Computational geometry: an introduction
Computational geometry: an introduction
Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
New upper bounds in Klee's measure problem
SIAM Journal on Computing
A generalized discrepancy and quadrature error bound
Mathematics of Computation
Renyi's entropy as an index of diversity in simple-stage cluster sampling
Information Sciences: an International Journal
On the L2-discrepancy for anchored boxes
Journal of Complexity
A minimum spanning tree algorithm with inverse-Ackermann type complexity
Journal of the ACM (JACM)
Minimum-weight spanning tree algorithms a survey and empirical study
Computers and Operations Research
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Measurement of Population Diversity
Selected Papers from the 5th European Conference on Artificial Evolution
Population Diversity and Fitness Measures Based on Genomic Distances
Population Diversity and Fitness Measures Based on Genomic Distances
GECCO '05 Proceedings of the 7th annual conference on Genetic and evolutionary computation
Particle swarms and population diversity
Soft Computing - A Fusion of Foundations, Methodologies and Applications
An analysis of diversity measures
Machine Learning
A note on optimal point distributions in [0,1)s
Journal of Computational and Applied Mathematics
Particle Swarm Optimization with Diversity-Controlled Acceleration Coefficients
ICNC '07 Proceedings of the Third International Conference on Natural Computation - Volume 04
A diversity maintaining population-based incremental learning algorithm
Information Sciences: an International Journal
Replacement strategies to preserve useful diversity in steady-state genetic algorithms
Information Sciences: an International Journal
A measure of variance for hierarchical nominal attributes
Information Sciences: an International Journal
Dynamic population variation in genetic programming
Information Sciences: an International Journal
Bio-inspired and gradient-based algorithms to train MLPs: The influence of diversity
Information Sciences: an International Journal
Diversity enhanced particle swarm optimizer for global optimization of multimodal problems
CEC'09 Proceedings of the Eleventh conference on Congress on Evolutionary Computation
CEC'09 Proceedings of the Eleventh conference on Congress on Evolutionary Computation
Good permutations for deterministic scrambled Halton sequences in terms of L2-discrepancy
Journal of Computational and Applied Mathematics
GECCO'03 Proceedings of the 2003 international conference on Genetic and evolutionary computation: PartII
The underlying similarity of diversity measures used in evolutionary computation
GECCO'03 Proceedings of the 2003 international conference on Genetic and evolutionary computation: PartII
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
Fundamental effects of clustering on the Euclidean embedding of internet hosts
NETWORKING'07 Proceedings of the 6th international IFIP-TC6 conference on Ad Hoc and sensor networks, wireless networks, next generation internet
Information Sciences: an International Journal
Multiobjective programming using uniform design and genetic algorithm
IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews
The balance between proximity and diversity in multiobjective evolutionary algorithms
IEEE Transactions on Evolutionary Computation
Diversity in genetic programming: an analysis of measures and correlation with fitness
IEEE Transactions on Evolutionary Computation
A faster algorithm for calculating hypervolume
IEEE Transactions on Evolutionary Computation
U-measure: a quality measure for multiobjective programming
IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans
A diversity-driven structure learning algorithm for building hierarchical neuro-fuzzy classifiers
Information Sciences: an International Journal
Using Voronoi diagrams to solve a hybrid facility location problem with attentive facilities
Information Sciences: an International Journal
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Measures to evaluate the diversity of a set of points (population) in Euclidean space play an important role in a variety of areas of science and engineering. Well-known measures are often used without a clear insight into their quality and many of them do not appropriately penalize populations with a few distant groups of collocated or closely located points. To the best of our knowledge, there is a lack of rigorous criteria to compare diversity measures and help select an appropriate one. In this work we define a mathematical notion of ectropy for classifying diversity measures in terms of the extent to which they tend to penalize point collocation, we investigate the advantages and disadvantages of several known measures and we propose some novel ones that exhibit a good ectropic behavior. In particular, we introduce a quasi-entropy measure based on a geometric covering problem, three measures based on discrepancy from uniform distribution and one based on Euclidean minimum spanning trees. All considered measures are tested and compared on a large set of random and structured populations. Special attention is also devoted to the complexity of computing the measures. Most of the novel measures compare favorably with the classical ones in terms of ectropy. The measure based on Euclidean minimum spanning trees turns out to be the most promising one in terms of the tradeoff between the ectropic behavior and the computational complexity.