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" = matrix(v[K:(K+d*K-1)], ncol=K),
 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       c(L$p[1:(K-1)], as.double(L$β), L$b)
 116     },
 117
 118     #TODO: compare with R version?
 119     #D <- diag(d) #matrix of ej vectors
 120     #Y * X
 121     #Y * ( t( apply(X, 1, function(row) row %o% row) ) - Reduce('+', lapply(1:d, function(j) as.double(D[j,] %o% D[j,])), rep(0, d*d)))
 122     #Y * ( t( apply(X, 1, function(row) row %o% row %*% row) ) - Reduce('+', lapply(1:d, function(j) ), rep(0, d*d*d)))
 123     computeW = function(θ)
 124     {
 125       #require(MASS)
 126       dd <- d + d^2 + d^3
 127       M <- Moments(θ)
 128       Omega <- matrix( .C("Compute_Omega",
 129         X=as.double(X), Y=as.double(Y), M=as.double(M),
 130         pn=as.integer(n), pd=as.integer(d),
 131         W=as.double(W), PACKAGE="morpheus")$W, nrow=dd, ncol=dd )
 132       W <<- MASS::ginv(Omega, tol=1e-4)
 133       NULL #avoid returning W
 134     },
 135
 136     Moments = function(θ)
 137     {
 138       "Vector of moments, of size d+d^2+d^3"
 139
 140       p <- θ$p
 141       β <- θ$β
 142       λ <- sqrt(colSums(β^2))
 143       b <- θ$b
 144
 145       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 146       β2 <- apply(β, 2, function(col) col %o% col)
 147       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 148
 149       c(
 150         β  %*% (p * .G(li,1,λ,b)),
 151         β2 %*% (p * .G(li,2,λ,b)),
 152         β3 %*% (p * .G(li,3,λ,b)))
 153     },
 154
 155     f = function(θ)
 156     {
 157       "Product t(hat_Mi - Mi) W (hat_Mi - Mi) with Mi(theta)"
 158
 159       L <- expArgs(θ)
 160       A <- as.matrix(Mhat - Moments(L))
 161       t(A) %*% W %*% A
 162     },
 163
 164     grad_f = function(θ)
 165     {
 166       "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 167
 168       L <- expArgs(θ)
 169       -2 * t(grad_M(L)) %*% W %*% as.matrix((Mhat - Moments(L)))
 170     },
 171
 172     grad_M = function(θ)
 173     {
 174       "Gradient of the vector of moments, size (dim=)d+d^2+d^3 x K-1+K+d*K"
 175
 176       p <- θ$p
 177       β <- θ$β
 178       λ <- sqrt(colSums(β^2))
 179       μ <- sweep(β, 2, λ, '/')
 180       b <- θ$b
 181
 182       res <- matrix(nrow=nrow(W), ncol=0)
 183
 184       # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 185       β2 <- apply(β, 2, function(col) col %o% col)
 186       β3 <- apply(β, 2, function(col) col %o% col %o% col)
 187
 188       # Some precomputations
 189       G1 = .G(li,1,λ,b)
 190       G2 = .G(li,2,λ,b)
 191       G3 = .G(li,3,λ,b)
 192       G4 = .G(li,4,λ,b)
 193       G5 = .G(li,5,λ,b)
 194
 195       # Gradient on p: K-1 columns, dim rows
 196       km1 = 1:(K-1)
 197       res <- cbind(res, rbind(
 198         sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K],
 199         sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K],
 200         sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ))
 201
 202       for (i in 1:d)
 203       {
 204         # i determines the derivated matrix dβ[2,3]
 205
 206         dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 207         dβ_right <- matrix(0, nrow=d, ncol=K)
 208         block <- i
 209         dβ_right[block,] <- dβ_right[block,] + 1
 210         dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 211
 212         dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 213         dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 214           sweep(dβ_right, 2, β[j,], '*')
 215         }) )
 216         block <- ((i-1)*d+1):(i*d)
 217         dβ2_right[block,] <- dβ2_right[block,] + β
 218         dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 219
 220         dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 221         dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 222           sweep(dβ2_right, 2, β[j,], '*')
 223         }) )
 224         block <- ((i-1)*d*d+1):(i*d*d)
 225         dβ3_right[block,] <- dβ3_right[block,] + β2
 226         dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 227
 228         res <- cbind(res, rbind(dβ, dβ2, dβ3))
 229       }
 230
 231       # Gradient on b
 232       res <- cbind(res, rbind(
 233         sweep(β,  2, p * G2, '*'),
 234         sweep(β2, 2, p * G3, '*'),
 235         sweep(β3, 2, p * G4, '*') ))
 236
 237       res
 238     },
 239
 240     run = function(θ0)
 241     {
 242       "Run optimization from θ0 with solver..."
