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         # Dimensions
  90         K = "integer",
  91         d = "integer"
  92     ),
  93
  94     methods = list(
  95         initialize = function(...)
  96         {
  97             "Check args and initialize K, d"
  98
  99             callSuper(...)
 100             if (!hasArg("li") || !hasArg("M1") || !hasArg("M2") || !hasArg("M3")
 101                 || !hasArg("K"))
 102             {
 103                 stop("Missing arguments")
 104             }
 105
 106             d <<- length(M1)
 107         },
 108
 109         expArgs = function(x)
 110         {
 111             "Expand individual arguments from vector x"
 112
 113             list(
 114                 # p: dimension K-1, need to be completed
 115                 "p" = c(x[1:(K-1)], 1-sum(x[1:(K-1)])),
 116                 "β" = matrix(x[K:(K+d*K-1)], ncol=K),
 117                 "b" = x[(K+d*K):(K+(d+1)*K-1)])
 118         },
 119
 120         linArgs = function(o)
 121         {
 122             " Linearize vectors+matrices into a vector x"
 123
 124             c(o$p[1:(K-1)], as.double(o$β), o$b)
 125         },
 126
 127         f = function(x)
 128         {
 129             "Sum of squares (Mi - hat_Mi)^2 where Mi is obtained from formula"
 130
 131             P <- expArgs(x)
 132             p <- P$p
 133             β <- P$β
 134             λ <- sqrt(colSums(β^2))
 135             b <- P$b
 136
 137             # Tensorial products β^2 = β2 and β^3 = β3 must be computed from current β1
 138             β2 <- apply(β, 2, function(col) col %o% col)
 139             β3 <- apply(β, 2, function(col) col %o% col %o% col)
 140
 141             return(
 142                 weights[1] * sum( ( β  %*% (p * .G(li,1,λ,b)) - M1 )^2 ) +
 143                 weights[2] * sum( ( β2 %*% (p * .G(li,2,λ,b)) - M2 )^2 ) +
 144                 weights[3] * sum( ( β3 %*% (p * .G(li,3,λ,b)) - M3 )^2 ) )
 145         },
 146
 147         grad_f = function(x)
 148         {
 149             "Gradient of f, dimension (K-1) + d*K + K = (d+2)*K - 1"
 150
 151             P <- expArgs(x)
 152             p <- P$p
 153             β <- P$β
 154             λ <- sqrt(colSums(β^2))
 155             μ <- sweep(β, 2, λ, '/')
 156             b <- P$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             # Some precomputations
 163             G1 = .G(li,1,λ,b)
 164             G2 = .G(li,2,λ,b)
 165             G3 = .G(li,3,λ,b)
 166             G4 = .G(li,4,λ,b)
 167             G5 = .G(li,5,λ,b)
 168
 169             # (Mi - hat_Mi)^2 ' == (Mi - hat_Mi)' 2(Mi - hat_Mi) = Mi' Fi
 170             F1 = as.double( 2 * ( β  %*% (p * G1) - M1 ) )
 171             F2 = as.double( 2 * ( β2 %*% (p * G2) - M2 ) )
 172             F3 = as.double( 2 * ( β3 %*% (p * G3) - M3 ) )
 173
 174             km1 = 1:(K-1)
 175             grad <- #gradient on p
 176                 weights[1] * t( sweep(as.matrix(β [,km1]), 2, G1[km1], '*') - G1[K] * β [,K] ) %*% F1 +
 177                 weights[2] * t( sweep(as.matrix(β2[,km1]), 2, G2[km1], '*') - G2[K] * β2[,K] ) %*% F2 +
 178                 weights[3] * t( sweep(as.matrix(β3[,km1]), 2, G3[km1], '*') - G3[K] * β3[,K] ) %*% F3
 179
 180             grad_β <- matrix(nrow=d, ncol=K)
 181             for (i in 1:d)
 182             {
 183                 # i determines the derivated matrix dβ[2,3]
 184
 185                 dβ_left <- sweep(β, 2, p * G3 * β[i,], '*')
