# How to calculate overlap between empirical probability densities?

I'm looking for a method to calculate the area of overlap between two kernel density estimates in R, as a measure of similarity between two samples. To clarify, in the following example, I would need to quantify the area of the purplish overlapping region:

library(ggplot2)
set.seed(1234)
d <- data.frame(variable=c(rep("a", 50), rep("b", 30)), value=c(rnorm(50), runif(30, 0, 3)))
ggplot(d, aes(value, fill=variable)) + geom_density(alpha=.4, color=NA) A similar question was discussed here, the difference being that I need to do this for arbitrary empirical data rather than predefined normal distributions. The overlap package addresses this question, but apparently only for timestamp data, which doesn't work for me. The Bray-Curtis index (as implemented in vegan package's vegdist(method="bray") function) also seems relevant but again for somewhat different data.

I'm interested in both the theoretical approach and the R functions I might employ to implement it.

• "quantify the purple area" is a problem in estimation, not in hypothesis testing, so you cannot hope to "accomplish this using a standard citable statistical test". You contradict yourself. Please clarify what you actually want. If all you want is an estimate of the area of overlap of two KDEs, that's a simple calculation. – Glen_b May 14 '14 at 0:39
• @Glen_b thanks for the comment, helped to clarify my non-statistician thinking. I believe the area of overlap between KDEs is indeed what I'm looking for -- I've edited the question to reflect that. – mmk May 14 '14 at 5:25
• I would be very concerned about the risk of arbitrariness in this method. Depending on the kernel bandwidth, the computed overlap between any two datasets could be made to equal any chosen value in the interval $(0,1)$. The default bandwidths are not optimized for this purpose and therefore conceivably could give surprising, arbitrary, or inconsistent results. Datasets with natural bounds (such as non-negative data or proportions, etc.) would further introduce unwanted edge effects. What to do instead? Start with the reason for this calculation: what is this "similarity" intended to mean? – whuber Jun 4 '15 at 19:26
• The same question appeared a few months later but referred to intersection points however there were some valid notes which could be taken into consideration. In the referred question is about two empirical distributions. I add the link as this post only answers this via kernel density estimation and for normal distributions. The link below I think extends on the question for pairs of empirical distributions. stats.stackexchange.com/questions/122857/… – Barnaby 7 hours ago – Barnaby Jul 27 '15 at 10:13

## 4 Answers

The area of overlap of two kernel density estimates may be approximated to any desired degree of accuracy.

1) Since the original KDEs have probably been evaluated over some grid, if the grid is the same for both (or can easily be made the same), the exercise could be as easy as simply taking $\min(K_1(x),K_2(x))$ at each point and then using the trapezoidal rule, or even a midpoint rule.

If the two are on different grids and can't easily be recalculated on the same grid, interpolation could be used.

2) You might find the point (or points) of intersection and integrate the lower of the two KDEs in each interval where each one is lower. In your diagram above you'd integrate the blue curve to the left of the intersection and the pink one to the right by whatever means you like/have available. This can be done essentially exactly by considering the area under each kernel component $\frac{1}{h}K(\frac{x-x_i}{h})$ to the left or right of that cut-off point.

However, whuber's comments above should be clearly borne in mind -- this is not necessarily a very meaningful thing to do.

• How do you calculate the error associated with method one and method 2? – olliepower May 14 '14 at 7:17
• In normal circumstances, both will be miniscule compared with the error in the kernel density estimates, so I wouldn't worry too much. Error bounds can be calculated on trapezoidal methods and other numerical integration of course - such calculations are pretty standard - but it's pointless worrying given that KDEs have large uncertainties. Method 2 will be accurate to accumulated rounding error of the calculations. – Glen_b May 14 '14 at 8:17
• These methodology suggestions makes sense, thanks very much for your answer. I will work on implementing this in R, but as a novice I would be interested in suggestions on how to code this cleanly. – mmk May 14 '14 at 17:21

For the sake of completeness, here's how I ended up doing this in R:

# simulate two samples
a <- rnorm(100)
b <- rnorm(100, 2)

# define limits of a common grid, adding a buffer so that tails aren't cut off
lower <- min(c(a, b)) - 1
upper <- max(c(a, b)) + 1

# generate kernel densities
da <- density(a, from=lower, to=upper)
db <- density(b, from=lower, to=upper)
d <- data.frame(x=da$x, a=da$y, b=db$y) # calculate intersection densities d$w <- pmin(d$a, d$b)

# integrate areas under curves
library(sfsmisc)
total <- integrate.xy(d$x, d$a) + integrate.xy(d$x, d$b)
intersection <- integrate.xy(d$x, d$w)

# compute overlap coefficient
overlap <- 2 * intersection / total


As noted, there is inherent uncertainty and subjectivity involved in the KDE generation and also in the integration.

