Towards Efficient Fuzzy Information Processing

When we learn from books or daily experience, we make associations and draw inferences on the basis of information that is insufficient for under­ standing. One example of insufficient information may be a small sample derived from observing experiments. With this perspective, the need for de­ veloping a better understanding of the behavior of a small sample presents a problem that is far beyond purely academic importance. During the past 15 years considerable progress has been achieved in the study of this issue in China. One distinguished result is the principle of in­ formation diffusion. According to this principle, it is possible to partly fill gaps caused by incomplete information by changing crisp observations into fuzzy sets so that one can improve the recognition of relationships between input and output. The principle of information diffusion has been proven suc­ cessful for the estimation of a probability density function. Many successful applications reflect the advantages of this new approach. It also supports an argument that fuzzy set theory can be used not only in "soft" science where some subjective adjustment is necessary, but also in "hard" science where all data are recorded.

I: Principle of Information Diffusion.- 1. Introduction.- 1.1 Information Sciences.- 1.2 Fuzzy Information.- 1.2.1 Some basic notions of fuzzy set theory.- 1.2.2 Fuzzy information defined by fuzzy entropy.- 1.2.3 Traditional fuzzy information without reference to entropy.- 1.2.4 Fuzzy information due to an incomplete data set.- 1.2.5 Fuzzy information and its properties.- 1.2.6 Fuzzy information processing.- 1.3 Fuzzy function approximation.- 1.4 Summary.- Referencess.- 2. Information Matrix.- 2.1 Small-Sample Problem.- 2.2 Information Matrix.- 2.3 Information Matrix on Crisp Intervals.- 2.4 Information Matrix on Fuzzy Intervals.- 2.5 Mechanism of Information Matrix.- 2.6 Some Approaches Describing or Producing Relationships.- 2.6.1 Equations of mathematical physics.- 2.6.2 Regression.- 2.6.3 Neural networks.- 2.6.4 Fuzzy graphs.- 2.7 Conclusion and Discussion.- References.- Appendix 2.A: Some Earthquake Data.- 3. Some Concepts From Probability and Statistics.- 3.1 Introduction.- 3.2 Probability.- 3.2.1 Sample spaces, outcomes, and events.- 3.2.2 Probability.- 3.2.3 Joint, marginal, and conditional probabilities.- 3.2.4 Random variables.- 3.2.5 Expectation value, variance, functions of random variables.- 3.2.6 Continuous random variables.- 3.2.7 Probability density function.- 3.2.8 Cumulative distribution function.- 3.3 Some Probability Density Functions.- 3.3.1 Uniform distribution.- 3.3.2 Normal distribution.- 3.3.3 Exponential distribution.- 3.3.4 Lognormal distribution.- 3.4 Statistics and Some Traditional Estimation Methods.- 3.4.1 Statistics.- 3.4.2 Maximum likelihood estimate.- 3.4.3 Histogram.- 3.4.4 Kernel method.- 3.5 Monte Carlo Methods.- 3.5.1 Pseudo-random numbers.- 3.5.2 Uniform random numbers.- 3.5.3 Normal random numbers.- 3.5.4 Exponential random numbers.- 3.5.5 Lognormal random numbers.- References.- 4. Information Distribution.- 4.1 Introduction.- 4.2 Definition of Information Distribution.- 4.3 1-Dimension Linear Information Distribution.- 4.4 Demonstration of Benefit for Probability Estimation.- 4.4.1 Model description.- 4.4.2 Normal experiment.- 4.4.3 Exponential experiment.- 4.4.4 Lognormal experiment.- 4.4.5 Comparison with maximum likelihood estimate.- 4.4.6 Results.- 4.5 Non-Linear Distribution.- 4.6 r-Dimension Distribution.- 4.7 Fuzzy Relation Matrix from Information Distribution.- 4.7.1 Rf based on fuzzy concepts.- 4.7.2 Rm based on fuzzy implication theory.- 4.7.3 Rc based on conditional falling shadow.- 4.8 Approximate Inference Based on Information Distribution.- 4.8.1 Max-min inference for Rf.- 4.8.2 Similarity inference for Rf.- 4.8.3 Max-min inference for Rm.- 4.8.4 Total-falling-shadow inference for Rc.- 4.9 Conclusion and Discussion.- References.- Appendix 4.A: Linear Distribution Program.- Appendix 4.B: Intensity Scale.- 5. Information Diffusion.- 5.1 Problems in Information Distribution.- 5.2 Definition of Incomplete-Data Set.- 5.2.1 Incompleteness.- 5.2.2 Correct-data set.- 5.2.3 Incomplete-data set.- 5.3 Fuzziness of a Given Sample.- 5.3.1 Fuzziness in terms of fuzzy sets.- 5.3.2 Fuzziness in terms of philosophy.- 5.3.3 Fuzziness of an incomplete sample.- 5.4 Information Diffusion.- 5.5 Random Sets and Covering Theory.- 5.5.1 Fuzzy logic and possibility theory.- 5.5.2 Random sets.- 5.5.3 Covering function.- 5.5.4 Set-valuedization of observation.- 5.6 Principle of Information Diffusion.