median {clue} | R Documentation |
Compute the median of an ensemble of partitions or hierarchies. The median minimizes the sum of dissimilarities between itself and the elements of the ensemble over a suitable class of partitions or hierarchies.
cl_median(x, method = NULL, weights = 1, control = list())
x |
an ensemble of partitions or hierarchies, or something
coercible to that (see cl_ensemble ). |
method |
a character string specifying one of the built-in
methods for computing medians, or a function to be taken as a
user-defined method, or NULL (default value). If a character
string, its lower-cased version is matched against the lower-cased
names of the available built-in methods using pmatch .
See Details for available built-in methods and defaults. |
weights |
a numeric vector with non-negative case weights.
Recycled to the number of elements in the ensemble given by x
if necessary. |
control |
a list of control parameters. See Details. |
Median clusterings are special cases of “consensus” clusterings characterized as the solutions of an optimization problem. See Gordon (2001) for more information.
If all elements of the ensemble are partitions, the built-in methods
for obtaining medians proceed by minimizing L(m) = sum w_b
d(x_b, m) for a suitable dissimilarity measure d (see
cl_dissimilarity
) over all soft partitions with k
classes, where w_b is the case weight given to element x_b
of the ensemble.
Available methods are as follows.
"DWH"
The following control parameters are available for this method.
k
order
"GV1"
The following control parameters are available for this method.
k
maxiter
reltol
sqrt(.Machine$double.eps)
.start
verbose
getOption("verbose")
.
"GV3"
ls_fit_ultrametric
for more information on the SUMT approach. This optimization
problem is equivalent to finding the membership matrix m for
which the sum of the squared differences between C(m) = m m'
and the weighted average co-membership matrix sum_b w_b
C(m_b) of the partitions is minimal.
Availabe control parameters are method
, control
,
eps
, q
, and verbose
, which have the same
roles as for ls_fit_ultrametric
, and the following.
k
start
By default, method "DWH"
is used.
If all elements of the ensemble are hierarchies, the built-in method
(named "cophenetic"
) for computing medians is based on
minimizing L(u) = sum w_b d(x_b, u) over all ultrametrics,
where d is Euclidean dissimilarity. This is equivalent to
finding the best least squares ultrametric approximation of the
weighted average d = sum w_b u_b of the ultrametrics u_b
of the hierarchies x_b, which is attempted by calling
ls_fit_ultrametric
on d with appropriate control
parameters.
If a user-defined agreement method is to be employed, it must be a function taking the cluster ensemble, the case weights, and a list of control parameters as its arguments.
All built-in methods use heuristics for solving hard optimization problems, and cannot be guaranteed to find a global minimum. Standard practice would recommend to use the best solution found in “sufficiently many” replications of the methods.
The median partition or hierarchy.
E. Dimitriadou and A. Weingessel and K. Hornik (2002). A combination scheme for fuzzy clustering. International Journal of Pattern Recognition and Artificial Intelligence, 16, 901–912.
A. D. Gordon and M. Vichi (2001). Fuzzy partition models for fitting a set of partitions. Psychometrika, 66, 229–248.
A. D. Gordon (1999). Classification (2nd edition). Boca Raton, FL: Chapman & Hall/CRC.
## Median partition for the Rosenberg-Kim kinship terms partition ## data based on co-membership dissimilarities. data("Kinship82") m1 <- cl_median(Kinship82, method = "GV3", control = list(k = 3, verbose = TRUE)) ## (Note that one should really use several replicates of this.) ## Total co-membership dissimilarity: sum(cl_dissimilarity(Kinship82, m1, "comem")) ## Compare to the consensus solution given in Gordon & Vichi (2001). data("Kinship82_Consensus") m2 <- Kinship82_Consensus[["JMF"]] sum(cl_dissimilarity(Kinship82, m2, "comem")) ## Seems we get a better solution ... ## How dissimilar are these solutions? cl_dissimilarity(m1, m2, "comem") ## How "fuzzy" are they? cl_fuzziness(cl_ensemble(m1, m2)) ## Do the "nearest" hard partitions fully agree? cl_dissimilarity(as.cl_hard_partition(m1), as.cl_hard_partition(m2)) ## Hmm ... ## Median partition for the Gordon and Vichi (2001) macroeconomic ## partition data based on Euclidean dissimilarities. data("Macro") set.seed(1) m1 <- cl_median(Macro, method = "GV1", control = list(k = 2, verbose = TRUE)) ## (Note that one should really use several replicates of this.) ## Total Euclidean dissimilarity: sum(cl_dissimilarity(Macro, m1)) ## Compare to the consensus solution given in Gordon & Vichi (2001). data("Macro_Consensus") m2 <- Macro_Consensus[["MF1"]] sum(cl_dissimilarity(Macro, m2)) ## Seems we get a better solution ... ## And in fact, it is qualitatively different: table(as.cl_hard_partition(m1), as.cl_hard_partition(m2)) ## Hmm ...