pkg/R/optimParams.R

   1 #' Optimize parameters
   2 #'
   3 #' Optimize the parameters of a mixture of logistic regressions model, possibly using
   4 #' \code{mu <- computeMu(...)} as a partial starting point.
   5 #'
   6 #' @param K Number of populations.
   7 #' @param link The link type, 'logit' or 'probit'.
   8 #' @param optargs a list with optional arguments:
   9 #'   \itemize{
  10 #'     \item 'M' : list of moments of order 1,2,3: will be computed if not provided.
  11 #'     \item 'X,Y' : input/output, mandatory if moments not given
  12 #'     \item 'exact': use exact formulas when available?
  13 #'   }
  14 #'
  15 #' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
  16 #'   outputs the list of optimized parameters
  17 #'   \itemize{
  18 #'     \item p: proportions, size K
  19 #'     \item β: regression matrix, size dxK
  20 #'     \item b: intercepts, size K
  21 #'   }
  22 #'   x0 is a vector containing respectively the K-1 first elements of p, then β by
  23 #'   columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
  24 #'
  25 #' @seealso \code{multiRun} to estimate statistics based on β, and
  26 #'   \code{generateSampleIO} for I/O random generation.
  27 #'
  28 #' @examples
  29 #' # Optimize parameters from estimated μ
  30 #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
  31 #' μ = computeMu(io$X, io$Y, list(K=2))
  32 #' M <- computeMoments(io$X, io$Y)
  33 #' o <- optimParams(2, "logit", list(M=M))
  34 #' x0 <- c(1/2, as.double(μ), c(0,0))
  35 #' par0 <- o$run(x0)
  36 #' # Compare with another starting point
  37 #' x1 <- c(1/2, 2*as.double(μ), c(0,0))
  38 #' par1 <- o$run(x1)
  39 #' o$f( o$linArgs(par0) )
  40 #' o$f( o$linArgs(par1) )
  41 #' @export
  42 optimParams = function(K, link=c("logit","probit"), optargs=list())
  43 {
  44     # Check arguments
  45     link <- match.arg(link)
  46     if (!is.list(optargs))
  47         stop("optargs: list")
  48     if (!is.numeric(K) || K < 2)
  49         stop("K: integer >= 2")
  50
  51     M <- optargs$M
  52     if (is.null(M))
  53     {
  54         if (is.null(optargs$X) || is.null(optargs$Y))
  55             stop("If moments are not provided, X and Y are required")
  56         M <- computeMoments(optargs$X,optargs$Y)
  57     }
  58
  59     # TODO: field?!
  60     exactComp <<- optargs$exact
  61     if (is.null(exactComp))
  62         exactComp <<- FALSE
  63
  64     # Build and return optimization algorithm object
  65     methods::new("OptimParams", "li"=link, "M1"=as.double(M[[1]]),
  66         "M2"=as.double(M[[2]]), "M3"=as.double(M[[3]]), "K"=as.integer(K))
  67 }
  68
  69 # Encapsulated optimization for p (proportions), β and b (regression parameters)
  70 #
  71 # @field li Link, 'logit' or 'probit'
  72 # @field M1 Estimated first-order moment
  73 # @field M2 Estimated second-order moment (flattened)
  74 # @field M3 Estimated third-order moment (flattened)
  75 # @field K Number of populations
  76 # @field d Number of dimensions
  77 #
  78 setRefClass(
  79     Class = "OptimParams",
  80
  81     fields = list(
  82         # Inputs
  83         li = "character", #link 'logit' or 'probit'
  84         M1 = "numeric", #order-1 moment (vector size d)
  85         M2 = "numeric", #M2 easier to process as a vector
  86         M3 = "numeric", #M3 easier to process as a vector
  87         # Dimensions
  88         K = "integer",
  89         d = "integer"
  90     ),
  91
  92     methods = list(
  93         initialize = function(...)
  94         {
  95             "Check args and initialize K, d"
  96
  97             callSuper(...)
