Density based fuzzy c-means clustering of non-convex patterns

We propose a new technique to perform unsupervised data classification (clustering) based on density induced metric and non-smooth optimization. Our goal is to automatically recognize multidimensional clusters of non-convex shape. We present a modification of the fuzzy c-means algorithm, which uses the data induced metric, defined with the help of Delaunay triangulation. We detail computation of the distances in such a metric using graph algorithms. To find optimal positions of cluster prototypes we employ the discrete gradient method of non-smooth optimization. The new clustering method is capable to identify non-convex overlapped d-dimensional clusters.

Fuzzy clustering of non-convex patterns using global optimization

This paper discusses various extensions of the classical within-group sum of squared errors functional, routinely used as the clustering criterion. Fuzzy c-means algorithm is extended to the case when clusters have irregular shapes, by representing the clusters with more than one prototype. The resulting minimization problem is non-convex and non-smooth. A recently developed cutting angle method of global optimization is applied to this difficult problem

Convex hulls as an hypothesis language bias

Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces which biases such systems towards particular hypothesis types. However, there is reason to believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. We propose the use of less strongly biased hypothesis languages which might be expected to model' concepts using a number of structures close to the number of actual structures in the domain. An instantiation of such a language, a convex hull based classifier, CHI, has been implemented to investigate modeling concepts as a small number of large geometric structures in n-dimensional space. A comparison of the number of regions induced is made against other well-known systems on a representative selection of largely or wholly continuous valued machine learning tasks. The convex hull system is shown to produce a number of induced regions about an order of magnitude less than well-known systems and very close to the number of actual concepts. This representation, as convex hulls, allows the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry.

Convex hulls in concept induction

Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces. This strongly biases such systems towards particular hypothesis types but there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. This thesis investigates modeling concepts as large geometric structures in n-dimensional space. Convex hulls are a superset of the set of axis orthogonal hyperrectangles into which axis orthogonal systems partition the instance space. In consequence, there is reason to believe that convex hulls might provide a more flexible and general learning bias than axis orthogonal regions. The formation of convex hulls around a group of points of the same class is shown to be a usable generalisation and is more general than generalisations produced by axis-orthogonal based classifiers, without constructive induction, like decision trees, decision lists and rules. The use of a small number of large hulls as a concept representation is shown to provide classification performance which can be better than that of classifiers which use a large number of small fragmentary regions for each concept. A convex hull based classifier, CH1, has been implemented and tested. CH1 can handle categorical and continuous data. Algorithms for two basic generalisation operations on hulls, inflation and facet deletion, are presented. The two operations are shown to improve the accuracy of the classifier and provide moderate classification accuracy over a representative selection of typical, largely or wholly continuous valued machine learning tasks. The classifier exhibits superior performance to well-known axis-orthogonal-based classifiers when presented with domains where the underlying decision surfaces are not axis parallel. The strengths and weaknesses of the system are identified. One particular advantage is the ability of the system to model domains with approximately the same number of structures as there are underlying concepts. This leads to the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry, which is not possible from a multiplicity of small regions.

Parallel bucket sorting on graphics processing units based on convex optimization

We found an interesting relation between convex optimization and sorting problem. We present a parallel algorithm to compute multiple order statistics of the data by minimizing a number of related convex functions. The computed order statistics serve as splitters that group the data into buckets suitable for parallel bitonic sorting. This led us to a parallel bucket sort algorithm, which we implemented for many-core architecture of graphics processing units (GPUs). The proposed sorting method is competitive to the state-of-the-art GPU sorting algorithms and is superior to most of them for long sorting keys.

A penalty-based aggregation operator for non-convex intervals

In the case of real-valued inputs, averaging aggregation functions have been studied extensively with results arising in fields including probability and statistics, fuzzy decision-making, and various sciences. Although much of the behavior of aggregation functions when combining standard fuzzy membership values is well established, extensions to interval-valued fuzzy sets, hesitant fuzzy sets, and other new domains pose a number of difficulties. The aggregation of non-convex or discontinuous intervals is usually approached in line with the extension principle, i.e. by aggregating all real-valued input vectors lying within the interval boundaries and taking the union as the final output. Although this is consistent with the aggregation of convex interval inputs, in the non-convex case such operators are not idempotent and may result in outputs which do not faithfully summarize or represent the set of inputs. After giving an overview of the treatment of non-convex intervals and their associated interpretations, we propose a novel extension of the arithmetic mean based on penalty functions that provides a representative output and satisfies idempotency.