 186                 dβ_right <- matrix(0, nrow=d, ncol=K)
 187                 block <- i
 188                 dβ_right[block,] <- dβ_right[block,] + 1
 189                 dβ <- dβ_left + sweep(dβ_right, 2,  p * G1, '*')
 190
 191                 dβ2_left <- sweep(β2, 2, p * G4 * β[i,], '*')
 192                 dβ2_right <- do.call( rbind, lapply(1:d, function(j) {
 193                     sweep(dβ_right, 2, β[j,], '*')
 194                 }) )
 195                 block <- ((i-1)*d+1):(i*d)
 196                 dβ2_right[block,] <- dβ2_right[block,] + β
 197                 dβ2 <- dβ2_left + sweep(dβ2_right, 2, p * G2, '*')
 198
 199                 dβ3_left <- sweep(β3, 2, p * G5 * β[i,], '*')
 200                 dβ3_right <- do.call( rbind, lapply(1:d, function(j) {
 201                     sweep(dβ2_right, 2, β[j,], '*')
 202                 }) )
 203                 block <- ((i-1)*d*d+1):(i*d*d)
 204                 dβ3_right[block,] <- dβ3_right[block,] + β2
 205                 dβ3 <- dβ3_left + sweep(dβ3_right, 2, p * G3, '*')
 206
 207                 grad_β[i,] <-
 208           weights[1] * t(dβ) %*% F1 +
 209           weights[2] * t(dβ2) %*% F2 +
 210           weights[3] * t(dβ3) %*% F3
 211             }
 212             grad <- c(grad, as.double(grad_β))
 213
 214             grad = c(grad, #gradient on b
 215                 weights[1] * t( sweep(β,  2, p * G2, '*') ) %*% F1 +
 216                 weights[2] * t( sweep(β2, 2, p * G3, '*') ) %*% F2 +
 217                 weights[3] * t( sweep(β3, 2, p * G4, '*') ) %*% F3 )
 218
 219             grad
 220         },
 221
 222         run = function(x0)
 223         {
 224             "Run optimization from x0 with solver..."
 225
 226         if (!is.list(x0))
 227             stop("x0: list")
 228       if (is.null(x0$β))
 229         stop("At least x0$β must be provided")
 230             if (!is.matrix(x0$β) || any(is.na(x0$β)) || ncol(x0$β) != K)
 231                 stop("x0$β: matrix, no NA, ncol == K")
 232       if (is.null(x0$p))
 233         x0$p = rep(1/K, K-1)
 234       else if (length(x0$p) != K-1 || sum(x0$p) > 1)
 235         stop("x0$p should contain positive integers and sum to < 1")
 236       # Next test = heuristic to detect missing b (when matrix is called "beta")
 237       if (is.null(x0$b) || all(x0$b == x0$β))
 238         x0$b = rep(0, K)
 239       else if (any(is.na(x0$b)))
 240         stop("x0$b cannot have missing values")
 241
 242             op_res = constrOptim( linArgs(x0), .self$f, .self$grad_f,
 243                 ui=cbind(
 244                     rbind( rep(-1,K-1), diag(K-1) ),
 245                     matrix(0, nrow=K, ncol=(d+1)*K) ),
 246                 ci=c(-1,rep(0,K-1)) )
 247
 248             expArgs(op_res$par)
 249         }
 250     )
 251 )
 252
 253 # Compute vectorial E[g^{(order)}(<β,x> + b)] with x~N(0,Id) (integral in R^d)
 254 #                 = E[g^{(order)}(z)] with z~N(b,diag(λ))
 255 #
 256 # @param link Link, 'logit' or 'probit'
 257 # @param order Order of derivative
 258 # @param λ Norm of columns of β
 259 # @param b Intercept
 260 #
 261 .G <- function(link, order, λ, b)
 262 {
 263     # NOTE: weird "integral divergent" error on inputs:
 264     # link="probit"; order=2; λ=c(531.8099,586.8893,523.5816); b=c(-118.512674,-3.488020,2.109969)
 265     # Switch to pracma package for that (but it seems slow...)
 266
 267     exactComp <- FALSE #TODO: global, or argument...