• There is now a package on CRAN called overlapping that estimates the area of the overlap of 2 (or more) empirical distributions. Check out the documentation here: rdocumentation.org/packages/overlapping/versions/1.5.0/topics/… – Stefan Avey Oct 31 '17 at 15:35
• Total should be: total = integrate.xy(d$x, d$a) + integrate.xy(d$x, d$b) - integrate.xy(d$x, d$w), which can be confirmed by using the package overlapping. – Rafael Jun 26 '19 at 14:27
• @mmk can you do this for 2D densities? – OverFlow Police Jul 1 '19 at 16:04

First, I might be wrong but I think your solution wouldn't work in case where there is multiples points where the Kernel Density Estimates (KDE) intersect. Second, although the overlap package was created for use with timestamp data, you can still use it to estimate the area of overlap of any two KDEs. You simply have to rescale your data so that it range from 0 to 2π.
For exemple :

# simulate two sample
a <- rnorm(100)
b <- rnorm(100, 2)

# To use overplapTrue(){overlap} the scale must be in radian (i.e. 0 to 2pi)
# To keep the *relative* value of a and b the same, combine a and b in the
# same dataframe before rescaling. You'll need to load the ‘scales‘ library.
# But first add a "Source" column to be able to distinguish between a and b
# after they are combined.
a = data.frame( value = a, Source = "a" )
b = data.frame( value = b, Source = "b" )
d = rbind(a, b)
library(scales)
d$value <- rescale( d$value, to = c(0,2*pi) )

# Now you can created the rescaled a and b vectors
a <- d[d$Source == "a", 1] b <- d[d$Source == "b", 1]

# You can then calculate the area of overlap as you did previously.
# It should give almost exactly the same answers.
# Or you can use either the overlapTrue() and overlapEst() function
# provided with the overlap packages.
# Note that with these function the KDE are fitted using von Mises kernel.
library(overlap)
# Using overlapTrue():
# define limits of a common grid, adding a buffer so that tails aren't cut off
lower <- min(d$value)-1 upper <- max(d$value)+1
# generate kernel densities
da <- density(a, from=lower, to=upper, adjust = 1)
db <- density(b, from=lower, to=upper, adjust = 1)
# Compute overlap coefficient
overlapTrue(da$y,db$y)

# Using overlapEst():
overlapEst(a, b, kmax = 3, adjust=c(0.8, 1, 4), n.grid = 500)

# You can also plot the two KDEs and the region of overlap using overlapPlot()
# but sadly I haven't found a way of changing the x scale so that the scale
# range correspond to the initial x value and not the rescaled value.
# You can only change the maximum value of the scale using the xscale argument
# (i.e. it always range from 0 to n, where n is set with xscale = n).
# So if some of your data take negative value, you're probably better off with
# a different plotting method. You can change the x label with the xlab
# argument.
overlapPlot(a, b, xscale = 10, xlab= "x metrics", rug=T)


An alternative method for empirical estimation is to use the ROC (Receiver Operating Curve) technology for the estimation. The Youden threshold gives us an empirical estimate for the main point of intersection (see https://journals.lww.com/epidem/Fulltext/2005/01000/Optimal_Cut_point_and_Its_Corresponding_Youden.11.aspx and https://math.stackexchange.com/questions/2404750/intersection-normal-distributions-and-minimal-decision-error/2435957#2435957).

The Youden threshold is the threshold where the sum of test sensitivity and specificity is maximized and the sum of the error rates (False positive Rate and False Negative Rate) is minimized. The overlap is equal to this minimal sum of error rates.

library(UncertainInterval)
simple_roc2 <- function(ref, test){
tab = table(test, ref) # head(tab)
data.frame(threshold=paste('>=',rownames(tab)),
ref0 = tab[,1],
ref1 = tab[,2],
FPR = rev(cumsum(rev(tab[,1])/sum(tab[,1]))), # 1-Sp
TPR = rev(cumsum(rev(tab[,2])/sum(tab[,2]))), # Se
row.names=1:nrow(tab))
}
a <- rnorm(10000)
b <- rnorm(10000, 2)
test=c(a,b)
ref=c(rep(0, length(a)), rep(1, length(b)))
# table(test, ref)
res = simple_roc2(ref, test)
res$FNR = 1-res$TPR # 1-Se
pos.optimal.threshold = which.min(res$FPR+res$FNR)
optimal.threshold=row.names(table(test, ref))[pos.optimal.threshold] # Youden threshold
plotMD(ref, test) # library(UncertainInterval) # includes kernel intersection estimate
abline(v=optimal.threshold, col='red')
overlap1(a, b)
(overlap2 = min(res$FPR+res$FNR))


In this case, this non-parametric estimate has a slight tendency of under estimation of the true value. This roc-technique only handles a single (main) point of intersection. It is not dependent on any specific distribution. Make sure that distribution b has the higher values (mean(b) > mean(a)).

Repeatedly eyeballing plots produced by plotMD shows that with 2 * 100 cases the sample overlap differs considerably. Most differences are due to sample differences, but, dependent on the distributions, all methods have conditions in which they do not work properly. Using a gaussian kernel density is sensitive to spikes in the data, which can be underestimated. Kernel density methods are dependent on the fine-tuning parameters given to the density function. The roc-method has no parameters, but it assumes a single point of intersection. Consequentially, it may overestimate overlap when an additional point of intersection is present (the critical point is the presence of more than one point of intersection, not variance). This overestimation may be negligible when this secondary point of intersection is at the tails of both distributions.

