daisy.RdCompute 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.
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:
"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.
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.
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.
Dissimilarities are used as inputs to cluster analysis and multidimensional scaling. The choice of metric may have a large impact.
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.
data(agriculture)
## Example 1 in ref:
## Dissimilarities using Euclidean metric and without standardization
d.agr <- daisy(agriculture, metric = "euclidean", stand = FALSE)
d.agr
#> Dissimilarities :
#> B DK D GR E F IRL
#> DK 5.408327
#> D 2.061553 3.405877
#> GR 22.339651 22.570113 22.661200
#> E 9.818350 11.182576 10.394710 12.567418
#> F 3.448188 3.512834 2.657066 20.100995 8.060397
#> IRL 12.747549 13.306014 13.080138 9.604166 3.140064 10.564563
#> I 5.803447 5.470832 5.423099 17.383325 5.727128 2.773085 7.920859
#> L 4.275512 2.220360 2.300000 24.035391 12.121056 4.060788 14.569145
#> NL 1.649242 5.096077 2.435159 20.752349 8.280097 2.202272 11.150785
#> P 17.236299 17.864490 17.664088 5.162364 7.430343 15.164432 4.601087
#> UK 2.828427 8.052950 4.850773 21.485344 8.984431 5.303772 12.103718
#> I L NL P
#> DK
#> D
#> GR
#> E
#> F
#> IRL
#> I
#> L 6.660330
#> NL 4.204759 4.669047
#> P 12.515990 19.168985 15.670673
#> UK 6.723095 7.102112 3.124100 16.323296
#>
#> Metric : euclidean
#> Number of objects : 12
as.matrix(d.agr)[,"DK"] # via as.matrix.dist(.)
#> B DK D GR E F IRL I
#> 5.408327 0.000000 3.405877 22.570113 11.182576 3.512834 13.306014 5.470832
#> L NL P UK
#> 2.220360 5.096077 17.864490 8.052950
## compare with
as.matrix(daisy(agriculture, metric = "gower"))
#> B DK D GR E F IRL
#> B 0.00000000 0.22148078 0.08178881 0.8438459 0.38135483 0.11538211 0.47547804
#> DK 0.22148078 0.00000000 0.13969197 0.9145729 0.45208184 0.12117405 0.54620505
#> D 0.08178881 0.13969197 0.00000000 0.8854337 0.42294264 0.09203485 0.51706585
#> GR 0.84384585 0.91457286 0.88543366 0.0000000 0.46249103 0.79339881 0.36836781
#> E 0.38135483 0.45208184 0.42294264 0.4624910 0.00000000 0.33090779 0.09412321
#> F 0.11538211 0.12117405 0.09203485 0.7933988 0.33090779 0.00000000 0.42503100
#> IRL 0.47547804 0.54620505 0.51706585 0.3683678 0.09412321 0.42503100 0.00000000
#> I 0.15222215 0.22294916 0.19380996 0.6916237 0.22913268 0.10177511 0.32325589
#> L 0.15646414 0.06501664 0.07467532 0.9601090 0.49761796 0.16671017 0.59174117
#> NL 0.05318802 0.19426679 0.09477583 0.7906578 0.32816681 0.08816811 0.42229002
#> P 0.66155453 0.73228154 0.70314234 0.1822913 0.28019970 0.61110749 0.18607649
#> UK 0.10095934 0.32244012 0.18274816 0.7629870 0.30049599 0.21634145 0.39461920
#> I L NL P UK
#> B 0.1522221 0.15646414 0.05318802 0.6615545 0.1009593
#> DK 0.2229492 0.06501664 0.19426679 0.7322815 0.3224401
#> D 0.1938100 0.07467532 0.09477583 0.7031423 0.1827482
#> GR 0.6916237 0.96010899 0.79065783 0.1822913 0.7629870
#> E 0.2291327 0.49761796 0.32816681 0.2801997 0.3004960
#> F 0.1017751 0.16671017 0.08816811 0.6111075 0.2163414
#> IRL 0.3232559 0.59174117 0.42229002 0.1860765 0.3946192
#> I 0.0000000 0.26848528 0.11202114 0.5093324 0.2401945
#> L 0.2684853 0.00000000 0.16945115 0.7778177 0.2574235
#> NL 0.1120211 0.16945115 0.00000000 0.6083665 0.1281733
#> P 0.5093324 0.77781766 0.60836651 0.0000000 0.5806957
#> UK 0.2401945 0.25742348 0.12817333 0.5806957 0.0000000
## Example 2 in reference, extended --- different ways of "mixed" / "gower":
example(flower) # -> data(flower) *and* provide 'flowerN'
#>
#> flower> data(flower)
#>
#> flower> str(flower) # factors, ordered, numeric
#> 'data.frame': 18 obs. of 8 variables:
#> $ V1: Factor w/ 2 levels "0","1": 1 2 1 1 1 1 1 1 2 2 ...
#> $ V2: Factor w/ 2 levels "0","1": 2 1 2 1 2 2 1 1 2 2 ...
#> $ V3: Factor w/ 2 levels "0","1": 2 1 1 2 1 1 1 2 1 1 ...
#> $ V4: Factor w/ 5 levels "1","2","3","4",..: 4 2 3 4 5 4 4 2 3 5 ...
