Compute all the pairwise dissimilarities (distances) between observations
in the data set. The original variables may be of mixed types. In
that case, or whenever metric = "gower" is set, a
generalization of Gower's formula is used, see ‘Details’
below.
daisy(x, metric = c("euclidean", "manhattan", "gower"),
stand = FALSE, type = list(), weights = rep.int(1, p),
warnBin = warnType, warnAsym = warnType, warnConst = warnType,
warnType = TRUE)numeric matrix or data frame, of dimension \(n\times p\),
say. Dissimilarities will be computed
between the rows of x. Columns of mode numeric
(i.e. all columns when x is a matrix) will be recognized as
interval scaled variables, columns of class factor will be
recognized as nominal variables, and columns of class ordered
will be recognized as ordinal variables. Other variable types
should be specified with the type argument. Missing values
(NAs) are allowed.
character string specifying the metric to be used.
The currently available options are "euclidean" (the default),
"manhattan" and "gower".
Euclidean distances are root sum-of-squares of differences, and
manhattan distances are the sum of absolute differences.
“Gower's distance” is chosen by metric "gower"
or automatically if some columns of x are not numeric. Also
known as Gower's coefficient (1971),
expressed as a dissimilarity, this implies that a particular
standardisation will be applied to each variable, and the
“distance” between two units is the sum of all the
variable-specific distances, see the details section.
logical flag: if TRUE, then the measurements in x
are standardized before calculating the
dissimilarities. Measurements are standardized for each variable
(column), by subtracting the variable's mean value and dividing by
the variable's mean absolute deviation.
If not all columns of x are numeric, stand will
be ignored and Gower's standardization (based on the
range) will be applied in any case, see argument
metric, above, and the details section.
list for specifying some (or all) of the types of the
variables (columns) in x. The list may contain the following
components: "ordratio" (ratio scaled variables to be treated as
ordinal variables), "logratio" (ratio scaled variables that
must be logarithmically transformed), "asymm" (asymmetric
binary) and "symm" (symmetric binary variables). Each
component's value is a vector, containing the names or the numbers
of the corresponding columns of x.
Variables not mentioned in the type list are interpreted as
usual (see argument x).
an optional numeric vector of length \(p\)(=ncol(x)); to
be used in “case 2” (mixed variables, or metric = "gower"),
specifying a weight for each variable (x[,k]) instead of
\(1\) in Gower's original formula.
logicals indicating if the corresponding type checking warnings should be signalled (when found).
logical indicating if all the type checking warnings should be active or not.
an object of class "dissimilarity" containing the
dissimilarities among the rows of x. This is typically the
input for the functions pam, fanny, agnes or
diana. For more details, see dissimilarity.object.
Dissimilarities are used as inputs to cluster analysis and multidimensional scaling. The choice of metric may have a large impact.
The original version of daisy is fully described in chapter 1
of Kaufman and Rousseeuw (1990).
Compared to dist whose input must be numeric
variables, the main feature of daisy is its ability to handle
other variable types as well (e.g. nominal, ordinal, (a)symmetric
binary) even when different types occur in the same data set.
The handling of nominal, ordinal, and (a)symmetric binary data is
achieved by using the general dissimilarity coefficient of Gower
(1971). If x contains any columns of these
data-types, both arguments metric and stand will be
ignored and Gower's coefficient will be used as the metric. This can
also be activated for purely numeric data by metric = "gower".
With that, each variable (column) is first standardized by dividing
each entry by the range of the corresponding variable, after
subtracting the minimum value; consequently the rescaled variable has
range \([0,1]\), exactly.
Note that setting the type to symm (symmetric binary) gives the
same dissimilarities as using nominal (which is chosen for
non-ordered factors) only when no missing values are present, and more
efficiently.
Note that daisy signals a warning when 2-valued numerical
variables do not have an explicit type specified, because the
reference authors recommend to consider using "asymm"; the
warning may be silenced by warnBin = FALSE.
In the daisy algorithm, missing values in a row of x are not
included in the dissimilarities involving that row. There are two
main cases,
If all variables are interval scaled (and metric is
not "gower"), the metric is "euclidean", and
\(n_g\) is the number of columns in which
neither row i and j have NAs, then the dissimilarity d(i,j) returned is
\(\sqrt{p/n_g}\) (\(p=\)ncol(x)) times the
Euclidean distance between the two vectors of length \(n_g\)
shortened to exclude NAs. The rule is similar for the "manhattan"
metric, except that the coefficient is \(p/n_g\). If \(n_g = 0\),
the dissimilarity is NA.
When some variables have a type other than interval scaled, or
if metric = "gower" is specified, the
dissimilarity between two rows is the weighted mean of the contributions of
each variable. Specifically,
$$d_{ij} = d(i,j) = \frac{\sum_{k=1}^p w_k \delta_{ij}^{(k)} d_{ij}^{(k)}}{
\sum_{k=1}^p w_k \delta_{ij}^{(k)}}.
$$
In other words, \(d_{ij}\) is a weighted mean of
\(d_{ij}^{(k)}\) with weights \(w_k \delta_{ij}^{(k)}\),
where \(w_k\)= weigths[k],
\(\delta_{ij}^{(k)}\) is 0 or 1, and
\(d_{ij}^{(k)}\), the k-th variable contribution to the
total distance, is a distance between x[i,k] and x[j,k],
see below.
The 0-1 weight \(\delta_{ij}^{(k)}\) becomes zero
when the variable x[,k] is missing in either or both rows
(i and j), or when the variable is asymmetric binary and both
values are zero. In all other situations it is 1.
The contribution \(d_{ij}^{(k)}\) of a nominal or binary variable to the total
dissimilarity is 0 if both values are equal, 1 otherwise.
The contribution of other variables is the absolute difference of
both values, divided by the total range of that variable. Note
that “standard scoring” is applied to ordinal variables,
i.e., they are replaced by their integer codes 1:K. Note
that this is not the same as using their ranks (since there
typically are ties).
As the individual contributions \(d_{ij}^{(k)}\) are in
\([0,1]\), the dissimilarity \(d_{ij}\) will remain in
this range.
If all weights \(w_k \delta_{ij}^{(k)}\) are zero,
the dissimilarity is set to NA.
Gower, J. C. (1971) A general coefficient of similarity and some of its properties, Biometrics 27, 857--874.
Kaufman, L. and Rousseeuw, P.J. (1990) Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.
Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997) Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis 26, 17--37.
dissimilarity.object, dist,
pam, fanny, clara,
agnes, diana.
# NOT RUN {
data(agriculture)
## Example 1 in ref:
## Dissimilarities using Euclidean metric and without standardization
d.agr <- daisy(agriculture, metric = "euclidean", stand = FALSE)
d.agr
as.matrix(d.agr)[,"DK"] # via as.matrix.dist(.)
## compare with
as.matrix(daisy(agriculture, metric = "gower"))
data(flower)
## Example 2 in ref
summary(dfl1 <- daisy(flower, type = list(asymm = 3)))
summary(dfl2 <- daisy(flower, type = list(asymm = c(1, 3), ordratio = 7)))
## this failed earlier:
summary(dfl3 <- daisy(flower,
type = list(asymm = c("V1", "V3"), symm= 2,
ordratio= 7, logratio= 8)))
# }
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