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 #'     \item weights Weights on moments when minimizing sum of squares
  14 #'   }
  15 #'
  16 #' @return An object 'op' of class OptimParams, initialized so that \code{op$run(x0)}
  17 #'   outputs the list of optimized parameters
  18 #'   \itemize{
  19 #'     \item p: proportions, size K
  20 #'     \item β: regression matrix, size dxK
  21 #'     \item b: intercepts, size K
  22 #'   }
  23 #'   x0 is a vector containing respectively the K-1 first elements of p, then β by
  24 #'   columns, and finally b: \code{x0 = c(p[1:(K-1)],as.double(β),b)}.
  25 #'
  26 #' @seealso \code{multiRun} to estimate statistics based on β, and
  27 #'   \code{generateSampleIO} for I/O random generation.
  28 #'
  29 #' @examples
  30 #' # Optimize parameters from estimated μ
  31 #' io = generateSampleIO(10000, 1/2, matrix(c(1,-2,3,1),ncol=2), c(0,0), "logit")
  32 #' μ = computeMu(io$X, io$Y, list(K=2))
  33 #' M <- computeMoments(io$X, io$Y)
  34 #' o <- optimParams(2, "logit", list(M=M))
  35 #' x0 <- c(1/2, as.double(μ), c(0,0))
  36 #' par0 <- o$run(x0)
  37 #' # Compare with another starting point
  38 #' x1 <- c(1/2, 2*as.double(μ), c(0,0))
  39 #' par1 <- o$run(x1)
  40 #' o$f( o$linArgs(par0) )
  41 #' o$f( o$linArgs(par1) )
  42 #' @export
  43 optimParams = function(K, link=c("logit","probit"), optargs=list())
  44 {
  45     # Check arguments
  46     link <- match.arg(link)
  47     if (!is.list(optargs))
  48         stop("optargs: list")
  49     if (!is.numeric(K) || K < 2)
  50         stop("K: integer >= 2")
  51
  52     M <- optargs$M
  53     if (is.null(M))
  54     {
  55         if (is.null(optargs$X) || is.null(optargs$Y))
  56             stop("If moments are not provided, X and Y are required")
  57         M <- computeMoments(optargs$X,optargs$Y)
  58     }
  59
  60   weights <- optargs$weights
  61   if (is.null(weights))
  62     weights <- rep(1, K)
  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]]),
  67     "weights"=weights, "K"=as.integer(K))
  68 }
  69
  70 # Encapsulated optimization for p (proportions), β and b (regression parameters)
  71 #
  72 # @field li Link, 'logit' or 'probit'
  73 # @field M1 Estimated first-order moment
  74 # @field M2 Estimated second-order moment (flattened)
  75 # @field M3 Estimated third-order moment (flattened)
  76 # @field weights Vector of moments' weights
  77 # @field K Number of populations
  78 # @field d Number of dimensions
  79 #
  80 setRefClass(
  81     Class = "OptimParams",
  82
  83     fields = list(
  84         # Inputs
  85         li = "character", #link 'logit' or 'probit'
  86         M1 = "numeric", #order-1 moment (vector size d)
  87         M2 = "numeric", #M2 easier to process as a vector
  88         M3 = "numeric", #M3 easier to process as a vector
  89         weights = "numeric", #weights on moments
  90         # Dimensions
  91         K = "integer",
  92         d = "integer"
  93     ),
  94
  95     methods = list(
  96         initialize = function(...)
  97         {
  98             "Check args and initialize K, d"
  99
 100             callSuper(...)
