In this exercise we’ll look at debugging R using the RStudio integration.

We’ll be debugging some code that I wrote (incorrectly!) for simple boundary correction of a univariate kernel density estimator.

A kernel density estimator is a way to estimate an unknown density function $$f$$ from which we have drawn a sample $$X = {x_1, x_2, \ldots x_n}$$ of size $$n$$. Given the sample $$X$$, we estimate $$f$$ using $\hat{f}(x) = \frac{1}{n}\sum_{i=1}^n K_h(x - x_i)$ where $$K_h$$ is a kernel function with associated bandwidth $$h$$. This estimator can be shown to be consistent and has bias of order $$O(h^2)$$ as $$n\rightarrow \infty$$ in the case where $$f$$ has support larger than the range of $$X$$.

However, if the support of $$f$$ is constrained, at the boundaries $$\hat{f}$$ will be not only biased, but will be inconsistent, as points close to the boundary will have some of the weight of the kernel function $$K_h$$ fall outside the support, thus when $$\hat{f}$$ is truncated to the known support, points close to the boundary will have less weight than others. In the simple case of a single boundary (e.g. at $$x=0$$) it can be shown that $$\mathsf{E}[\hat{f}] = f/2$$ in the limit at $$n \rightarrow 0$$.

A simple correction to this is to divide the estimate $$\hat{f}$$ by the integral of the kernel $$K_h$$ centered at $$x$$ over the support. i.e. $b(x) = \int_{\mathsf{supp}(f)} K_h(z - x)dz.$ For the gaussian univariate kernel, $$b(x)$$ is just an application of pnorm, so relatively trivial to implement.

However, a few common R coding errors has meant that I didn’t get it right!

My first attempt can be found here

Download and load it up into R studio, then hit the Source button to source the file so R knows about the function. Then work through the following points.

1. By default, it applies no boundary correction as the limits are set at x0=-Inf and x1=Inf. Try running the function on some uniform data as follows

x <- runif(100)
d1 <- density(x)
d2 <- density_bc(x)
plot(d1)
lines(d2, col="red")

This should result in no red, as I had an error in the code.

2. Try debugging this by placing a breakpoint (click in the inner margin to the left of the line number, or use Shift F9 while on the line) next to the density(x) line in the file. Then re-run the line d2 <- density_bc(x). You’ll then be in the debugger. Step through the code using the Next button (or F10). At any point you can evaluate the values of the local variables in the usual way. You can also change variables if you like (or run any other R functions on them).

3. The first thing you should notice is that I incorrectly used && when I should have used the component-wise & operator. Fix that up, stop the debugger (F8), re-source the file (Ctrl S) and re-run the d2 <- density_nc(x). If you hit the breakpoint, just hit Continue (Shift F5) to continue. Hopefully it should give you something sensible now!

4. Now let’s run the same test, this time restricting our density estimate to the interval $$[0,1]$$, the support of $$f$$ as follows

x <- runif(100)
d1 <- density(x)
d2 <- density_bc(x, 0, 1)
plot(d1)
lines(d2, col="red")

You should notice that it still isn’t right - it isn’t restricting the range of x at all! Debug it again and fix up the further issue with the computation of inside. Once done, you should get it correctly restricted to between 0 and 1.

5. You should notice, however, that the estimate still doesn’t look right - it should look relatively flat, like the uniform distribution we sampled x from. Debug further to figure out what is going wrong.

6. Once you have it all working, you might think we’ve got a pretty good function. However, there’s still problems. In particular, it’s not very robust. Make sure you remove your breakpoint before continuing, then try this

x[50] <- NA
d1 <- density_bc(x)

Bam! You should hit an error. Notice that RStudio gives you some options. Click Show Traceback to see the call stack (the order of function calls). Click Rerun with Debug to re-run it. You’ll see it’s dying in the density function as x contains missing values. A simple way to fix this up is to notice that density has a na.rm parameter available to it. Fix it up and re-run.

7. Let’s try testing robustness further with

x <- rexp(100, 0.1)
d1 <- density(x)
d2 <- density_bc(x, 0, NA)
plot(d1)
liens(d2, col="red")

This will fail - figure out why using breakpoints, and fix it up so that both x0 and x1 might be able to take NA (and perhaps NULL?) parameters.

8. Finally, we’d expect the function to work even if end up with points outside the support of $$f$$. Try this:

x <- runif(100)
d1 <- density(x)
d2 <- density_bc(x, 0, 0.5)
plot(d1)
lines(d2, col="red")

Figure out why it’s failing, and fix it up.

Well done - you debuggered it!