dist                  package:stats                  R Documentation

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:

       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 square distance between the two vectors (2
          norm).

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

     'manhattan': Absolute distance between the two vectors (1 norm).

     'canberra': sum(|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.


     '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 pth root of the sum of the pth 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 <= 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"'): 

    Size: integer, the number of observations in the dataset.

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

Diag, Upper: logicals corresponding to the arguments 'diag' and 'upper'
          above, specifying how the object should be printed.

    call: optionally, the 'call' used to create the object.

  method: 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:

     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