 101             if (!hasArg("li") || !hasArg("M1") || !hasArg("M2") || !hasArg("M3")
 102                 || !hasArg("K"))
 103             {
 104                 stop("Missing arguments")
 105             }
 106
 107             d <<- length(M1)
 108         },
 109
 110         expArgs = function(x)
 111         {
 112             "Expand individual arguments from vector x"
 113
 114             list(
 115                 # p: dimension K-1, need to be completed
 116                 "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
 117                 "β" = matrix(x[K:(K+d*K-1)], ncol=K),
 118                 "b" = x[(K+d*K):(K+(d+1)*K-1)])
 119         },
 120
 121         linArgs = function(o)
 122         {
 123             " Linearize vectors+matrices into a vector x"
 124
 125             c(o$p[1:(K-1)], as.double(o$β), o$b)
 126         },
 127
 128         f = function(x)
 129         {
 130             "Sum of squares (Mi - hat_Mi)^2 where Mi is obtained from formula"
 131
 132             P <- expArgs(x)
 133             p <- P$p
 134             β <- P$β
 135             λ <- sqrt(colSums(β^2))
 136             b <- P$b
 137
 138             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 139             β2 <- apply(β, 2, function(col) col %o% col)
 140             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 141
 142             return(
 143                 weights[1] * sum( ( β  %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
 144                 weights[2] * sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
 145                 weights[3] * sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
 146         },
 147
 148         grad_f = function(x)
 149         {
 150             "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 151
 152             P <- expArgs(x)
 153             p <- P$p
 154             β <- P$β
 155             λ <- sqrt(colSums(β^2))
 156             μ <- sweep(β, 2, λ, '/')
 157             b <- P$b
 158
 159             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 160             β2 <- apply(β, 2, function(col) col %o% col)
 161             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 162
 163             # Some precomputations
 164             G1 = .G(li,1,λ,b)
 165             G2 = .G(li,2,λ,b)
 166             G3 = .G(li,3,λ,b)
 167             G4 = .G(li,4,λ,b)
 168             G5 = .G(li,5,λ,b)
 169
 170             # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
 171             F1 = as.double( 2 * ( β  %*% (p * G1) - M1 ) )
 172             F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
 173             F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
 174
 175             km1 = 1:(K-1)
 176             grad <- #gradient on p
 177                 weights[1] * t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
 178                 weights[2] * t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
 179                 weights[3] * t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
 180
 181             grad_β <- matrix(nrow=d, ncol=K)
 182             for (i in 1:d)
 183             {
 184                 # i determines the derivated matrix dβ[2,3]
 185
 186                 dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 187                 dβ_right <- matrix(0, nrow=d, ncol=K)
 188                 block <- i
 189                 dβ_right[block,] <- dβ_right[block,] + 1
 190                 dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 191
 192                 dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 193                 dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 194                     sweep(dβ_right, 2, β[j,], '*')
 195                 }) )
 196                 block <- ((i-1)*d+1):(i*d)
 197                 dβ2_right[block,] <- dβ2_right[block,] + β
 198                 dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 199
 200                 dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 201                 dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 202                     sweep(dβ2_right, 2, β[j,], '*')
 203                 }) )
 204                 block <- ((i-1)*d*d+1):(i*d*d)
 205                 dβ3_right[block,] <- dβ3_right[block,] + β2
 206                 dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 207
 208                 grad_β[i,] <-
 209           weights[1] * t(dβ) %*% F1 +
 210           weights[2] * t(dβ2) %*% F2 +
 211           weights[3] * t(dβ3) %*% F3
 212             }
 213             grad <- c(grad, as.double(grad_β))
 214
 215             grad = c(grad, #gradient on b
 216                 weights[1] * t( sweep(β,  2, p * G2, '*') ) %*% F1 +
 217                 weights[2] * t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
 218                 weights[3] * t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
 219
 220             grad
 221         },
 222
 223         run = function(x0)
 224         {
 225             "Run optimization from x0 with solver..."
