# Distance Measures

## How to use `distance()`

The `distance()` function implemented in `philentropy` is able to compute 46 different distances/similarities between probability density functions (see `?philentropy::distance` for details).

### Simple Example

The `distance()` function is implemented using the same logic as R’s base functions `stats::dist()` and takes a `matrix` or `data.frame` as input. The corresponding `matrix` or `data.frame` should store probability density functions (as rows) for which distance computations should be performed.

    # define a probability density function P
    P <- 1:10/sum(1:10)
    # define a probability density function Q
    Q <- 20:29/sum(20:29)

    # combine P and Q as matrix object
    x <- rbind(P,Q)

Please note that when defining a `matrix` from vectors, probability vectors should be combined as rows (`rbind()`).

    library(philentropy)

    # compute the Euclidean Distance with default parameters
    distance(x, method = "euclidean")

    euclidean
    0.1280713

For this simple case you can compare the results with R’s base function to compute the euclidean distance `stats::dist()`.

    # compute the Euclidean Distance using R's base function
    stats::dist(x, method = "euclidean")

              P
    Q 0.1280713

However, the R base function `stats::dist()` only computes the following distance measures: `"euclidean", "maximum", "manhattan", "canberra", "binary" or "minkowski"`, whereas `distance()` allows you to choose from 46 distance/similarity measures.

To find out which `method`s are implemented in `distance()` you can consult the `getDistMethods()` function.

    # names of implemented distance/similarity functions
    getDistMethods()

     [1] "euclidean"         "manhattan"         "minkowski"         "chebyshev"
     [5] "sorensen"          "gower"             "soergel"           "kulczynski_d"
     [9] "canberra"          "lorentzian"        "intersection"      "non-intersection"
    [13] "wavehedges"        "czekanowski"       "motyka"            "kulczynski_s"
    [17] "tanimoto"          "ruzicka"           "inner_product"     "harmonic_mean"
    [21] "cosine"            "hassebrook"        "jaccard"           "dice"
    [25] "fidelity"          "bhattacharyya"     "hellinger"         "matusita"
    [29] "squared_chord"     "squared_euclidean" "pearson"           "neyman"
    [33] "squared_chi"       "prob_symm"         "divergence"        "clark"
    [37] "additive_symm"     "kullback-leibler"  "jeffreys"          "k_divergence"
    [41] "topsoe"            "jensen-shannon"    "jensen_difference" "taneja"
    [45] "kumar-johnson"     "avg"

Now you can choose any distance/similarity `method` that serves you.

    # compute the Jaccard Distance with default parameters
    distance(x, method = "jaccard")

     jaccard
    0.133869

Analogously, in case a probability matrix is specified the following output is generated.

    # combine three probabilty vectors to a probabilty matrix
    ProbMatrix <- rbind(1:10/sum(1:10), 20:29/sum(20:29),30:39/sum(30:39))
    rownames(ProbMatrix) <- paste0("Example", 1:3)

    # compute the euclidean distance between all
    # pairwise comparisons of probability vectors
    distance(ProbMatrix, method = "euclidean")

    #> Metric: 'euclidean'; comparing: 3 vectors.
              v1         v2         v3
    v1 0.0000000 0.12807130 0.13881717
    v2 0.1280713 0.00000000 0.01074588
    v3 0.1388172 0.01074588 0.00000000

Alternatively, users can specify the argument `use.row.names = TRUE` to maintain the rownames of the input matrix and pass them as rownames and colnames to the output distance matrix.

    # compute the euclidean distance between all
    # pairwise comparisons of probability vectors
    distance(ProbMatrix, method = "euclidean", use.row.names = TRUE)

    #> Metric: 'euclidean'; comparing: 3 vectors.
              Example1   Example2   Example3
    Example1 0.0000000 0.12807130 0.13881717
    Example2 0.1280713 0.00000000 0.01074588
    Example3 0.1388172 0.01074588 0.00000000

This output differs from the output of `stats::dist()`.

    # compute the euclidean distance between all
    # pairwise comparisons of probability vectors
    # using stats::dist()
    stats::dist(ProbMatrix, method = "euclidean")

               1          2
    2 0.12807130
    3 0.13881717 0.01074588

Whereas `distance()` returns a symmetric distance matrix, `stats::dist()` returns only one part of the symmetric matrix.

However, users can also specify the argument `as.dist.obj = TRUE` in `philentropy::distance()` to retrieve a `philentropy::distance()` output which is an object of type `stats::dist()`.

    ProbMatrix <- rbind(1:10/sum(1:10), 20:29/sum(20:29),30:39/sum(30:39))
    rownames(ProbMatrix) <- paste0("test", 1:3)
    distance(ProbMatrix, method = "euclidean", use.row.names = TRUE, as.dist.obj = TRUE)

    Metric: 'euclidean'; comparing: 3 vectors.
               test1      test2
    test2 0.12807130
    test3 0.13881717 0.01074588

Now let’s compare the run times of base R and `philentropy`. For this purpose you need to install the `microbenchmark` package.

    # install.packages("microbenchmark")
    library(microbenchmark)

    microbenchmark(
      distance(x,method = "euclidean", test.na = FALSE),
      dist(x,method = "euclidean"),
      euclidean(x[1 , ], x[2 , ], FALSE)
    )

    Unit: microseconds
                                                   expr    min      lq     mean  median      uq    max neval
     distance(x, method = "euclidean", test.na = FALSE) 26.518 28.3495 29.73174 29.2210 30.1025 62.096   100
                          dist(x, method = "euclidean") 11.073 12.9375 14.65223 14.3340 15.1710 65.130   100
                       euclidean(x[1, ], x[2, ], FALSE)  4.329  4.9605  5.72378  5.4815  6.1240 22.510   100

As you can see, although the `distance()` function is quite fast, the internal checks cause it to be 2x slower than the base `dist()` function (for the `euclidean` example). Nevertheless, in case you need to implement a faster version of the corresponding distance measure you can type `philentropy::` and then `TAB` allowing you to select the base distance computation functions (written in C++), e.g. `philentropy::euclidean()` which is almost 3x faster than the base `dist()` function.

The advantage of `distance()` is that it implements 46 distance measures based on base C++ functions that can be accessed individually by typing `philentropy::` and then `TAB`. In future versions of `philentropy` I will optimize the `distance()` function so that internal checks for data type correctness and correct input data will take less termination time than the base `dist()` function.

## Detailed assessment of individual similarity and distance metrics

The vast amount of available similarity metrics raises the immediate question which metric should be used for which application. Here, I will review the origin of each individual metric and will discuss the most recent literature that aims to compare these measures. I hope that users will find valuable insights and might be stimulated to conduct their own comparative research since this is a field of ongoing research.

### \(L_p\) Minkowski Family

#### Euclidean distance

The euclidean distance is (named after Euclid) a straight line distance between two points. Euclid argued that that the shortest distance between two points is always a line.

> \(d = \sqrt{\sum_{i = 1}^N | P_i - Q_i |^2)}\)

#### Manhattan distance

> \(d = \sum_{i = 1}^N | P_i - Q_i |\)

#### Minkowski distance

> \(d = ( \sum_{i = 1}^N | P_i - Q_i |^p)^{1/p}\)

#### Chebyshev distance

> \(d = max | P_i - Q_i |\)