Divisive Hierarchical Algorithm Research Articles

Finite mixtures of multivariate skew Student’s t distributions with independent logistic skewing functions

This paper extends the multivariate skew t distributions with independent logistic skewing functions (MSTIL) introduced in Kwong and Nadarajah (Methodology and Computing in Applied Probability 24 (2022) 1669–1691) to finite mixture models (FM-MSTIL). A stochastic EM-type algorithm is proposed for fitting the FM-MSTIL, and a divisive hierarchical algorithm is proposed for initialisations and model selections. We show that the model can outperform other finite mixture models in the literature for some simulated data sets. The performance of the FM-MSTIL in cluster analysis is also investigated. We show that the FM-MSTIL-R, a nested version of the FM-MSTIL, performs well for automatic gating tasks on some flow cytometry data sets in the FlowCap-I challenge. The FM-MSTIL-R achieved a better overall score than all other competing algorithms in the original challenge. An efficient implementation of the FM-MSTIL is available as an R package in GitHub.

Prediction of the fate of Hg and other contaminants in soil around a former chlor-alkali plant using Fuzzy Hierarchical Cross-Clustering approach

An associative simultaneous fuzzy divisive hierarchical algorithm was used to predict the fate of Hg and other contaminants in soil around a former chlor-alkali plant. The algorithm was applied on several natural and anthropogenic characteristics of soil including water leachable, mobile, semi-mobile, non-mobile fractions and total Hg, Al, Ba, Ca, Cr, Cu, Fe, K, Li, Mg, Mn, Na, Sr, Zn, water leachable fraction of Cl−, NO3− and SO42−, pH and total organic carbon. The cross-classification algorithm provided a divisive fuzzy partition of the soil samples and associated characteristics. Soils outside the perimeter of the former chlor-alkali plant were clustered based on the natural characteristics and total Hg. In contaminated zones Hg speciation becomes relevant and the assessment of species distribution is necessary. The descending order of concentration of Hg species in the test site was semi-mobile>mobile>non-mobile>water-leachable. Physico-chemical features responsible for similarities or differences between uncontaminated soil samples or contaminated with Hg, Cu, Zn, Ba and NO3− were also highlighted. Other characteristics of the contaminated soil were found to be Ca, sulfate, Na and chloride, some of which with influence on Hg fate. The presence of Ca and sulfate in soil induced a higher water leachability of Hg, while Cu had an opposite effect by forming amalgam. The used algorithm provided an in-deep understanding of processes involving Hg species and allowed to make prediction of the fate of Hg and contaminants linked to chlor-alkali-industry.

Applications of reformulations in mathematical programming

Mathematical programming is a technique that can be used to solve real-world optimization problems, where one wants to maximize, or minimize, an objective function subject to some constraints on the decision variables. The key features of mathematical programming are the creation of a model for describing the problem (the so called formulation), and the implementation of efficient algorithms to solve it (also called solvers). In this thesis, we focus on the first point. More precisely, we study some problems arising from different domains, and starting from the most natural models for describing them, we propose alternative formulations, which share some properties with the original models but are somehow better (for instance in terms of computational time needed to obtain the solution by the solver). These new models are called reformulations. We follow the classification of reformulations proposed by Liberti in [Reformulations in Mathematical Programming: Definitions and Systematics, RAIRO-OR, 43(1):55-86, 2009]: exact reformulations (also called opt-reformulations), narrowings, relaxations. This thesis is concerned with three mathematical programming applications where the reformulation was crucial to obtain a good solution. The first problem tackled herein is graph clustering by means of modularity maximization. Since this problem is NP-hard, several heuristics are proposed. We focus on a divisive hierarchical algorithm which works by recursively splitting a cluster into two new clusters in an optimal way. This splitting step is performed by solving a convex binary quadratic program. This is reformulated exactly to a more compact form without changing the optimal solutions set (exact reformulation). We also evaluate the impact provided by the reduction of the number of symmetric global optima of the problem, which is also an important topic of the next part of this thesis. The computational times are considerably reduced with respect to the original formulation. The second problem tackled in the thesis is the Packing of Equal Circles in a Square (PECS), where one wants to place non-overlapping equal circles in a unit square in such a way as to maximize the common radius. One of the reasons why the problem is hard to solve is the presence of several symmetric optimal solutions, and consequently a very large Branch-and-Bound tree. Some of the symmetric optima are made infeasible by adjoining some Symmetry Breaking Constraints (SBCs) to the formulation, thereby obtaining a narrowing. Both computational time and size of the Branch-and-Bound tree outperform the ones provided by the original formulation. The third application considered in the thesis is that of computing the convex relaxation for multilinear problems, and to compare the primal formulation and another one obtained using a representation. Although these two relaxations are both already known in the literature, we make a striking observation, i.e., that the dual relaxation leads to a faster and more stable solution process as regards CPU time.

ROCKET: A Robust Parallel Algorithm for Clustering Large-Scale Transaction Databases

SUMMARY We propose a robust and efficient algorithm called ROCKET for clustering large-scale transaction databases. ROCKET is a divisive hierarchical algorithm that makes the most of recent hardware architecture. ROCKET handles the cases with the small and the large number of similar transaction pairs separately and efficiently. Through experiments, we show that ROCKET achieves high-quality clustering with a dramatic performance improvement.