to be completed
 112         "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
 113         "β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
 114         "b" = v[(K+d*K):(K+(d+1)*K-1)])
 115     },
 116
 117     linArgs = function(L)
 118     {
 119       "Linearize vectors+matrices from list L into a vector"
 120
 121       # β linearized row by row, to match derivatives order
 122       c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
 123     },
 124
 125     computeW = function(θ)
 126     {
 127       require(MASS)
 128       dd <- d + d^2 + d^3
 129       M <- Moments(θ)
 130       Omega <- matrix( .C("Compute_Omega",
 131         X=as.double(X), Y=as.integer(Y), M=as.double(M),
 132         pn=as.integer(n), pd=as.integer(d),
 133         W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
 134       MASS::ginv(Omega)
 135     },
 136
 137     Moments = function(θ)
 138     {
 139       "Vector of moments, of size d+d^2+d^3"
 140
 141       p <- θ$p
 142       β <- θ$β
 143       λ <- sqrt(colSums(β^2))
 144       b <- θ$b
 145
 146       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 147       β2 <- apply(β, 2, function(col) col %o% col)
 148       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 149
 150       c(
 151         β  %*% (p * .G(li,1,λ,b)),
 152         β2 %*% (p * .G(li,2,λ,b)),
 153         β3 %*% (p * .G(li,3,λ,b)))
 154     },
 155
 156     f = function(θ)
 157     {
 158       "Product t(hat_Mi - Mi) W (hat_Mi - Mi) with Mi(theta)"
 159
 160       L <- expArgs(θ)
 161       A <- as.matrix(Mhat - Moments(L))
 162       t(A) %*% W %*% A
 163     },
 164
 165     grad_f = function(θ)
 166     {
 167       "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 168
 169       L <- expArgs(θ)
 170       -2 * t(grad_M(L)) %*% W %*% as.matrix(Mhat - Moments(L))
 171     },
 172
 173     grad_M = function(θ)
 174     {
 175       "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
 176
 177       p <- θ$p
 178       β <- θ$β
 179       λ <- sqrt(colSums(β^2))
 180       μ <- sweep(β, 2, λ, '/')
 181       b <- θ$b
 182
 183       res <- matrix(nrow=nrow(W), ncol=0)
 184
 185       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 186       β2 <- apply(β, 2, function(col) col %o% col)
 187       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 188
 189       # Some precomputations
 190       G1 = .G(li,1,λ,b)
 191       G2 = .G(li,2,λ,b)
 192       G3 = .G(li,3,λ,b)
 193       G4 = .G(li,4,λ,b)
 194       G5 = .G(li,5,λ,b)
 195
 196       # Gradient on p: K-1 columns, dim rows
 197       km1 = 1:(K-1)
 198       res <- cbind(res, rbind(
 199         sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
 200         sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
 201         sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ))
 202
 203       for (i in 1:d)
 204       {
 205         # i determines the derivated matrix dβ[2,3]
 206
 207         dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 208         dβ_right <- matrix(0, nrow=d, ncol=K)
 209         block <- i
 210         dβ_right[block,] <- dβ_right[block,] + 1
 211         dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 212
 213         dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 214         dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 215           sweep(dβ_right, 2, β[j,], '*')
 216         }) )
 217         block <- ((i-1)*d+1):(i*d)
 218         dβ2_right[block,] <- dβ2_right[block,] + β
 219         dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 220
 221         dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 222         dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 223           sweep(dβ2_right, 2, β[j,], '*')
 224         }) )
 225         block <- ((i-1)*d*d+1):(i*d*d)
 226         dβ3_right[block,] <- dβ3_right[block,] + β2
 227         dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 228
 229         res <- cbind(res, rbind(dβ, dβ2, dβ3))
 230       }
 231
 232       # Gradient on b
 233       res <- cbind(res, rbind(
 234         sweep(β,  2, p * G2, '*'),
 235         sweep(β2, 2, p * G3, '*'),
 236         sweep(β3, 2, p * G4, '*') ))
 237
 238       res
 239     },
 240
 241     run = function(θ0)
 242     {
 243       "Run optimization from θ0 with solver..."
 244
 245       if (!is.list(θ0))
 246         stop("θ0: list")
 247       if (is.null(θ0$β))
 248         stop("At least θ0$β must be provided")
 249       if (!is.matrix(θ0$β) || any(is.na(θ0$β))
 250         || nrow(θ0$β) != d || ncol(θ0$β) != K)
 251       {
 252         stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
 253       }
 254       if (is.null(θ0$p))
 255         θ0$p = rep(1/K, K-1)
 256       else if (!is.numeric(θ0$p) || length(θ0$p) != K-1
 257         || any(is.na(θ0$p)) || sum(θ0$p) > 1)
 258       {
 259         stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
 260       }
 261       if (is.null(θ0$b))
 262         θ0$b = rep(0, K)
 263       else if (!is.numeric(θ0$b) || length(θ0$b) != K || any(is.na(θ0$b)))
 264         stop("θ0$b: length K, no NA")
 265       # TODO: stopping condition? N iterations? Delta <= epsilon ?
 266       loopMax <- 2
 267       for (loop in 1:loopMax)
 268       {
 269         op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
 270           ui=cbind(
 271             rbind( rep(-1,K-1), diag(K-1) ),
 272             matrix(0, nrow=K, ncol=(d+1)*K) ),
 273           ci=c(-1,rep(0,K-1)) )
 274         if (loop < loopMax) #avoid computing an extra W
 275           W <<- computeW(expArgs(op_res$par))
 276         #print(op_res$value) #debug
 277         #print(expArgs(op_res$par)) #debug
 278       }
 279
 280       expArgs(op_res$par)
 281     }
 282   )
 283 )
 284
 285 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 286 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 287 # by numerically evaluating the integral.
 288 #
 289 # @param link Link, 'logit' or 'probit'
 290 # @param order Order of derivative
 291 # @param λ Norm of columns of β
 292 # @param b Intercept
 293 #
 294 .G <- function(link, order, λ, b)
 295 {
 296   # NOTE: weird "integral divergent" error on inputs:
 297   # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 298   # Switch to pracma package for that (but it seems slow...)
 299   sapply( seq_along(λ), function(k) {
 300     res <- NULL
 301     tryCatch({
 302       # Fast code, may fail:
 303       res <- stats::integrate(
 304         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 305         lower=-Inf, upper=Inf )$value
 306     }, error = function(e) {
 307       # Robust slow code, no fails observed:
 308       sink("/dev/null") #pracma package has some useless printed outputs...
 309       res <- pracma::integral(
 310         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 311         xmin=-Inf, xmax=Inf, method="Kronrod")
 312       sink()
 313     })
 314     res
 315   })
 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     # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
 324     function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 325     function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 326     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 327     function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 328     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( 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 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 336     function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 337     function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 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 }