Cluster Analysis Problem Research Articles

Parametric linear programming and cluster analysis

In the cluster analysis problem one seeks to partition a finite set of objects into disjoint groups (or clusters) such that each group contains relatively similar objects and, relatively dissimilar objects are placed in different groups. For certain classes of the problem or, under certain assumptions, the partitioning exercise can be formulated as a sequence of linear programs (LPs), each with a parametric objective function. Such LPs can be solved using the parametric linear programming procedure developed by Gass and Saaty [(Gass, S., Saaty, T. (1955), Naval Research Logistics Quarterly 2, 39–45)]. In this paper, a parametric linear programming model for solving cluster analysis problems is presented. We show how this model can be used to find optimal solutions for certain variations of the clustering problem or, in other cases, for an approximation of the general clustering problem.

Kohonen maps for solving a class of location-allocation problems

Location-Allocation problems occur whenever more than one facility need be located to serve a set of demand centers and it is not known or fixed a priori their allocation to the supply centers. This paper deals with a continuous space problem in which demand centers are independently served from a given number of independent, uncapacitated supply centers. Installation costs are assumed not to depend on neither the actual location nor the actual throughput of the supply centers. Transportation costs are considered to be proportional to the square Euclidean distance travelled and a minisum criterium is adopted. The problem is recognized as identical to certain Cluster Analysis and Vector Quantization problems. Such a relationship leads to applying Kohonen Maps, which are Artificial Neural Networks capable of extracting the main features, i.e. the structure, of the input data through a self-organizing process based on local adaptation rules. This approach has previously been applied to other combinatorial problems such as the Travelling Salesperson Problem.

Clustering data using a modified integer genetic algorithm (IGA)

This paper developed a modified genetic algorithm with integer representation (IGA) for cluster analysis problem. The IGA method expands the basic concepts of conventional GAs to include fitness scaling, a modified selection operator, and three newly proposed genetic operators, i.e., competition, self-reproduction and diversification. Moreover, a new clustering criterion was introduced and compared with the commonly used square-error criterion. Clustering of simulated and real chemical data showed that IGA consistently outperformed conventional GAs both in search efficiency and in search precision, and the introduced criterion provided better performance than the square-error criterion.

Optimal clustering: A model and method

Classifying items into distinct groupings is fundamental in scientific inquiry. The objective of cluster analysis is to assign n objects to up to K mutually exclusive groups while minimizing some measure of dissimilarity among the items. Few mathematical programming approaches have been applied to these problems. Most clustering methods to date only consider lowering the amount of interaction between each observation and the group mean or median. Clustering used in information systems development to determine groupings of modules requires a model that will account for the total group interaction. We formulate a mixed-integer programming model for optimal clustering based upon scaled distance measures to account for this total group interaction. We discuss an efficient, implicit enumeration algorithm along with some implementation issues, a method for computing tight bounds for each node in the solution tree, and a small example. A computational example problem, taken from the computer-assisted process organization (CAPO) literature, is presented. Detailed computational results indicate that the method is effective for solving this type of cluster analysis problem.

FORCOD: A coding and classification system for formed parts

Abstract Part coding and classification systems are required to apply the group technology philosophy. Thus far, most coding and classification systems are for metal removal processes. This paper proposes a coding and classification scheme for the metal forming process. Coding details based on part geometry, tolerance level, surface finish, material, heat treatment, and defect level are defined. As an example, based on the proposed coding scheme, a multi-objective cluster analysis problem is formulated and solved for bosses and connecting rods, which are typical examples of forged parts.

Cluster Analysis of Antigenic Profiles of Tumors: Selection of Number of Clusters Using Akaike’s Information Criterion

A basic problem of cluster analysis is the determination or selection of the number of clusters evinced in any set of data. We address this issue with multinomial data using Akaike's information criterion and demonstrate its utility in identifying an appropriate number of clusters of tumor types with similar profiles of cell surface antigens.

