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$run(x0)}
   9 #'   outputs the list of optimized parameters
  10 #'   \itemize{
  11 #'     \item p: proportions, size K
  12 #'     \item β: regression matrix, size dxK
  13 #'     \item b: intercepts, size K
  14 #'   }
  15 #'   x0 is a vector containing respectively the K-1 first elements of p, then β by
  16 #'   columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
  17 #'
  18 #' @seealso \code{multiRun} to estimate statistics based on β, and
  19 #'   \code{generateSampleIO} for I/O random generation.
  20 #'
  21 #' @examples
  22 #' # Optimize parameters from estimated μ
  23 #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
  24 #' μ = computeMu(io$X, io$Y, list(K=2))
  25 #' o <- optimParams(io$X, io$Y, 2, "logit")
  26 #' x0 <- list(p=1/2, β=μ, b=c(0,0))
  27 #' par0 <- o$run(x0)
  28 #' # Compare with another starting point
  29 #' x1 <- list(p=1/2, β=2*μ, b=c(0,0))
  30 #' par1 <- o$run(x1)
  31 #' o$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         M1 = "numeric",
  71         M2 = "numeric", #M2 easier to process as a vector
  72         M3 = "numeric", #same for M3
  73         # Dimensions
  74         K = "integer",
  75     n = "integer",
  76         d = "integer",
  77     # Weights matrix (generalized least square)
  78     W = "matrix"
  79     ),
  80
  81     methods = list(
  82         initialize = function(...)
  83         {
  84             "Check args and initialize K, d, W"
  85
  86       callSuper(...)
  87             if (!hasArg("X") || !hasArg("Y") || !hasArg("K") || !hasArg("li"))
  88                 stop("Missing arguments")
  89
  90       # Precompute empirical moments
  91       M <- computeMoments(optargs$X,optargs$Y)
  92       M1 <<- as.double(M[[1]])
  93       M2 <<- as.double(M[[2]])
  94       M3 <<- as.double(M[[3]])
  95
  96             n <<- nrow(X)
  97             d <<- length(M1)
  98       W <<- diag(d+d^2+d^3) #initialize at W = Identity
  99         },
 100
 101         expArgs = function(x)
 102         {
 103             "Expand individual arguments from vector x"
 104
 105             list(
 106                 # p: dimension K-1, need to be completed
 107                 "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
 108                 "β" = matrix(x[K:(K+d*K-1)], ncol=K),
 109                 "b" = x[(K+d*K):(K+(d+1)*K-1)])
 110         },
 111
 112         linArgs = function(o)
 113         {
 114             " Linearize vectors+matrices into a vector x"
 115
 116             c(o$p[1:(K-1)], as.double(o$β), o$b)
 117         },
 118
 119     getOmega = function(theta)
 120     {
 121       dim <- d + d^2 + d^3
 122       matrix( .C("Compute_Omega",
 123         X=as.double(X), Y=as.double(Y), pn=as.integer(n), pd=as.integer(d),
 124         p=as.double(theta$p), β=as.double(theta$β), b=as.double(theta$b),
 125         W=as.double(W), PACKAGE="morpheus")$W, nrow=dim, ncol=dim)
 126     },
 127
 128     f = function(theta)
 129     {
 130             "Product t(Mi - hat_Mi) W (Mi - hat_Mi) with Mi(theta)"
 131
 132       p <- theta$p
 133             β <- theta$β
 134             λ <- sqrt(colSums(β^2))
 135             b <- theta$b
 136
 137             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 138             β2 <- apply(β, 2, function(col) col %o% col)
 139             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 140
 141             A <- matrix(c(
 142                 β  %*% (p * .G(li,1,λ,b)) - M1,
 143                 β2 %*% (p * .G(li,2,λ,b)) - M2,
 144                 β3 %*% (p * .G(li,3,λ,b)) - M3), ncol=1)
 145       t(A) %*% W %*% A
 146     },
 147
 148         grad_f = function(x)
 149         {
 150             "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 151
 152       # TODO: formula -2 t(grad M(theta)) . W . (Mhat - M(theta))
 153     }
 154
 155     grad_M = function(theta)
 156     {
 157       # TODO: adapt code below for grad of d+d^2+d^3 vector of moments,
 158       # instead of grad (sum(Mhat-M(theta)^2)) --> should be easier
 159
 160       P <- expArgs(x)
 161             p <- P$p
 162             β <- P$β
 163             λ <- sqrt(colSums(β^2))
 164             μ <- sweep(β, 2, λ, '/')
 165             b <- P$b
 166
 167             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 168             β2 <- apply(β, 2, function(col) col %o% col)
 169             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 170
 171             # Some precomputations
 172             G1 = .G(li,1,λ,b)
 173             G2 = .G(li,2,λ,b)
 174             G3 = .G(li,3,λ,b)
 175             G4 = .G(li,4,λ,b)
 176             G5 = .G(li,5,λ,b)
 177
 178             # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
 179             F1 = as.double( 2 * ( β  %*% (p * G1) - M1 ) )
 180             F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
 181             F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
 182
 183             km1 = 1:(K-1)
 184             grad <- #gradient on p
 185               t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
 186                 t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
 187                 t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
 188
 189             grad_β <- matrix(nrow=d, ncol=K)
 190             for (i in 1:d)
 191             {
 192                 # i determines the derivated matrix dβ[2,3]
 193
 194                 dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 195                 dβ_right <- matrix(0, nrow=d, ncol=K)