  98             if (!hasArg("li") || !hasArg("M1") || !hasArg("M2") || !hasArg("M3")
  99                 || !hasArg("K"))
 100             {
 101                 stop("Missing arguments")
 102             }
 103
 104             d <<- length(M1)
 105         },
 106
 107         expArgs = function(x)
 108         {
 109             "Expand individual arguments from vector x"
 110
 111             list(
 112                 # p: dimension K-1, need to be completed
 113                 "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
 114                 "β" = matrix(x[K:(K+d*K-1)], ncol=K),
 115                 "b" = x[(K+d*K):(K+(d+1)*K-1)])
 116         },
 117
 118         linArgs = function(o)
 119         {
 120             " Linearize vectors+matrices into a vector x"
 121
 122             c(o$p[1:(K-1)], as.double(o$β), o$b)
 123         },
 124
 125         f = function(x)
 126         {
 127             "Sum of squares (Mi - hat_Mi)^2 where Mi is obtained from formula"
 128
 129             P <- expArgs(x)
 130             p <- P$p
 131             β <- P$β
 132             λ <- sqrt(colSums(β^2))
 133             b <- P$b
 134
 135             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 136             β2 <- apply(β, 2, function(col) col %o% col)
 137             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 138
 139             return(
 140                 sum( ( β  %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
 141                 sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
 142                 sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
 143         },
 144
 145         grad_f = function(x)
 146         {
 147             "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 148
 149             P <- expArgs(x)
 150             p <- P$p
 151             β <- P$β
 152             λ <- sqrt(colSums(β^2))
 153             μ <- sweep(β, 2, λ, '/')
 154             b <- P$b
 155
 156             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 157             β2 <- apply(β, 2, function(col) col %o% col)
 158             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 159
 160             # Some precomputations
 161             G1 = .G(li,1,λ,b)
 162             G2 = .G(li,2,λ,b)
 163             G3 = .G(li,3,λ,b)
 164             G4 = .G(li,4,λ,b)
 165             G5 = .G(li,5,λ,b)
 166
 167             # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
 168             F1 = as.double( 2 * ( β  %*% (p * G1) - M1 ) )
 169             F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
 170             F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
 171
 172             km1 = 1:(K-1)
 173             grad <- #gradient on p
 174                 t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
 175                 t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
 176                 t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
 177
 178             grad_β <- matrix(nrow=d, ncol=K)
 179             for (i in 1:d)
 180             {
 181                 # i determines the derivated matrix dβ[2,3]
 182
 183                 dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 184                 dβ_right <- matrix(0, nrow=d, ncol=K)
 185                 block <- i
 186                 dβ_right[block,] <- dβ_right[block,] + 1
 187                 dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 188
 189                 dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 190                 dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 191                     sweep(dβ_right, 2, β[j,], '*')
 192                 }) )
 193                 block <- ((i-1)*d+1):(i*d)
 194                 dβ2_right[block,] <- dβ2_right[block,] + β
 195                 dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 196
 197                 dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 198                 dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 199                     sweep(dβ2_right, 2, β[j,], '*')
 200                 }) )
 201                 block <- ((i-1)*d*d+1):(i*d*d)
 202                 dβ3_right[block,] <- dβ3_right[block,] + β2
 203                 dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 204
 205                 grad_β[i,] <- t(dβ) %*% F1 + t(dβ2) %*% F2 + t(dβ3) %*% F3
 206             }
 207             grad <- c(grad, as.double(grad_β))
 208
 209             grad = c(grad, #gradient on b
 210                 t( sweep(β,  2, p * G2, '*') ) %*% F1 +
 211                 t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
 212                 t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
 213
 214             grad
 215         },
 216
 217         run = function(x0)
 218         {
 219             "Run optimization from x0 with solver..."
 220
 221             if (!is.numeric(x0) || any(is.na(x0)) || length(x0) != (d+2)*K-1
 222                 || any(x0[1:(K-1)] < 0) || sum(x0[1:(K-1)]) > 1)
 223             {
 224                 stop("x0: numeric vector, no NA, length (d+2)*K-1, sum(x0[1:(K-1) >= 0]) <= 1")
 225             }
 226
 227             op_res = constrOptim( x0, .self$f, .self$grad_f,
 228                 ui=cbind(
 229                     rbind( rep(-1,K-1), diag(K-1) ),
 230                     matrix(0, nrow=K, ncol=(d+1)*K) ),
 231                 ci=c(-1,rep(0,K-1)) )
 232
 233             expArgs(op_res$par)
 234         }
 235     )
 236 )
 237
 238 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 239 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 240 #
 241 # @param link Link, 'logit' or 'probit'
 242 # @param order Order of derivative
 243 # @param λ Norm of columns of β
 244 # @param b Intercept
 245 #
 246 .G <- function(link, order, λ, b)
 247 {
 248     # NOTE: weird "integral divergent" error on inputs:
 249     # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 250     # Switch to pracma package for that (but it seems slow...)
 251
 252     if (exactComp && link == "probit")
 253     {
 254         # Use exact computations
 255         sapply( seq_along(λ), function(k) {
 256             .exactProbitIntegral(order, λ[k], b[k])
 257         })
 258     }
 259
 260     else
 261     {
 262         # Numerical integration
 263         sapply( seq_along(λ), function(k) {
 264             res <- NULL
 265             tryCatch({
 266                 # Fast code, may fail:
 267                 res <- stats::integrate(
 268                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 269                     lower=-Inf, upper=Inf )$value
 270             }, error = function(e) {
 271                 # Robust slow code, no fails observed:
 272                 sink("/dev/null") #pracma package has some useless printed outputs...
 273                 res <- pracma::integral(
 274                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 275                     xmin=-Inf, xmax=Inf, method="Kronrod")
 276                 sink()
 277             })
 278             res
 279         })
 280     }
 281 }
 282
 283 # TODO: check these computations (wrong atm)
 284 .exactProbitIntegral <- function(order, λ, b)
 285 {
 286     c1 = (1/sqrt(2*pi)) * exp( -.5 * b/((λ^2+1)^2) )
 287     if (order == 1)
 288         return (c1)
 289     c2 = b - λ^2 / (λ^2+1)
 290     if (order == 2)
 291         return (c1 * c2)
 292     if (order == 3)
 293         return (c1 * (λ^2 - 1 + c2^2))
 294     if (order == 4)
 295         return ( (c1*c2/((λ^2+1)^2)) * (-λ^4*((b+1)^2+1) -
 296             2*λ^3 + λ^2*(2-2*b*(b-1)) + 6*λ + 3 - b^2) )
 297     if (order == 5) #only remaining case...
 298         return ( c1 * (3*λ^4+c2^4+6*c1^2*(λ^2-1) - 6*λ^2 + 6) )
 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         # TODO: exact values for the integral E[g^(k)(λz+b)]
 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 }