Mathematics of uncertainty: ideas, methods, application problems
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Series: | Studies in fuzziness and soft computing
189 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 177 - 182 |
| Physical Description: | XIV, 190 S. Ill., graph. Darst. |
| ISBN: | 9783540284574 3540284575 |
Staff View
MARC
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| 100 | 1 | |a Bandemer, Hans |d 1932-2009 |e Verfasser |0 (DE-588)115503897 |4 aut | |
| 240 | 1 | 0 | |a Ratschläge zum mathematischen Umgang mit Ungewißheit |
| 245 | 1 | 0 | |a Mathematics of uncertainty |b ideas, methods, application problems |c Hans-Walter Bandemer |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XIV, 190 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Studies in fuzziness and soft computing |v 189 | |
| 500 | |a Literaturverz. S. 177 - 182 | ||
| 650 | 4 | |a Incertitude (Théorie de l'information) | |
| 650 | 4 | |a Mathématiques floues | |
| 650 | 4 | |a Fuzzy mathematics | |
| 650 | 4 | |a Uncertainty (Information theory) | |
| 650 | 0 | 7 | |a Datenanalyse |0 (DE-588)4123037-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Ungewissheit |0 (DE-588)4137198-7 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804135112095301632 |
|---|---|
| adam_text | CONTENTS
1
INTRODUCTION
...............................................
1
1.1
APPLICATION
OF
MATHEMATICS...............................
1
1.2
THE
QUALITY
OF
THE
MATHEMATICAL
TREATMENT
................
2
1.2.1
THE
QUALITY
OF
THE
MODEL
..........................
2
1.2.2
THE
QUALITY
OF
THE
SOLVING
PROCEDURE
................
5
1.2.3
THE
QUALITY
OF
THE
DATA............................
6
1.3
ONMODELHARMONY
.....................................
7
1.4
ON
INFORMATION
BALANCE..................................
8
2
MATHEMATICAL
REPRESENTATION
OF
SIMPLE
DATA
AND
CONNECTIONS
...........................
11
2.1
SOME
ELEMENTARY
PROCEDURES
OF
DATA
ANALYSIS
..............
11
2.1.1
DATA
AND
THEIR
REPRESENTATION
.....................
11
2.1.2
SIMPLE
PROCEDURES
OF
DATA
ANALYSIS
.................
13
2.2
REPRESENTATION
OF
FUNCTIONAL
RELATIONSHIPS
BASING
ON
DATA
...
17
2.2.1
RELATIONSHIPS
AND
DATA
............................
18
2.2.2
INTERPOLATION......................................
19
2.3
LOCAL
APPROXIMATION
....................................
21
2.3.1
APPROXIMATION....................................
22
2.3.2
OTHER
APPROXIMATION
PRINCIPLES.....................
28
2.3.3
EMPIRICAL
SMOOTHING
..............................
30
2.4
GLOBAL
APPROXIMATION
...................................
31
2.4.1
APPROXIMATING
FUNCTIONAL
RELATIONSHIPS
.............
32
2.4.2
APPROXIMATION
WITH
LOCALLY
VARIABLE
SETUPS..........
33
2.4.3
APPROXIMATION
IN
DIFFERENTIAL
AND
INTEGRAL
EQUATIONS
..
37
2.5
APPROXIMATE
OPTIMIZATION
OF
EMPIRICAL
FUNCTIONS...........
37
3
SPECIFICATION
AND
USE
OF
OBSERVATION
FUZZINESS
............
39
3.1
SPECIFICATION
BY
INTERVALS
.................................
39
3.1.1
SIMPLE
ERROR
PROPAGATION
..........................
39
3.1.2
BASIC
IDEAS
OF
INTERVAL
MATHEMATICS
.................
40
VIII
CONTENTS
3.2
SPECIFICATION
BY
FUZZY
SETS
...............................
43
3.2.1
THE
IDEA
OF
A
FUZZY
SET
............................
44
3.2.2
SPECIFICATION
OF
FUZZY
SETS..........................
46
3.2.3
OPERATIONS
WITH
FUZZY
SETS.........................
52
3.2.4
CONNECTIONS
VIA
FUNCTIONS
AND
OF
FUZZY
NUMBERS
.....
54
3.2.5
FUZZY
RELATIONS
...................................
59
4
SPECIFICATION
AND
USE
OF
UNCERTAIN
VARIABILITY
.............
63
4.1
CHANCE
AND
PROBABILITY
..................................
63
4.1.1
MODEL
IDEAS
FOR
THE
NOTION
CHANCE
..................
63
4.1.2
PROBABILITY
.......................................
65
4.1.3
RANDOM
VARIABLES
AND
THEIR
DISTRIBUTIONS
...........
69
4.1.4
ASYMPTOTIC
STATEMENTS
............................
76
4.2
PROBABILISTIC
INFERENCE
...................................
77
4.2.1
SAMPLES..........................................
78
4.2.2
PARAMETER
ESTIMATION..............................
81
4.2.3
TESTING
OF
HYPOTHESES..............................
85
4.2.4
PROBLEMS
WITH
IMPRECISE
DATA
......................
87
4.3
BAYESIANTHEORY
........................................
89
4.3.1
BAYESIAN
INFERENCE.................................
90
4.3.2
HIERARCHICAL
INFERENCE
AND
ROBUSTNESS
...............
92
4.3.3
NUMERICAL
PROBLEMS
...............................
94
5
SPECIFICATION
OF
VAGUENESS
OF
STATEMENTS
ON
SETS
......................................
97
5.1
FUZZY
MEASURES
.........................................
97
5.1.1
THE
IDEA
OF
A
FUZZY
MEASURE
.......................
97
5.1.2
FORMS
AND
PROPERTIES
OF
FUZZY
MEASURES
.............
99
5.2
SIMPLE
INFERENCE
WITH
FUZZY
MEASURES
.....................101
5.2.1
SPECIFICATION
OF
PARTIAL
IGNORANCE....................102
5.2.2
POSSIBILISTIC
INFERENCE
..............................105
5.3
PROBABILITY
AND
FUZZINESS
................................107
5.3.1
PROBABILITY
OF
FUZZY
EVENTS.........................108
5.3.2
RANDOM
FUZZY
SETS................................111
6
METHODS
FROM
QUALITATIVE
DATA
ANALYSIS
..................113
6.1
CRISP
CLASSIFICATION
OF
CRISP
DATA
.........................113
6.1.1
THE
PROBLEM
OF
CLUSTER
ANALYSIS
....................113
6.1.2
MATHEMATICAL
FORMULATION
OF
THE
PROBLEM............115
6.1.3
SOME
PROCEDURES
FOR
CRISP
CLUSTER
PARTITION
..........117
6.1.4
CLUSTERING
WITH
A
MATHEMATICAL-STATISTICAL
BACKGROUND
120
6.1.5
BASIC
IDEAS
OF
NEURAL
NETWORKS
.....................122
6.2
FUZZY
CLASSIFICATION
OF
CRISP
DATA
.........................129
6.2.1
FUZZY
CLUSTER.....................................129
6.2.2
PROCEDURES
OF
FUZZY
CLUSTER
ANALYSIS
................130
CONTENTS
IX
6.3
FUZZY
CLASSIFICATION
OF
FUZZY
DATA.........................133
6.3.1
FUZZY
SIMILARITY
OF
FUZZY
DATA......................133
6.3.2
THE
USE
OF
THE
CONCEPT
FOR
CLASSIFICATION
............137
7
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
.....................141
7.1
STATISTICAL
REGRESSION
ANALYSIS
............................142
7.1.1
MODEL
ASSUMPTIONS
WITH
RANDOM
DEPENDENT
VARIABLES
142
7.1.2
THE
PROBLEM
OF
ESTIMATION
.........................144
7.1.3
DISCUSSION
OF
THE
MODEL
ASSUMPTIONS
................147
7.1.4
FURTHER
A-PRIORI
KNOWLEDGE
AND
ASSUMPTIONS.........151
7.1.5
RANDOM
INFLUENCE
IN
ALL
VARIABLES...................153
7.1.6
LOCAL
REGRESSION
IN
A
RANDOM
FIELD
.................154
7.2
FUZZY
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
...............159
7.2.1
CRISP
DATA
ANALYSIS
AS
A
STARTING
POINT
..............160
7.2.2
EXPLORATIVE
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
....162
7.2.3
EVALUATION
WITH
ADDITIONAL
ASSUMPTIONS
.............169
7.2.4
INFERENCE
WITH
FUZZY
PARAMETER
VALUES...............170
8
OUTLOOK
AND
CONCLUSIONS
..................................173
REFERENCES
.....................................................177
INDEX
..........................................................183
|
| adam_txt |
CONTENTS
1
INTRODUCTION
.
1
1.1
APPLICATION
OF
MATHEMATICS.
1
1.2
THE
QUALITY
OF
THE
MATHEMATICAL
TREATMENT
.
2
1.2.1
THE
QUALITY
OF
THE
MODEL
.
2
1.2.2
THE
QUALITY
OF
THE
SOLVING
PROCEDURE
.
5
1.2.3
THE
QUALITY
OF
THE
DATA.
6
1.3
ONMODELHARMONY
.
7
1.4
ON
INFORMATION
BALANCE.
8
2
MATHEMATICAL
REPRESENTATION
OF
SIMPLE
DATA
AND
CONNECTIONS
.
11
2.1
SOME
ELEMENTARY
PROCEDURES
OF
DATA
ANALYSIS
.
11
2.1.1
DATA
AND
THEIR
REPRESENTATION
.
11
2.1.2
SIMPLE
PROCEDURES
OF
DATA
ANALYSIS
.
13
2.2
REPRESENTATION
OF
FUNCTIONAL
RELATIONSHIPS
BASING
ON
DATA
.
17
2.2.1
RELATIONSHIPS
AND
DATA
.
18
2.2.2
INTERPOLATION.
19
2.3
LOCAL
APPROXIMATION
.
21
2.3.1
APPROXIMATION.
22
2.3.2
OTHER
APPROXIMATION
PRINCIPLES.
28
2.3.3
EMPIRICAL
SMOOTHING
.
30
2.4
GLOBAL
APPROXIMATION
.
31
2.4.1
APPROXIMATING
FUNCTIONAL
RELATIONSHIPS
.
32
2.4.2
APPROXIMATION
WITH
LOCALLY
VARIABLE
SETUPS.
33
2.4.3
APPROXIMATION
IN
DIFFERENTIAL
AND
INTEGRAL
EQUATIONS
.
37
2.5
APPROXIMATE
OPTIMIZATION
OF
EMPIRICAL
FUNCTIONS.
37
3
SPECIFICATION
AND
USE
OF
OBSERVATION
FUZZINESS
.
39
3.1
SPECIFICATION
BY
INTERVALS
.
39
3.1.1
SIMPLE
ERROR
PROPAGATION
.
39
3.1.2
BASIC
IDEAS
OF
INTERVAL
MATHEMATICS
.
40
VIII
CONTENTS
3.2
SPECIFICATION
BY
FUZZY
SETS
.
43
3.2.1
THE
IDEA
OF
A
FUZZY
SET
.
44
3.2.2
SPECIFICATION
OF
FUZZY
SETS.
46
3.2.3
OPERATIONS
WITH
FUZZY
SETS.
52
3.2.4
CONNECTIONS
VIA
FUNCTIONS
AND
OF
FUZZY
NUMBERS
.
54
3.2.5
FUZZY
RELATIONS
.
59
4
SPECIFICATION
AND
USE
OF
UNCERTAIN
VARIABILITY
.
63
4.1
CHANCE
AND
PROBABILITY
.
63
4.1.1
MODEL
IDEAS
FOR
THE
NOTION
CHANCE
.
63
4.1.2
PROBABILITY
.
65
4.1.3
RANDOM
VARIABLES
AND
THEIR
DISTRIBUTIONS
.
69
4.1.4
ASYMPTOTIC
STATEMENTS
.
76
4.2
PROBABILISTIC
INFERENCE
.
77
4.2.1
SAMPLES.
78
4.2.2
PARAMETER
ESTIMATION.
81
4.2.3
TESTING
OF
HYPOTHESES.
85
4.2.4
PROBLEMS
WITH
IMPRECISE
DATA
.
87
4.3
BAYESIANTHEORY
.
89
4.3.1
BAYESIAN
INFERENCE.
90
4.3.2
HIERARCHICAL
INFERENCE
AND
ROBUSTNESS
.
92
4.3.3
NUMERICAL
PROBLEMS
.
94
5
SPECIFICATION
OF
VAGUENESS
OF
STATEMENTS
ON
SETS
.
97
5.1
FUZZY
MEASURES
.
97
5.1.1
THE
IDEA
OF
A
FUZZY
MEASURE
.
97
5.1.2
FORMS
AND
PROPERTIES
OF
FUZZY
MEASURES
.
99
5.2
SIMPLE
INFERENCE
WITH
FUZZY
MEASURES
.101
5.2.1
SPECIFICATION
OF
PARTIAL
IGNORANCE.102
5.2.2
POSSIBILISTIC
INFERENCE
.105
5.3
PROBABILITY
AND
FUZZINESS
.107
5.3.1
PROBABILITY
OF
FUZZY
EVENTS.108
5.3.2
RANDOM
FUZZY
SETS.111
6
METHODS
FROM
QUALITATIVE
DATA
ANALYSIS
.113
6.1
CRISP
CLASSIFICATION
OF
CRISP
DATA
.113
6.1.1
THE
PROBLEM
OF
CLUSTER
ANALYSIS
.113
6.1.2
MATHEMATICAL
FORMULATION
OF
THE
PROBLEM.115
6.1.3
SOME
PROCEDURES
FOR
CRISP
CLUSTER
PARTITION
.117
6.1.4
CLUSTERING
WITH
A
MATHEMATICAL-STATISTICAL
BACKGROUND
120
6.1.5
BASIC
IDEAS
OF
NEURAL
NETWORKS
.122
6.2
FUZZY
CLASSIFICATION
OF
CRISP
DATA
.129
6.2.1
FUZZY
CLUSTER.129
6.2.2
PROCEDURES
OF
FUZZY
CLUSTER
ANALYSIS
.130
CONTENTS
IX
6.3
FUZZY
CLASSIFICATION
OF
FUZZY
DATA.133
6.3.1
FUZZY
SIMILARITY
OF
FUZZY
DATA.133
6.3.2
THE
USE
OF
THE
CONCEPT
FOR
CLASSIFICATION
.137
7
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
.141
7.1
STATISTICAL
REGRESSION
ANALYSIS
.142
7.1.1
MODEL
ASSUMPTIONS
WITH
RANDOM
DEPENDENT
VARIABLES
142
7.1.2
THE
PROBLEM
OF
ESTIMATION
.144
7.1.3
DISCUSSION
OF
THE
MODEL
ASSUMPTIONS
.147
7.1.4
FURTHER
A-PRIORI
KNOWLEDGE
AND
ASSUMPTIONS.151
7.1.5
RANDOM
INFLUENCE
IN
ALL
VARIABLES.153
7.1.6
LOCAL
REGRESSION
IN
A
RANDOM
FIELD
.154
7.2
FUZZY
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
.159
7.2.1
CRISP
DATA
ANALYSIS
AS
A
STARTING
POINT
.160
7.2.2
EXPLORATIVE
EVALUATION
OF
FUNCTIONAL
RELATIONSHIPS
.162
7.2.3
EVALUATION
WITH
ADDITIONAL
ASSUMPTIONS
.169
7.2.4
INFERENCE
WITH
FUZZY
PARAMETER
VALUES.170
8
OUTLOOK
AND
CONCLUSIONS
.173
REFERENCES
.177
INDEX
.183 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bandemer, Hans 1932-2009 |
| author_GND | (DE-588)115503897 |
| author_facet | Bandemer, Hans 1932-2009 |
| author_role | aut |
| author_sort | Bandemer, Hans 1932-2009 |
| author_variant | h b hb |
| building | Verbundindex |
| bvnumber | BV021309294 |
| callnumber-first | Q - Science |
| callnumber-label | Q375 |
| callnumber-raw | Q375 |
| callnumber-search | Q375 |
| callnumber-sort | Q 3375 |
| callnumber-subject | Q - General Science |
| classification_rvk | SK 840 |
| ctrlnum | (OCoLC)62472441 (DE-599)BVBBV021309294 |
| dewey-full | 511.33 003/.54 |
| dewey-hundreds | 500 - Natural sciences and mathematics 000 - Computer science, information, general works |
| dewey-ones | 511 - General principles of mathematics 003 - Systems |
| dewey-raw | 511.33 003/.54 |
| dewey-search | 511.33 003/.54 |
| dewey-sort | 3511.33 |
| dewey-tens | 510 - Mathematics 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV021309294 |
| illustrated | Illustrated |
| index_date | 2024-07-02T13:55:26Z |
| indexdate | 2024-07-09T20:35:19Z |
| institution | BVB |
| isbn | 9783540284574 3540284575 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014629880 |
| oclc_num | 62472441 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-384 |
| owner_facet | DE-703 DE-824 DE-384 |
| physical | XIV, 190 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Studies in fuzziness and soft computing |
| series2 | Studies in fuzziness and soft computing |
| spelling | Bandemer, Hans 1932-2009 Verfasser (DE-588)115503897 aut Ratschläge zum mathematischen Umgang mit Ungewißheit Mathematics of uncertainty ideas, methods, application problems Hans-Walter Bandemer Berlin [u.a.] Springer 2006 XIV, 190 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Studies in fuzziness and soft computing 189 Literaturverz. S. 177 - 182 Incertitude (Théorie de l'information) Mathématiques floues Fuzzy mathematics Uncertainty (Information theory) Datenanalyse (DE-588)4123037-1 gnd rswk-swf Ungewissheit (DE-588)4137198-7 gnd rswk-swf Datenanalyse (DE-588)4123037-1 s Ungewissheit (DE-588)4137198-7 s DE-604 Studies in fuzziness and soft computing 189 (DE-604)BV021858135 189 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2665371&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014629880&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bandemer, Hans 1932-2009 Mathematics of uncertainty ideas, methods, application problems Studies in fuzziness and soft computing Incertitude (Théorie de l'information) Mathématiques floues Fuzzy mathematics Uncertainty (Information theory) Datenanalyse (DE-588)4123037-1 gnd Ungewissheit (DE-588)4137198-7 gnd |
| subject_GND | (DE-588)4123037-1 (DE-588)4137198-7 |
| title | Mathematics of uncertainty ideas, methods, application problems |
| title_alt | Ratschläge zum mathematischen Umgang mit Ungewißheit |
| title_auth | Mathematics of uncertainty ideas, methods, application problems |
| title_exact_search | Mathematics of uncertainty ideas, methods, application problems |
| title_exact_search_txtP | Mathematics of uncertainty ideas, methods, application problems |
| title_full | Mathematics of uncertainty ideas, methods, application problems Hans-Walter Bandemer |
| title_fullStr | Mathematics of uncertainty ideas, methods, application problems Hans-Walter Bandemer |
| title_full_unstemmed | Mathematics of uncertainty ideas, methods, application problems Hans-Walter Bandemer |
| title_short | Mathematics of uncertainty |
| title_sort | mathematics of uncertainty ideas methods application problems |
| title_sub | ideas, methods, application problems |
| topic | Incertitude (Théorie de l'information) Mathématiques floues Fuzzy mathematics Uncertainty (Information theory) Datenanalyse (DE-588)4123037-1 gnd Ungewissheit (DE-588)4137198-7 gnd |
| topic_facet | Incertitude (Théorie de l'information) Mathématiques floues Fuzzy mathematics Uncertainty (Information theory) Datenanalyse Ungewissheit |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2665371&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014629880&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021858135 |
| work_keys_str_mv | AT bandemerhans ratschlagezummathematischenumgangmitungewißheit AT bandemerhans mathematicsofuncertaintyideasmethodsapplicationproblems |