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