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Computing Statistics under Interval and Fuzzy Uncertainty

Applications to Computer Science and Engineering

  • Book
  • © 2012

Overview

Authors:
  1. Hung T. Nguyen

    1. Department of Economics, ChiangMai University, Chiangmai, Thailand

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  2. Vladik Kreinovich

    1. Department of Computer Science, University of Texas at El Paso, El Paso, USA

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  3. Berlin Wu

    1. Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan

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  4. Gang Xiang

    1. Applied Biomathematics, Inc., Setauket, USA

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  • Recent advances in Computing Statistics under Interval and Fuzzy Uncertainty
  • Presents various Applications to Computer Science and Engineering

Part of the book series: Studies in Computational Intelligence (SCI, volume 393)

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Table of contents (47 chapters)

  1. Front Matter

    Download chapter PDF

  2. Computing Statistics under Interval and Fuzzy Uncertainty: Formulation of the Problem and an Overview of General Techniques Which Can Be Used for Solving This Problem

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Formulation of the Problem

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 3-8

    3. Computing Statistics under Probabilistic and Interval Uncertainty: A Brief Description

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 9-10

    4. Computing Statistics under Fuzzy Uncertainty: Formulation of the Problem

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 11-17

    5. Computing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 19-24

    6. Computing under Interval Uncertainty: Traditional Approach Based on Uniform Distributions

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 25-27

    7. Computing under Interval Uncertainty: When Measurement Errors Are Small

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 29-33

    8. Computing under Interval Uncertainty: General Algorithms

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 35-45

    9. Computing under Interval Uncertainty: Computational Complexity

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 47-50

    10. Towards Selecting Appropriate Statistical Characteristics: The Basics of Decision Theory and the Notion of Utility

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 51-54

    11. How to Select Appropriate Statistical Characteristics

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 55-58

  3. Algorithms for Computing Statistics under Interval and Fuzzy Uncertainty

    1. Front Matter

      Pages 59-59
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    2. Computing under Fuzzy Uncertainty Can Be Reduced to Computing under Interval Uncertainty: Reminder

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 61-61

    3. Computing Mean under Interval Uncertainty

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 63-63

    4. Computing Median (and Quantiles) under Interval Uncertainty

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 65-66

    5. Computing Variance under Interval Uncertainty: An Example of an NP-Hard Problem

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 67-78

    6. Types of Interval Data Sets: Towards Feasible Algorithms

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 79-94

    7. Computing Variance under Interval Uncertainty: Efficient Algorithms

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 95-117

    8. Computing Variance under Hierarchical Privacy-Related Interval Uncertainty

      • Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang
      Pages 119-127
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Keywords

About this book

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.

Reviews

From the reviews:

“This book is a research exposition by Kreinovich and coworkers. … The main goal is to present algorithms for computation of statistical characteristics (like variance) but under interval and fuzzy uncertainty of the available data. In this book, fuzzy uncertainty is reduced to interval uncertainty by alpha-cutwise consideration of (convex) fuzzy uncertainty. … For increase of readability, mathematical proofs are presented always at the end of the chapters.” (Wolfgang Näther, Zentralblatt MATH, Vol. 1238, 2012)

Authors and Affiliations

  • Department of Economics, ChiangMai University, Chiangmai, Thailand

    Hung T. Nguyen

  • Department of Computer Science, University of Texas at El Paso, El Paso, USA

    Vladik Kreinovich

  • Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan

    Berlin Wu

  • Applied Biomathematics, Inc., Setauket, USA

    Gang Xiang

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