Solved – Why is the R density plot a bell curve when all datapoints are 0

density functionkernel-smoothingr

When I graph a density plot in R, and all the numbers are slightly greater than 0, I get essentially a vertical line at x = 0. But when all the numbers are exactly equal to 0, I get some sort of bell curve. Why is that? It seems counterintuitive.

The command I used to plot the curves was

for (i in 0:60) {
    cur_data = subset(data, time == i)
    p <- ggplot(cur_data, aes(x=error)) +
                geom_density() +
                theme_bw() +
                xlab(paste("Error distribution (minute ", i, ")", sep="")) +
                xlim(0, 1)
    ggsave(...)
}

At i = 60, cur_data should be entirely populated by the values 0.0.

Density plot, entries slightly greater than 0

Density plot, entries all 0

(Originally posted on Stack Overflow; was told to post here.)

Best Answer

You should explain what the intuition is that you have that the behavior runs counter to - it would make it easier to focus the explanation to address that.

A kernel density estimate is the convolution of the sample probability function ($n$ point masses of size $\frac{1}{n}$) and the kernel function (itself, by default, a normal density).

The result in the default case is a mixture of normal (Gaussian) densities, each with center at the data values, each with standard deviation $h$ (the bandwidth of the kernel), and weight $\frac{1}{n}$.

When all the data are coincident, the resulting mixture density is a sum of $n$ weighted densities, all with the same mean and standard deviation ... which is just the kernel itself, centered at that data value.

The difference in behavior you see might relate to the trim argument in ggplot2::stat_density. When the range of values is exactly zero, my guess is that it's setting trim to FALSE (or at least something other than TRUE), but when it's even a little larger than 0 it's at the default (TRUE). You'd need to look into the source to double check, but that would be my guess. If that's what's happening, you should be able to modify that behavior.

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