MATHEMATICS OF DATA FUSION

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1 MATHEMATICS OF DATA FUSION by I. R. GOODMAN NCCOSC RDTE DTV, San Diego, California, U.S.A. RONALD P. S. MAHLER Lockheed Martin Tactical Defences Systems, Saint Paul, Minnesota, U.S.A. and HUNG T. NGUYEN Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, U.S.A. KLUWER ACADEMIC PUBLISHERS DORDRECHT/ BOSTON / LONDON

2 Preface xi 1 Introduction What is Data Fusion? Random Set Theory Conditional and Relational Event Algebra 10 1 Introduction to Data Fusion 15 2 Data Fusion and Standard Techniques Data Fusion Chapter Summary What is Data Fusion? The Subdisciplines of Data Fusion Central vs. Distributed Fusion Some Major Problems of Data Fusion How Does One Fuse Data? Multisensor, Multitarget Estimation Indirect Estimation Direct Estimation Expert Systems ' Imprecise Evidence " Vague Evidence: Fuzzy Logic Contingent Evidence: Conditional Event Algebra Partial-Probabilistic Evidence Random Sets ; "Finite-Set Statistics" Random Set Formulation of Data Fusion Problems An Integral and Differential Calculus for Data Fusion The Global Density of a Sensor Suite A Simple Illustration The Parallelism Between Point- and Finite-Set Statistics Data Fusion Using Ambiguous Evidence 68

3 vi Possible Objections to "Finite-Set Statistics" Conditional and Relational Event Algebra Bibliography 79 II The Random Set Approach to Data Fusion 91 3 Foundations of Random Sets Distributions of Random Sets Radon-Nikodym Derivatives Mobius Transforms of Set-Functions Random Sets in Decision-Making Confidence Region Estimation Imprecise Probabilities Uncertainty Modeling Bibliography Finite Random Sets Mathematical Preliminaries Relationship Between the Euclidean-Space and Hit-or-Miss Topologies Hybrid Spaces The Hausdorff Metric A Calculus of Set Functions Discrete Case: Difference Calculus on Finite Sets The Set Integral The Generalized Radon-Nikodym Derivative Set Derivatives Basic Properties of Finite Random Subsets Belief Measures of Finite Random Subsets Existence of Set Derivatives Global Probability Density Functions Global Covering Densities Relationship to Other Approaches Bibliography Finite-Set Statistics Basic Statistical Concepts Expected Values Covariances Prior and Posterior Global Densities Global Parametric Estimation Global Estimators of Vector-Valued Functions The Global ML and MAP Estimators "Set Parameters" and the Statistical Consistency of the Global ML and MAP Estimators 194

4 vii 5.3 Information Theory and Information Fusion Global Information Theory Global Best-Performance Inequalities Bibliography Fusion of Unambiguous Observations The Central-Fusion Problem Random Set Measurement Models Single-Sensor, Single-Target Measurement Models Multisensor, Multitarget Measurement Models Conventional Interpretation of Global Measurement Models Modeling Prior Knowledge Random Set Motion Models Bayesian Recursive Nonlinear Filtering Global Nonlinear Filtering Constructing Global Motion Models Closure Properties of Canonical Global Densities Random Set Outputs of Estimators Simple Examples Two Targets in One Dimension Multiple Targets in Two Dimensions Bibliography Fusion of Ambiguous Observations Overview of the Approach: The Finite Universe Case Evidence as a Constraint on Data Measurement Models for Data and Evidence The Strong-Consistency Measurement Model The Data-Dependent Measurement Model Weak-Consistency Measurement Models Signature-Based Measurement JVIodels for Evidence Unified Data Fusion Modeling Ambiguous Evidence Using DRACS Conditioning on Ambiguous Evidence: Single Sensor, Single Target Case Conditioning on Ambiguous Evidence: Single Sensor, Multitarget Case. -. v Multisensor, Multitarget Case Bayesian Characterization of Rules of Evidential Combination Nonlinear Filtering With Data and Evidence Bibliography 293

