validity: Validity Measures for Partitions and Hierarchies

Description

Compute validity measures for partitions and hierarchies, attempting to measure how well these clusterings capture the underlying structure in the data they were obtained from.

Usage

cl_validity(x, ...)
## S3 method for class 'default':
cl_validity(x, d, ...)

Arguments

an object representing a partition or hierarchy.

a dissimilarity object from which x was obtained.

...

arguments to be passed to or from methods.

Value

A list of class "cl_validity" with the computed validity measures.

Details

cl_validity is a generic function.

For partitions, its default method gives the dissimilarity accounted for, defined as $1 - a_w / a_t$, where $a_t$ is the average total dissimilarity, and the average within dissimilarity $a_w$ is given by $$\sum_{i,j} \sum_k m_{ik}m_{jk} d_{ij} / \sum_{i,j} \sum_k m_{ik}m_{jk}$$ where $d$ and $m$ are the dissimilarities and memberships, respectively, and the sums are over all pairs of objects and all classes.

For hierarchies, the validity measures computed by default are variance accounted for (VAF, e.g., Hubert, Arabie & Meulman, 2006) and deviance accounted for (DEV, e.g., Smith, 2001). If u is the ultrametric corresponding to the hierarchy x and d the dissimilarity x was obtained from, these validity measures are given by $$\mathrm{VAF} = \max\left(1 - \sum_{i,j} (d_{ij} - u_{ij})^2 / \sum_{i,j} (d_{ij} - \mathrm{mean}(d)) ^ 2, 0\right)$$ and $$\mathrm{DEV} = \max\left(1 - \sum_{i,j} |d_{ij} - u_{ij}| / \sum_{i,j} |d_{ij} - \mathrm{median}(d)| ^ 2, 0\right)$$ respectively. Note that VAF and DEV are not invariant under rescaling u, and may be arbitrarily small (i.e., 0 using the above definitions) even though u and d are structurally close in some sense. For the results of using agnes and diana, the agglomerative and divisive coefficients are provided in addition to the default ones.

References

L. Hubert and P. Arabie and J. Meulman (2006). The structural representation of proximity matrices with MATLAB. Philadelphia, PA: SIAM.

T. J. Smith (2001). Constructing ultrametric and additive trees based on the $L_1$ norm. Journal of Classification, 18/2, 185--207.