# What you get and what you should be getting: checking numerical code

**dahtah » R**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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*[Update 24.08.12: added missing alias following comment by brobar]*

Whenever I write numerical code I spend half my time debugging my algebra, painstakingly uncovering one sign mistake after another in my calculations. Usually I have computed by hand the gradient or the integral of some nasty function, and I have to check it against a numerical estimate (for example returned by the wonderful numDeriv package). I have written a small utility called **cmpplot**, for “comparison plot”, that allows me to quickly plot what I’m getting from my hand-computed function against what I should be getting (the numerical estimate). It’s rather useful if you need to diagnose sign mistakes, forgotten factors, etc. It’s too small to be a R package so I’m just going to put the code below, with an example.

Let’s say that after lengthy calculations we have decided that the derivative of cos(x) is sin(x):

fun <- function(x) cos(x) dfun <- function(x) sin(x)

Although quite confident in our result, we decide to check against a numerical derivative anyway:

require(numDeriv) xv <- seq(0,2*pi,l=100) d.num <- sapply(xv,function(v) grad(fun,v)) d.th <- dfun(xv)

A call to** cmpplot** reveals an embarassing mistake:

cmpplot(d.num,d.th)

The points should all fall on the diagonal line, and most obviously they don’t.** cmpplot** also displays the result of a regression of d.th on d.num, which tells us we’ve made a sign mistake and that the derivative of cos(x) is actually -sin(x).

Sometimes a function is only correct for some arguments: for example

fun <- function(x) sign(x)*x^2 dfun <- function(x) 2*x ##WRONG for x < 0

in cases like this calling cmpplot with argument “type = 2″ provides a better way of visualising what’s going on:

cmpplot(d.num,d.th,type=2)

The black dots should all align with the blue dots and only half of them do so.

**cmpplot** also has methods for matrices: this time we take cos(x) as a function that takes a vector argument and returns a vector argument, and we want to compute its Jacobian. Again we have made a mistake in our theoretical calculations:

fun <- function(x) cos(2*x) Jfun <- function(x) -diag(sin(2*x))

Checking against a numerical Jacobian:

<pre>x <- rnorm(10) J.num <- jacobian(fun,x) J.th <- Jfun(x) cmpplot(J.th,J.num)

**cmpplot** produces an image plot of the absolute error, and we see that we have made a mistake on the diagonal.

Other options include:

cmpplot(J.th,J.num,type=2)

to plot just the eigen- or singular values of the two matrices and

cmpplot(J.th,J.num,flatten=T)

to treat the matrices as vectors.

Here’s the full code for cmpplot, just copy and paste!

#Comparison Plots #Some visual tools to compare what you got and what you should be getting #Simon Barthelmé (University of Geneva), 2012 cmpplot <- function(x,...) UseMethod("cmpplot",x) cmpplot.matrix <- function(A,B,flatten=F,type=1,...) { name.a <- deparse(substitute(A)) name.b <- deparse(substitute(B)) if (flatten) { cmpplot.numeric(as.numeric(A),as.numeric(B),type=type,...) } else { if (type == 1) { A <- Matrix(A) B <- Matrix(B) ttl <- sprintf("Absolute difference between %s and %s",name.a,name.b) image(abs(A-B),sub=ttl,colorkey=T) } else if (type == 2) { if (is.symmetric(A) & is.symmetric(B)) { sym <- T vA <- eigen(A)$val vB <- eigen(B)$val ylab <- "Eigenvalue" } else { sym <- F vA <- svd(A)$d vB <- svd(B)$d ylab <- "Singular value" } ttl <- sprintf('Spectrum of %s vs. spectrum of %s',name.a,name.b) ind <- 1:length(vA) ylim <- range(c(vA,vB)) plot(ind,vA,ylim=ylim,pch=20,xlab="Index",ylab=ylab,main=ttl) points(ind,vB,pch=3,col="red") legend("topright",c(name.a,name.b),pch=c(20,3),col=c('black','blue'),fill="white") } } } cmpplot.numeric <- function(a,b,type=1,stretch=F) { name.a <- deparse(substitute(a)) name.b <- deparse(substitute(b)) if (type==1) { if (stretch) { xlim <- range(c(a,b)) ylim <- xlim } else { xlim <- range(a) ylim <- range(b) } plot(a,b,xlab=name.a,ylab=name.b,xlim=xlim,ylim=ylim,col="blue",pch=20) abline(0,1) m <- lm(b ~ a) msg <- sprintf('Regression line: y = %.2f * x + %.2f',coef(m)[2],coef(m)[1]) mtext(msg) } else if (type == 2) { ylim <- range(c(a,b)) plot(a,xlab="Index",ylab="Value",pch=20,ylim=ylim) pointsb(b,pch=3) legend("topright",c(name.a,name.b),pch=c(20,3),col=c('black','blue'),bg="white") } else if (type == 3) { a <- rank(a);b <- rank(b); name.a <- paste(name.a,"(rank)") name.b <- paste(name.b,"(rank)") plot(a,b,xlab=name.a,ylab=name.b) abline(0,1) } } is.symmetric <- function(A) { if (nrow(A) == ncol(A)) { all(A==t(A)) } else { FALSE } } pointsb <- function(...) { points(...,col="blue") }

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