Comparing many probability density functions

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
##  13.174 13.9965 16.40060 14.5220 15.1030 174.956   100
##  19.316 20.0175 29.36207 20.5985 21.2040 855.084   100
##   1.163  1.2720  1.62433  1.4530  1.6580  15.458   100
##   1.843  2.0580  2.86564  2.2945  2.5045  53.300   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
##    139.159   147.630   170.26379   160.1935   186.1270   333.121   100
##  15798.455 16305.129 17371.60482 17672.7980 18313.0510 19345.575   100
##     21.169    22.482    27.19098    25.9585    29.3695    93.744   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       lq
##                                            many_dists(M1, M2) 1974.489 2047.050
##  dist_many_many(M1, M2, method = "euclidean", testNA = FALSE)   32.530   35.682
##        mean    median       uq      max neval
##  2251.90401 2073.3635 2097.693 7984.547   100
##    39.25039   37.1095   38.963   64.480   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.