#> $ V5: Ord.factor w/ 3 levels "1"<"2"<"3": 3 1 3 2 2 3 3 2 1 2 ...
#> $ V6: Ord.factor w/ 18 levels "1"<"2"<"3"<"4"<..: 15 3 1 16 2 12 13 7 4 14 ...
#> $ V7: num 25 150 150 125 20 50 40 100 25 100 ...
#> $ V8: num 15 50 50 50 15 40 20 15 15 60 ...
#>
#> flower> ## "Nicer" version (less numeric more self explainable) of 'flower':
#> flower> flowerN <- flower
#>
#> flower> colnames(flowerN) <- c("winters", "shadow", "tubers", "color",
#> flower+ "soil", "preference", "height", "distance")
#>
#> flower> for(j in 1:3) flowerN[,j] <- (flowerN[,j] == "1")
#>
#> flower> levels(flowerN$color) <- c("1" = "white", "2" = "yellow", "3" = "pink",
#> flower+ "4" = "red", "5" = "blue")[levels(flowerN$color)]
#>
#> flower> levels(flowerN$soil) <- c("1" = "dry", "2" = "normal", "3" = "wet")[levels(flowerN$soil)]
#>
#> flower> flowerN
#> winters shadow tubers color soil preference height distance
#> 1 FALSE TRUE TRUE red wet 15 25 15
#> 2 TRUE FALSE FALSE yellow dry 3 150 50
#> 3 FALSE TRUE FALSE pink wet 1 150 50
#> 4 FALSE FALSE TRUE red normal 16 125 50
#> 5 FALSE TRUE FALSE blue normal 2 20 15
#> 6 FALSE TRUE FALSE red wet 12 50 40
#> 7 FALSE FALSE FALSE red wet 13 40 20
#> 8 FALSE FALSE TRUE yellow normal 7 100 15
#> 9 TRUE TRUE FALSE pink dry 4 25 15
#> 10 TRUE TRUE FALSE blue normal 14 100 60
#> 11 TRUE TRUE TRUE blue wet 8 45 10
#> 12 TRUE TRUE TRUE white normal 9 90 25
#> 13 TRUE TRUE FALSE white normal 6 20 10
#> 14 TRUE TRUE TRUE red normal 11 80 30
#> 15 TRUE FALSE FALSE pink normal 10 40 20
#> 16 TRUE FALSE FALSE red normal 18 200 60
#> 17 TRUE FALSE FALSE yellow normal 17 150 60
#> 18 FALSE FALSE TRUE yellow dry 5 25 10
#>
#> flower> ## ==> example(daisy) on how it is used
#> flower>
#> flower>
#> flower>
summary(d0 <- daisy(flower)) # -> the first 3 {0,1} treated as *N*ominal
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1418 0.3904 0.4829 0.4865 0.5865 0.8875
#> Metric : mixed ; Types = N, N, N, N, O, O, I, I
#> Number of objects : 18
summary(dS123 <- daisy(flower, type = list(symm = 1:3))) # first 3 treated as *S*ymmetric
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1418 0.3904 0.4829 0.4865 0.5865 0.8875
#> Metric : mixed ; Types = S, S, S, N, O, O, I, I
#> Number of objects : 18
stopifnot(dS123 == d0) # i.e., *S*ymmetric <==> *N*ominal {for 2-level factor}
summary(dNS123<- daisy(flowerN, type = list(symm = 1:3)))
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1418 0.3904 0.4829 0.4865 0.5865 0.8875
#> Metric : mixed ; Types = S, S, S, N, O, O, I, I
#> Number of objects : 18
stopifnot(dS123 == d0)
## by default, however ...
summary(dA123 <- daisy(flowerN)) # .. all 3 logicals treated *A*symmetric binary (w/ warning)
#> Warning: setting 'logical' variables 1, 2, 3 to type 'asymm'
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1592 0.4358 0.5341 0.5347 0.6291 0.8910
#> Metric : mixed ; Types = A, A, A, N, O, O, I, I
#> Number of objects : 18
summary(dA3 <- daisy(flower, type = list(asymm = 3)))
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1418 0.4164 0.5101 0.5098 0.6051 0.8875
#> Metric : mixed ; Types = N, N, A, N, O, O, I, I
#> Number of objects : 18
summary(dA13 <- daisy(flower, type = list(asymm = c(1, 3), ordratio = 7)))
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1647 0.4387 0.5265 0.5293 0.6252 0.9007
#> Metric : mixed ; Types = A, N, A, N, O, O, T, I
#> Number of objects : 18
## 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)))
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1649 0.4378 0.5350 0.5288 0.6318 0.8972
#> Metric : mixed ; Types = A, S, A, N, O, O, T, I
#> Number of objects : 18
## If we'd treat the first 3 as simple {0,1}
Nflow <- flower
Nflow[,1:3] <- lapply(flower[,1:3], function(f) as.integer(as.character(f)))
summary(dN <- daisy(Nflow)) # w/ warning: treated binary .. 1:3 as interval
#> Warning: binary variable(s) 1, 2, 3 treated as interval scaled
#> 153 dissimilarities, summarized :
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.1418 0.3904 0.4829 0.4865 0.5865 0.8875
#> Metric : mixed ; Types = I, I, I, N, O, O, I, I
#> Number of objects : 18
## 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"