daisy: Dissimilarity Matrix Calculation

In cluster: "Finding Groups in Data": Cluster Analysis Extended Rousseeuw et al.

Dissimilarity Matrix Calculation

Description

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.

Usage

daisy(x, metric = c("euclidean", "manhattan", "gower"),
      stand = FALSE, type = list(), weights = rep.int(1, p),
      warnBin = warnType, warnAsym = warnType, warnConst = warnType,
      warnType = TRUE)

Arguments

x	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.
metric	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.
stand	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.
type	list for specifying some (or all) of the types of the variables (columns) in x. The list may contain the following components: "asymm" Asymmetric binary variable, aka "A" in result Types, see dissimilarity.object. "symm" Symmetric binary variable, aka "S". "factor" Nominal – the default for factor variables, aka "N". When the factor has 2 levels, this is equivalent to type = "S" for a (symmetric) binary variable. "ordered" Ordinal – the default for ordered (factor) variables, aka "O", see dissimilarity.object. "logratio" ratio scaled numeric variables that are to be logarithmically transformed (log10) and then treated as numeric ("I"): must be positive numeric variable. "ordratio" “raTio”-like variable to be treated as ordered (using the factor codes unclass(as.ordered(x[,j]))), aka "T". "numeric"/"integer" Interval scaled – the default for all numeric (incl integer) columns of x, aka "I" in result Types, see dissimilarity.object. Each component is a (character or numeric) vector, containing either the names or the numbers of the corresponding columns of x. Variables not mentioned in type are interpreted as usual, see argument x, and also ‘default’ above. Consequently, the default type = list() may often be sufficient.
weights	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.
warnBin, warnAsym, warnConst	logicals indicating if the corresponding type checking warnings should be signalled (when found).
warnType	logical indicating if all the type checking warnings should be active or not.

Details

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,

1. 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.

2. 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.

Value

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.

Background

Dissimilarities are used as inputs to cluster analysis and multidimensional scaling. The choice of metric may have a large impact.

Author(s)

Anja Struyf, Mia Hubert, and Peter and Rousseeuw, for the original version.
Martin Maechler improved the NA handling and type specification checking, and extended functionality to metric = "gower" and the optional weights argument.

References

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.

See Also

dissimilarity.object, dist, pam, fanny, clara, agnes, diana.

Examples

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"))

## Example 2 in reference, extended  ---  different ways of "mixed" / "gower":

example(flower) # -> data(flower) *and* provide 'flowerN'

summary(d0    <- daisy(flower))  # -> the first 3 {0,1} treated as *N*ominal
summary(dS123 <- daisy(flower,  type = list(symm = 1:3))) # first 3 treated as *S*ymmetric
stopifnot(dS123 == d0) # i.e.,  *S*ymmetric <==> *N*ominal {for 2-level factor}
summary(dNS123<- daisy(flowerN, type = list(symm = 1:3)))
stopifnot(dS123 == d0)
## by default, however ...
summary(dA123 <- daisy(flowerN)) # .. all 3 logicals treated *A*symmetric binary (w/ warning)
summary(dA3  <- daisy(flower, type = list(asymm = 3)))
summary(dA13 <- daisy(flower, type = list(asymm = c(1, 3), ordratio = 7)))
## Mixing variable *names* and column numbers (failed in the past):
summary(dfl3 <- daisy(flower, type = list(asymm = c("V1", "V3"), symm= 2,
                                          ordratio= 7, logratio= 8)))

## If we'd treat the first 3 as simple {0,1}
Nflow <- flower
Nflow[,1:3] <- lapply(flower[,1:3], \(f) as.integer(as.character(f)))
summary(dN <- daisy(Nflow)) # w/ warning: treated binary .. 1:3 as interval
## Still, using Euclidean/Manhattan distance for {0-1} *is* identical to treating them as "N" :
stopifnot(dN == d0)
stopifnot(dN == daisy(Nflow, type = list(symm = 1:3))) # or as "S"

cluster documentation built on Nov. 28, 2023, 1:07 a.m.