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