compareDerivatives: function to compare analytic and numeric derivatives

Description

This function compares analytic and numerical derivative and prints related diagnostics information. It is intended for testing and debugging code for analytic derivatives for maximization algorithms.

Usage

compareDerivatives(f, grad, hess=NULL, t0, eps=1e-6,
                   printLevel=1, print=printLevel > 0,
                   max.rows=getOption("max.rows", 20),
                   max.cols=getOption("max.cols", 7),
                   ...)

Arguments

function to be differentiated. The parameter (vector) of interest must be the first argument. The function may return a vector, in that case the derivative will be a matrix.

grad

analytic gradient. This may be either a function, returning the analytic gradient, or a numeric vector, the pre-computed gradient. The function must use the same set of parameters as f. If f is a vector-valued function, grad must return/be a matrix where the number of rows equals the number of components of f, and the number of columns must equal to the number of components in t0.

hess

function returning the analytic hessian. If present, hessian matrices are compared too. Only appropriate for scalar-valued functions.

numeric vector, parameter at which the derivatives are compared. The derivative is taken with respect to this vector. both fm grad (if function) and hess (if present) must accept this value as the first parameter.

eps

numeric. Step size for numeric differentiation. Central derivative is used.

printLevel

numeric: a positive number prints summary of the comparison. 0 does not do any printing, only returns the comparison results (invisibly).

deprecated (for backward compatibility only).

max.rows

maximum number of matrix rows to be printed.

max.cols

maximum number of columns to be printed.

…

further arguments to f, grad and hess.

Value

A list with following components:

the input argument t0

f.t0

f(t0)

compareGrad

a list with components analytic = grad(t0), nmeric = numericGradient(f, t0), and their rel.diff.

maxRelDiffGrad

max(abs(rel.diff))

If hess is also provided, the following optional components are also present:

compareHessian

a list with components analytic = hess(t0), numeric = numericGradient(grad, t0), and their rel.diff.

maxRelDiffHess

max(abs(rel.diff)) for the Hessian

Details

Analytic derivatives (and Hessian) substantially improve the estimation speed and reliability. However, these are typically hard to program. This utility compares the programmed result and the (internally calculated) numeric derivative. For every component of f, it prints the parameter value, analytic and numeric derivative, and their relative difference $$\textrm{rel.diff} = \frac{\textrm{analytic} - \textrm{numeric}}{\frac{1}{2}(|\textrm{analytic}| + |\textrm{numeric}|)}.$$ If $\textrm{analytic} = 0$ and $\textrm{numeric} = 0$, then rel.diff is also set to 0. If analytic derivatives are correct and the function is sufficiently smooth, expect the relative differences to be less than $10^{-7}$.

Examples

Run this code

# NOT RUN {
## A simple example with sin(x)' = cos(x)
f <- function(x) c(sin=sin(x))
Dsin <- compareDerivatives(f, cos, t0=c(angle=1))
##
## Example of normal log-likelihood.  Two-parameter
## function.
##
x <- rnorm(100, 1, 2) # generate rnorm x
l <- function(b) sum(dnorm(x, mean=b[1], sd=b[2], log=TRUE))
gradl <- function(b) {
    c(mu=sum(x - b[1])/b[2]^2,
    sigma=sum((x - b[1])^2/b[2]^3 - 1/b[2]))
}
gradl. <- compareDerivatives(l, gradl, t0=c(mu=1,sigma=2))

##
## An example with f returning a vector, t0 = a scalar
##
trig <- function(x)c(sin=sin(x), cos=cos(x))
Dtrig <- function(x)c(sin=cos(x), cos=-sin(x))
Dtrig. <- compareDerivatives(trig, Dtrig, t0=1)
# }