 243
 244       if (!is.list(θ0))
 245         stop("θ0: list")
 246       if (is.null(θ0$β))
 247         stop("At least θ0$β must be provided")
 248       if (!is.matrix(θ0$β) || any(is.na(θ0$β)) || ncol(θ0$β) != K)
 249         stop("θ0$β: matrix, no NA, ncol == K")
 250       if (is.null(θ0$p))
 251         θ0$p = rep(1/K, K-1)
 252       else if (length(θ0$p) != K-1 || sum(θ0$p) > 1)
 253         stop("θ0$p should contain positive integers and sum to < 1")
 254       # Next test = heuristic to detect missing b (when matrix is called "beta")
 255       if (is.null(θ0$b) || all(θ0$b == θ0$β))
 256         θ0$b = rep(0, K)
 257       else if (any(is.na(θ0$b)))
 258         stop("θ0$b cannot have missing values")
 259       # TODO: stopping condition? N iterations? Delta <= epsilon ?
 260       for (loop in 1:10)
 261       {
 262         op_res = constrOptim( linArgs(θ0), .self$f, .self$grad_f,
 263           ui=cbind(
 264             rbind( rep(-1,K-1), diag(K-1) ),
 265             matrix(0, nrow=K, ncol=(d+1)*K) ),
 266           ci=c(-1,rep(0,K-1)) )
 267
 268         computeW(expArgs(op_res$par))
 269         # debug:
 270         #print(W)
 271         print(op_res$value)
 272         print(expArgs(op_res$par))
 273       }
 274
 275       expArgs(op_res$par)
 276     }
 277   )
 278 )
 279
 280 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 281 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 282 # by numerically evaluating the integral.
 283 #
 284 # @param link Link, 'logit' or 'probit'
 285 # @param order Order of derivative
 286 # @param λ Norm of columns of β
 287 # @param b Intercept
 288 #
 289 .G <- function(link, order, λ, b)
 290 {
 291   # NOTE: weird "integral divergent" error on inputs:
 292   # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 293   # Switch to pracma package for that (but it seems slow...)
 294   sapply( seq_along(λ), function(k) {
 295     res <- NULL
 296     tryCatch({
 297       # Fast code, may fail:
 298       res <- stats::integrate(
 299         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 300         lower=-Inf, upper=Inf )$value
 301     }, error = function(e) {
 302       # Robust slow code, no fails observed:
 303       sink("/dev/null") #pracma package has some useless printed outputs...
 304       res <- pracma::integral(
 305         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 306         xmin=-Inf, xmax=Inf, method="Kronrod")
 307       sink()
 308     })
 309     res
 310   })
 311 }
 312
 313 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 314 #
 315 .deriv <- list(
 316   "probit"=list(
 317     # 'probit' derivatives list;
 318     # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
 319     function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 320     function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 321     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 322     function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 323     function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 324   ),
 325   "logit"=list(
 326     # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 327     # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 328     function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 329     function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 330     function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 331     function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 332     function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 333   )
 334 )
 335
 336 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 337 #
 338 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 339 #
 340 .zin <- function(x)
 341 {
 342   x[is.nan(x)] <- 0.
 343   x
 344 }