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.