 268
 269     if (exactComp && link == "probit")
 270     {
 271         # Use exact computations
 272         sapply( seq_along(λ), function(k) {
 273             .exactProbitIntegral(order, λ[k], b[k])
 274         })
 275     }
 276
 277     else
 278     {
 279         # Numerical integration
 280         sapply( seq_along(λ), function(k) {
 281             res <- NULL
 282             tryCatch({
 283                 # Fast code, may fail:
 284                 res <- stats::integrate(
 285                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 286                     lower=-Inf, upper=Inf )$value
 287             }, error = function(e) {
 288                 # Robust slow code, no fails observed:
 289                 sink("/dev/null") #pracma package has some useless printed outputs...
 290                 res <- pracma::integral(
 291                     function(z) .deriv[[link]][[order]](λ[k]*z+b[k]) * exp(-z^2/2) / sqrt(2*pi),
 292                     xmin=-Inf, xmax=Inf, method="Kronrod")
 293                 sink()
 294             })
 295             res
 296         })
 297     }
 298 }
 299
 300 # TODO: check these computations (wrong atm)
 301 .exactProbitIntegral <- function(order, λ, b)
 302 {
 303     c1 = (1/sqrt(2*pi)) * exp( -.5 * b/((λ^2+1)^2) )
 304     if (order == 1)
 305         return (c1)
 306     c2 = b - λ^2 / (λ^2+1)
 307     if (order == 2)
 308         return (c1 * c2)
 309     if (order == 3)
 310         return (c1 * (λ^2 - 1 + c2^2))
 311     if (order == 4)
 312         return ( (c1*c2/((λ^2+1)^2)) * (-λ^4*((b+1)^2+1) -
 313             2*λ^3 + λ^2*(2-2*b*(b-1)) + 6*λ + 3 - b^2) )
 314     if (order == 5) #only remaining case...
 315         return ( c1 * (3*λ^4+c2^4+6*c1^2*(λ^2-1) - 6*λ^2 + 6) )
 316 }
 317
 318 # Derivatives list: g^(k)(x) for links 'logit' and 'probit'
 319 #
 320 .deriv <- list(
 321     "probit"=list(
 322         # 'probit' derivatives list;
 323         # TODO: exact values for the integral E[g^(k)(λz+b)]
 324         function(x) exp(-x^2/2)/(sqrt(2*pi)),                     #g'
 325         function(x) exp(-x^2/2)/(sqrt(2*pi)) *  -x,               #g''
 326         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^2 - 1),        #g^(3)
 327         function(x) exp(-x^2/2)/(sqrt(2*pi)) * (-x^3 + 3*x),      #g^(4)
 328         function(x) exp(-x^2/2)/(sqrt(2*pi)) * ( x^4 - 6*x^2 + 3) #g^(5)
 329     ),
 330     "logit"=list(
 331         # Sigmoid derivatives list, obtained with http://www.derivative-calculator.net/
 332         # @seealso http://www.ece.uc.edu/~aminai/papers/minai_sigmoids_NN93.pdf
 333         function(x) {e=exp(x); .zin(e                                    /(e+1)^2)}, #g'
 334         function(x) {e=exp(x); .zin(e*(-e   + 1)                         /(e+1)^3)}, #g''
 335         function(x) {e=exp(x); .zin(e*( e^2 - 4*e    + 1)                /(e+1)^4)}, #g^(3)
 336         function(x) {e=exp(x); .zin(e*(-e^3 + 11*e^2 - 11*e   + 1)       /(e+1)^5)}, #g^(4)
 337         function(x) {e=exp(x); .zin(e*( e^4 - 26*e^3 + 66*e^2 - 26*e + 1)/(e+1)^6)}  #g^(5)
 338     )
 339 )
 340
 341 # Utility for integration: "[return] zero if [argument is] NaN" (Inf / Inf divs)
 342 #
 343 # @param x Ratio of polynoms of exponentials, as in .S[[i]]
 344 #
 345 .zin <- function(x)
 346 {
 347     x[is.nan(x)] <- 0.
 348     x
 349 }