Buckley / AL / USA Fuzzy Probability and Statistics

1. Auflage 2008
ISBN: 978-3-540-33190-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 196, 296 Seiten

Reihe: Studies in Fuzziness and Soft Computing

This book combines material from our previous books FP (Fuzzy Probabilities: New Approach and Applications,Physica-Verlag, 2003) and FS (Fuzzy Statistics, Springer, 2004), plus has about one third new results. From FP we have material on basic fuzzy probability, discrete (fuzzy Poisson,binomial) and continuous (uniform, normal, exponential) fuzzy random variables. From FS we included chapters on fuzzy estimation and fuzzy hypothesis testing related to means, variances, proportions, correlation and regression. New material includes fuzzy estimators for arrival and service rates, and the uniform distribution, with applications in fuzzy queuing theory. Also, new to this book, is three chapters on fuzzy maximum entropy (imprecise side conditions) estimators producing fuzzy distributions and crisp discrete/continuous distributions. Other new results are: (1) two chapters on fuzzy ANOVA (one-way and two-way), (2) random fuzzy numbers with applications to fuzzy Monte Carlo studies, and (3) a fuzzy nonparametric estimator for the median.

Written for:
Engineers, researchers, and students in Fuzziness and Applied Mathematics.

Keywords:
Fuzzy Probabilities
Fuzzy Statistics

Autoren/Hrsg.

Buckley, James J.

