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
##  21.017 22.6785 33.91872 23.9470 26.1130  305.616   100
##  27.555 28.9300 67.36313 29.9755 33.1265 2190.872   100
##   2.548  2.9290  3.72877  3.1580  3.7405   24.948   100
##   3.645  4.2180  6.32354  4.7135  5.1485  127.471   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
##    389.581   528.141   705.30695   703.8735   799.2435  2660.721   100
##  33281.702 39837.296 47609.73996 45371.5355 52265.8085 92813.572   100
##     34.407    40.242    56.62543    49.6750    59.3605   169.750   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) 3049.288 3479.443
##  dist_many_many(M1, M2, method = "euclidean", testNA = FALSE)   48.604   57.196
##        mean   median       uq       max neval
##  5008.11392 4017.414 5288.151 24289.867   100
##    77.55257   64.302   78.925   507.448   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.

Jakub Nowosad

2021-08-20

`dist_one_one()`

`dist_one_many()`

`dist_many_many()`