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