A convex geometry-based blind source separation method for separating nonnegative sources

This paper presents a convex geometry (CG)-based method for blind separation of nonnegative sources. First, the unaccessible source matrix is normalized to be column-sum-to-one by mapping the available observation matrix. Then, its zero-samples are found by searching the facets of the convex hull spanned by the mapped observations. Considering these zero-samples, a quadratic cost function with respect to each row of the unmixing matrix, together with a linear constraint in relation to the involved variables, is proposed. Upon which, an algorithm is presented to estimate the unmixing matrix by solving a classical convex optimization problem. Unlike the traditional blind source separation (BSS) methods, the CG-based method does not require the independence assumption, nor the uncorrelation assumption. Compared with the BSS methods that are specifically designed to distinguish between nonnegative sources, the proposed method requires a weaker sparsity condition. Provided simulation results illustrate the performance of our method.

On the proximal Landweber Newton method for a class of nonsmooth convex problems

We consider a class of nonsmooth convex optimization problems where the objective function is a convex differentiable function regularized by the sum of the group reproducing kernel norm and (Formula presented.)-norm of the problem variables. This class of problems has many applications in variable selections such as the group LASSO and sparse group LASSO. In this paper, we propose a proximal Landweber Newton method for this class of convex optimization problems, and carry out the convergence and computational complexity analysis for this method. Theoretical analysis and numerical results show that the proposed algorithm is promising.

Fuzzy c-means density based clustering using data induced metric

We propose a new data induced metric to perform un supervised data classification (clustering). Our goal is to automatically recognize clusters of non-convex shape. We present a new version of fuzzy c-means al gorithm, based on the data induced metric, which is capable to identify non-convex d-dimensional clusters.

Robust strictly positive real synthesis of polynomial segments for discrete time systems

By using the result of robust strictly positive real synthesis of polynomial segments for continuous time systems, it is proved that, for any two n-th order polynomials a(z) and b(z), the Schur stability of their convex combination is necessary and sufficient for the existence of an n-th order polynomial c(z) such that c(z)/a(z) and c(z)/b(z) are both strictly positive real. We also provide the construction method of c(z). Illustrative examples are provided to show the effectiveness of this method.

Review of applications of the Cutting Angle methods

The theory of abstract convexity provides us with the necessary tools for building accurate one-sided approximations of functions. Cutting angle methods have recently emerged as a tool for global optimization of families of abstract convex functions. Their applicability have been subsequently extended to other problems, such as scattered data interpolation. This paper reviews three different applications of cutting angle methods, namely global optimization, generation of nonuniform random variates and multivatiate interpolation.

Non-smooth optimization methods for computation of the conditional value-at-risk and portfolio optimization

We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21-41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters.

Application of the strain localization CAFE model to investigate extrusion with various die shapes

Multi scale CAFE model for the prediction of initiation and propagation of the micro shear bands and shear bands in metallic materials subjected to plastic deformation is presented. The CAFE approach is the combination of the Cellular Automata (CA) and the Finite Element (FE) methods. The application of the developed CAFE model to analyze material flow during extrusion is the objective of the present work. The proposed CAFE approach is applied in this work to simulation of the extrusion with flat face and convex dies and to investigate differences in the material flow. The initial FE meshes with the set of the CA point are generated for the numerical tests and the results of the metal flow predicted by the CAFE method are presented in the paper.

Construction of k-Lipschitz triangular norms and conorms from empirical data

This paper examines the practical construction of k-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on k-convex additive generators and translate k-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees k-Lipschitz property of the resulting triangular norms and conorms.

Vehicle detection and classification by measuring and processing magnetic signal

This paper presents novel vehicle detection and classification method by measuring and processing magnetic signal based on single micro-electro- mechanical system (MEMS) magnetic sensor. When a vehicle moves over the ground, it generates a succession of impacts on the earth's magnetic field, which can be detected by single magnetic sensor. The magnetic signal measured by the magnetic sensor is related to the moving direction and the type of the vehicle. Generally, the recognition rate using single sensor detector is not high. In order to improve the recognition rate, a novel feature extraction algorithm and a novel vehicle classification and recognition algorithm are presented. The concavity and convexity areas, and the angles of concave and convex parts of the waveform are extracted. An improved support vector machine (ISVM) classifier is developed to perform vehicle classification and recognition. The effectiveness of the proposed approach is verified by outdoor experiments.