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"))
  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   # Build and return optimization algorithm object
  46   methods::new("OptimParams", "li"=link, "X"=X,
  47     "Y"=as.integer(Y), "K"=as.integer(K))
  48 }
  49
  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 #'
  62 setRefClass(
  63   Class = "OptimParams",
  64
  65   fields = list(
  66     # Inputs
  67     li = "character", #link function
  68     X = "matrix",
  69     Y = "numeric",
  70     Mhat = "numeric", #vector of empirical moments
  71     # Dimensions
  72     K = "integer",
  73     n = "integer",
  74     d = "integer",
  75     # Weights matrix (generalized least square)
  76     W = "matrix"
  77   ),
  78
  79   methods = list(
  80     initialize = function(...)
  81     {
  82       "Check args and initialize K, d, W"
  83
  84       callSuper(...)
  85       if (!hasArg("X") || !hasArg("Y") || !hasArg("K") || !hasArg("li"))
  86         stop("Missing arguments")
  87
  88       # Precompute empirical moments
  89       M <- computeMoments(X, Y)
  90       M1 <- as.double(M[[1]])
  91       M2 <- as.double(M[[2]])
  92       M3 <- as.double(M[[3]])
  93       Mhat <<- c(M1, M2, M3)
  94
  95       n <<- nrow(X)
  96       d <<- length(M1)
  97       W <<- diag(d+d^2+d^3) #initialize at W = Identity
  98     },
  99
 100     expArgs = function(v)
 101     {
 102       "Expand individual arguments from vector v into a list"
 103
 104       list(
 105         # p: dimension K-1, need to be completed
 106         "p" = c(v[1:(K-1)], 1-sum(v[1:(K-1)])),
 107         "β" = t(matrix(v[K:(K+d*K-1)], ncol=d)),
 108         "b" = v[(K+d*K):(K+(d+1)*K-1)])
 109     },
 110
 111     linArgs = function(L)
 112     {
 113       "Linearize vectors+matrices from list L into a vector"
 114
 115       # β linearized row by row, to match derivatives order
 116       c(L$p[1:(K-1)], as.double(t(L$β)), L$b)
 117     },
 118
 119     computeW = function(θ)
 120     {
 121       require(MASS)
 122       dd <- d + d^2 + d^3
 123       M <- Moments(θ)
 124       Omega <- matrix( .C("Compute_Omega",
 125         X=as.double(X), Y=as.integer(Y), M=as.double(M),
 126         pn=as.integer(n), pd=as.integer(d),
 127         W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
 128       MASS::ginv(Omega)
 129     },
 130
 131     Moments = function(θ)
 132     {
 133       "Vector of moments, of size d+d^2+d^3"
 134
 135       p <- θ$p
 136       β <- θ$β
 137       λ <- sqrt(colSums(β^2))
 138       b <- θ$b
 139
 140       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 141       β2 <- apply(β, 2, function(col) col %o% col)
 142       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 143
 144       c(
 145         β  %*% (p * .G(li,1,λ,b)),
 146         β2 %*% (p * .G(li,2,λ,b)),
 147         β3 %*% (p * .G(li,3,λ,b)))
 148     },
 149
 150     f = function(θ)
 151     {
 152       "Product t(hat_Mi - Mi) W (hat_Mi - Mi) with Mi(theta)"
 153
 154       L <- expArgs(θ)
 155       A <- as.matrix(Mhat - Moments(L))
 156       t(A) %*% W %*% A
 157     },
 158
 159     grad_f = function(θ)
 160     {
 161       "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 162
 163       L <- expArgs(θ)
 164       -2 * t(grad_M(L)) %*% W %*% as.matrix(Mhat - Moments(L))
 165     },
 166
 167     grad_M = function(θ)
 168     {
 169       "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
 170
 171       p <- θ$p
 172       β <- θ$β
 173       λ <- sqrt(colSums(β^2))
 174       μ <- sweep(β, 2, λ, '/')
 175       b <- θ$b
 176
 177       res <- matrix(nrow=nrow(W), ncol=0)
 178
 179       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 180       β2 <- apply(β, 2, function(col) col %o% col)
 181       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 182
 183       # Some precomputations
 184       G1 = .G(li,1,λ,b)
 185       G2 = .G(li,2,λ,b)
 186       G3 = .G(li,3,λ,b)
 187       G4 = .G(li,4,λ,b)
 188       G5 = .G(li,5,λ,b)
 189
 190       # Gradient on p: K-1 columns, dim rows
 191       km1 = 1:(K-1)
 192       res <- cbind(res, rbind(
 193         sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
 194         sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
 195         sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ))
 196
 197       for (i in 1:d)
 198       {
 199         # i determines the derivated matrix dβ[2,3]
 200
 201         dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 202         dβ_right <- matrix(0, nrow=d, ncol=K)
 203         block <- i
 204         dβ_right[block,] <- dβ_right[block,] + 1
 205         dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 206
 207         dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 208         dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 209           sweep(dβ_right, 2, β[j,], '*')
 210         }) )
 211         block <- ((i-1)*d+1):(i*d)
 212         dβ2_right[block,] <- dβ2_right[block,] + β
 213         dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 214
 215         dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 216         dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 217           sweep(dβ2_right, 2, β[j,], '*')
 218         }) )
 219         block <- ((i-1)*d*d+1):(i*d*d)
 220         dβ3_right[block,] <- dβ3_right[block,] + β2
 221         dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 222
 223         res <- cbind(res, rbind(dβ, dβ2, dβ3))
 224       }
 225
 226       # Gradient on b
 227       res <- cbind(res, rbind(
 228         sweep(β,  2, p * G2, '*'),
 229         sweep(β2, 2, p * G3, '*'),
 230         sweep(β3, 2, p * G4, '*') ))
 231
 232       res
 233     },
 234
 235     run = function(θ0)
 236     {
 237       "Run optimization from θ0 with solver..."
