Richard J. Hathaway’s research while affiliated with Mathematical Sciences Research Institute and other places

The fuzzy c-means (FCM) clustering algorithm has long been used to cluster numerical data. More recently FCM has found application in certain schemes for clustering heterogenous data sets consisting of mixtures of numerical, interval, and fuzzy data. The range of applicability of FCM is extended here to include clustering data whose features are continuous random variables. Parametric, nonparametric and empirical models are presented, and in each case, the distributional laws of the random variables are encoded to give a real, finite-dimensional representation to which FCM can be applied. Subsequent decoding of the results yields cluster prototypes that are similar in nature to the original data themselves. Some properties of the approach are noted and the results of preliminary computational experimentation are given.

In this short paper, a unified framework for performing density-weighted fuzzy c-means (FCM) clustering of feature and relational datasets is presented. The proposed approach consists of reducing the original dataset to a smaller one, assigning each selected datum a weight reflecting the number of nearby data, clustering the weighted reduced dataset using a weighted version of the feature or relational data FCM algorithm, and if desired, extending the reduced data results back to the original dataset. Several methods are given for each of the tasks of data subset selection, weight assignment, and extension of the weighted clustering results. The newly proposed weighted version of the non-Euclidean relational FCM algorithm is proved to produce the identical results as its feature data analog for a certain type of relational data. Artificial and real data examples are used to demonstrate and contrast various instances of this general approach.

An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P∗ in three steps: 1. recognizing that the problem cannot be solved using simple (non-calculus) techniques, 2. approximating the solution of P∗ using simple techniques, and 3. perfecting the approximation by taking an appropriate limit. The modest contribution of this note attempts to package this point-of-view in a form that can be easily remembered and appreciated by students. The acronym is presented, explained, and illustrated using several important calculus topics.

The visual assessment of tendency (VAT) technique, developed by J.C. Bezdek, R.J. Hathaway and J.M. Huband, uses a visual approach to find the number of clusters in data. In this paper, we develop a new algorithm that processes the numeric output of VAT programs, other than gray level images as in VAT, and produces the tendency curves. Possible cluster borders will be seen as high-low patterns on the curves, which can be caught not only by human eyes but also by the computer. Our numerical results are very promising. The program caught cluster structures even in cases where the visual outputs of VAT are virtually useless.

We improve the visual assessment of tendency (VAT) technique, which, developed by J.C. Bezdek, R.J. Hathaway and J.M. Huband, uses a visual approach to find the number of clusters in data. Instead of using square gray level images of dissimilarity matrices as in VAT, we further process the matrices and produce the tendency curves. Possible cluster structure will be shown as peak-valley patterns on the curves, which can be caught not only by human eyes but also by the computer. Our numerical experiments showed that the computer can catch cluster structures from the tendency curves even in cases where the visual outputs of VAT are virtually useless.

We have an m times n matrix D, and assume that its entries correspond to pair wise dissimilarities between m row objects O_r and n column objects O_c, which, taken together (as a union), comprise a set O of N = m + n objects. This paper develops a new visual approach that applies to four different cluster assessment problems associated with O. The problems are the assessment of cluster tendency: PI) amongst the row objects O_r; P2) amongst the column objects O_c; P3) amongst the union of the row and column objects O_r U O_c; and P4) amongst the union of the row and column objects that contain at least one object of each type (co-clusters). The basis of the method is to regard D as a subset of known values that is part of a larger, unknown N times N dissimilarity matrix, and then impute the missing values from D. This results in estimates for three square matrices (D_r, D_c, D_rUc) that can be visually assessed for clustering tendency using the previous VAT or sVAT algorithms. The output from assessment of D_rUc ultimately leads to a rectangular coVAT image which exhibits clustering tendencies in D. Five examples are given to illustrate the new method. Two important points: i) because VAT is scalable by sVAT to data sets of arbitrary size, and because coVAT depends explicitly (and only) on VAT, this new approach is immediately scalable to, say, the scoVAT model, which works for even very large (unloadable) data sets without alteration; and ii) VAT, sVAT and coVAT are autonomous, parameter free models - no "hidden values" are needed to make them work.

Most iterative clustering algorithms require a good initialization to achieve accurate results. A new initialization procedure for all such algorithms is given that is exact when the data contain compact, separated clusters. Our examples use c-means clustering.

Approximating clusters in very large (VL=unloadable) data sets has been considered from many angles. The proposed approach has three basic steps: (i) progressive sampling of the VL data, terminated when a sample passes a statistical goodness of fit test; (ii) clustering the sample with a literal (or exact) algorithm; and (iii) non-iterative extension of the literal clusters to the remainder of the data set. Extension accelerates clustering on all (loadable) data sets. More importantly, extension provides feasibility—a way to find (approximate) clusters—for data sets that are too large to be loaded into the primary memory of a single computer. A good generalized sampling and extension scheme should be effective for acceleration and feasibility using any extensible clustering algorithm. A general method for progressive sampling in VL sets of feature vectors is developed, and examples are given that show how to extend the literal fuzzy (c-means) and probabilistic (expectation-maximization) clustering algorithms onto VL data. The fuzzy extension is called the generalized extensible fast fuzzy c-means (geFFCM) algorithm and is illustrated using several experiments with mixtures of five-dimensional normal distributions.