pkg/R/optimParams.R

   1 #' optimParams
   2 #'
   3 #' Wrapper function for OptimParams class
   4 #'
   5 #' @name optimParams
   6 #'
   7 #' @param X Data matrix of covariables
   8 #' @param Y Output as a binary vector
   9 #' @param K Number of populations.
  10 #' @param link The link type, 'logit' or 'probit'.
  11 #' @param M the empirical cross-moments between X and Y (optional)
  12 #' @param nc Number of cores (default: 0 to use all)
  13 #'
  14 #' @return An object 'op' of class OptimParams, initialized so that
  15 #'   \code{op$run(θ0)} outputs the list of optimized parameters
  16 #'   \itemize{
  17 #'     \item p: proportions, size K
  18 #'     \item β: regression matrix, size dxK
  19 #'     \item b: intercepts, size K
  20 #'   }
  21 #'   θ0 is a list containing the initial parameters. Only β is required
  22 #'   (p would be set to (1/K,...,1/K) and b to (0,...0)).
  23 #'
  24 #' @seealso \code{multiRun} to estimate statistics based on β, and
  25 #'   \code{generateSampleIO} for I/O random generation.
  26 #'
  27 #' @examples
  28 #' # Optimize parameters from estimated μ
  29 #' io <- generateSampleIO(100,
  30 #'   1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
  31 #' μ <- computeMu(io$X, io$Y, list(K=2))
  32 #' o <- optimParams(io$X, io$Y, 2, "logit")
  33 #' \dontrun{
  34 #' θ0 <- list(p=1/2, β=μ, b=c(0,0))
  35 #' par0 <- o$run(θ0)
  36 #' # Compare with another starting point
  37 #' θ1 <- list(p=1/2, β=2*μ, b=c(0,0))
  38 #' par1 <- o$run(θ1)
  39 #' # Look at the function values at par0 and par1:
  40 #' o$f( o$linArgs(par0) )
  41 #' o$f( o$linArgs(par1) )}
  42 #'
  43 #' @export
  44 optimParams <- function(X, Y, K, link=c("logit","probit"), M=NULL, nc=0)
  45 {
  46   # Check arguments
  47   if (!is.matrix(X) || any(is.na(X)))
  48     stop("X: numeric matrix, no NAs")
  49   if (!is.numeric(Y) || any(is.na(Y)) || any(Y!=0 & Y!=1))
  50     stop("Y: binary vector with 0 and 1 only")
  51   link <- match.arg(link)
  52   if (!is.numeric(K) || K!=floor(K) || K < 2 || K > ncol(X))
  53     stop("K: integer >= 2, <= d")
  54
  55   if (is.null(M))
  56   {
  57     # Precompute empirical moments
  58     Mtmp <- computeMoments(X, Y)
  59     M1 <- as.double(Mtmp[[1]])
  60     M2 <- as.double(Mtmp[[2]])
  61     M3 <- as.double(Mtmp[[3]])
  62     M <- c(M1, M2, M3)
  63   }
  64   else
  65     M <- c(M[[1]], M[[2]], M[[3]])
  66
  67   # Build and return optimization algorithm object
  68   methods::new("OptimParams", "li"=link, "X"=X,
  69     "Y"=as.integer(Y), "K"=as.integer(K), "Mhat"=as.double(M), "nc"=as.integer(nc))
  70 }
  71
  72 # Encapsulated optimization for p (proportions), β and b (regression parameters)
  73 #
  74 # Optimize the parameters of a mixture of logistic regressions model, possibly using
  75 # \code{mu <- computeMu(...)} as a partial starting point.
  76 #
  77 # @field li Link function, 'logit' or 'probit'
  78 # @field X Data matrix of covariables
  79 # @field Y Output as a binary vector
  80 # @field Mhat Vector of empirical moments
  81 # @field K Number of populations
  82 # @field n Number of sample points
  83 # @field d Number of dimensions
  84 # @field nc Number of cores (OpenMP //)
  85 # @field W Weights matrix (initialized at identity)
  86 #
  87 setRefClass(
  88   Class = "OptimParams",
  89
  90   fields = list(
  91     # Inputs
  92     li = "character", #link function
  93     X = "matrix",
  94     Y = "numeric",
  95     Mhat = "numeric", #vector of empirical moments
  96     # Dimensions
  97     K = "integer",
  98     n = "integer",
  99     d = "integer",
 100     nc = "integer",
 101     # Weights matrix (generalized least square)
 102     W = "matrix"
 103   ),
 104
 105   methods = list(
 106     initialize = function(...)
