39 research outputs found
A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid
The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.Choquet integral, Sugeno integral, capacity, bipolarity, preferences
How to score alternatives when criteria are scored on an ordinal scale
We address in this paper the problem of scoring alternatives when they are evaluated with respect to several criteria on a finite ordinal scale . We show that in general, the ordinal scale has to be refined or shrunk in order to be able to represent the preference of the decision maker by an aggregation operator belonging to the family of mean operators. The paper recalls previous theoretical results of the author giving necessary and sufficient conditions for a representation of preferences, and then focusses on describing practical algorithms and examples.ordinal scale, aggregation of scores, mean operator, refinement of scale
Fuzzy measures and integrals in MCDA
This chapter aims at a unified presentation of various methods of MCDA based onfuzzy measures (capacity) and fuzzy integrals, essentially the Choquet andSugeno integral. A first section sets the position of the problem ofmulticriteria decision making, and describes the various possible scales ofmeasurement (difference, ratio, and ordinal). Then a whole section is devotedto each case in detail: after introducing necessary concepts, the methodologyis described, and the problem of the practical identification of fuzzy measuresis given. The important concept of interaction between criteria, central inthis chapter, is explained in details. It is shown how it leads to k-additivefuzzy measures. The case of bipolar scales leads to thegeneral model based on bi-capacities, encompassing usual models based oncapacities. A general definition of interaction for bipolar scales isintroduced. The case of ordinal scales leads to the use of Sugeno integral, andits symmetrized version when one considers symmetric ordinal scales. Apractical methodology for the identification of fuzzy measures in this contextis given. Lastly, we give a short description of some practical applications.Choquet integral; fuzzy measure; interaction; bi-capacities
Robust Integrals
In decision analysis and especially in multiple criteria decision analysis,
several non additive integrals have been introduced in the last years. Among
them, we remember the Choquet integral, the Shilkret integral and the Sugeno
integral. In the context of multiple criteria decision analysis, these
integrals are used to aggregate the evaluations of possible choice
alternatives, with respect to several criteria, into a single overall
evaluation. These integrals request the starting evaluations to be expressed in
terms of exact-evaluations. In this paper we present the robust Choquet,
Shilkret and Sugeno integrals, computed with respect to an interval capacity.
These are quite natural generalizations of the Choquet, Shilkret and Sugeno
integrals, useful to aggregate interval-evaluations of choice alternatives into
a single overall evaluation. We show that, when the interval-evaluations
collapse into exact-evaluations, our definitions of robust integrals collapse
into the previous definitions. We also provide an axiomatic characterization of
the robust Choquet integral.Comment: 24 page
Representation of preferences over a finite scale by a mean operator
Suppose that a decision maker provides a weak order on a given set of alternatives, each alternative being described by a vector of scores, which are given on a finite ordinal scale . The paper addresses the question of the representation of this weak order by some mean operator, and gives necessary and sufficient conditions for such a representation, with possible shrinking and/or refinement of the scale .preference representation, finite scale, meanoperator, aggregation of scores, refinement of scale
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented.
The model includes in particular the classical model on the system of all
subsets of a finite set. The general model associates canonical non-negative
and positively homogeneous superadditive functionals with generalized belief
functions relative to an ordered system, which are then extended to arbitrary
valuations on the set system. It is shown that the general Choquet integral can
be computed by a simple Monge-type algorithm for so-called intersection
systems, which include as a special case weakly union-closed families.