AL,

USA,

Inhaltsverzeichnis

1;Contents;7
2;Chapter 1 Introduction;14
2.1;1.1 Introduction;14
2.2;1.2 Notation;16
2.3;1.3 Previous Research;17
2.4;1.4 Figures;17
2.5;1.5 Maple/Solver Commands;18
2.6;1.6 References;18
3;Chapter 2 Fuzzy Sets;20
3.1;2.1 Introduction;20
3.2;2.2 Fuzzy Sets;20
3.3;2.3 Fuzzy Arithmetic;24
3.4;2.4 Fuzzy Functions;26
3.5;2.5 Ordering Fuzzy Numbers;30
3.6;2.6 References;31
4;Chapter 3 Fuzzy Probability Theory;33
4.1;3.1 Introduction;33
4.2;3.2 Fuzzy Probabilities from Con . dence Intervals;33
4.3;3.3 Fuzzy Probabilities from Expert Opinion;35
4.4;3.4 Restricted Fuzzy Arithmetic;36
4.5;3.5 Fuzzy Probability;44
4.6;3.6 Fuzzy Conditional Probability;48
4.7;3.7 Fuzzy Independence;50
4.8;3.8 Fuzzy Bayes’ Formula;52
4.9;3.9 Applications;53
4.10;3.10 References;60
5;Chapter 4 Discrete Fuzzy Random Variables;62
5.1;4.1 Introduction;62
5.2;4.2 Fuzzy Binomial;62
5.3;4.3 Fuzzy Poisson;65
5.4;4.4 Applications;68
5.5;4.5 References;71
6;Chapter 5 Continuous Fuzzy Random Variables;72
6.1;5.1 Introduction;72
6.2;5.2 Fuzzy Uniform;72
6.3;5.3 Fuzzy Normal;74
6.4;5.4 Fuzzy Negative Exponential;76
6.5;5.5 Applications;78
6.6;5.6 References;85
7;Chapter 6 Estimate µ, Variance Known;86
7.1;6.1 Introduction;86
7.2;6.2 Fuzzy Estimation;86
7.3;6.3 Fuzzy Estimator of;87
7.4;6.4 References;90
8;Chapter 7 Estimate µ, Variance Unknown;91
8.1;7.1 Fuzzy Estimator of;91
8.2;7.2 References;93
9;Chapter 8 Estimate p, Binomial Population;94
9.1;8.1 Fuzzy Estimator of;94
9.2;8.2 References;96
10;Chapter 9 Estimate s2 from a Normal Population;97
10.1;9.1 Introduction;97
10.2;9.2 Biased Fuzzy Estimator;97
10.3;9.3 Unbiased Fuzzy Estimator;98
10.4;9.4 References;102
11;Chapter 10 Fuzzy Arrival/Service Rates;103
11.1;10.1 Introduction;103
11.2;10.2 Fuzzy Arrival Rate;103
11.3;10.3 Fuzzy Service Rate;105
11.4;10.4 References;107
12;Chapter 11 Fuzzy Uniform;108
12.1;11.1 Introduction;108
12.2;11.2 Fuzzy Estimators;108
12.3;11.3 References;112
13;Chapter 12 Fuzzy Max Entropy Principle;113
13.1;12.1 Introduction;113
13.2;12.2 Maximum Entropy Principle;113
13.3;12.3 Imprecise Side-Conditions;117
13.4;12.4 Summary and Conclusions;119
13.5;12.5 References;120
14;Chapter 13 Max Entropy: Crisp Discrete Solutions;121
14.1;13.1 Introduction;121
14.2;13.2 Max Entropy: Discrete Distributions;121
14.3;13.3 Max Entropy: Imprecise Side-Conditions;122
14.4;13.4 Summary and Conclusions;129
14.5;13.5 References;129
15;Chapter 14 Max Entropy: Crisp Continuous Solutions;131
15.1;14.1 Introduction;131
15.2;14.2 Max Entropy: Probability Densities;132
15.3;14.3 Max Entropy: Imprecise Side-Conditions;133
15.4;14.4 E = [0,M];133
15.5;14.5 E = [0,8);141
15.6;14.6 E = (.8,8);145
15.7;14.7 Summary and Conclusions;146
15.8;14.8 References;146
16;Chapter 15 Tests on µ, Variance Known;148
16.1;15.1 Introduction;148
16.2;15.2 Non-Fuzzy Case;148
16.3;15.3 Fuzzy Case;149
16.4;15.4 One-Sided Tests;153
16.5;15.5 References;154
17;Chapter 16 Tests on µ, Variance Unknown;155
17.1;16.1 Introduction;155
17.2;16.2 Crisp Case;155
17.3;16.3 Fuzzy Model;156
17.4;16.4 References;161
18;Chapter 17 Tests on p for a Binomial Population;162
18.1;17.1 Introduction;162
18.2;17.2 Non-Fuzzy Test;162
18.3;17.3 Fuzzy Test;163
18.4;17.4 References;165
19;Chapter 18 Tests on s2, Normal Population;166
19.1;18.1 Introduction;166
19.2;18.2 Crisp Hypothesis Test;166
19.3;18.3 Fuzzy Hypothesis Test;167
19.4;18.4 References;169
20;Chapter 19 Fuzzy Correlation;170
20.1;19.1 Introduction;170
20.2;19.2 Crisp Results;170
20.3;19.3 Fuzzy Theory;171
20.4;19.4 References;173
21;Chapter 20 Estimation in Simple Linear Regression;174
21.1;20.1 Introduction;174
21.2;20.2 Fuzzy Estimators;175
21.3;20.3 References;178
22;Chapter 21 Fuzzy Prediction in Linear Regression;179
22.1;21.1 Prediction;179
22.2;21.2 References;181
23;Chapter 22 Hypothesis Testing in Regression;182
23.1;22.1 Introduction;182
23.2;22.2 Tests on;182
23.3;22.3 Tests on;184
23.4;22.4 References;186
24;Chapter 23 Estimation in Multiple Regression;187
24.1;23.1 Introduction;187
24.2;23.2 Fuzzy;188
24.3;23.3 References;192
25;Chapter 24 Fuzzy Prediction in Regression;193
25.1;24.1 Prediction;193
25.2;24.2 References;195
26;Chapter 25 Hypothesis Testing in Regression;196
26.1;25.1 Introduction;196
26.2;25.2 Tests on;196
26.3;25.3 Tests on;198
26.4;25.4 References;200
27;Chapter 26 Fuzzy One-Way ANOVA;201
27.1;26.1 Introduction;201
27.2;26.2 Crisp Hypothesis Test;201
27.3;26.3 Fuzzy Hypothesis Test;202
27.4;26.4 References;205
28;Chapter 27 Fuzzy Two-Way ANOVA;206
28.1;27.1 Introduction;206
28.2;27.2 Crisp Hypothesis Tests;206
28.3;27.3 Fuzzy Hypothesis Tests;208
28.4;27.4 References;214
29;Chapter 28 Fuzzy Estimator for the Median;215
29.1;28.1 Introduction;215
29.2;28.2 Crisp Estimator for the Median;215
29.3;28.3 Fuzzy Estimator;216
29.4;28.4 Reference;217
30;Chapter 29 Random Fuzzy Numbers;218
30.1;29.1 Introduction;218
30.2;29.2 Random Fuzzy Numbers;219
30.3;29.3 Tests for Randomness;220
30.4;29.4 Monte Carlo Study;225
30.5;29.5 References;228
31;Chapter 30 Selected Maple/Solver Commands;230
31.1;30.1 Introduction;230
31.2;30.2 SOLVER;230
31.3;30.3 Maple;232
31.4;30.4 References;246
32;Chapter 31 Summary and Future Research;247
32.1;31.1 Summary;247
32.2;31.2 Future Research;248
32.3;31.3 References;250
33;Index;251
34;List of Figures;258
35;List of Tables;261

