Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems

We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These values provide a discrete approximation of of minimum variance problems for discrete distributions, a problem motivated by the question how to estimate the $\alpha$-quantile of an aggregate random variable with unknown dependence structure given the marginals of the constituent random variables. We relate this problem to the multidimensional bottleneck assignment problem and show that there exists a polynomial $2$-approximation algorithm if the matrix has only $3$ columns. In general, deciding complete mixability is $\mathcal{NP}$-complete. In particular the swapping algorithm of Puccetti et al. is not an exact method unless $\mathcal{NP}\subseteq\mathcal{ZPP}$. For a fixed number of columns it remains $\mathcal{NP}$-complete, but there exists a PTAS. The problem can be solved in pseudopolynomial time for a fixed number of rows, and even in polynomial time if all columns furthermore contain entries from the same multiset

Current Open Questions in Complete Mixability

Complete and joint mixability has raised considerable interest in recent few years, in both the theory of distributions with given margins, and applications in discrete optimization and quantitative risk management. We list various open questions in the theory of complete and joint mixability, which are mathematically concrete, and yet accessible to a broad range of researchers without specific background knowledge. In addition to the discussions on open questions, some results contained in this paper are new

Centers of probability measures without the mean

In the recent years, the notion of mixability has been developed with applications to operations research, optimal transportation, and quantitative finance. An n-tuple of distributions is said to be jointly mixable if there exist n random variables following these distributions and adding up to a constant, called center, with probability one. When the n distributions are identical, we speak of complete mixability. If each distribution has finite mean, the center is obviously the sum of the means. In this paper, we investigate the set of centers of completely and jointly mixable distributions not having a finite mean. In addition to several results, we show the (possibly counterintuitive) fact that, for each (Formula presented.), there exist n standard Cauchy random variables adding up to a constant C if and only if (Formula presented.

Extremal Dependence Concepts

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent devel- opments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open ques- tions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative de- pendence given in literature and introduce a novel notion, which in a gen- eral sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Re- arrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems

Computational Methods in Finance Related to Distributions with Known Marginals

Model uncertainty and the dependence structures of various risk factors are important components of measuring and managing financial risk, such as market, credit and operational risks. In this thesis we provide a systematic investigation into these issues by studying their impacts on Credit Value Adjustment (CVA), Counterparty Credit Risk (CCR), and estimating Value-at-Risk for a portfolio of financial instruments. In particular we address the numerical issues of finding an unknown (worst-case) copula that ties marginal distributions of risk factors together given partial information about them

A Practical Investigation into Achieving Bio-Plausibility in Evo-Devo Neural Microcircuits Feasible in an FPGA

Many researchers has conjectured, argued, or in some cases demonstrated, that bio-plausibility can bring about emergent properties such as adaptability, scalability, fault-tolerance, self-repair, reliability, and autonomy to bio-inspired intelligent systems. Evolutionary-developmental (evo-devo) spiking neural networks are a very bio-plausible mixture of such bio-inspired intelligent systems that have been proposed and studied by a few researchers. However, the general trend is that the complexity and thus the computational cost grow with the bio-plausibility of the system. FPGAs (Field- Programmable Gate Arrays) have been used and proved to be one of the flexible and cost efficient hardware platforms for research' and development of such evo-devo systems. However, mapping a bio-plausible evo-devo spiking neural network to an FPGA is a daunting task full of different constraints and trade-offs that makes it, if not infeasible, very challenging. This thesis explores the challenges, trade-offs, constraints, practical issues, and some possible approaches in achieving bio-plausibility in creating evolutionary developmental spiking neural microcircuits in an FPGA through a practical investigation along with a series of case studies. In this study, the system performance, cost, reliability, scalability, availability, and design and testing time and complexity are defined as measures for feasibility of a system and structural accuracy and consistency with the current knowledge in biology as measures for bio-plausibility. Investigation of the challenges starts with the hardware platform selection and then neuron, cortex, and evo-devo models and integration of these models into a whole bio-inspired intelligent system are examined one by one. For further practical investigation, a new PLAQIF Digital Neuron model, a novel Cortex model, and a new multicellular LGRN evo-devo model are designed, implemented and tested as case studies. Results and their implications for the researchers, designers of such systems, and FPGA manufacturers are discussed and concluded in form of general trends, trade-offs, suggestions, and recommendations

Conception des outils pour le suivi des activités et l’aide au pilotage dans le secteur médico-social

Improving the performance of medico-social sector requires management tools capable of quickly and synthetically provide a clear and quantified view of activities towards the users. Unfortunally, today does not exist a satisfactory tools to control activities in each structure, as neither at the level of the managing organization. The object of the research work is to develop tools to aid management and monitoring of activities in the medico-social sector in order to improve the current vision of the medico-social sector too compartmentalized. Closely related to Ressourcial (former information system department of OVE Foundation), this work is part of a CIFRE thesis and is one of the first theses of engineering sciences that address to organizational problems in the medical-social sector. The first scientific challenge of this research is to propose a formal and structured approach to monitoring activities in these institutions. The proposed approach should be able to realize a typology of activities in a very broad spectrum; activities performed by institutions with operating modes, and business cultures, highly heterogeneous. On this aspect, the scientist lock of this work is that the aggregation on the same model, intrinsically heterogeneous operating modes. The second scientific challenge of this thesis is the development of innovative approaches for analyzing these monitoring data and to aid management of these structures from the same data.L’amélioration de la performance du secteur médico-social nécessite des outils de pilotage aptes à donner rapidement et de façon synthétique une vision claire et quantifiée des activités réalisées en direction des usagers. Or, il n’existe pas aujourd’hui d’outils satisfaisants permettant un pilotage des activités au niveau de chaque structure, mais également au niveau de l'organisme gestionnaire. L’objet du travail de recherche est de concevoir des outils pour l’aide au pilotage et au suivi des activités dans le secteur médico-social permettant d’améliorer la vision actuelle du secteur médico-social trop cloisonné. En relation étroite avec Ressourcial (ancien département du système d’information de la Fondation OVE), ce travail s’inscrit dans le cadre d’une thèse CIFRE et constitue l'une des premières thèses des sciences de l'ingénieur qui s'attaquent aux problèmes d'organisation dans le secteur médico-social. Le premier enjeu scientifique de cette recherche est de proposer une démarche formelle et structurée de suivi des activités réalisées dans ces établissements. L’approche proposée devra être capable de rendre compte d’une typologie d’activités au spectre très large ; activités réalisées par des établissements ayant des modes de fonctionnements, et des cultures métiers, fortement hétérogènes. Sur ce volet, le verrou scientifique de ce travail est celui l’agrégation sur un même modèle, de modes de fonctionnement intrinsèquement hétérogènes. Le second enjeu scientifique de cette thèse est celui du développement d’approches innovantes pour l’analyse de ces données de suivi et pour l’aide au management de ces structures à partir de ces mêmes données