 107     {
 108       "Check args and initialize K, d, W"
 109
 110       callSuper(...)
 111       if (!hasArg("X") || !hasArg("Y") || !hasArg("K")
 112         || !hasArg("li") || !hasArg("Mhat") || !hasArg("nc"))
 113       {
 114         stop("Missing arguments")
 115       }
 116
 117       n <<- nrow(X)
 118       d <<- ncol(X)
 119       # W will be initialized when calling run()
 120     },
 121
 122     expArgs = function(v)
 123     {
 124       "Expand individual arguments from vector v into a list"
 125
 126       list(
 127         # p: dimension K-1, need to be completed
 128         "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
 129         "β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
 130         "b" = v[(K+d*K):(K+(d+1)*K-1)])
 131     },
 132
 133     linArgs = function(L)
 134     {
 135       "Linearize vectors+matrices from list L into a vector"
 136
 137       # β linearized row by row, to match derivatives order
 138       c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
 139     },
 140
 141     # TODO: relocate computeW in utils.R
 142     computeW = function(θ)
 143     {
 144       "Compute the weights matrix from a parameters list"
 145
 146       require(MASS)
 147       dd <- d + d^2 + d^3
 148       M <- Moments(θ)
 149       Omega <- matrix( .C("Compute_Omega",
 150         X=as.double(X), Y=as.integer(Y), M=as.double(M),
 151         pnc=as.integer(nc), pn=as.integer(n), pd=as.integer(d),
 152         W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
 153       MASS::ginv(Omega)
 154     },
 155
 156     Moments = function(θ)
 157     {
 158       "Compute the vector of theoretical moments (size d+d^2+d^3)"
 159
 160       p <- θ$p
 161       β <- θ$β
 162       λ <- sqrt(colSums(β^2))
 163       b <- θ$b
 164
 165       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 166       β2 <- apply(β, 2, function(col) col %o% col)
 167       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 168
 169       c(
 170         β  %*% (p * .G(li,1,λ,b)),
 171         β2 %*% (p * .G(li,2,λ,b)),
 172         β3 %*% (p * .G(li,3,λ,b)))
 173     },
 174
 175     f = function(θ)
 176     {
 177       "Function to minimize: t(hat_Mi - Mi(θ)) . W . (hat_Mi - Mi(θ))"
 178
 179       L <- expArgs(θ)
 180       A <- as.matrix(Mhat - Moments(L))
 181       t(A) %*% W %*% A
 182     },
 183
 184     grad_f = function(θ)
 185     {
 186       "Gradient of f: vector of size (K-1) + d*K + K = (d+2)*K - 1"
 187
 188       L <- expArgs(θ)
 189       -2 * t(grad_M(L)) %*% W %*% as.matrix(Mhat - Moments(L))
 190     },
 191
 192     grad_M = function(θ)
 193     {
 194       "Gradient of the moments vector: matrix of size d+d^2+d^3 x K-1+K+d*K"
 195
 196       p <- θ$p
 197       β <- θ$β
 198       λ <- sqrt(colSums(β^2))
 199       μ <- sweep(β, 2, λ, '/')
 200       b <- θ$b
 201
 202       res <- matrix(nrow=nrow(W), ncol=0)
 203
 204       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 205       β2 <- apply(β, 2, function(col) col %o% col)
 206       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 207
 208       # Some precomputations
 209       G1 = .G(li,1,λ,b)
 210       G2 = .G(li,2,λ,b)
 211       G3 = .G(li,3,λ,b)
 212       G4 = .G(li,4,λ,b)
 213       G5 = .G(li,5,λ,b)
 214
 215       # Gradient on p: K-1 columns, dim rows
 216       km1 = 1:(K-1)
 217       res <- cbind(res, rbind(
 218         sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
 219         sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
 220         sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ))
 221
 222       for (i in 1:d)
 223       {
 224         # i determines the derivated matrix dβ[2,3]
 225
 226         dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 227         dβ_right <- matrix(0, nrow=d, ncol=K)
 228         block <- i
 229         dβ_right[block,] <- dβ_right[block,] + 1
 230         dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 231
 232         dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 233         dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 234           sweep(dβ_right, 2, β[j,], '*')
 235         }) )
 236         block <- ((i-1)*d+1):(i*d)
 237         dβ2_right[block,] <- dβ2_right[block,] + β
 238         dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 239
 240         dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 241         dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 242           sweep(dβ2_right, 2, β[j,], '*')
 243         }) )
 244         block <- ((i-1)*d*d+1):(i*d*d)
 245         dβ3_right[block,] <- dβ3_right[block,] + β2
 246         dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 247
 248         res <- cbind(res, rbind(dβ, dβ2, dβ3))
 249       }
 250
 251       # Gradient on b
 252       res <- cbind(res, rbind(
 253         sweep(β,  2, p * G2, '*'),
 254         sweep(β2, 2, p * G3, '*'),
 255         sweep(β3, 2, p * G4, '*') ))
 256
 257       res
 258     },
 259
 260     # userW allows to bypass the W optimization by giving a W matrix
 261     run = function(θ0, userW=NULL)
 262     {
 263       "Run optimization from θ0 with solver..."