Generalizing Lov\'asz' classical characterization, we give a characterization
of the superadditivity of the Choquet integral relative to a capacity on a
union-closed system in terms of an appropriate model of supermodularity of such
capacities
Efficient Data Driven Multi Source Fusion
Data/information fusion is an integral component of many existing and emerging applications; e.g., remote sensing, smart cars, Internet of Things (IoT), and Big Data, to name a few. While fusion aims to achieve better results than what any one individual input can provide, often the challenge is to determine the underlying mathematics for aggregation suitable for an application. In this dissertation, I focus on the following three aspects of aggregation: (i) efficient data-driven learning and optimization, (ii) extensions and new aggregation methods, and (iii) feature and decision level fusion for machine learning with applications to signal and image processing. The Choquet integral (ChI), a powerful nonlinear aggregation operator, is a parametric way (with respect to the fuzzy measure (FM)) to generate a wealth of aggregation operators. The FM has 2N variables and N(2N − 1) constraints for N inputs. As a result, learning the ChI parameters from data quickly becomes impractical for most applications. Herein, I propose a scalable learning procedure (which is linear with respect to training sample size) for the ChI that identifies and optimizes only data-supported variables. As such, the computational complexity of the learning algorithm is proportional to the complexity of the solver used. This method also includes an imputation framework to obtain scalar values for data-unsupported (aka missing) variables and a compression algorithm (lossy or losselss) of the learned variables. I also propose a genetic algorithm (GA) to optimize the ChI for non-convex, multi-modal, and/or analytical objective functions. This algorithm introduces two operators that automatically preserve the constraints; therefore there is no need to explicitly enforce the constraints as is required by traditional GA algorithms. In addition, this algorithm provides an efficient representation of the search space with the minimal set of vertices. Furthermore, I study different strategies for extending the fuzzy integral for missing data and I propose a GOAL programming framework to aggregate inputs from heterogeneous sources for the ChI learning. Last, my work in remote sensing involves visual clustering based band group selection and Lp-norm multiple kernel learning based feature level fusion in hyperspectral image processing to enhance pixel level classification
Insights and Characterization of l1-norm Based Sparsity Learning of a Lexicographically Encoded Capacity Vector for the Choquet Integral
This thesis aims to simultaneously minimize function error and model complexity for data fusion via the Choquet integral (CI). The CI is a generator function, i.e., it is parametric and yields a wealth of aggregation operators based on the specifics of the underlying fuzzy measure. It is often the case that we desire to learn a fusion from data and the goal is to have the smallest possible sum of squared error between the trained model and a set of labels. However, we also desire to learn as “simple’’ of solutions as possible. Herein, L1-norm regularization of a lexicographically encoded capacity vector relative to the CI is explored. The impact of regularization is explored in terms of what capacities and aggregation operators it induces under different common and extreme scenarios. Synthetic experiments are provided in order to illustrate the propositions and concepts put forth
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lovász' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities.Choquet integral, belief function, measurability, set systems, Monge algorithm, supermodularity
Lexicographic refinements in possibilistic sequential decision-making models
Ce travail contribue à la théorie de la décision possibiliste et plus précisément à la prise de décision séquentielle dans le cadre de la théorie des possibilités, à la
fois au niveau théorique et pratique. Bien qu'attrayante pour sa capacité à résoudre les problèmes de décision qualitatifs, la théorie de la décision possibiliste souffre d'un
inconvénient important : les critères d'utilité qualitatives possibilistes comparent les actions avec les opérateurs min et max, ce qui entraîne un effet de noyade. Pour
surmonter ce manque de pouvoir décisionnel, plusieurs raffinements ont été proposés dans la littérature. Les raffinements lexicographiques sont particulièrement intéressants
puisqu'ils permettent de bénéficier de l'arrière-plan de l'utilité espérée, tout en restant "qualitatifs". Cependant, ces raffinements ne sont définis que pour les problèmes de
décision non séquentiels.
Dans cette thèse, nous présentons des résultats sur l'extension des raffinements lexicographiques aux problèmes de décision séquentiels, en particulier aux Arbres de
Décision et aux Processus Décisionnels de Markov possibilistes. Cela aboutit à des nouveaux algorithmes de planification plus "décisifs" que leurs contreparties possibilistes.