 196                 block <- i
 197                 dβ_right[block,] <- dβ_right[block,] + 1
 198                 dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 199
 200                 dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 201                 dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 202                     sweep(dβ_right, 2, β[j,], '*')
 203                 }) )
 204                 block <- ((i-1)*d+1):(i*d)
 205                 dβ2_right[block,] <- dβ2_right[block,] + β
 206                 dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 207
 208                 dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 209                 dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 210                     sweep(dβ2_right, 2, β[j,], '*')
 211                 }) )
 212                 block <- ((i-1)*d*d+1):(i*d*d)
 213                 dβ3_right[block,] <- dβ3_right[block,] + β2
 214                 dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 215
 216                 grad_β[i,] <- t(dβ) %*% F1 + t(dβ2) %*% F2 + t(dβ3) %*% F3
 217             }
 218             grad <- c(grad, as.double(grad_β))
 219
 220             grad = c(grad, #gradient on b
 221                 t( sweep(β,  2, p * G2, '*') ) %*% F1 +
 222                 t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
 223                 t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
 224
 225             grad
 226         },
 227
 228     # TODO: rename x(0) into theta(0) --> θ
 229         run = function(x0)
 230         {
 231             "Run optimization from x0 with solver..."
 232
 233         if (!is.list(x0))
 234             stop("x0: list")
 235       if (is.null(x0$β))
 236         stop("At least x0$β must be provided")
 237             if (!is.matrix(x0$β) || any(is.na(x0$β)) || ncol(x0$β) != K)
 238                 stop("x0$β: matrix, no NA, ncol == K")
 239       if (is.null(x0$p))
 240         x0$p = rep(1/K, K-1)
 241       else if (length(x0$p) != K-1 || sum(x0$p) > 1)
 242         stop("x0$p should contain positive integers and sum to < 1")
 243       # Next test = heuristic to detect missing b (when matrix is called "beta")
 244       if (is.null(x0$b) || all(x0$b == x0$β))
 245         x0$b = rep(0, K)
 246       else if (any(is.na(x0$b)))
 247         stop("x0$b cannot have missing values")
 248
 249             op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
 250                 ui=cbind(
 251                     rbind( rep(-1,K-1), diag(K-1) ),
 252                     matrix(0, nrow=K, ncol=(d+1)*K) ),
 253                 ci=c(-1,rep(0,K-1)) )
 254
 255       # We get a first non-trivial estimation of W: getOmega(theta)^{-1}
 256       # TODO: loop, this redefine f, so that we can call constrOptim again...
 257       # Stopping condition? N iterations? Delta <= ε ?
 258
 259             expArgs(op_res$par)
 260         }
 261     )
 262 )
 263
 264 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 265 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 266 # by numerically evaluating the integral.
 267 #
 268 # @param link Link, 'logit' or 'probit'
 269 # @param order Order of derivative
 270 # @param λ Norm of columns of β
 271 # @param b Intercept
 272 #
 273 .G <- function(link, order, λ, b)
 274 {
 275     # NOTE: weird "integral divergent" error on inputs:
 276     # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 277     # Switch to pracma package for that (but it seems slow...)
 278   sapply( seq_along(λ), function(k) {
 279     res <- NULL
 280     tryCatch({
 281       # Fast code, may fail:
 282       res <- stats::integrate(
 283         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 284         lower=-Inf, upper=Inf )$value
 285     }, error = function(e) {
 286       # Robust slow code, no fails observed:
 287       sink("/dev/null") #pracma package has some useless printed outputs...
 288       res <- pracma::integral(
 289         function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 290         xmin=-Inf, xmax=Inf, method="Kronrod")
 291       sink()
 292     })
 293     res
 294   })
 295 }
 296
 297 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 298 #
 299 .deriv <- list(
 300     "probit"=list(
 301         # 'probit' derivatives list;
 302         # NOTE: exact values for the integral E[g^(k)(λz+b)] could be computed
 303         function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 304         function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 305         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 306         function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 307         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 308     ),
 309     "logit"=list(
 310         # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 311         # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 312         function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 313         function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 314         function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 315         function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 316         function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 317     )
 318 )
 319
 320 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 321 #
 322 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 323 #
 324 .zin <- function(x)
 325 {
 326     x[is.nan(x)] <- 0.
 327     x
 328 }