CLUSTERT — A simulation-based expert

Cluster analysis consists of applying statistical and heuris tic methods in an attempt to discover the group composition of a data set. Users of traditional clustering software often find non-intelligent packages difficult to use because of the many parameters that must be specified before an analysis can begin; these include a distance or similarity metric, a method for stopping group amalgama tion or division, and a particular clustering algorithm. CLUSTERT is an expert system for solving cluster analysis problems that was developed because of this difficulty. Unlike most expert systems, however, CLUSTERT is primarily based on knowledge generated by simulation experiments. The system also provides suggestions and an intelligent environment for conducting additional simulations that may reinforce

On the partitioning of squared Euclidean distance and its applications in cluster analysis

The partitioning of squared Eucliean distance between two vectors in M-dimensional space into the sum of squared lengths of vectors in mutually orthogonal subspaces is discussed and applications given to specific cluster analysis problems. Examples of how the partitioning idea can be used to help describe and interpret derived clusters, derive similarity measures for use in cluster analysis, and to design Monte Carlo studies with carefully specified types and magnitudes of differences between the underlying population mean vectors are presented. Most of the example applications presented in this paper involve the clustering of longitudinal data, but their use in cluster analysis need not be limited to this arena.

Quotients with respect to similarity relations

The problem of the quotient w.r.t. a similarity relation is closely related to Poincaré's paradox and central problems in cluster analysis. Amazingly this problem remains open for more than 30 years. As a generalization of strong M-categories we introduce the concept of M-valued sets, which leads to the definition of the category M-SET. In M-SET we give a complete solution of the quotient problem.

On the meaning of Dunn's partition coefficient for fuzzy clusters

A constantly recurring problem in cluster analysis is that of evaluating the number of clusters that are present in a data set. The development of fuzzy clustering has brought no definite answer to this question, although some new means have been provided offering new ways to tackle this problem. One of the promising approaches was Dunn's partition coefficient F k ( U) that has been shown to vary between 1, for hard clusters, and 1/ k, for completely fuzzy sets of objects: hence the idea that F k ( U) expresses a measure of how far a given fuzzy partition is from a hard one. Assuming moreover that an optimal partition in an optimal number of clusters will have a ‘harder’ look than any other partition, F k ( U) could be considered as a cluster validity index. That this procedure is unjustified is shown by the study of the ( D, F)-diagram in which, by analogy to the partition coefficient F, a fuzziness coefficient D is defined. An important result of this study is that a higher F k ( U) value does not always correspond to a better allocation than a partition with a lower value: this observation contradicts the role of the partition coefficient as cluster validity measurement. These results are emply confirmed by various artificial and classical numerical examples.

Cluster Analysis of Marine Nematodes

A small set of real data on marine nematode densities was analysed using 100 different combinations of methods of cluster analysis. The properties of 25 different similarity coefficients (sensu lato) were investigated in combination with four different clustering algorithms for three different measures of the nematode populations: species presence or absence, species abundance and species densities. The results of analysis by each combination of methods were compared directly with the individual species records. The best representation of the data was produced by the Bray-Curtis similarity coefficient and group average clustering algorithm. Species presences and absences were unsatisfactory in measuring nematode populations with all combinations of similarity coefficients and clustering algorithms. Percentage abundances of species produced generally good results with all combinations of methods but there were difficulties in interpreting results. Species densities were generally the best measure of nematode populations. Of the similarity coefficients, the Gower metric, euclidean distance squared and Bray-Curtis coefficients were generally the best but the former two were better with percentages and the latter with densities. The group average clustering algorithm was always the best. The results of other studies are considered in the light of the present results. The practical choices and problems of cluster analysis are discussed, including: the emphasis on rare and common species, the influence of the number of species and the distribution of individuals among species and the distortion of dendrograms by different clustering algorithms. Also discussed are more theoretical matters such as the statistical considerations of cluster analysis, the philosophy of cluster analysis and the relationship between species percentage abundances and densities.

A bootstrap testing procedure for investigating the number of subpopulations

Determining the number of subpopulations from sample data is a major problem in cluster analysis. We assume in this study that the subpopulations correspond to modes of the population density function f. We then propose using test statistics based on the kth nearest neighbor clustering method to investigate the modality of f. A modified bootstrap procedure for estimating the sample significance levels of these statistics in the univariate case is described. The performance of this procedure in determining the number of subpopulations will be illustrated by generated and real data sets.

