\HeaderA{dist}{Distance Matrix Computation}{dist}
\aliasA{as.dist}{dist}{as.dist}
\methaliasA{as.dist.default}{dist}{as.dist.default}
\aliasA{as.matrix.dist}{dist}{as.matrix.dist}
\aliasA{format.dist}{dist}{format.dist}
\aliasA{labels.dist}{dist}{labels.dist}
\aliasA{print.dist}{dist}{print.dist}
\keyword{multivariate}{dist}
\keyword{cluster}{dist}
\begin{Description}\relax
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.
\end{Description}
\begin{Usage}
\begin{verbatim}
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, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a numeric matrix, data frame or \code{"dist"} object.
\item[\code{method}] the distance measure to be used. This must be one of
\code{"euclidean"}, \code{"maximum"}, \code{"manhattan"},
\code{"canberra"}, \code{"binary"} or \code{"minkowski"}.
Any unambiguous substring can be given.
\item[\code{diag}] logical value indicating whether the diagonal of the
distance matrix should be printed by \code{print.dist}.
\item[\code{upper}] logical value indicating whether the upper triangle of the
distance matrix should be printed by \code{print.dist}.
\item[\code{p}] The power of the Minkowski distance.
\item[\code{m}] An object with distance information to be converted to a
\code{"dist"} object.  For the default method, a \code{"dist"}
object, or a matrix (of distances) or an object which can be coerced
to such a matrix using \code{\LinkA{as.matrix}{as.matrix}()}.  (Only the lower
triangle of the matrix is used, the rest is ignored).
\item[\code{digits, justify}] passed to \code{\LinkA{format}{format}} inside of
\code{print()}.
\item[\code{right, ...}] further arguments, passed to other methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Available distance measures are (written for two vectors \eqn{x}{} and
\eqn{y}{}):
\describe{
\item[\code{euclidean}:] Usual square distance between the two
vectors (2 norm).

\item[\code{maximum}:] Maximum distance between two components of \eqn{x}{}
and \eqn{y}{} (supremum norm)

\item[\code{manhattan}:] Absolute distance between the two vectors
(1 norm).

\item[\code{canberra}:] \eqn{\sum_i |x_i - y_i| / |x_i + y_i|}{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.


\item[\code{binary}:] (aka \emph{asymmetric binary}): The vectors
are regarded as binary bits, so non-zero elements are \sQuote{on}
and zero elements are \sQuote{off}.  The distance is the
\emph{proportion} of bits in which only one is on amongst those in
which at least one is on.

\item[\code{minkowski}:] The \eqn{p}{} norm, the \eqn{p}{}th root of the
sum of the \eqn{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 \code{Inf} values are involved, all pairs of values are
excluded when their contribution to the distance gave \code{NaN} or
\code{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 \code{NA}.

The \code{"dist"} method of \code{as.matrix()} and \code{as.dist()}
can be used for conversion between objects of class \code{"dist"}
and conventional distance matrices.

\code{as.dist()} is a generic function.  Its default method handles
objects inheriting from class \code{"dist"}, or coercible to matrices
using \code{\LinkA{as.matrix}{as.matrix}()}.  Support for classes representing
distances (also known as dissimilarities) can be added by providing an
\code{\LinkA{as.matrix}{as.matrix}()} or, more directly, an \code{as.dist} method
for such a class.
\end{Details}
\begin{Value}
\code{dist} returns an object of class \code{"dist"}.

The lower triangle of the distance matrix stored by columns in a
vector, say \code{do}. If \code{n} is the number of
observations, i.e., \code{n <- attr(do, "Size")}, then
for \eqn{i < j <= n}{}, the dissimilarity between (row) i and j is
\code{do[n*(i-1) - i*(i-1)/2 + j-i]}.
The length of the vector is \eqn{n*(n-1)/2}{}, i.e., of order \eqn{n^2}{}.

The object has the following attributes (besides \code{"class"} equal
to \code{"dist"}):
\begin{ldescription}
\item[\code{Size}] integer, the number of observations in the dataset.
\item[\code{Labels}] optionally, contains the labels, if any, of the
observations of the dataset.
\item[\code{Diag, Upper}] logicals corresponding to the arguments \code{diag}
and \code{upper} above, specifying how the object should be printed.
\item[\code{call}] optionally, the \code{\LinkA{call}{call}} used to create the
object.
\item[\code{method}] optionally, the distance method used; resulting from
\code{\LinkA{dist}{dist}()}, the (\code{\LinkA{match.arg}{match.arg}()}ed) \code{method}
argument.
\end{ldescription}
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979)
\emph{Multivariate Analysis.} Academic Press.

Borg, I. and Groenen, P. (1997)
\emph{Modern Multidimensional Scaling.  Theory and Applications.}
Springer.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{daisy}{daisy}} in the \pkg{cluster} package with more
possibilities in the case of \emph{mixed} (continuous / categorical)
variables.
\code{\LinkA{hclust}{hclust}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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
\end{ExampleCode}
\end{Examples}

