clue (version 0.2-1)

median: Median Partitions and Hierarchies

Description

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.

Usage

cl_median(x, method = NULL, weights = 1, control = list())

Arguments

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 l

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.

Value

• The median partition or hierarchy.

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. [object Object],[object Object],[object Object] 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.

References

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.

See Also

cl_medoid

Examples

## 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 ...