 264
 265       if (!is.list(θ0))
 266         stop("θ0: list")
 267       if (is.null(θ0$β))
 268         stop("At least θ0$β must be provided")
 269       if (!is.matrix(θ0$β) || any(is.na(θ0$β))
 270         || nrow(θ0$β) != d || ncol(θ0$β) != K)
 271       {
 272         stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
 273       }
 274       if (is.null(θ0$p))
 275         θ0$p = rep(1/K, K-1)
 276       else if (!is.numeric(θ0$p) || length(θ0$p) != K-1
 277         || any(is.na(θ0$p)) || sum(θ0$p) > 1)
 278       {
 279         stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
 280       }
 281       # NOTE: [["b"]] instead of $b because $b would match $beta (in pkg-cran)
 282       if (is.null(θ0[["b"]]))
 283         θ0$b = rep(0, K)
 284       else if (!is.numeric(θ0$b) || length(θ0$b) != K || any(is.na(θ0$b)))
 285         stop("θ0$b: length K, no NA")
 286
 287       # (Re)Set W to identity, to allow several run from the same object
 288       W <<- if (is.null(userW)) diag(d+d^2+d^3) else userW
 289
 290       # NOTE: loopMax = 3 seems to not improve the final results.
 291       loopMax <- ifelse(is.null(userW), 2, 1)
 292       x_init <- linArgs(θ0)
 293       for (loop in 1:loopMax)
 294       {
 295         op_res <- constrOptim( x_init, .self$f, .self$grad_f,
 296           ui=cbind(
 297             rbind( rep(-1,K-1), diag(K-1) ),
 298             matrix(0, nrow=K, ncol=(d+1)*K) ),
 299           ci=c(-1,rep(0,K-1)) )
 300         if (loop < loopMax) #avoid computing an extra W
 301           W <<- computeW(expArgs(op_res$par))
 302         #x_init <- op_res$par #degrades performances (TODO: why?)
 303       }
 304
 305       expArgs(op_res$par)
 306     }
 307   )
 308 )
 309
 310 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 311 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 312 # by numerically evaluating the integral.
 313 #
 314 # @param link Link, 'logit' or 'probit'
 315 # @param order Order of derivative
 316 # @param λ Norm of columns of β
 317 # @param b Intercept
 318 #
 319 .G <- function(link, order, λ, b)
 320 {
 321   # NOTE: weird "integral divergent" error on inputs:
 322   # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 323   # Switch to pracma package for that (but it seems slow...)
 324   sapply( seq_along(λ), function(k) {
 325     res <- NULL
 326     tryCatch({
 327       # Fast code, may fail:
 328       res <- stats::integrate(
 329         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 330         lower=-Inf, upper=Inf )$value
 331     }, error = function(e) {
 332       # Robust slow code, no fails observed:
 333       sink("/dev/null") #pracma package has some useless printed outputs...
 334       res <- pracma::integral(
 335         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 336         xmin=-Inf, xmax=Inf, method="Kronrod")
 337       sink()
 338     })
 339     res
 340   })
 341 }
 342
 343 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 344 #
 345 .deriv <- list(
 346   "probit"=list(
 347     # 'probit' derivatives list;
 348     # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
 349     function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 350     function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 351     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 352     function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 353     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 354   ),
 355   "logit"=list(
 356     # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 357     # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 358     function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 359     function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 360     function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 361     function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 362     function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 363   )
 364 )
 365
 366 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 367 #
 368 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 369 #
 370 .zin <- function(x)
 371 {
 372   x[is.nan(x)] <- 0.
 373   x
 374 }