The p-Median Problem for Cluster Analysis: A Comparative Test Using the Mixture Model Approach

Recently, Mulvey and Crowder (Mulvey, J., H. Crowder. 1979. Cluster analysis: an application of Lagrangian relaxation. Management Sci. 25 329–340.) suggested that the p-median problem might be useful for cluster analysis problems (where the goal is to group objects described by a vector of characteristics in such a way that objects in the same group are somehow more alike than objects in different groups). The intent of this paper is to test Mulvey and Crowder's proposal using the mixture model approach; i.e., by applying a number of algorithms (including one for the p-median problem) to a set of objects randomly sampled from a number of known multivariate populations and comparing the ability of each algorithm to detect the original populations. In order to evaluate the results, a generalized partition comparison measure and its distribution are developed. Using this measure, results from various algorithms are compared.

Multidimensional Sorting

We introduce a process called geometric sorting, which can be applied to an arbitrary configuration of points in d space, and which encodes in compact form the order properties of the configuration, just as the arrangement of a set of numbers in size place encodes its order properties. We give an algorithm for carrying out this sorting procedure in time $O(n^d \log n)$, which generalizes the optimum sorting time of $O(n\log n)$ for the linear case; In addition, we give an efficient algorithm for determining whether two randomly numbered configurations in $\mathbb{R}^d $ have the same order type, using a distinguished family of orderings of each. Finally, we indicate how this new concept of sorting can be applied to problems in pattern recognition, stereochemistry, and cluster analysis.

A Probability theory of hierarchic clustering using random dendrograms

It is widely recognized that a major, current problem in cluster analysis is that of validating results. This paper looks at one possible approach to the validation of results. It considers the structure of (unlabelled) dendrograms and proposes a model of random dendrograms. Algorithms for calculating the probability distribution of a coefficient of structure of a dendrogram are discussed, —both by enumeration of distinct dendrograms, and by Monte Carlo generation of dendrograms

A combined algorithm for weighting the variables and clustering in the clustering problem

One problem in clustering (classification) analysis relates to whether or not the original variables should be transformed in some way before they are used by the clustering algorithm. More often than not, the original variables do require some transformation. The purpose of the transformation may be a desire to have more compact clusters in the space of the transformed variables, to take into account the different nature and/or units of the variables involved, to allow for the different or equal ‘importance’ of different variables, to minimize the number of variables used, etc. Among the linear transformations of variables we distinguish two groups - those which change only the scales of the variables (they are often called weighting procedures), and those which also rotate the space of variables (a good example would be the method of principal components (1)). This paper addresses the former group of transformations. One strong reason for using the weighted variables (as opposed to their linear combinations) is that when using them one can interpret the results of the classification in terms of the original (physical) variables. Unfortunately, weighting the variables can result in ‘spoiling’ the compactness of the clusters in the space of the weighted variables if the weighting procedure being used ‘does not care’ about the results of clustering (in other words if the weighting is done prior to and independently of the clustering). A method of weighting the variables which is a part of the classification procedure and thus guarantees an improvement of the cluster clarity is suggested in this paper. The weights of variables and the clusters of objects produced by the algorithm correspond to a local minimum of some classification criterion. Because of this, the resultant weights can be interpreted as a measure of ‘importance’ of the variables for the classification purpose. These weights are compared with such popular weighting procedures as equal variance (6) and Mahalanobis distance (7) methods. Two examples of the performance of the algorithm are presented.

Applications of cluster analysis algorithm to geostatistical series

The common problems of cluster analysis are briefly reviewed and discussed. A cluster analysis algorithm for partitioning geostatistical series is extensively described and applied to examples of various visual and statistical types. The utilization of the algorithm in measuring the divergence between geostatistical distributions is shown. The usefulness of the algorithm is demonstrated in solving the problem of optimal location of m facilities. A method for reconstructing territories by simple geometric shapes via cluster analysis is also given.

Standardization of Measures Prior to Cluster Analysis

A common problem in cluster analysis is the determination of a scale-free measure of distance between individuals. This paper presents a procedure for scaling measurements using a reference individual as a standard of comparison. The procedure is particularly useful in classification of the results of laboratory procedures, where a reference standard is routinely produced. An example is the clustering of patterns that result from crossed antigen-antibody electrophoresis for determining the phenotype of the serum protein alpha 1-antitrypsin. The procedure appears to remove extraneous variability while retaining the information necessary for classification.

Quantitation of the PAS reaction in skeletal muscle of man.

Quantitation of the PAS reaction in skeletal muscle of man.

Unresolved Problems in Cluster Analysis

Unresolved Problems in Cluster Analysis