# dist

##### Distance Matrix Computation

This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix.

- Keywords
- multivariate, cluster

##### Usage

`dist(x, method = "euclidean", diag = FALSE, upper = FALSE, p = 2)`as.dist(m, diag = FALSE, upper = FALSE)
# S3 method for default
as.dist(m, diag = FALSE, upper = FALSE)

# S3 method for dist
print(x, diag = NULL, upper = NULL,
digits = getOption("digits"), justify = "none",
right = TRUE, …)

# S3 method for dist
as.matrix(x, …)

##### Arguments

- x
a numeric matrix, data frame or

`"dist"`

object.- method
the distance measure to be used. This must be one of

`"euclidean"`

,`"maximum"`

,`"manhattan"`

,`"canberra"`

,`"binary"`

or`"minkowski"`

. Any unambiguous substring can be given.- diag
logical value indicating whether the diagonal of the distance matrix should be printed by

`print.dist`

.- upper
logical value indicating whether the upper triangle of the distance matrix should be printed by

`print.dist`

.- p
The power of the Minkowski distance.

- m
An object with distance information to be converted to a

`"dist"`

object. For the default method, a`"dist"`

object, or a matrix (of distances) or an object which can be coerced to such a matrix using`as.matrix()`

. (Only the lower triangle of the matrix is used, the rest is ignored).- digits, justify
passed to

`format`

inside of`print()`

.- right, …
further arguments, passed to other methods.

##### Details

Available distance measures are (written for two vectors \(x\) and \(y\)):

`euclidean`

:Usual distance between the two vectors (2 norm aka \(L_2\)), \(\sqrt{\sum_i (x_i - y_i)^2}\).

`maximum`

:Maximum distance between two components of \(x\) and \(y\) (supremum norm)

`manhattan`

:Absolute distance between the two vectors (1 norm aka \(L_1\)).

`canberra`

:\(\sum_i |x_i - y_i| / (|x_i| + |y_i|)\). Terms with zero numerator and denominator are omitted from the sum and treated as if the values were missing.

This is intended for non-negative values (e.g., counts), in which case the denominator can be written in various equivalent ways; Originally, R used \(x_i + y_i\), then from 1998 to 2017, \(|x_i + y_i|\), and then the correct \(|x_i| + |y_i|\).

`binary`

:(aka

*asymmetric binary*): The vectors are regarded as binary bits, so non-zero elements are ‘on’ and zero elements are ‘off’. The distance is the*proportion*of bits in which only one is on amongst those in which at least one is on.`minkowski`

:The \(p\) norm, the \(p\)th root of the sum of the \(p\)th powers of the differences of the components.

Missing values are allowed, and are excluded from all computations
involving the rows within which they occur.
Further, when `Inf`

values are involved, all pairs of values are
excluded when their contribution to the distance gave `NaN`

or
`NA`

.
If some columns are excluded in calculating a Euclidean, Manhattan,
Canberra or Minkowski distance, the sum is scaled up proportionally to
the number of columns used. If all pairs are excluded when
calculating a particular distance, the value is `NA`

.

The `"dist"`

method of `as.matrix()`

and `as.dist()`

can be used for conversion between objects of class `"dist"`

and conventional distance matrices.

`as.dist()`

is a generic function. Its default method handles
objects inheriting from class `"dist"`

, or coercible to matrices
using `as.matrix()`

. Support for classes representing
distances (also known as dissimilarities) can be added by providing an
`as.matrix()`

or, more directly, an `as.dist`

method
for such a class.

##### Value

`dist`

returns an object of class `"dist"`

.

The lower triangle of the distance matrix stored by columns in a
vector, say `do`

. If `n`

is the number of
observations, i.e., `n <- attr(do, "Size")`

, then
for \(i < j \le n\), the dissimilarity between (row) i and j is
`do[n*(i-1) - i*(i-1)/2 + j-i]`

.
The length of the vector is \(n*(n-1)/2\), i.e., of order \(n^2\).

The object has the following attributes (besides `"class"`

equal
to `"dist"`

):

integer, the number of observations in the dataset.

optionally, contains the labels, if any, of the observations of the dataset.

logicals corresponding to the arguments `diag`

and `upper`

above, specifying how the object should be printed.

optionally, the `call`

used to create the
object.

optionally, the distance method used; resulting from
`dist()`

, the (`match.arg()`

ed) `method`

argument.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979)
*Multivariate Analysis.* Academic Press.

Borg, I. and Groenen, P. (1997)
*Modern Multidimensional Scaling. Theory and Applications.*
Springer.

##### See Also

`daisy`

in the cluster package with more
possibilities in the case of *mixed* (continuous / categorical)
variables.
`hclust`

.

##### Examples

`library(stats)`

```
# NOT RUN {
require(graphics)
x <- matrix(rnorm(100), nrow = 5)
dist(x)
dist(x, diag = TRUE)
dist(x, upper = TRUE)
m <- as.matrix(dist(x))
d <- as.dist(m)
stopifnot(d == dist(x))
## Use correlations between variables "as distance"
dd <- as.dist((1 - cor(USJudgeRatings))/2)
round(1000 * dd) # (prints more nicely)
plot(hclust(dd)) # to see a dendrogram of clustered variables
## example of binary and canberra distances.
x <- c(0, 0, 1, 1, 1, 1)
y <- c(1, 0, 1, 1, 0, 1)
dist(rbind(x, y), method = "binary")
## answer 0.4 = 2/5
dist(rbind(x, y), method = "canberra")
## answer 2 * (6/5)
## To find the names
labels(eurodist)
## Examples involving "Inf" :
## 1)
x[6] <- Inf
(m2 <- rbind(x, y))
dist(m2, method = "binary") # warning, answer 0.5 = 2/4
## These all give "Inf":
stopifnot(Inf == dist(m2, method = "euclidean"),
Inf == dist(m2, method = "maximum"),
Inf == dist(m2, method = "manhattan"))
## "Inf" is same as very large number:
x1 <- x; x1[6] <- 1e100
stopifnot(dist(cbind(x, y), method = "canberra") ==
print(dist(cbind(x1, y), method = "canberra")))
## 2)
y[6] <- Inf #-> 6-th pair is excluded
dist(rbind(x, y), method = "binary" ) # warning; 0.5
dist(rbind(x, y), method = "canberra" ) # 3
dist(rbind(x, y), method = "maximum") # 1
dist(rbind(x, y), method = "manhattan") # 2.4
# }
```

*Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0*