# Comparing many probability density functions

#### 2021-08-20

The philentropy package has several mechanisms to calculate distances between probability density functions. The main one is to use the the distance() function, which enables to compute 46 different distances/similarities between probability density functions (see ?philentropy::distance and a companion vignette for details). Alternatively, it is possible to call each distance/dissimilarity function directly. For example, the euclidean() function will compute the euclidean distance, while jaccard - the Jaccard distance. The complete list of available distance measures are available with the philentropy::getDistMethods() function.

Both of the above approaches have their pros and cons. The distance() function is more flexible as it allows users to use any distance measure and can return either a matrix or a dist object. It also has several defensive programming checks implemented, and thus, it is more appropriate for regular users. Single distance functions, such as euclidean() or jaccard(), can be, on the other hand, slightly faster as they directly call the underlining C++ code.

Now, we introduce three new low-level functions that are intermediaries between distance() and single distance functions. They are fairly flexible, allowing to use of any implemented distance measure, but also usually faster than calling the distance() functions (especially, if it is needed to use many times). These functions are:

• dist_one_one() - expects two vectors (probability density functions), returns a single value
• dist_one_many() - expects one vector (a probability density function) and one matrix (a set of probability density functions), returns a vector of values
• dist_many_many() - expects two matrices (two sets of probability density functions), returns a matrix of values

Let’s start testing them by attaching the philentropy package.

library(philentropy)

## dist_one_one()

dist_one_one() is a lower level equivalent to distance(). However, instead of accepting a numeric data.frame or matrix, it expects two vectors representing probability density functions. In this example, we create two vectors, P and Q.

P <- 1:10 / sum(1:10)
Q <- 20:29 / sum(20:29)

To calculate the euclidean distance between them we can use several approaches - (a) build-in R dist() function, (b) philentropy::distance(), (c) philentropy::euclidean(), or the new dist_one_one().

# install.packages("microbenchmark")
microbenchmark::microbenchmark(
dist(rbind(P, Q), method = "euclidean"),
distance(rbind(P, Q), method = "euclidean", test.na = FALSE, mute.message = TRUE),
euclidean(P, Q, FALSE),
dist_one_one(P, Q, method = "euclidean", testNA = FALSE)
)
## Unit: microseconds
##                                                                                    expr
##                                                 dist(rbind(P, Q), method = "euclidean")
##  distance(rbind(P, Q), method = "euclidean", test.na = FALSE,      mute.message = TRUE)
##                                                                  euclidean(P, Q, FALSE)
##                                dist_one_one(P, Q, method = "euclidean", testNA = FALSE)
##     min      lq     mean  median     uq      max neval
##  21.024 22.0665 26.83100 23.4125 23.901  336.156   100
##  32.786 33.7415 58.98310 34.5680 35.239 2315.590   100
##   2.586  2.8385  3.17071  3.0570  3.464    4.778   100
##   3.871  4.4115  5.46040  4.9085  5.213   56.764   100

All of them return the same, single value. However, as you can see in the benchmark above, some are more flexible, and others are faster.

## dist_one_many()

The role of dist_one_many() is to calculate distances between one probability density function (in a form of a vector) and a set of probability density functions (as rows in a matrix).

Firstly, let’s create our example data.

set.seed(2020-08-20)
P <- 1:10 / sum(1:10)
M <- t(replicate(100, sample(1:10, size = 10) / 55))

P is our input vector and M is our input matrix.

Distances between the P vector and probability density functions in M can be calculated using several approaches. For example, we could write a for loop (adding a new code) or just use the existing distance() function and extract only one row (or column) from the results. The dist_one_many() allows for this calculation directly as it goes through each row in M and calculates a given distance measure between P and values in this row.

# install.packages("microbenchmark")
microbenchmark::microbenchmark(
as.matrix(dist(rbind(P, M), method = "euclidean"))[1, ][-1],
distance(rbind(P, M), method = "euclidean", test.na = FALSE, mute.message = TRUE)[1, ][-1],
dist_one_many(P, M, method = "euclidean", testNA = FALSE)
)
## Unit: microseconds
##                                                                                             expr
##                                      as.matrix(dist(rbind(P, M), method = "euclidean"))[1, ][-1]
##  distance(rbind(P, M), method = "euclidean", test.na = FALSE,      mute.message = TRUE)[1, ][-1]
##                                        dist_one_many(P, M, method = "euclidean", testNA = FALSE)
##        min        lq        mean    median        uq       max neval
##    316.244   397.361   494.36541   491.927   568.745   849.122   100
##  26182.286 28366.181 32239.31384 30350.948 35339.433 50017.425   100
##     27.124    31.942    39.40121    37.929    43.306   127.129   100

The dist_one_many() returns a vector of values. It is, in this case, much faster than distance(), and visibly faster than dist() while allowing for more possible distance measures to be used.

## dist_many_many()

dist_many_many() calculates distances between two sets of probability density functions (as rows in two matrix objects).

Let’s create two new matrix example data.

set.seed(2020-08-20)
M1 <- t(replicate(10, sample(1:10, size = 10) / 55))
M2 <- t(replicate(10, sample(1:10, size = 10) / 55))

M1 is our first input matrix and M2 is our second input matrix. I am not aware of any function build-in R that allows calculating distances between rows of two matrices, and thus, to solve this problem, we can create our own - many_dists()

many_dists = function(m1, m2){
r = matrix(nrow = nrow(m1), ncol = nrow(m2))
for (i in seq_len(nrow(m1))){
for (j in seq_len(nrow(m2))){
x = rbind(m1[i, ], m2[j, ])
r[i, j] = distance(x, method = "euclidean", mute.message = TRUE)
}
}
r
}

… and compare it to dist_many_many().

# install.packages("microbenchmark")
microbenchmark::microbenchmark(
many_dists(M1, M2),
dist_many_many(M1, M2, method = "euclidean", testNA = FALSE)
)
## Unit: microseconds
##                                                          expr      min
##                                            many_dists(M1, M2) 2850.561
##  dist_many_many(M1, M2, method = "euclidean", testNA = FALSE)   40.507
##         lq       mean    median       uq      max neval
##  3218.5515 3782.73070 3417.8890 3681.904 16620.19   100
##    46.2875   53.14176   50.5715   54.483   172.86   100

Both many_dists()and dist_many_many() return a matrix. The above benchmark concludes that dist_many_many() is about 30 times faster than our custom many_dists() approach.