A practical guide to averaging functions:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cham
Springer International Publishing
[2016]
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| Schriftenreihe: | Studies in fuzziness and soft computing
volume 329 |
| Schlagworte: | |
| Online-Zugang: | BTU01 FAB01 FAW01 FHA01 FHI01 FHN01 FHR01 FKE01 FRO01 FWS01 FWS02 UBY01 Volltext Abstract Inhaltsverzeichnis |
| Beschreibung: | 1 Online Ressource (xix, 352 p. 44 illus. in color) |
| ISBN: | 9783319247533 |
| DOI: | 10.1007/978-3-319-24753-3 |
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| adam_text | A PRACTICAL GUIDE TO AVERAGING FUNCTIONS
/ BELIAKOV, GLEB
: 2016
ABSTRACT / INHALTSTEXT
THIS BOOK OFFERS AN EASY-TO-USE AND PRACTICE-ORIENTED REFERENCE GUIDE TO
MATHEMATICAL AVERAGES. IT PRESENTS DIFFERENT WAYS OF AGGREGATING INPUT
VALUES GIVEN ON A NUMERICAL SCALE, AND OF CHOOSING AND/OR CONSTRUCTING
AGGREGATING FUNCTIONS FOR SPECIFIC APPLICATIONS. BUILDING ON A PREVIOUS
MONOGRAPH BY BELIAKOV ET AL. PUBLISHED BY SPRINGER IN 2007, IT OUTLINES
NEW AGGREGATION METHODS DEVELOPED IN THE INTERIM, WITH A SPECIAL FOCUS
ON THE TOPIC OF AVERAGING AGGREGATION FUNCTIONS. IT EXAMINES RECENT
ADVANCES IN THE FIELD, SUCH AS AGGREGATION ON LATTICES, PENALTY-BASED
AGGREGATION AND WEAKLY MONOTONE AVERAGING, AND EXTENDS MANY OF THE
ALREADY EXISTING METHODS, SUCH AS: ORDERED WEIGHTED AVERAGING (OWA),
FUZZY INTEGRALS AND MIXTURE FUNCTIONS. A SUBSTANTIAL MATHEMATICAL
BACKGROUND IS NOT CALLED FOR, AS ALL THE RELEVANT MATHEMATICAL NOTIONS
ARE EXPLAINED HERE AND REPORTED ON TOGETHER WITH A WEALTH OF GRAPHICAL
ILLUSTRATIONS OF DISTINCT FAMILIES OF AGGREGATION FUNCTIONS. THE AUTHORS
MAINLY FOCUS ON PRACTICAL APPLICATIONS AND GIVE CENTRAL IMPORTANCE TO
THE CONCISENESS OF EXPOSITION, AS WELL AS THE RELEVANCE AND
APPLICABILITY OF THE REPORTED METHODS, OFFERING A VALUABLE RESOURCE FOR
COMPUTER SCIENTISTS, IT SPECIALISTS, MATHEMATICIANS, SYSTEM ARCHITECTS,
KNOWLEDGE ENGINEERS AND PROGRAMMERS, AS WELL AS FOR ANYONE FACING THE
ISSUE OF HOW TO COMBINE VARIOUS INPUTS INTO A SINGLE OUTPUT VALUE
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
Titel: A practical guide to averaging functions
Autor: Beliakov, Gleb
Jahr: 2016
Contents
1 Review of Aggregation Functions..................................................1
1.1 Aggregation Functions..........................................................1
1.2 Applications of Aggregation Functions....................................5
1.3 Classification and General Properties......................................9
1.3.1 Main Classes............................................................9
1.3.2 Main Properties........................................................10
1.3.3 Duality....................................................................18
1.3.4 Comparability..........................................................21
1.3.5 Continuity and Stability............................................21
1.4 Main Families and Prototypical Examples................................24
1.4.1 Min and Max..........................................................24
1.4.2 Means......................................................................24
1.4.3 Medians..................................................................25
1.4.4 Ordered Weighted Averaging......................................26
1.4.5 Choquet and Sugeno Integrals....................................26
1.4.6 Conjunctive and Disjunctive Functions........................27
1.4.7 Mixed Aggregation....................................................28
1.5 Composition and Transformation of Aggregation Functions.... 29
1.6 How to Choose an Aggregation Function................................32
1.7 Supplementary Material: Some Methods for Approximation
and Optimization..................................................................35
1.7.1 Univariate Approximation and Smoothing....................35
1.7.2 Approximation with Constraints..................................37
1.7.3 Multivariate Approximation........................................38
1.7.4 Convex and Non-convex Optimization........................41
1.7.5 Main Tools and Libraries..........................................47
References....................................................................................50
xi
xii Contents
2 Classical Averaging Functions........................................................55
2.1 Semantics............................................................................55
2.1.1 Measure of Orness....................................................56
2.2 Classical Means....................................................................58
2.2.1 Arithmetic Mean......................................................58
2.3 Weighted Quasi-arithmetic Means..........................................65
2.3.1 Definitions..............................................................65
2.3.2 Main Properties........................................................66
2.3.3 Examples................................................................69
2.3.4 Calculation..............................................................73
2.3.5 Weighting Triangles..................................................73
2.3.6 Weights Dispersion..................................................76
2.3.7 How to Choose Weights............................................77
2.4 Other Means........................................................................82
2.4.1 Gini Means..............................................................82
2.4.2 Bonferroni Means....................................................83
2.4.3 Heronian Mean........................................................85
2.4.4 Generalized Logarithmic Means..................................86
2.4.5 Cauchy and Lagrangean Means..................................88
2.4.6 Mean of Bajraktarevic..............................................89
2.4.7 Mixture Functions....................................................89
2.4.8 Compound Means....................................................90
2.4.9 Extending Bivariate Means to More Than
Two Arguments........................................................91
References....................................................................................97
3 Ordered Weighted Averaging........................................................101
3.1 Definitions..........................................................................101
3.2 Main Properties....................................................................102
3.2.1 Orness Measure........................................................103
3.2.2 Entropy....................................................................105
3.3 Other Types of OWA Functions..............................................106
3.3.1 Neat OWA..............................................................106
3.3.2 Generalized OWA....................................................107
