pkg/R/optimParams.R

   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+d*K):(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) + d*K + 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):(i*d)
 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)*d*d+1):(i*d*d)
 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(2*pi)) *  -x,               #g''
 344     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 345     function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 346     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( 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 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 354     function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 355     function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 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 }