Nguyen H.T., Kreinovich V., Wu B., Xiang G. Computing Statistics under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering

Berlin: Springer, 2012. - 443 p.In many practical situations, we are interested in statistics characterizing a population of objects: e.g. in the mean height of people from a certain area.
Most algorithms for estimating such statistics assume that the sample values are exact. In practice, sample values come from measurements, and measurements are never absolutely accurate. Sometimes, we know the exact probability distribution of the measurement inaccuracy, but often, we only know the upper bound on this inaccuracy. In this case, we have interval uncertainty: e.g. if the measured value is 1.0, and inaccuracy is bounded by 0.1, then the actual (unknown) value of the quantity can be anywhere between 1.0 - 0.1 = 0.9 and 1.0 + 0.1 = 1.1. In other cases, the values are expert estimates, and we only have fuzzy information about the estimation inaccuracy.
This book shows how to compute statistics under such interval and fuzzy uncertainty. The resulting methods are applied to computer science (optimal scheduling of different processors), to information technology (maintaining privacy), to computer engineering (design of computer chips), and to data processing in geosciences, radar imaging, and structural mechanics.Formulation of the Problem
Computing Statistics under Probabilistic and Interval Uncertainty: A Brief Description
Computing Statistics under Fuzzy Uncertainty: Formulation of the Problem
Computing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty
Computing under Interval Uncertainty: Traditional Approach Based on Uniform Distributions
Computing under Interval Uncertainty: When Measurement Errors Are Small
Computing under Interval Uncertainty: General Algorithms
Computing under Interval Uncertainty: Computational Complexity
Towards Selecting Appropriate Statistical Characteristics: The Basics of Decision Theory and the Notion of Utility
How to Select Appropriate Statistical CharacteristicsComputing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty: Reminder
Computing Mean under Interval Uncertainty
Computing Median (and Quantiles) under Interval Uncertainty
Computing Variance under Interval Uncertainty: An Example of an NP-Hard Problem
Types of Interval Data Sets: Towards Feasible Algorithms
Computing Variance under Interval Uncertainty: Efficient Algorithms
Computing Variance under Hierarchical Privacy-Related Interval Uncertainty
Computing Outlier Thresholds under Interval Uncertainty
Computing Higher Moments under Interval Uncertainty
Computing Mean, Variance, Higher Moments, and Their Linear Combinations under Interval Uncertainty: A Brief SummaryComputing Covariance under Interval Uncertainty
Computing Correlation under Interval Uncertainty
Computing Expected Value under Interval Uncertainty
Computing Entropy under Interval Uncertainty. I
Computing Entropy under Interval Uncertainty. II
Computing the Range of Convex Symmetric Functions under Interval Uncertainty
Computing Statistics under Interval Uncertainty: Possibility of Parallelization
Computing Statistics under Interval Uncertainty: Case of Relative AccuracyHow Reliable Is the Input Data?
How Accurate Is the Input Data?From Computing Statistics under Interval and Fuzzy Uncertainty to Practical Applications: Need to Propagate the Statistics through Data Processing
Applications to Bioinformatics
Applications to Computer Science: Optimal Scheduling for Global Computing
Applications to Information Management: How to Estimate Degree of Trust
Applications to Information Management: How to Measure Loss of Privacy
Application to Signal Processing: Using 1-D Radar Observations to Detect a Space Explosion Core among the Explosion Fragments
Applications to Computer Engineering: Timing Analysis of Computer Chips
Applications to Mechanical Engineering: Failure Analysis under Interval and Fuzzy Uncertainty
Applications to Geophysics: Inverse ProblemNeed to Go Beyond Interval and Fuzzy UncertaintyBeyond Interval Uncertainty: Taking Constraints into Account
Beyond Interval Uncertainty: Case of Discontinuous Processes (Phase Transitions)
Beyond Interval Uncertainty in Describing Statistical Characteristics: Case of Smooth Distributions and Info-Gap Decision Theory
Beyond Traditional Interval Uncertainty in Describing Statistical Characteristics: Case of Interval Bounds on the Probability Density Function
Beyond Interval Uncertainty in Describing Statistical Characteristics: Case of Normal Distributions
Beyond Traditional Fuzzy Uncertainty: Interval-Valued Fuzzy Techniques
Beyond Traditional Fuzzy Uncertainty: Type-2 Fuzzy Techniques