R/derivatives.R

In ordinal: Regression Models for Ordinal Data

#############################################################################
##    Copyright (c) 2010-2022 Rune Haubo Bojesen Christensen
##
##    This file is part of the ordinal package for R (*ordinal*)
##
##    *ordinal* is free software: you can redistribute it and/or modify
##    it under the terms of the GNU General Public License as published by
##    the Free Software Foundation, either version 2 of the License, or
##    (at your option) any later version.
##
##    *ordinal* is distributed in the hope that it will be useful,
##    but WITHOUT ANY WARRANTY; without even the implied warranty of
##    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
##    GNU General Public License for more details.
##
##    A copy of the GNU General Public License is available at
##    <https://www.r-project.org/Licenses/> and/or
##    <http://www.gnu.org/licenses/>.
#############################################################################
## This file contains:
## Functions for finite difference computations of derivatives
## (gradient and Hessian) of user-specified functions.

deriv12 <- function(fun, x, delta=1e-4, fx=NULL, ...) {
### Compute gradient and Hessian at the same time (to save computing
### time)
    nx <- length(x)
    fx <- if(!is.null(fx)) fx else fun(x, ...)
    stopifnot(length(fx) == 1)
    H <- array(NA, dim=c(nx, nx))
    g <- numeric(nx)
    for(j in 1:nx) {
        ## Diagonal elements:
        xadd <- xsub <- x
        xadd[j] <- x[j] + delta
        xsub[j] <- x[j] - delta
        fadd <- fun(xadd, ...)
        fsub <- fun(xsub, ...)
        H[j, j] <- (fadd - 2 * fx + fsub) / delta^2
        g[j] <- (fadd - fsub) / (2 * delta)
        ## Off diagonal elements:
        for(i in 1:nx) {
            if(i >= j) break
            ## Compute upper triangular elements:
            xaa <- xas <- xsa <- xss <- x
            xaa[c(i, j)] <- x[c(i, j)] + c(delta, delta)
            xas[c(i, j)] <- x[c(i, j)] + c(delta, -delta)
            xsa[c(i, j)] <- x[c(i, j)] + c(-delta, delta)
            xss[c(i, j)] <- x[c(i, j)] - c(delta, delta)
            H[i, j] <- H[j, i] <-
                (fun(xaa, ...) - fun(xas, ...) -
                 fun(xsa, ...) + fun(xss, ...)) /
                     (4 * delta^2)
        }
    }
    list(gradient = g, Hessian = H)
}

myhess <- function(fun, x, fx=NULL, delta=1e-4, ...) {
    nx <- length(x)
    fx <- if(!is.null(fx)) fx else fun(x, ...)
    stopifnot(length(fx) == 1)
    H <- array(NA, dim=c(nx, nx))
    for(j in 1:nx) {
        ## Diagonal elements:
        xadd <- xsub <- x
        xadd[j] <- x[j] + delta
        xsub[j] <- x[j] - delta
        H[j, j] <- (fun(xadd, ...) - 2 * fx +
                    fun(xsub, ...)) / delta^2
        ## Upper triangular (off diagonal) elements:
        for(i in 1:nx) {
            if(i >= j) break
            xaa <- xas <- xsa <- xss <- x
            xaa[c(i, j)] <- x[c(i, j)] + c(delta, delta)
            xas[c(i, j)] <- x[c(i, j)] + c(delta, -delta)
            xsa[c(i, j)] <- x[c(i, j)] + c(-delta, delta)
            xss[c(i, j)] <- x[c(i, j)] - c(delta, delta)
            H[j, i] <- H[i, j] <-
                (fun(xaa, ...) - fun(xas, ...) -
                 fun(xsa, ...) + fun(xss, ...)) /
                     (4 * delta^2)
        }
    }
    H
}

mygrad <-
    function(fun, x, delta = 1e-4,
             method = c("central", "forward", "backward"), ...)
{
    method <- match.arg(method)
    nx <- length(x)
    if(method %in% c("central", "forward")) {
        Xadd <- matrix(rep(x, nx), nrow=nx, byrow=TRUE) + diag(delta, nx)
        fadd <- apply(Xadd, 1, fun, ...)
    }
    if(method %in% c("central", "backward")) {
        Xsub <- matrix(rep(x, nx), nrow=nx, byrow=TRUE) - diag(delta, nx)
        fsub <- apply(Xsub, 1, fun, ...) ## eval.parent perhaps?
    }
    res <- switch(method,
           "forward" = (fadd - fun(x, ...)) / delta,
           "backward" = (fun(x, ...) - fsub) / delta,
           "central" = (fadd - fsub) / (2 * delta)
           )
    res
}

grad.ctr3 <- function(fun, x, delta=1e-4, ...) {
    nx <- length(x)
    Xadd <- matrix(rep(x, nx), nrow=nx, byrow=TRUE) + diag(delta, nx)
    Xsub <- matrix(rep(x, nx), nrow=nx, byrow=TRUE) - diag(delta, nx)
    fadd <- apply(Xadd, 1, fun, ...)
    fsub <- apply(Xsub, 1, fun, ...) ## eval.parent perhaps?
    (fadd - fsub) / (2 * delta)
}

grad.ctr2 <- function(fun, x, delta=1e-4, ...) {
    ans <- x
    for(i in seq_along(x)) {
        xadd <- xsub <- x
        xadd[i] <- x[i] + delta
        xsub[i] <- x[i] - delta
        ans[i] <- (fun(xadd, ...) - fun(xsub, ...)) / (2 * delta)
    }
    ans
}

grad.ctr <- function(fun, x, delta=1e-4, ...) {
    sapply(seq_along(x), function(i) {
        xadd <- xsub <- x
        xadd[i] <- x[i] + delta
        xsub[i] <- x[i] - delta
        (fun(xadd, ...) - fun(xsub, ...)) / (2 * delta)
    })
}

grad <- grad.ctr

grad.ctr4 <- function(fun, x, delta=1e-4, ...) {
### - checking finiteness of x and fun-values
### - taking care to avoid floating point errors
### - not using h=x*delta rather than h=delta (important for small or
###   large x?)
    if(!all(is.finite(x)))
        stop("Cannot compute gradient: non-finite argument")
    ans <- x ## return values
    for(i in seq_along(x)) {
        xadd <- xsub <- x ## reset fun arguments
        xadd[i] <- x[i] + delta
        xsub[i] <- x[i] - delta
        ans[i] <- (fun(xadd, ...) - fun(xsub, ...)) / (xadd[i] - xsub[i])
### NOTE: xadd[i] - xsub[i] != 2*delta with floating point arithmetic.
    }
    if(!all(is.finite(ans))) {
        warning("cannot compute gradient: non-finite function values occured")
        ans[!is.finite(ans)] <- Inf
    }
    ans
}