---
title: "Distances"
output: rmarkdown::html_vignette
vignette: >
%\VignetteIndexEntry{Distances}
%\VignetteEngine{knitr::rmarkdown}
\usepackage[utf8]{inputenc}
---
## 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 function `stats::dist()` and takes a `matrix` or `data.frame` object as input. The corresponding `matrix` or `data.frame` should store probability density functions (as rows) for which distance computations should be performed.
```r
# 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()`).
```r
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()`.
```r
# 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 and when selecting the native distance functions underlying `distance()` users can speed up their computations 3-5x.
To find out which `method`s are implemented in `distance()` you can consult
the `getDistMethods()` function.
```r
# 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.
```r
# 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.
```r
# 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.
```r
# 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()`.
```r
# 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()`.
```r
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.
> Note: Please make sure to insert vector objects (in our example P, Q) when directly running the low-level functions such as `euclidean()` etc. Otherwise, computational overheads are produced that
[significantly slow down computations when using large vectors](https://github.com/drostlab/philentropy/issues/30).
```r
# install.packages("microbenchmark")
library(microbenchmark)
microbenchmark(
distance(x,method = "euclidean", test.na = FALSE),
dist(x,method = "euclidean"),
euclidean(P, Q, 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(P, Q, 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](https://en.wikipedia.org/wiki/Euclidean_distance) is (named after [Euclid](https://en.wikipedia.org/wiki/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
> $d = max | P_i - Q_i |$
-->