Relational visual cluster validity

The assessment of cluster validity plays a very important role in cluster analysis. Most commonly used cluster validity methods are based on statistical hypothesis testing or finding the best clustering scheme by computing a number of different cluster validity indices. A number of visual methods of cluster validity have been produced to display directly the validity of clusters by mapping data into two- or three-dimensional space. However, these methods may lose too much information to correctly estimate the results of clustering algorithms. Although the visual cluster validity (VCV) method of Hathaway and Bezdek can successfully solve this problem, it can only be applied for object data, i.e. feature measurements. There are very few validity methods that can be used to analyze the validity of data where only a similarity or dissimilarity relation exists – relational data. To tackle this problem, this paper presents a relational visual cluster validity (RVCV) method to assess the validity of clustering relational data. This is done by combining the results of the non-Euclidean relational fuzzy c-means (NERFCM) algorithm with a modification of the VCV method to produce a visual representation of cluster validity. RVCV can cluster complete and incomplete relational data and adds to the visual cluster validity theory. Numeric examples using synthetic and real data are presente

Fuzzy cluster validation using the partition negentropy criterion

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-04277-5_24Proceedings of the 19th International Conference, Limassol, Cyprus, September 14-17, 2009We introduce the Partition Negentropy Criterion (PNC) for cluster validation. It is a cluster validity index that rewards the average normality of the clusters, measured by means of the negentropy, and penalizes the overlap, measured by the partition entropy. The PNC is aimed at finding well separated clusters whose shape is approximately Gaussian. We use the new index to validate fuzzy partitions in a set of synthetic clustering problems, and compare the results to those obtained by the AIC, BIC and ICL criteria. The partitions are obtained by fitting a Gaussian Mixture Model to the data using the EM algorithm. We show that, when the real clusters are normally distributed, all the criteria are able to correctly assess the number of components, with AIC and BIC allowing a higher cluster overlap. However, when the real cluster distributions are not Gaussian (i.e. the distribution assumed by the mixture model) the PNC outperforms the other indices, being able to correctly evaluate the number of clusters while the other criteria (specially AIC and BIC) tend to overestimate it.This work has been partially supported with funds from MEC BFU2006-07902/BFI, CAM S-SEM-0255-2006 and CAM/UAM project CCG08-UAM/TIC-442

The effect of low number of points in clustering validation via the negentropy increment

This is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Neurocomputing, 74, 16, (2011) DOI: 10.1016/j.neucom.2011.03.023Selected papers of the 10th International Work-Conference on Artificial Neural Networks (IWANN2009)We recently introduced the negentropy increment, a validity index for crisp clustering that quantifies the average normality of the clustering partitions using the negentropy. This index can satisfactorily deal with clusters with heterogeneous orientations, scales and densities. One of the main advantages of the index is the simplicity of its calculation, which only requires the computation of the log-determinants of the covariance matrices and the prior probabilities of each cluster. The negentropy increment provides validation results which are in general better than those from other classic cluster validity indices. However, when the number of data points in a partition region is small, the quality in the estimation of the log-determinant of the covariance matrix can be very poor. This affects the proper quantification of the index and therefore the quality of the clustering, so additional requirements such as limitations on the minimum number of points in each region are needed. Although this kind of constraints can provide good results, they need to be adjusted depending on parameters such as the dimension of the data space. In this article we investigate how the estimation of the negentropy increment of a clustering partition is affected by the presence of regions with small number of points. We find that the error in this estimation depends on the number of points in each region, but not on the scale or orientation of their distribution, and show how to correct this error in order to obtain an unbiased estimator of the negentropy increment. We also quantify the amount of uncertainty in the estimation. As we show, both for 2D synthetic problems and multidimensional real benchmark problems, these results can be used to validate clustering partitions with a substantial improvement.This work has been funded by DGUI-CAM/UAM (Project CCG10-UAM/TIC-5864

Normality-based validation for crisp clustering

This is the author’s version of a work that was accepted for publication in Pattern Recognition. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Pattern Recognition, 43, 36, (2010) DOI 10.1016/j.patcog.2009.09.018We introduce a new validity index for crisp clustering that is based on the average normality of the clusters. Unlike methods based on inter-cluster and intra-cluster distances, this index emphasizes the cluster shape by using a high order characterization of its probability distribution. The normality of a cluster is characterized by its negentropy, a standard measure of the distance to normality which evaluates the difference between the cluster's entropy and the entropy of a normal distribution with the same covariance matrix. The definition of the negentropy involves the distribution's differential entropy. However, we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution, where all the points are assumed to belong to the same cluster. The resulting negentropy increment validity index only requires the computation of covariance matrices. We have applied the new index to an extensive set of artificial and real problems where it provides, in general, better results than other indices, both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred.This work has been partially supported with funds from MEC BFU2006-07902/BFI, CAM S-SEM-0255-2006 and CAM/UAM CCG08-UAM/TIC-442

Data mining methodologies for supporting engineers during system identification

Data alone are worth almost nothing. While data collection is increasing exponentially worldwide, a clear distinction between retrieving data and obtaining knowledge has to be made. Data are retrieved while measuring phenomena or gathering facts. Knowledge refers to data patterns and trends that are useful for decision making. Data interpretation creates a challenge that is particularly present in system identification, where thousands of models may explain a given set of measurements. Manually interpreting such data is not reliable. One solution is to use data mining. This thesis thus proposes an integration of techniques from data mining, a field of research where the aim is to find knowledge from data, into an existing multiple-model system identification methodology. It is shown that, within a framework for decision support, data mining techniques constitute a valuable tool for engineers performing system identification. For example, clustering techniques group similar models together in order to guide subsequent decisions since they might indicate possible states of a structure. A main issue concerns the number of clusters, which, usually, is unknown. For determining the correct number of clusters in data and estimating the quality of a clustering algorithm, a score function is proposed. The score function is a reliable index for estimating the number of clusters in a given data set, thus increasing understanding of results. Furthermore, useful information for engineers who perform system identification is achieved through the use of feature selection techniques. They allow selection of relevant parameters that explain candidate models. The core algorithm is a feature selection strategy based on global search. In addition to providing information about the candidate model space, data mining is found to be a valuable tool for supporting decisions related to subsequent sensor placement. When integrated into a methodology for iterative sensor placement, clustering is found to provide useful support through providing a rational basis for decisions related to subsequent sensor placement on existing structures. Greedy and global search strategies should be selected according to the context. Experiments show that whereas global search is more efficient for initial sensor placement, a greedy strategy is more suitable for iterative sensor placement

The Austronesian languages

This is a revised edition of the 2009 The Austronesian languages, which was published as a paperback in the then Pacific Linguistics series (ISBN 9780858836020). This revision includes typographical corrections, an improved index, and various minor content changes. The release of the open access edition serves to meet the strong ongoing demand for this important handbook, of which only 200 copies of the first edition were printed. This is the first single-authored book that attempts to describe the Austronesian language family in its entirety. Topics covered include: the physical and cultural background, official and national languages, largest and smallest languages in all major geographical regions, language contact, sound systems, linguistic palaeontology, morphology, syntax, the history of scholarship on Austronesian languages, and a critical assessment of the reconstruction of Proto Austronesian phonology.Australian National University, College of Asia and the Pacifi