# Comparing many probability density functions

#### Jakub Nowosad

#### 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
    ##  20.581 21.7560 25.74881 23.1155 23.8055  284.644   100
    ##  28.083 28.7445 51.33078 29.8440 31.6055 2132.509   100
    ##   2.509  2.7240  3.39208  3.0130  3.5095    8.403   100
    ##   3.708  4.0495  6.02215  4.7150  5.0955  117.806   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
    ##    284.485   370.5515   456.4489   448.7075   533.3575   776.544   100
    ##  23620.943 25740.1705 29194.4739 27281.6785 32796.1225 45305.363   100
    ##     22.739    28.7505    34.3501    34.4265    37.6905   124.308   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) 2310.314
    ##  dist_many_many(M1, M2, method = "euclidean", testNA = FALSE)   37.081
    ##         lq       mean    median        uq       max neval
    ##  2569.2030 2878.19842 2668.8310 2813.3190 12068.858   100
    ##    42.0015   97.64644   45.9455   49.4415  5060.589   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.