- 5.6.1 Associated characteristic function and relationships.- 5.6.2 Allocation function.- 5.6.3 Diffusion estimate.- 5.6.4 Principle of Information Diffusion.- 5.7 Estimating Probability by Information Diffusion.- 5.7.1 Asymptotically unbiased property.- 5.7.2 Mean squared consistent property.- 5.7.3 Asymptotically property of mean square error.- 5.7.4 Empirical distribution function, histogram and diffusion estimate.- 5.8 Conclusion and Discussion.- References.- 6. Quadratic Diffusion.- 6.1 Optimal Diffusion Function.- 6.2 Choosing ? Based on Kernel Theory.- 6.2.1 Mean integrated square error.- 6.2.2 References to a standard distribution.- 6.2.3 Least-squares cross-validation.- 6.2.4 Discussion.- 6.3 Searching for ? by Golden Section Method.- 6.4 Comparison with Other Estimates.- 6.5 Conclusion.- References.- 7. Normal Diffusion.- 7.1 Introduction.- 7.2 Molecule Diffusion Theory.- 7.2.1 Diffusion.- 7.2.2 Diffusion equation.- 7.3 Information Diffusion Equation.- 7.3.1 Similarities of molecule diffusion and information diffusion.- 7.3.2 Partial differential equation of information diffusion.- 7.4 Nearby Criteria of Normal Diffusion.- 7.5 The 0.618 Algorithm for Getting h.- 7.6 Average Distance Model.- 7.7 Conclusion and Discussion.- References.- II: Applications.- 8. Estimation of Epicentral Intensity.- 8.1 Introduction.- 8.2 Classical Methods.- 8.2.1 Linear regression.- 8.2.2 Fuzzy inference based on normal assumption.- 8.3 Self-Study Discrete Regression.- 8.3.1 Discrete regression.- 8.3.2 r-dimension diffusion.- 8.3.3 Self-study discrete regression.- 8.4 Linear Distribution Self-Study.- 8.5 Normal Diffusion Self-Study.- 8.6 Conclusion and Discussion.- References.- Appendix 8.A: Real and Estimated Epicentral Intensities.- Appendix 8.B: Program of NDSS.- 9. Estimation of Isoseismal Area.- 9.1 Introduction.- 9.2 Some Methods for Constructing Fuzzy Relationships.- 9.2.1 Fuzzy relation and fuzzy relationship.- 9.2.2 Multivalued logical-implication operator.- 9.2.3 Fuzzy associative memories.- 9.2.4 Self-study discrete regression.- 9.3 Multitude Relationships Given by Information Diffusion.- 9.4 Patterns Smoothening.- 9.5 Learning Relationships by BP Neural Networks.- 9.6 Calculation.- 9.7 Conclusion and Discussion.- References.- 10. Fuzzy Risk Analysis.- 10.1 Introduction.- 10.2 Risk Recognition and Management for Environment, Health, and Safety.- 10.3 A Survey of Fuzzy Risk Analysis.- 10.4 Risk Essence and Fuzzy Risk.- 10.5 Some Classical Models.- 10.5.1 Histogram.- 10.5.2 Maximum likelihood method.- 10.5.3 Kernel estimation.- 10.6 Model of Risk Assessment by Diffusion Estimate.- 10.7 Application in Risk Assessment of Flood Disaster.- 10.7.1 Normalized normal-diffusion estimate.- 10.7.2 Histogram estimate.- 10.7.3 Soft histogram estimate.- 10.7.4 Maximum likelihood estimate.- 10.7.5 Gaussian kernel estimate.- 10.7.6 Comparison.- 10.8 Conclusion and Discussion.- References.- 11. System Analytic Model for Natural Disasters.- 11.1 Classical System Model for Risk Assessment of Natural Disasters.- 11.1.1 Risk assessment of hazard.- 11.1.2 From magnitude to site intensity.- 11.1.3 Damage risk.- 11.1.4 Loss risk.- 11.2 Fuzzy Model for Hazard Analysis.- 11.2.1 Calculating primary information distribution.- 11.2.2 Calculating exceeding frequency distribution.- 11.2.3 Calculating fuzzy relationship between magnitude and probability.- 11.3 Fuzzy Systems Analytic Model.- 11.3.1 Fuzzy attenuation relationship.- 11.3.2 Fuzzy dose-response relationship.- 11.3.3 Fuzzy loss risk.- 11.4 Application in Risk Assessment of Earthquake Disaster.- 11.4.1 Fuzzy relationship between magnitude and probability.- 11.4.2 Intensity risk.- 11.4.3 Earthquake damage risk.- 11.4.4 Earthquake loss risk.- 11.5 Conclusion and Discussion.- References.- 12. Fuzzy Risk Calculation.- 12.1 Introduction.- 12.1.1 Fuzziness and probability.- 12.1.2 Possibility-probability distribution.- 12.2 Interior-outer-set Model.- 12.2.1 Model description.- 12.2.2 Calculation case.- 12.2.3 Algorithm and Fortran program.- 12.3 Ranking Alternatives Based on a PPD.- 12.3.1 Classical model of ranking alternatives.- 12.3.2 Fuzzy expected value.- 12.3.3 Center of gravity of a fuzzy expected value.- 12.3.4 Ranking alternatives by FEV.- 12.4 Application in Risk Management of Flood Disaster.- 12.4.1 Outline of Huarong county.- 12.4.2 PPD of flood in Huarong county.- 12.4.3 Benefit-output functions of farming alternatives.- 12.4.4 Ranking farming alternative based on the PPD.- 12.4.5 Comparing with the traditional probability method.- 12.5 Conclusion and Discussion.- References.- Appendix 12.A: Algorithm Program for Interior-outer-set Model.- List of Special Symbols.