How to make sense of the different methods and suggestions? Devising a test is most simple when we know the true value of two distributions. The true value of the overlap for the two normal distributions is easy to calculate. The point of intersection is simply the mean of the two means of the distributions, as they have equal variance: 1. The true overlap is then 0.3173105:

(true.overlap = pnorm(1,2,1)+ 1-pnorm(1,0,1))


See https://stackoverflow.com/questions/16982146/point-of-intersection-2-normal-curves/45184024#45184024 for a general method to calculate the point of intersection for two normal distributions.

In the original problem, there is a mix of a normal and a uniform distribution. The true value is in that case:

    true.value=sum(pmin(diff(pnorm(0:3)),1/3))


Running a simulation can show us which estimation method produces estimates that are closest to the true value:

library(sfsmisc)
overlap1 <- function(a,b){
lower <- min(c(a, b)) - 1
upper <- max(c(a, b)) + 1

# generate kernel densities
da <- density(a, from=lower, to=upper)
db <- density(b, from=lower, to=upper)
d <- data.frame(x=da$$x, a=da$$y, b=db$y) # calculate intersection densities d$$w <- pmin(d$$a, d$b)

# integrate areas under curves
total <- integrate.xy(d$$x, d$$a) + integrate.xy(d$$x, d$$b)
intersection <- integrate.xy(d$$x, d$$w)

# compute overlap coefficient
2 * intersection / total
}

library(overlap)
library(scales)
# For explanation of the next function see the answer of S. Venne
overlapEstimates =function(a, b){

a = data.frame( value = a, Source = "a" )
b = data.frame( value = b, Source = "b" )
d = rbind(a, b)

d$$value <- scales::rescale( d$$value, to = c(0,2*pi) )

a <- d[d$$Source == "a", 1] b <- d[d$$Source == "b", 1]

overlapEst(a, b, kmax = 3, adjust=c(0.8, 1, 4), n.grid = 500)
}

nsim=1000; nobs=100; m1=4; sd1=1; m2=6; sd2=1; poi=5
(true.overlap= 1-pnorm(poi, m1, sd1)+pnorm(poi,m2,sd2))
out=matrix(NA,nrow=nsim,ncol=4)
set.seed(0)
for (i in 1:nsim){
x <- rnorm( nobs, m1, sd1 )
y <- rnorm( nobs, m2, sd2 )

out[i,1] = overlap1(x,y)
out[i,2] = overlapping::overlap(list( x = x, y = y ))\$OV
out[i,3] = overlapEstimates(x,y)['Dhat4']
out[i,4] = roc.overlap(x,y)
}
(true.overlap=pnorm(poi,m2,sd2)+1-pnorm(poi,m1,sd1))
colMeans(out-true.overlap) # estimation errors
apply(out, 2, sd) # # sd of the estimation errors
apply(out, 2, range)-true.overlap
par(mfrow=c(2,2))
br = seq(-.33,+.33,by=0.05)
hist(out[,1]-true.overlap, breaks=br, ylim=c(0,500),
xlim=c(-.33,.33), main='overlap1');
abline(v=0, col='red')
hist(out[,2]-true.overlap, breaks=br, ylim=c(0,500),
xlim=c(-.33,.33), main='overlapping::overlap')
abline(v=0, col='red')
hist(out[,3]-true.overlap, breaks=br, ylim=c(0,600),
xlim=c(-.33,.33), main='overlap::overlapEst')
abline(v=0, col='red')
hist(out[,4]-true.overlap, breaks=br, ylim=c(0,500),
xlim=c(-.33,.33), main="ROC estimate");
abline(v=0, col='red') In this case, especially the function overlapping::overlap has a tendency of (slight) under-estimation, while overlap1 shows the least estimation error. Estimates that use the density function in one way or another, can produce better or worse results dependent on the parameters given to the density function. The roc method does not have parameters, which can be an advantage.

It is always wise to carefully look at a plot of the overlapping distributions and to devise a relevant testing method whether the technique for overlap estimation works as expected for the kind of data that you have. Especially techniques that systematically produce estimates that are often too low or to high are better not used.

• (1) Your initial analysis concerns theoretical distributions, not empirical ones; and it implicitly assumes they have equal variances (as well as being Normal). (2) This post does not appear to answer the question. – whuber Jan 14 at 14:10
• @whuber Thank you for your comment. You are right, the first analysis is additional, and shows a simple way to evaluate the different estimate methods. However, the roc method is an assumption free method for estimating the overlap of two empirical distributions and offers a complete answer on the original question. Not even a gaussian kernel has to be used, as is implicitly applied in the first method (overlap1). You may want to evaluate this method and compare it with the other ... – tak101 Jan 16 at 14:01