Leseproben

Chapter 1
Introduction (p. 1-2)

1.1 Introduction

This book is written in the following divisions: (1) the introductory chapters consisting of Chapters 1 and 2, (2) introduction to fuzzy probability in Chapters 3-5, (3) introduction to fuzzy estimation in Chapters 6-11, (4) fuzzy/crisp estimators of probability density (mass) functions based on a fuzzy maximum entropy principle in Chapters 12-14, (5) introduction to fuzzy hypothesis testing in Chapters 15-18, (6) fuzzy correlation and regression in Chapters 19-25, (7) Chapters 26 and 27 are about a fuzzy ANOVA model, (8) a fuzzy estimator of the median in nonparametric statistics in Chapter 28, and (9) random fuzzy numbers with applications to Monte Carlo studies in Chapter 29. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. For a beginning introduction to fuzzy sets and fuzzy logic see [8]. One other item relating to fuzzy sets, needed in fuzzy hypothesis testing, is also in Chapter 2: how we will determine which of the following three possibilities is true M <, N, M >, N or M . N, for two fuzzy numbers M, N.

The introduction to fuzzy probability in Chapters 3-5 is based on the book [1] and the reader is referred to that book for more information, especially applications. What is new here is: (1) using a nonlinear optimization program in Maple [13] to solve certain optimization problems in fuzzy probability, where previously we used a graphical method, and (2) a new algorithm, suitable for using only pencil and paper, for solving some restricted fuzzy arithmetic problems.

The introduction to fuzzy estimation is based on the book [3] and we refer the interested reader to that book for more about fuzzy estimators. The fuzzy estimators omitted from this book are those for µ1 . µ2, p1 . p2, s1/s2, etc. Fuzzy estimators for arrival and service rates is from [2] and [4]. The reader should see those book for applications in queuing networks. Also, fuzzy estimators for the uniform probability density can be found in [4], but the derivation of these fuzzy estimators is new to this book. The fuzzy uniform distribution was used for arrival/service rates in queuing models in [4].

The fuzzy/crisp probability density estimators based on a fuzzy maximum entropy principle are based on the papers [5],[6] and [7] and are new to this book. In Chapter 12 we obtain fuzzy results but in Chapters 13 and 14 we determine crisp discrete and crisp continuous probability densities. The introduction to fuzzy hypothesis testing in Chapters 15-18 is based on the book [3] and the reader needs to consult that book for more fuzzy hypothesis testing. What we omitted are tests on µ1 = µ2, p1 = p2, s1 = s2, etc.

The chapters on fuzzy correlation and regression come from [3]. The results on the fuzzy ANOVA (Chapters 26 and 27) and a fuzzy estimator for the median (Chapter 28) are new and have not been published before. The chapter on random fuzzy numbers (Chapter 29) is also new to this book and these results have not been previously published. Applications of crisp random numbers to Monte Carlo studies are well known and we also plan to use random fuzzy numbers in Monte Carlo studies. Our first use of random fuzzy numbers will be to get approximate solutions to fuzzy optimization problems whose solution is unknown or computationally very difficult. However, this becomes a rather large project and will probably be the topic of a future book.

Chapter 30 contains selected Maple/Solver ([11],[13],[20]) commands used in the book to solve optimization problems or to generate the figures. The final chapter has a summary and suggestions for future research. All chapters can be read independently. This means that some material is repeated in a sequence of chapters. For example, in Chapters 15-18 on fuzzy hypothesis testing in each chapter we first review the crisp case, then fuzzify to obtain our fuzzy statistic which is then used to construct the fuzzy critical values and we finally present a numerical example. However, you should first know about fuzzy estimators (Chapters 6-11) before going on to fuzzy hypothesis testing.

A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data. Other authors discussing fuzzy statistics usually begin with fuzzy data. We assume we have a random sample giving real number data x1, x2, ..., xn which is then used to generate our fuzzy estimators. Using fuzzy estimators in hypothesis testing and regression obviously leads to fuzzy hypothesis testing and fuzzy regression.