metric = "gower" is set, a
generalization of Gower's formula is used, see daisy(x, metric = c("euclidean", "manhattan", "gower"),
stand = FALSE, type = list(), weights = rep.int(1, p))x. Columns of mode numeric
(i.e. all columns when x is a matrix) will be recognized as
"euclidean" (the default),
"manhattan" and "gower".
Euclidean distances are root sum-of-squares of differences, and
mx
are standardized before calculating the
dissimilarities. Measurements are standardized for each variable
(column), by subtracting the variable's mean value and dividing by
thex. The list may contain the following
components: "ordratio" (ratio scaled variables to be treated as
ordinal variables), "logratio"<ncol(x)); to
be used in metric = "gower"),
specifying a weight for each variable (x[,k]) instead of
$1$ in Gower's or"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.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 now gives a warning when 2-valued numerical
variables do not have an explicit type specified, because the
reference authors recommend to consider using "asymm".
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,
metricisnot"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.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 betweenx[i,k]andx[j,k],
see below. The 0-1 weight$\delta_{ij}^{(k)}$becomes zero
when the variablex[,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
that1:K. Note
that this is not the same as using their ranks (since there
typically are ties).
% contrary to what Kaufman & Rousseeuw write in their book, and
% the original help page.
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 toNA.
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.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)))Run the code above in your browser using DataLab