cl_agreement: Agreement Between Partitions or Hierarchies

Description

Compute the agreement between (ensembles) of partitions or hierarchies.

Usage

cl_agreement(x, y = NULL, method = "euclidean", …)

Arguments

an ensemble of partitions or hierarchies and dissimilarities, or something coercible to that (see cl_ensemble).

NULL (default), or as for x.

method

a character string specifying one of the built-in methods for computing agreement, or a function to be taken as a user-defined method. 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.

…

further arguments to be passed to methods.

Value

If y is NULL, an object of class "cl_agreement" containing the agreements between the all pairs of components of x. Otherwise, an object of class "cl_cross_agreement" with the agreements between the components of x and the components of y.

Details

If y is given, its components must be of the same kind as those of x (i.e., components must either all be partitions, or all be hierarchies or dissimilarities).

If all components are partitions, the following built-in methods for measuring agreement between two partitions with respective membership matrices $u$ and $v$ (brought to a common number of columns) are available:

"euclidean": $1 - d / m$ , where $d$ is the Euclidean dissimilarity of the memberships, i.e., the square root of the minimal sum of the squared differences of $u$ and all column permutations of $v$ , and $m$ is an upper bound for the maximal Euclidean dissimilarity. See Dimitriadou, Weingessel and Hornik (2002).
"manhattan": $1 - d / m$ , where $d$ is the Manhattan dissimilarity of the memberships, i.e., the minimal sum of the absolute differences of $u$ and all column permutations of $v$ , and $m$ is an upper bound for the maximal Manhattan dissimilarity.
"Rand": the Rand index (the rate of distinct pairs of objects both in the same class or both in different classes in both partitions), see Rand (1971) or Gordon (1999), page 198. For soft partitions, (currently) the Rand index of the corresponding nearest hard partitions is used.
"cRand": the Rand index corrected for agreement by chance, see Hubert and Arabie (1985) or Gordon (1999), page 198. Can only be used for hard partitions.
"NMI": Normalized Mutual Information, see Strehl and Ghosh (2002). For soft partitions, (currently) the NMI of the corresponding nearest hard partitions is used.
"KP": the Katz-Powell index, i.e., the product-moment correlation coefficient between the elements of the co-membership matrices $C (u) = u u^{'}$ and $C (v)$ , respectively, see Katz and Powell (1953). For soft partitions, (currently) the Katz-Powell index of the corresponding nearest hard partitions is used. (Note that for hard partitions, the $(i, j)$ entry of $C (u)$ is one iff objects $i$ and $j$ are in the same class.)
"angle": the maximal cosine of the angle between the elements of $u$ and all column permutations of $v$ .
"diag": the maximal co-classification rate, i.e., the maximal rate of objects with the same class ids in both partitions after arbitrarily permuting the ids.
"FM": the index of Fowlkes and Mallows (1983), i.e., the ratio $N_{x y} / \sqrt{N_{x} N_{y}}$ of the number $N_{x y}$ of distinct pairs of objects in the same class in both partitions and the geometric mean of the numbers $N_{x}$ and $N_{y}$ of distinct pairs of objects in the same class in partition $x$ and partition $y$ , respectively. For soft partitions, (currently) the Fowlkes-Mallows index of the corresponding nearest hard partitions is used.
"Jaccard": the Jaccard index, i.e., the ratio of the numbers of distinct pairs of objects in the same class in both partitions and in at least one partition, respectively. For soft partitions, (currently) the Jaccard index of the corresponding nearest hard partitions is used.
"purity": the purity of the classes of x with respect to those of y, i.e., $\sum_{j} max_{i} n_{i j} / n$ , where $n_{i j}$ is the joint frequency of objects in class $i$ for x and in class $j$ for y, and $n$ is the total number of objects.
"PS": Prediction Strength, see Tibshirani and Walter (2005): the minimum, over all classes $j$ of y, of the maximal rate of objects in the same class for x and in class $j$ for y.

If all components are hierarchies, available built-in methods for measuring agreement between two hierarchies with respective ultrametrics $u$ and $v$ are as follows.