 238
 239       if (!is.list(θ0))
 240         stop("θ0: list")
 241       if (is.null(θ0$β))
 242         stop("At least θ0$β must be provided")
 243       if (!is.matrix(θ0$β) || any(is.na(θ0$β))
 244         || nrow(θ0$β) != d || ncol(θ0$β) != K)
 245       {
 246         stop("θ0$β: matrix, no NA, nrow = d, ncol = K")
 247       }
 248       if (is.null(θ0$p))
 249         θ0$p = rep(1/K, K-1)
 250       else if (!is.numeric(θ0$p) || length(θ0$p) != K-1
 251         || any(is.na(θ0$p)) || sum(θ0$p) > 1)
 252       {
 253         stop("θ0$p: length K-1, no NA, positive integers, sum to <= 1")
 254       }
 255       if (is.null(θ0$b))
 256         θ0$b = rep(0, K)
 257       else if (!is.numeric(θ0$b) || length(θ0$b) != K || any(is.na(θ0$b)))
 258         stop("θ0$b: length K, no NA")
 259       # TODO: stopping condition? N iterations? Delta <= epsilon ?
 260       loopMax <- 2
 261       for (loop in 1:loopMax)
 262       {
 263         op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
 264           ui=cbind(
 265             rbind( rep(-1,K-1), diag(K-1) ),
 266             matrix(0, nrow=K, ncol=(d+1)*K) ),
 267           ci=c(-1,rep(0,K-1)) )
 268         if (loop < loopMax) #avoid computing an extra W
 269           W <<- computeW(expArgs(op_res$par))
 270         #print(op_res$value) #debug
 271         #print(expArgs(op_res$par)) #debug
 272       }
 273
 274       expArgs(op_res$par)
 275     }
 276   )
 277 )
 278
 279 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 280 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 281 # by numerically evaluating the integral.
 282 #
 283 # @param link Link, 'logit' or 'probit'
 284 # @param order Order of derivative
 285 # @param λ Norm of columns of β
 286 # @param b Intercept
 287 #
 288 .G <- function(link, order, λ, b)
 289 {
 290   # NOTE: weird "integral divergent" error on inputs:
 291   # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 292   # Switch to pracma package for that (but it seems slow...)
 293   sapply( seq_along(λ), function(k) {
 294     res <- NULL
 295     tryCatch({
 296       # Fast code, may fail:
 297       res <- stats::integrate(
 298         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 299         lower=-Inf, upper=Inf )$value
 300     }, error = function(e) {
 301       # Robust slow code, no fails observed:
 302       sink("/dev/null") #pracma package has some useless printed outputs...
 303       res <- pracma::integral(
 304         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 305         xmin=-Inf, xmax=Inf, method="Kronrod")
 306       sink()
 307     })
 308     res
 309   })
 310 }
 311
 312 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 313 #
 314 .deriv <- list(
 315   "probit"=list(
 316     # 'probit' derivatives list;
 317     # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
 318     function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 319     function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 320     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 321     function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 322     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 323   ),
 324   "logit"=list(
 325     # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 326     # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 327     function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 328     function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 329     function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 330     function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 331     function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 332   )
 333 )
 334
 335 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 336 #
 337 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 338 #
 339 .zin <- function(x)
 340 {
 341   x[is.nan(x)] <- 0.
 342   x
 343 }