5 viii 8 Output Measurement Performance Evaluation Information Measured With Respect to Ground Truth Relative Information Components of Information Information With Constraints Nonparametric Estimation Review of Nonparametric Estimation "Global" Nonparametric Estimation Global Reproducing-Kernel Estimation Bibliography 337 III Use of Conditional and Relational Events in Data Fusion 339 Scope of Work Introduction to the Conditional and Relational Event Algebra Aspects of Data Fusion Philosophy of Approach Overview of the Problem and the Need for an Algebraic Basis Preceding Numerical Calculations Algebraic Approach to Treating Information Partitioning of Information Boolean Algebra, Probability Spaces, and Deduction and Enduction Algebraic Combining of Information The Algebraic Decision Theory Problem, Relational Event Algebra, and Introduction to Measures of Similarity Bibliography Potential Application of Conditional Event Algebra to Combining Conditional Information Modeling Inference Rules.. : Conditional Event Algebra Problem and Connections with Similarity Measures Application of Conditional Event Algebra to the Determination of Constant-Probability Events Bibliography Three Particular Conditional Event Algebras General Remarks on Conditional Event Algebra DeFinetti-Goodman-Nguyen-Walker Conditional Event Algebra Adams-Calabrese (AC) Conditional Event Algebra Some Comparisons of DGNW and AC Conditional Event Algebras376

6 ix 11.5 Lewis' Negative Result Concerning Forced Boolean Conditional Event Algebras Introduction to Product Space Conditional Event Algebra Bibliography Further Development of Product Space Conditional Event Algebra Equivalence, Partial Ordering, and Calculus of Logical Operations Additional Important Properties of PS Comparison of [...] and (...) Type of Events Lewis' Theorem and PS Higher Order Conditionals for PS Other Properties of PS Conditional Events and Their Relation to Conditioning of Random Variables for PS Boolean Conditional Event Algebras Distinct from PS Fundamental Characterization of PS Bibliography Product Space Conditional Event Algebra as a Tool for Further Analysis of Conditional Event Algebra Issues Direct Connections between PS and both DGNW and AC via Partial Ordering and Deduction-Enduction Relations Direct Application of PS to Motivating Example Rigorous Formulation of Constant Probability Events and Intervals within a PS Framework Bibliography Testing of Hypotheses for Distinctness of Events and Event Similarity Issues 425 General Remarks Classical Testing of Statistical Hypotheses and Estimation / Regression Applied to the Comparison of Different Probability Distributions Testing Hypotheses of Different Events Relative to a Common Probability Measure Testing Hypotheses under a Higher Order Probability Assumption on the Relative Atoms Probability Distance Functions and PS Conditional Event Algebra Numerical-Valued Metrics on a Probability Space Algebraic Metrics on a Probability Space Development of Probability Distance Functions using Algebraic Metrics 440

7 x 14.8 Additional Relations among the Basic Probability Distance Functions and Open Issues Bibliography Testing Hypotheses And Estimation Relative To Natural Language Descriptions Motivating Example Copulas, Cocopulas, and Fuzzy Logic Operators Numerically-Based Measures of Similarity and Metrics for the Problem One-Point Random Set Coverage Representations of Fuzzy Sets and Fuzzy Logic Use of One-Point Coverages with Probability Distance Functions Incorporation of Fuzzy Logic Modifiers and Use of Relational Event Algebra in Example Additional Analysis of Example Bibliography Development of Relational Event Algebra Proper to Address Data Fusion Problems 481 Overview Use of Relational Event Algebra and Probability Distances in Comparing Opinions of Two Experts' Weighted Combinations of Probabilities Comparing and Combining Polynomials or Analytic Functions of Probabilities with Weighted Coefficients Using Relational Event Algebra and Probability Distances Comparison of Models Whose Uncertainties Are Two Argument Quadratic Functions General Relational Event Algebra Problem and a Modification -" for Relational Events Having Constant-Probability Event Coefficients Possibly Dependent upon Probabilities Concluding Remarks Bibliography 501 Index 503

Part III. U se of Conditional and Relational Events in Data Fusion

Part III. U se of Conditional and Relational Events in Data Fusion Part III U se of Conditional and Relational Events in Data Fusion 341 Scope of Work In Part III here, we consider a number of topics associated with conditional and relational event algebra, focusing on