"euclidean": $1 / (1 + d)$ , where $d$ is the Euclidean dissimilarity of the ultrametrics (i.e., the square root of the sum of the squared differences of $u$ and $v$ ).
"manhattan": $1 / (1 + d)$ , where $d$ is the Manhattan dissimilarity of the ultrametrics (i.e., the sum of the absolute differences of $u$ and $v$ ).
"cophenetic": The cophenetic correlation coefficient. (I.e., the product-moment correlation of the ultrametrics.)
"angle": the cosine of the angle between the ultrametrics.
"gamma": $1 - d$ , where $d$ is the rate of inversions between the associated ultrametrics (i.e., the rate of pairs $(i, j)$ and $(k, l)$ for which $u_{i j} < u_{k l}$ and $v_{i j} > v_{k l}$ ). (This agreement measure is a linear transformation of Kruskal's $γ$ .)

The measures based on ultrametrics also allow computing agreement with “raw” dissimilarities on the underlying objects (R objects inheriting from class "dist").

If a user-defined agreement method is to be employed, it must be a function taking two clusterings as its arguments.

Symmetric agreement objects of class "cl_agreement" are implemented as symmetric proximity objects with self-proximities identical to one, and inherit from class "cl_proximity". They can be coerced to dense square matrices using as.matrix. It is possible to use 2-index matrix-style subscripting for such objects; unless this uses identical row and column indices, this results in a (non-symmetric agreement) object of class "cl_cross_agreement".

References

E. Dimitriadou, A. Weingessel and K. Hornik (2002). A combination scheme for fuzzy clustering. International Journal of Pattern Recognition and Artificial Intelligence, 16, 901--912. 10.1142/S0218001402002052.

E. B. Fowlkes and C. L. Mallows (1983). A method for comparing two hierarchical clusterings. Journal of the American Statistical Association, 78, 553--569. 10.1080/01621459.1983.10478008.

A. D. Gordon (1999). Classification (2nd edition). Boca Raton, FL: Chapman & Hall/CRC.

L. Hubert and P. Arabie (1985). Comparing partitions. Journal of Classification, 2, 193--218. 10.1007/bf01908075.

W. M. Rand (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846--850. 10.2307/2284239.

L. Katz and J. H. Powell (1953). A proposed index of the conformity of one sociometric measurement to another. Psychometrika, 18, 249--256. 10.1007/BF02289063.

A. Strehl and J. Ghosh (2002). Cluster ensembles --- A knowledge reuse framework for combining multiple partitions. Journal of Machine Learning Research, 3, 583--617. https://www.jmlr.org/papers/volume3/strehl02a/strehl02a.pdf.

R. Tibshirani and G. Walter (2005). Cluster validation by Prediction Strength. Journal of Computational and Graphical Statistics, 14/3, 511--528. 10.1198/106186005X59243.

Examples

Run this code

# NOT RUN {
## An ensemble of partitions.
data("CKME")
pens <- CKME[1 : 20]		# for saving precious time ...
summary(c(cl_agreement(pens)))
summary(c(cl_agreement(pens, method = "Rand")))
summary(c(cl_agreement(pens, method = "diag")))
cl_agreement(pens[1:5], pens[6:7], method = "NMI")
## Equivalently, using subscripting.
cl_agreement(pens, method = "NMI")[1:5, 6:7]

## An ensemble of hierarchies.
d <- dist(USArrests)
hclust_methods <-
    c("ward", "single", "complete", "average", "mcquitty")
hclust_results <- lapply(hclust_methods, function(m) hclust(d, m))
names(hclust_results) <- hclust_methods 
hens <- cl_ensemble(list = hclust_results)
summary(c(cl_agreement(hens)))
## Note that the Euclidean agreements are *very* small.
## This is because the ultrametrics differ substantially in height:
u <- lapply(hens, cl_ultrametric)
round(sapply(u, max), 3)
## Rescaling the ultrametrics to [0, 1] gives:
u <- lapply(u, function(x) (x - min(x)) / (max(x) - min(x)))
shens <- cl_ensemble(list = lapply(u, as.cl_dendrogram))
summary(c(cl_agreement(shens)))
## Au contraire ...
summary(c(cl_agreement(hens, method = "cophenetic")))
cl_agreement(hens[1:3], hens[4:5], method = "gamma")
# }

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