Computing Statistics under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering

✍ Scribed by Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2012
• Tongue: English
• Leaves: 443
• Series: Studies in Computational Intelligence 393
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages -
Front Matter....Pages 1-1
Formulation of the Problem....Pages 3-8
Computing Statistics under Probabilistic and Interval Uncertainty: A Brief Description....Pages 9-10
Computing Statistics under Fuzzy Uncertainty: Formulation of the Problem....Pages 11-17
Computing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty....Pages 19-24
Computing under Interval Uncertainty: Traditional Approach Based on Uniform Distributions....Pages 25-27
Computing under Interval Uncertainty: When Measurement Errors Are Small....Pages 29-33
Computing under Interval Uncertainty: General Algorithms....Pages 35-45
Computing under Interval Uncertainty: Computational Complexity....Pages 47-50
Towards Selecting Appropriate Statistical Characteristics: The Basics of Decision Theory and the Notion of Utility....Pages 51-54
How to Select Appropriate Statistical Characteristics....Pages 55-58
Front Matter....Pages 59-59
Computing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty: Reminder....Pages 61-61
Computing Mean under Interval Uncertainty....Pages 63-63
Computing Median (and Quantiles) under Interval Uncertainty....Pages 65-66
Computing Variance under Interval Uncertainty: An Example of an NP-Hard Problem....Pages 67-78
Types of Interval Data Sets: Towards Feasible Algorithms....Pages 79-94
Computing Variance under Interval Uncertainty: Efficient Algorithms....Pages 95-117
Computing Variance under Hierarchical Privacy-Related Interval Uncertainty....Pages 119-127
Computing Outlier Thresholds under Interval Uncertainty....Pages 129-151
Computing Higher Moments under Interval Uncertainty....Pages 153-166
Computing Mean, Variance, Higher Moments, and Their Linear Combinations under Interval Uncertainty: A Brief Summary....Pages 167-168
Front Matter....Pages 59-59
Computing Covariance under Interval Uncertainty....Pages 169-171
Computing Correlation under Interval Uncertainty....Pages 173-176
Computing Expected Value under Interval Uncertainty....Pages 177-180
Computing Entropy under Interval Uncertainty. I....Pages 181-191
Computing Entropy under Interval Uncertainty. II....Pages 193-209
Computing the Range of Convex Symmetric Functions under Interval Uncertainty....Pages 211-219
Computing Statistics under Interval Uncertainty: Possibility of Parallelization....Pages 221-224
Computing Statistics under Interval Uncertainty: Case of Relative Accuracy....Pages 225-234
Front Matter....Pages 235-235
How Reliable Is the Input Data?....Pages 237-242
How Accurate Is the Input Data?....Pages 243-248
Front Matter....Pages 249-249
From Computing Statistics under Interval and Fuzzy Uncertainty to Practical Applications: Need to Propagate the Statistics through Data Processing....Pages 251-259
Applications to Bioinformatics....Pages 261-264
Applications to Computer Science: Optimal Scheduling for Global Computing....Pages 265-275
Applications to Information Management: How to Estimate Degree of Trust....Pages 277-281
Applications to Information Management: How to Measure Loss of Privacy....Pages 283-287
Application to Signal Processing: Using 1-D Radar Observations to Detect a Space Explosion Core among the Explosion Fragments....Pages 289-298
Applications to Computer Engineering: Timing Analysis of Computer Chips....Pages 299-304
Applications to Mechanical Engineering: Failure Analysis under Interval and Fuzzy Uncertainty....Pages 305-316
Applications to Geophysics: Inverse Problem....Pages 317-329
Front Matter....Pages 331-331
Need to Go Beyond Interval and Fuzzy Uncertainty....Pages 333-334
Front Matter....Pages 331-331
Beyond Interval Uncertainty: Taking Constraints into Account....Pages 335-347
Beyond Interval Uncertainty: Case of Discontinuous Processes (Phase Transitions)....Pages 349-355
Beyond Interval Uncertainty in Describing Statistical Characteristics: Case of Smooth Distributions and Info-Gap Decision Theory....Pages 357-366
Beyond Traditional Interval Uncertainty in Describing Statistical Characteristics: Case of Interval Bounds on the Probability Density Function....Pages 367-378
Beyond Interval Uncertainty in Describing Statistical Characteristics: Case of Normal Distributions....Pages 379-389
Beyond Traditional Fuzzy Uncertainty: Interval-Valued Fuzzy Techniques....Pages 391-393
Beyond Traditional Fuzzy Uncertainty: Type-2 Fuzzy Techniques....Pages 395-399
Back Matter....Pages -

✦ Subjects

Appl.Mathematics/Computational Methods of Engineering; Artificial Intelligence (incl. Robotics); Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences