Limit theorems and applications of set-valued and fuzzy set-valued random variables:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Academic Publishers
2002
|
| Series: | Theory and decision library
Series B, Mathematical and statistical methods ; 43 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XII, 391 S. 25 cm |
| ISBN: | 1402009186 |
Staff View
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Record in the Search Index
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| adam_text | LIMIT THEOREMS AND APPLICATIONS OF SET-VALUED AND FUZZY SET-VALUED
RANDOM VARIABLES BY SHOUMEI LI BEIJING POLYTECHNIC UNIVERSITY, BEIJING,
THE PEOPLES REPUBLIC OF CHINA YUKIO OGURA SAGA UNIVERSITY, SAGA, JAPAN
AND VLADIK KREINOVICH UNIVERSITY OF TEXAS EL PASO, EL PASO, USA. KLUWER
ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON CONTENTS PREFACE IX PART
I LIMIT THEOREMS OF SET-VALUED AND FUZZY SET- VALUED RANDOM VARIABLES 1.
THE SPACE OF SET-VALUED RANDOM VARIABLES 1 1 HYPERSPACES OF A BANACH
SPACE 1 1.1 THE HAUSDORFF METRIC IN HYPERSPACES AND AN EM- BEDDING
THEOREM 1 1.2 CONVERGENCES IN HYPERSPACES 12 2 SET-VALUED RANDOM
VARIABLES 20 3 THE SET OF INTEGRABLE SELECTIONS 26 4 THE SPACES OF
INTEGRABLY BOUNDED SET-VALUED RANDOM VARIABLES 34 2. THE AUMANN INTEGRAL
AND THE CONDITIONAL EXPECTATION OF A SET-VALUED RANDOM VARIABLE 41 1 THE
AUMANN INTEGRAL AND ITS PROPERTIES 41 2 SUFFICIENT CONDITIONS FOR THE
AUMANN INTEGRALS TO BE CLOSED 47 3 CONDITIONAL EXPECTATION AND ITS
PROPERTIES 54 4 FATOU S LEMMAS AND LEBESGUE S DOMINATED CONVERGENCE
THEOREMS 67 5 RADON-NIKODYM THEOREMS FOR SET-VALUED MEASURES 73 5.1
SET-VALUED MEASURES 74 5.2 RADON-NIKODYM THEOREMS FOR SET-VALUED MEA-
SURES 81 VI LIMIT THEOREMS AND APPLICATIONS 3. STRONG LAWS OF LARGE
NUMBERS AND CENTRAL LIMIT THEOREMS FOR SET-VALUED RANDOM VARIABLES 87 1
LIMIT THEOREMS FOR SET-VALUED RANDOM VARIABLES IN THE HAUSDORFF METRIC
87 1.1 STRONG LAWS OF LARGE NUMBERS IN THE HAUSDORFF METRIC 87 1.2
CENTRAL LIMIT THEOREMS 96 2 STRONG LAWS OF LARGE NUMBERS FOR SET-VALUED
RANDOM VARIABLES IN THE KURATOWSKI-MOSCO SENSE 100 3 GAUSSIAN SET-VALUED
RANDOM VARIABLES 105 4 APPENDIX A OF SUBSECTION 3.1.2 108 4. CONVERGENCE
THEOREMS FOR SET-VALUED MARTINGALES 117 1 SET-VALUED MARTINGALES 117 2
REPRESENTATION THEOREMS FOR CLOSED CONVEX SET-VALUED MARTINGALES 126 3
CONVERGENCE OF CLOSED CONVEX SET-VALUED MARTINGALES IN THE
KURATOWSKI-MOSCO SENSE 134 4 CONVERGENCE OF CLOSED CONVEX SET-VALUED
SUBMARTINGALES AND SUPERMARTINGALES IN THE KURATOWSKI-MOSCO SENSE 138
4.1 CONVERGENCE OF CLOSED CONVEX SET-VALUED SUB- MARTINGALES IN THE
KURATOWSKI-MOSCO SENSE 138 4.2 CONVERGENCE OF CLOSED CONVEX SET-VALUED
SUPER- MARTINGALES IN THE KURATOWSKI-MOSCO SENSE 142 5 CONVERGENCE OF
CLOSED CONVEX SET-VALUED SUPERMARTINGALES (MARTINGALES) WHOSE VALUES MAY
BE UNBOUNDED 143 6 OPTIONAL SAMPLING THEOREMS FOR CLOSED CONVEX
SET-VALUED MARTINGALES 150 7 DOOB DECOMPOSITION OF SET-VALUED
SUBMARTINGALES 155 5. FUZZY SET-VALUED RANDOM VARIABLES 161 1 FUZZY SETS
162 2 THE SPACE OF FUZZY SET-VALUED RANDOM VARIABLES 171 3 EXPECTATIONS
OF FUZZY SET-VALUED RANDOM VARIABLES 181 4 CONDITIONAL EXPECTATIONS OF
FUZZY RANDOM SETS 184 5 THE RADON-NIKODYM THEOREM FOR FUZZY SET-VALUED
MEA- SURES 187 CONTENTS VII 6. CONVERGENCE THEOREMS FOR FUZZY SET-VALUED
RANDOM VARIABLES 191 1 EMBEDDING THEOREMS AND GAUSSIAN FUZZY RANDOM SETS
191 1.1 EMBEDDING THEOREMS 191 1.2 GAUSSIAN FUZZY SET-VALUED RANDOM
VARIABLES 195 2 STRONG LAWS OF LARGE NUMBERS FOR FUZZY SET-VALUED RAN-
DOM VARIABLES 197 3 CENTRAL LIMIT THEOREMS FOR FUZZY SET-VALUED RANDOM
VARIABLES 205 4 FUZZY SET-VALUED MARTINGALES 214 7. CONVERGENCES IN THE
GRAPHICAL SENSE FOR FUZZY SET-VALUED RANDOM VARIABLES 221 1 CONVERGENCES
IN THE GRAPHICAL SENSE FOR FUZZY SETS 221 2 SEPARABILITY FOR THE
GRAPHICAL CONVERGENCES AND APPLICA- TIONS TO STRONG LAWS OF LARGE
NUMBERS 226 3 CONVERGENCE IN THE GRAPHICAL SENSE FOR FUZZY SET-VALUED
MARTINGALES AND SMARTINGALES 231 REFERENCES FOR PART I 235 PART II
PRACTICAL APPLICATIONS OF SET-VALUED RANDOM VARIABLES 8. MATHEMATICAL
FOUNDATIONS FOR THE APPLICATIONS OF SET-VALUED RANDOM VARIABLES 253 1
HOW CAN LIMIT THEOREMS BE APPLIED? 253 2 RELEVANT OPTIMIZATION
TECHNIQUES 257 2.1 INTRODUCTION: OPTIMIZATION OF SET FUNCTIONS IS A
PRACTICALLY IMPORTANT BUT DIFFICULT PROBLEM 257 2.2 THE EXISTING METHODS
OF OPTIMIZING SET FUNCTIONS: THEIR SUCCESSES (IN BRIEF) AND THE
TERRITORIAL DIVI- SION PROBLEM AS A CHALLENGE 261 2.3 A DIFFERENTIAL
FORMALISM FOR SET FUNCTIONS 264 2.4 FIRST APPLICATION OF THE NEW
FORMALISM: TERRITORIAL DIVISION PROBLEM 274 2.5 SECOND APPLICATION OF
THE NEW FORMALISM: STATISTI- CAL EXAMPLE * EXCESS MASS METHOD 282 2.6
FURTHER DIRECTIONS, RELATED RESULTS, AND OPEN PROBLEMS 284 VIII LIMIT
THEOREMS AND APPLICATIONS 3 OPTIMIZATION UNDER UNCERTAINTY AND RELATED
SYMMETRY TECHNIQUES 286 3.1 CASE STUDY: SELECTING ZONES IN A PLANE 286
3.2 GENERAL CASE 290 9. APPLICATIONS TO IMAGING 295 1 APPLICATIONS TO
ASTRONOMY 295 2 APPLICATIONS TO AGRICULTURE 306 2.1 DETECTING TRASH IN
GINNED COTTON 306 2.2 CLASSIFICATION OF INSECTS IN THE COTTON FIELD 313
3 APPLICATIONS TO MEDICINE 322 3.1 TOWARDS FOUNDATIONS FOR TRADITIONAL
ORIENTAL MEDI- CINE 322 3.2 TOWARDS OPTIMAL PAIN RELIEF: ACUPUNCTURE AND
SPINAL CORD STIMULATION 325 4 APPLICATIONS TO MECHANICAL FRACTURES 336
4.1 FAULT SHAPES 336 4.2 BEST SENSOR LOCATIONS FOR DETECTING SHAPES 337
5 WHAT SEGMENTS ARE THE BEST IN REPRESENTING CONTOURS? 342 6 SEARCHING
FOR A TYPICAL IMAGE 345 6.1 AVERAGE SET 345 6.2 AVERAGE SHAPE 351 10.
APPLICATIONS TO DATA PROCESSING 355 1 1-D CASE: WHY INTERVALS? A SIMPLE
LIMIT THEOREM 355 2 2-D CASE: CANDIDATE SETS FOR COMPLEX INTERVAL
ARITHME- TIC 360 3 MULTI-D CASE: WHY ELLIPSOIDS? 362 4 CONCLUSIONS 372
REFERENCES FOR PART II 373 INDEX 387
|
| adam_txt |
LIMIT THEOREMS AND APPLICATIONS OF SET-VALUED AND FUZZY SET-VALUED
RANDOM VARIABLES BY SHOUMEI LI BEIJING POLYTECHNIC UNIVERSITY, BEIJING,
THE PEOPLES REPUBLIC OF CHINA YUKIO OGURA SAGA UNIVERSITY, SAGA, JAPAN
AND VLADIK KREINOVICH UNIVERSITY OF TEXAS EL PASO, EL PASO, USA. KLUWER
ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON CONTENTS PREFACE IX PART
I LIMIT THEOREMS OF SET-VALUED AND FUZZY SET- VALUED RANDOM VARIABLES 1.
THE SPACE OF SET-VALUED RANDOM VARIABLES 1 1 HYPERSPACES OF A BANACH
SPACE 1 1.1 THE HAUSDORFF METRIC IN HYPERSPACES AND AN EM- BEDDING
THEOREM 1 1.2 CONVERGENCES IN HYPERSPACES 12 2 SET-VALUED RANDOM
VARIABLES 20 3 THE SET OF INTEGRABLE SELECTIONS 26 4 THE SPACES OF
INTEGRABLY BOUNDED SET-VALUED RANDOM VARIABLES 34 2. THE AUMANN INTEGRAL
AND THE CONDITIONAL EXPECTATION OF A SET-VALUED RANDOM VARIABLE 41 1 THE
AUMANN INTEGRAL AND ITS PROPERTIES 41 2 SUFFICIENT CONDITIONS FOR THE
AUMANN INTEGRALS TO BE CLOSED 47 3 CONDITIONAL EXPECTATION AND ITS
PROPERTIES 54 4 FATOU'S LEMMAS AND LEBESGUE'S DOMINATED CONVERGENCE
THEOREMS 67 5 RADON-NIKODYM THEOREMS FOR SET-VALUED MEASURES 73 5.1
SET-VALUED MEASURES 74 5.2 RADON-NIKODYM THEOREMS FOR SET-VALUED MEA-
SURES 81 VI LIMIT THEOREMS AND APPLICATIONS 3. STRONG LAWS OF LARGE
NUMBERS AND CENTRAL LIMIT THEOREMS FOR SET-VALUED RANDOM VARIABLES 87 1
LIMIT THEOREMS FOR SET-VALUED RANDOM VARIABLES IN THE HAUSDORFF METRIC
87 1.1 STRONG LAWS OF LARGE NUMBERS IN THE HAUSDORFF METRIC 87 1.2
CENTRAL LIMIT THEOREMS 96 2 STRONG LAWS OF LARGE NUMBERS FOR SET-VALUED
RANDOM VARIABLES IN THE KURATOWSKI-MOSCO SENSE 100 3 GAUSSIAN SET-VALUED
RANDOM VARIABLES 105 4 APPENDIX A OF SUBSECTION 3.1.2 108 4. CONVERGENCE
THEOREMS FOR SET-VALUED MARTINGALES 117 1 SET-VALUED MARTINGALES 117 2
REPRESENTATION THEOREMS FOR CLOSED CONVEX SET-VALUED MARTINGALES 126 3
CONVERGENCE OF CLOSED CONVEX SET-VALUED MARTINGALES IN THE
KURATOWSKI-MOSCO SENSE 134 4 CONVERGENCE OF CLOSED CONVEX SET-VALUED
SUBMARTINGALES AND SUPERMARTINGALES IN THE KURATOWSKI-MOSCO SENSE 138
4.1 CONVERGENCE OF CLOSED CONVEX SET-VALUED SUB- MARTINGALES IN THE
KURATOWSKI-MOSCO SENSE 138 4.2 CONVERGENCE OF CLOSED CONVEX SET-VALUED
SUPER- MARTINGALES IN THE KURATOWSKI-MOSCO SENSE 142 5 CONVERGENCE OF
CLOSED CONVEX SET-VALUED SUPERMARTINGALES (MARTINGALES) WHOSE VALUES MAY
BE UNBOUNDED 143 6 OPTIONAL SAMPLING THEOREMS FOR CLOSED CONVEX
SET-VALUED MARTINGALES 150 7 DOOB DECOMPOSITION OF SET-VALUED
SUBMARTINGALES 155 5. FUZZY SET-VALUED RANDOM VARIABLES 161 1 FUZZY SETS
162 2 THE SPACE OF FUZZY SET-VALUED RANDOM VARIABLES 171 3 EXPECTATIONS
OF FUZZY SET-VALUED RANDOM VARIABLES 181 4 CONDITIONAL EXPECTATIONS OF
FUZZY RANDOM SETS 184 5 THE RADON-NIKODYM THEOREM FOR FUZZY SET-VALUED
MEA- SURES 187 CONTENTS VII 6. CONVERGENCE THEOREMS FOR FUZZY SET-VALUED
RANDOM VARIABLES 191 1 EMBEDDING THEOREMS AND GAUSSIAN FUZZY RANDOM SETS
191 1.1 EMBEDDING THEOREMS 191 1.2 GAUSSIAN FUZZY SET-VALUED RANDOM
VARIABLES 195 2 STRONG LAWS OF LARGE NUMBERS FOR FUZZY SET-VALUED RAN-
DOM VARIABLES 197 3 CENTRAL LIMIT THEOREMS FOR FUZZY SET-VALUED RANDOM
VARIABLES 205 4 FUZZY SET-VALUED MARTINGALES 214 7. CONVERGENCES IN THE
GRAPHICAL SENSE FOR FUZZY SET-VALUED RANDOM VARIABLES 221 1 CONVERGENCES
IN THE GRAPHICAL SENSE FOR FUZZY SETS 221 2 SEPARABILITY FOR THE
GRAPHICAL CONVERGENCES AND APPLICA- TIONS TO STRONG LAWS OF LARGE
NUMBERS 226 3 CONVERGENCE IN THE GRAPHICAL SENSE FOR FUZZY SET-VALUED
MARTINGALES AND SMARTINGALES 231 REFERENCES FOR PART I 235 PART II
PRACTICAL APPLICATIONS OF SET-VALUED RANDOM VARIABLES 8. MATHEMATICAL
FOUNDATIONS FOR THE APPLICATIONS OF SET-VALUED RANDOM VARIABLES 253 1
HOW CAN LIMIT THEOREMS BE APPLIED? 253 2 RELEVANT OPTIMIZATION
TECHNIQUES 257 2.1 INTRODUCTION: OPTIMIZATION OF SET FUNCTIONS IS A
PRACTICALLY IMPORTANT BUT DIFFICULT PROBLEM 257 2.2 THE EXISTING METHODS
OF OPTIMIZING SET FUNCTIONS: THEIR SUCCESSES (IN BRIEF) AND THE
TERRITORIAL DIVI- SION PROBLEM AS A CHALLENGE 261 2.3 A DIFFERENTIAL
FORMALISM FOR SET FUNCTIONS 264 2.4 FIRST APPLICATION OF THE NEW
FORMALISM: TERRITORIAL DIVISION PROBLEM 274 2.5 SECOND APPLICATION OF
THE NEW FORMALISM: STATISTI- CAL EXAMPLE * EXCESS MASS METHOD 282 2.6
FURTHER DIRECTIONS, RELATED RESULTS, AND OPEN PROBLEMS 284 VIII LIMIT
THEOREMS AND APPLICATIONS 3 OPTIMIZATION UNDER UNCERTAINTY AND RELATED
SYMMETRY TECHNIQUES 286 3.1 CASE STUDY: SELECTING ZONES IN A PLANE 286
3.2 GENERAL CASE 290 9. APPLICATIONS TO IMAGING 295 1 APPLICATIONS TO
ASTRONOMY 295 2 APPLICATIONS TO AGRICULTURE 306 2.1 DETECTING TRASH IN
GINNED COTTON 306 2.2 CLASSIFICATION OF INSECTS IN THE COTTON FIELD 313
3 APPLICATIONS TO MEDICINE 322 3.1 TOWARDS FOUNDATIONS FOR TRADITIONAL
ORIENTAL MEDI- CINE 322 3.2 TOWARDS OPTIMAL PAIN RELIEF: ACUPUNCTURE AND
SPINAL CORD STIMULATION 325 4 APPLICATIONS TO MECHANICAL FRACTURES 336
4.1 FAULT SHAPES 336 4.2 BEST SENSOR LOCATIONS FOR DETECTING SHAPES 337
5 WHAT SEGMENTS ARE THE BEST IN REPRESENTING CONTOURS? 342 6 SEARCHING
FOR A 'TYPICAL' IMAGE 345 6.1 AVERAGE SET 345 6.2 AVERAGE SHAPE 351 10.
APPLICATIONS TO DATA PROCESSING 355 1 1-D CASE: WHY INTERVALS? A SIMPLE
LIMIT THEOREM 355 2 2-D CASE: CANDIDATE SETS FOR COMPLEX INTERVAL
ARITHME- TIC 360 3 MULTI-D CASE: WHY ELLIPSOIDS? 362 4 CONCLUSIONS 372
REFERENCES FOR PART II 373 INDEX 387 |
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| ctrlnum | (OCoLC)50495432 (DE-599)GBV352958766 |
| dewey-full | 519.2 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023203148 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:08:52Z |
| indexdate | 2024-07-09T21:12:58Z |
| institution | BVB |
| isbn | 1402009186 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016389354 |
| oclc_num | 50495432 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM |
| owner_facet | DE-91G DE-BY-TUM |
| physical | XII, 391 S. 25 cm |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Kluwer Academic Publishers |
| record_format | marc |
| series | Theory and decision library |
| series2 | Theory and decision library : Series B, Mathematical and statistical methods |
| spelling | Li, Shoumei Verfasser aut Limit theorems and applications of set-valued and fuzzy set-valued random variables by Shoumei Li ; Yukio Ogura and Vladik Kreinovich Dordrecht [u.a.] Kluwer Academic Publishers 2002 XII, 391 S. 25 cm txt rdacontent n rdamedia nc rdacarrier Theory and decision library : Series B, Mathematical and statistical methods 43 Includes bibliographical references and index Limit theorems (Probability theory) Random variables Fuzzy-Zufallsvariable (DE-588)4201987-4 gnd rswk-swf Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf Grenzwertsatz (DE-588)4158163-5 s Zufallsvariable (DE-588)4129514-6 s Fuzzy-Zufallsvariable (DE-588)4201987-4 s DE-604 Ogura, Yukio Verfasser aut Krejnovič, Vladik Ja. Verfasser aut Theory and decision library Series B, Mathematical and statistical methods ; 43 (DE-604)BV000021513 43 http://www.gbv.de/dms/goettingen/352958766.pdf lizenzfrei Inhaltsverzeichnis GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016389354&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Li, Shoumei Ogura, Yukio Krejnovič, Vladik Ja Limit theorems and applications of set-valued and fuzzy set-valued random variables Theory and decision library Limit theorems (Probability theory) Random variables Fuzzy-Zufallsvariable (DE-588)4201987-4 gnd Zufallsvariable (DE-588)4129514-6 gnd Grenzwertsatz (DE-588)4158163-5 gnd |
| subject_GND | (DE-588)4201987-4 (DE-588)4129514-6 (DE-588)4158163-5 |
| title | Limit theorems and applications of set-valued and fuzzy set-valued random variables |
| title_auth | Limit theorems and applications of set-valued and fuzzy set-valued random variables |
| title_exact_search | Limit theorems and applications of set-valued and fuzzy set-valued random variables |
| title_exact_search_txtP | Limit theorems and applications of set-valued and fuzzy set-valued random variables |
| title_full | Limit theorems and applications of set-valued and fuzzy set-valued random variables by Shoumei Li ; Yukio Ogura and Vladik Kreinovich |
| title_fullStr | Limit theorems and applications of set-valued and fuzzy set-valued random variables by Shoumei Li ; Yukio Ogura and Vladik Kreinovich |
| title_full_unstemmed | Limit theorems and applications of set-valued and fuzzy set-valued random variables by Shoumei Li ; Yukio Ogura and Vladik Kreinovich |
| title_short | Limit theorems and applications of set-valued and fuzzy set-valued random variables |
| title_sort | limit theorems and applications of set valued and fuzzy set valued random variables |
| topic | Limit theorems (Probability theory) Random variables Fuzzy-Zufallsvariable (DE-588)4201987-4 gnd Zufallsvariable (DE-588)4129514-6 gnd Grenzwertsatz (DE-588)4158163-5 gnd |
| topic_facet | Limit theorems (Probability theory) Random variables Fuzzy-Zufallsvariable Zufallsvariable Grenzwertsatz |
| url | http://www.gbv.de/dms/goettingen/352958766.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016389354&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000021513 |
| work_keys_str_mv | AT lishoumei limittheoremsandapplicationsofsetvaluedandfuzzysetvaluedrandomvariables AT ogurayukio limittheoremsandapplicationsofsetvaluedandfuzzysetvaluedrandomvariables AT krejnovicvladikja limittheoremsandapplicationsofsetvaluedandfuzzysetvaluedrandomvariables |
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