Beschreibung
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.
Autorenporträt
InhaltsangabeI: 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 Re