 226
 227         if (!is.list(x0))
 228             stop("x0: list")
 229       if (is.null(x0$β))
 230         stop("At least x0$β must be provided")
 231             if (!is.matrix(x0$β) || any(is.na(x0$β)) || ncol(x0$β) != K)
 232                 stop("x0$β: matrix, no NA, ncol == K")
 233       if (is.null(x0$p))
 234         x0$p = rep(1/K, K-1)
 235       else if (length(x0$p) != K-1 || sum(x0$p) > 1)
 236         stop("x0$p should contain positive integers and sum to < 1")
 237       # Next test = heuristic to detect missing b (when matrix is called "beta")
 238       if (is.null(x0$b) || all(x0$b == x0$β))
 239         x0$b = rep(0, K)
 240       else if (any(is.na(x0$b)))
 241         stop("x0$b cannot have missing values")
 242
 243             op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
 244                 ui=cbind(
 245                     rbind( rep(-1,K-1), diag(K-1) ),
 246                     matrix(0, nrow=K, ncol=(d+1)*K) ),
 247                 ci=c(-1,rep(0,K-1)) )
 248
 249             expArgs(op_res$par)
 250         }
 251     )
 252 )
 253
 254 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 255 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 256 #
 257 # @param link Link, 'logit' or 'probit'
 258 # @param order Order of derivative
 259 # @param λ Norm of columns of β
 260 # @param b Intercept
 261 #
 262 .G <- function(link, order, λ, b)
 263 {
 264     # NOTE: weird "integral divergent" error on inputs:
 265     # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 266     # Switch to pracma package for that (but it seems slow...)
 267
 268     exactComp <- FALSE #TODO: global, or argument...
 269
 270     if (exactComp && link == "probit")
 271     {
 272         # Use exact computations
 273         sapply( seq_along(λ), function(k) {
 274             .exactProbitIntegral(order, λ[k], b[k])
 275         })
 276     }
 277
 278     else
 279     {
 280         # Numerical integration
 281         sapply( seq_along(λ), function(k) {
 282             res <- NULL
 283             tryCatch({
 284                 # Fast code, may fail:
 285                 res <- stats::integrate(
 286                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 287                     lower=-Inf, upper=Inf )$value
 288             }, error = function(e) {
 289                 # Robust slow code, no fails observed:
 290                 sink("/dev/null") #pracma package has some useless printed outputs...
 291                 res <- pracma::integral(
 292                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 293                     xmin=-Inf, xmax=Inf, method="Kronrod")
 294                 sink()
 295             })
 296             res
 297         })
 298     }
 299 }
 300
 301 # TODO: check these computations (wrong atm)
 302 .exactProbitIntegral <- function(order, λ, b)
 303 {
 304     c1 = (1/sqrt(2*pi)) * exp( -.5 * b/((λ^2+1)^2) )
 305     if (order == 1)
 306         return (c1)
 307     c2 = b - λ^2 / (λ^2+1)
 308     if (order == 2)
 309         return (c1 * c2)
 310     if (order == 3)
 311         return (c1 * (λ^2 - 1 + c2^2))
 312     if (order == 4)
 313         return ( (c1*c2/((λ^2+1)^2)) * (-λ^4*((b+1)^2+1) -
 314             2*λ^3 + λ^2*(2-2*b*(b-1)) + 6*λ + 3 - b^2) )
 315     if (order == 5) #only remaining case...
 316         return ( c1 * (3*λ^4+c2^4+6*c1^2*(λ^2-1) - 6*λ^2 + 6) )
 317 }
 318
 319 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 320 #
 321 .deriv <- list(
 322     "probit"=list(
 323         # 'probit' derivatives list;
 324         # TODO: exact values for the integral E[g^(k)(λz+b)]
 325         function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 326         function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 327         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 328         function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 329         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 330     ),
 331     "logit"=list(
 332         # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 333         # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 334         function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 335         function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 336         function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 337         function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 338         function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 339     )
 340 )
 341
 342 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 343 #
 344 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 345 #
 346 .zin <- function(x)
 347 {
 348     x[is.nan(x)] <- 0.
 349     x
 350 }