71 research outputs found
Using Multiattribute Utility Copulas in Support of UAV Search and Destroy Operations
The multiattribute utility copula is an emerging form of utility function used by decision analysts to study decisions with dependent attributes. Failure to properly address attribute dependence may cause errors in selecting the optimal policy. This research examines two scenarios of interest to the modern warfighter. The first scenario employs a utility copula to determine the type, quantity, and altitude of UAVs to be sent to strike a stationary target. The second scenario employs a utility copula to examine the impact of attribute dependence on the optimal routing of UAVs in a contested operational environment when performing a search and destroy mission against a Markovian target. Routing decisions involve a tradeoff between risk of UAV exposure to the enemy and the ability to strike the target. This research informs decision makers and analysts with respect to the tactics, techniques, and procedures employed in UAV search and destroy missions. An ever increasing UAV operations tempo suggests such research becoming increasingly relevant to the warfighter
An information theory for preferences
Recent literature in the last Maximum Entropy workshop introduced an analogy
between cumulative probability distributions and normalized utility functions.
Based on this analogy, a utility density function can de defined as the
derivative of a normalized utility function. A utility density function is
non-negative and integrates to unity. These two properties form the basis of a
correspondence between utility and probability. A natural application of this
analogy is a maximum entropy principle to assign maximum entropy utility
values. Maximum entropy utility interprets many of the common utility functions
based on the preference information needed for their assignment, and helps
assign utility values based on partial preference information. This paper
reviews maximum entropy utility and introduces further results that stem from
the duality between probability and utility
A methodology for probabilistic aircraft technology assessment and selection under uncertainty
The high degree of complexity and uncertainty associated with aerospace engineering applications has driven designers and engineers towards the use of probabilistic and statistical analysis tools in order to understand and design for that uncertainty. As a result, probabilistic methods have permeated the aerospace field to the extent that single point deterministic designs are no longer credible, particularly in systems analysis, performance assessment, technology impact quantification, etc. However, as statistics theory is not the primary focus of most aerospace practitioners, incorrect assumptions and flawed methods are often unknowingly used in design. A common assumption of probabilistic assessments in the field of aerospace is the independence of random variables. These random variables represent design variables, noise variables, technology impacts, etc., which can be difficult to correlate but do have underlying relationships. The justification for the assumed independence is usually not discussed in the literature even though this can have a substantial effect on probabilistic assessment and uncertainty quantification results. In other cases the dependence between random variables is acknowledged but intentionally ignored on the basis of difficulty in characterizing underlying random variable relationships, a strong bias towards methodological simplicity and low computational expense, and the expectation of modest strength in random variable dependence. Probabilistic assessments also yield large amounts of data which is not effectively used due to the sheer volume of data and poor traceability to the drivers of uncertainty. The literature shows optimization techniques are resorted to in order to select from competing alternatives in multiobjective spaces, however, these techniques generally do not handle uncertainty well. The motivating question is, how can improvements be made to the probabilistic assessment process for aircraft technology assessments that capture technology impact tradeoffs and dependencies, and ultimately enable decision makers to make an axiomatic and rational selection under uncertainty? This question leads to the research objective of this work which is to develop a methodology ``to quantify and characterize aviation's environmental impact, uncertainties, and the trade-offs and interdependencies among various impacts'' \cite{Council2010}, in order to assess and select future aircraft technologies. Copula theory is suggested to address the problem of assumed independence on the input side of probabilistic assessments in aerospace applications. Copulas are functions that can be used to define probabilistic relationships between random variables. They are well documented in the literature and have been used in many fields such as the statistics, finance, and insurance industries. They can be used to quantify complex relationships, even if that is only qualitatively or notionally understood. In this way a designer's knowledge regarding uncertainty can be better represented and propagated to system level metrics through the probabilistic assessment. Utility theory is proposed as a solution to the challenge of effectively using output data from probabilistic assessments. Utility theory is a powerful tool used in economics, marketing, psychiatry, etc., to express preferences among competing alternatives. Utility theory can provide combined valuation to each alternative in a multiobjective design space while incorporating the uncertainty associated with each alternative. This can enable designers to rationally and axiomatically make selections consistent with their preferences, between complex solutions with varying degrees of uncertainty. This work provides an introduction to copula and utility theories for the aerospace audience. It also demonstrates how these theories can be applied in canonical problems to bridge gaps currently found in the literature with regards to probabilistic assessments of aircraft technologies. The key contributions of this research are (1) an Archimedean copula selection tree enabling practitioners to rapidly translate their qualitative understanding of dependence into copula families that can represent it quantitatively (2) estimation of the quantified effect of using copulas to capture probabilistic dependence in three representative aerospace applications (3) an expected utility formulation for axiomatically ranking and selecting aircraft technology packages under uncertainty and (4) a strategic elicitation procedure for multiattribute utility functions that does not need assumptions of independence conditions on preferences between the attributes. The proposed FAAST methodology is shown as an encompassing framework for the aircraft technology assessment and selection problem that fills capability gaps from the literature and supports the decision maker in a rational and axiomatic manner.Ph.D
The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures
My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case.
Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid.
Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence.
The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times
to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of
the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components
Preference Models for Creative Artifacts and Systems
Abstract Although there is vigorous debate around definitions of creativity, there is general consensus that creativity i) has multiple facets, and ii) inherently involves a subjective value judgment by an evaluator. In this paper, we present evaluation of creative artifacts and computational creativity systems through a multiattribute preference modeling lens. Specifically, we introduce the use of multiattribute value functions for creativity evaluation and argue that there are significant benefits to explicitly representing creativity judgments as subjective preferences using formal mathematical models. Various implications are illustrated with the help of examples from and inspired by the creativity literature
The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures
My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case.
Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid.
Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence.
The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times
to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of
the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components
Approximating multivariate distributions with cumulative residual entropy: a study on dynamic integrated climate-economy model
The complexity of real world decision problems is exacerbated by the need to make decisions with only partial information. How to model and make decisions in situations where only partial preference information is available is a significant challenge in decision analysis practice. In most of the studies, the probability distributions are approximated by using the mass function or density function of the decision maker. In this dissertation, our aim is to approximate representative probability and utility functions by using cumulative distribution functions instead of density/mass functions. This dissertation consists of four main sections. The first two sections introduce the proposed methods based on cumulative residual entropy, the third section compares the proposed approximation methods with the methods in information theory literature, and the final section of the dissertation discusses the cumulative impact of integrating uncertainty into the DICE model.
In the first section of the dissertation, we approximate discrete joint probability distributions using first-order dependence trees as well as the recent concept of cumulative residual entropy. We formulate the cumulative residual Kullback-Leibler (KL)-divergence and the cumulative residual mutual information measures in terms of the survival function. We then show that the optimal first-order dependence tree approximation of the joint distribution using the cumulative Kullback-Leibler divergence is the one with the largest sum of cumulative residual mutual information pairs.
In the second part of the dissertation, we approximate multivariate probability distributions with cumulative probability distributions rather than density functions in maximum entropy formulation. We use the discrete form of maximum cumulative residual entropy to approximate joint probability distributions to elicit multivariate probability distributions using their lower order assessments.
In the third part of the dissertation, we compare several approximation methods to test the accuracy of different approximations of joint distributions with respect to the true distribution from the set of all possible distributions that match the available information. A number of methods have beeb presented in the literature for joint probability distribution approximations and we specifically compare those approximation methods that use information theory to approximate multivariate probability distributions.
Finally, we study whether uncertainty significantly affects decision making especially in global warming policy decisions and integrate climatic and economic uncertainties into the DICE model to ascertain the cumulative impact of integrating uncertainty on climate change by applying cumulative residual entropy into the DICE model
A Decision Analysis Perspective on Multiple Response Robust Optimization
Decisions in which multiple objectives must be optimized simultaneously occur frequently in government, military, and industrial settings. One method a decision maker may use to assist in such decisions is the application of a desirability function. An informed specification of the desirability function\u27s parameters is essential to accurately describe the decision maker\u27s value trade-offs and risk preference. This thesis uses utility transversality to analyze the implicit trade-off and risk attitude assumptions attendant to the desirability function. The desirability function does not explicitly account for response variability. A robust solution takes not only the expected response into account, but also its variance. Assessing a utility function over desirability as a means to describe the decision maker\u27s risk attitude produces a robust operating solution consistent with those preferences. This thesis examines robustness as it applies to the desirability function in a manufacturing experiment example. Different levels of diplomatic, informational, military, and economic (DIME) instruments of national policy are investigated to examine their effect on the political, military, economic, social, infrastructure, and information (PMESII) systems of a nation. AFRL\u27s National Operational Environment Model (NOEM) serves as a basis for identifying a robust national policy in a scenario involving the Democratic Republic of Congo
Approximate uncertainty modeling in risk analysis with vine copulas
Many applications of risk analysis require us to jointly model multiple uncertain quantities. Bayesian networks and copulas are two common approaches to modelling joint uncertainties with probability distributions. This paper focuses on new methodologies for copulas by developing work of Cooke, Bedford, Kurowica and others on vines as a way of constructing higher dimensional distributions which do not suffer from some of the restrictions of alternatives such as the multivariate Gaussian copula. The paper provides a fundamental approximation result, demonstrating that we can approximate any density as closely as we like using vines. It further operationalizes this result by showing how minimum information copulas can be used to provide parametric classes of copulas which have such good levels of approximation. We extend previous approaches using vines by considering non-constant conditional dependencies which are particularly relevant in financial risk modelling. We discuss how such models may be quantified, in terms of expert judgement or by fitting data, and illustrate the approach by modelling two financial datasets
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