Dans un premier temps, nous présentons des relations de préférence lexicographiques optimistes et pessimistes entre les politiques avec et sans utilités intermédiaires, qui
raffinent respectivement les utilités possibilistes optimistes et pessimistes. Nous prouvons que les critères proposés satisfont le principe de l'efficacité de Pareto ainsi que
la propriété de monotonie stricte. Cette dernière garantit la possibilité d'application d'un algorithme de programmation dynamique pour calculer des politiques optimales. Nous
étudions tout d'abord l'optimisation lexicographique des politiques dans les Arbres de Décision possibilistes et les Processus Décisionnels de Markov à horizon fini. Nous
fournissons des adaptations de l'algorithme de programmation dynamique qui calculent une politique optimale en temps polynomial. Ces algorithmes sont basés sur la comparaison
lexicographique des matrices de trajectoires associées aux sous-politiques. Ce travail algorithmique est complété par une étude expérimentale qui montre la faisabilité et
l'intérêt de l'approche proposée. Ensuite, nous prouvons que les critères lexicographiques bénéficient toujours d'une fondation en termes d'utilité espérée, et qu'ils peuvent
être capturés par des utilités espérées infinitésimales.
La dernière partie de notre travail est consacrée à l'optimisation des politiques dans les Processus Décisionnels de Markov (éventuellement infinis) stationnaires. Nous
proposons un algorithme d'itération de la valeur pour le calcul des politiques optimales lexicographiques. De plus, nous étendons ces résultats au cas de l'horizon infini. La
taille des matrices augmentant exponentiellement (ce qui est particulièrement problématique dans le cas de l'horizon infini), nous proposons un algorithme d'approximation qui se
limite à la partie la plus intéressante de chaque matrice de trajectoires, à savoir les premières lignes et colonnes. Enfin, nous rapportons des résultats expérimentaux qui
prouvent l'efficacité des algorithmes basés sur la troncation des matrices.This work contributes to possibilistic decision theory and more specifically to sequential decision-making under possibilistic uncertainty, at both the theoretical and
practical levels. Even though appealing for its ability to handle qualitative decision problems, possibilisitic decision theory suffers from an important drawback: qualitative
possibilistic utility criteria compare acts through min and max operators, which leads to a drowning effect. To overcome this lack of decision power, several refinements have
been proposed in the literature. Lexicographic refinements are particularly appealing since they allow to benefit from the expected utility background, while remaining
"qualitative". However, these refinements are defined for the non-sequential decision problems only.
In this thesis, we present results on the extension of the lexicographic preference relations to sequential decision problems, in particular, to possibilistic Decision
trees and Markov Decision Processes. This leads to new planning algorithms that are more "decisive" than their original possibilistic counterparts. We first present optimistic
and pessimistic lexicographic preference relations between policies with and without intermediate utilities that refine the optimistic and pessimistic qualitative utilities
respectively. We prove that these new proposed criteria satisfy the principle of Pareto efficiency as well as the property of strict monotonicity. This latter guarantees that
dynamic programming algorithm can be used for calculating lexicographic optimal policies. Considering the problem of policy optimization in possibilistic decision trees and
finite-horizon Markov decision processes, we provide adaptations of dynamic programming algorithm that calculate lexicographic optimal policy in polynomial time. These
algorithms are based on the lexicographic comparison of the matrices of trajectories associated to the sub-policies. This algorithmic work is completed with an experimental
study that shows the feasibility and the interest of the proposed approach. Then we prove that the lexicographic criteria still benefit from an Expected Utility grounding, and
can be represented by infinitesimal expected utilities.
The last part of our work is devoted to policy optimization in (possibly infinite) stationary Markov Decision Processes. We propose a value iteration algorithm for the
computation of lexicographic optimal policies. We extend these results to the infinite-horizon case. Since the size of the matrices increases exponentially (which is especially
problematic in the infinite-horizon case), we thus propose an approximation algorithm which keeps the most interesting part of each matrix of trajectories, namely the first
lines and columns. Finally, we reports experimental results that show the effectiveness of the algorithms based on the cutting of the matrices