Distance Matrix Computation

Description

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.

Usage

dist(x, method = "euclidean", diag = FALSE, upper = FALSE, p = 2)

as.dist(m, diag = FALSE, upper = FALSE)
## Default S3 method:
as.dist(m, diag = FALSE, upper = FALSE)

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

## S3 method for class 'dist'
as.matrix(x, ...)

Arguments

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. This also called “Jaccard” distance in some contexts. Here, two all-zero observations have distance 0, whereas in traditional Jaccard definitions, the distance would be undefined for that case and give NaN numerically.

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"):

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

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