3.3.3 Weighted OWA........................................................109
3.4 How to Choose Weights in OWA..........................................114
3.4.1 Methods Based on Data............................................114
3.4.2 Methods Based on a Measure of Dispersion................117
3.4.3 Methods Based on Weight Generating Functions..........120
3.4.4 Fitting Weight Generating Functions............................123
3.4.5 Choosing Parameters of Generalized OWA..................126
3.5 Induced OWA......................................................................130
3.5.1 Definition................................................................130
3.5.2 Properties................................................................130
Contents xiii
3.5.3 Induced Generalized OWA........................................133
3.5.4 Choices for the Inducing Variable..............................134
3.6 Medians and Order Statistics..................................................137
3.6.1 Median....................................................................137
3.6.2 Order Statistics........................................................141
References....................................................................................141
4 Fuzzy Integrals............................................................................145
4.1 Choquet Integral..................................................................145
4.1.1 Semantics................................................................145
4.1.2 Definidons and Properties..........................................147
4.1.3 Types of Fuzzy Measures..........................................153
4.1.4 Interaction, Importance and Other Indices....................160
4.1.5 Special Cases of the Choquet Integral..........................164
4.1.6 Fitdng Fuzzy Measures..............................................166
4.1.7 Generalized Choquet Integral......................................173
4.2 Sugeno Integral....................................................................174
4.2.1 Definition and Properties............................................174
4.2.2 Special Cases............................................................175
4.3 Induced Fuzzy Integrals........................................................177
References....................................................................................178
5 Penalty Based Averages................................................................183
5.1 Motivation and Definitions....................................................183
5.2 Types of Penalty Functions....................................................186
5.2.1 Faithful Penalty Functions..........................................186
5.2.2 Restricted Dissimilarity Functions .... !....................188
5.2.3 Minkowski Gauge Based Penalties..............................191
5.3 Examples............................................................................193
5.3.1 Quasi-arithmetic Means, OWA and Choquet
Integral....................................................................193
5.3.2 Deviation Means......................................................194
5.3.3 Entropie Means........................................................195
5.3.4 Bregman Loss Functions............................................195
5.4 New Penalty Based Aggregation Functions..............................197
5.5 Relation to the Maximum Likelihood Principle........................201
5.6 Representation of Averages....................................................203
References....................................................................................204
6 More Types of Averaging and Construction Methods....................207
6.1 Some Construction Methods..................................................207
6.1.1 Idempotization..........................................................207
6.1.2 Means Defined by Using Graduation Curves................208
6.1.3 Aggregation Functions with Flying Parameter..............210
6.1.4 Construction of Shift-Invariant Functions....................212
6.1.5 Interpolator Constructions........................................212
6.2 Other Types of Aggregation and Properties..............................216
6.2.1 Bi-capacities............................................................216
6.2.2 Linguistic Aggregation Functions................................217
6.2.3 Multistage Aggregation..............................................217
6.2.4 Migrativity..............................................................218
6.3 Overlap and Grouping Functions............................................218
6.3.1 Definition of Overlap Functions
and Basic Properties..................................................219
6.3.2 Characterization of Overlap Functions..........................220
6.3.3 Homogeneous Overlap Functions................................221
6.3.4 A-IJpschitz Overlap Functions....................................222
6.3.5 «-dimensional Overlap Functions................................224
6.3.6 Grouping Functions..................................................225
6.4 Generalized Bonferroni Mean................................................227
6.4.1 Main Definitions......................................................227
6.4.2 Properties of the Generalized Bonferroni Mean............229
6.4.3 Replacing the Outer Mean..........................................230
6.4.4 Replacing the Inner Mean..........................................231
6.4.5 Replacing the Product Operation................................232
6.4.6 Extensions to ......................................................233
6.4.7 Boundcdness of the Generalized Bonferroni Mean .... 234
6.4.8 ¿-intolerance Boundedness........................................237
6.4.9 Generated t-norm and Generated Quasi-arithmetic
Means as Components of Bm......................................238
6.5 Consistency and Stability......................................................239
6.5.1 Motivation..............................................................239
6.5.2 Strictly Stable Families..............................................240
6.5.3 R-strict Stability........................................................241
6.5.4 Learning Consistent Weights......................................245
6.5.5 Consistency and Global Monotonicity..........................246
References....................................................................................247
7 Non-monotone Averages................................................................251
7.1 Motivation..........................................................................251
7.2 Weakly Monotone Functions..................................................252
7.2.1 Basic Properties of Weakly Monotone Functions..........254
7.3 Robust Estimators of Location................................................256
7.3.1 Mode......................................................................257
7.3.2 Shorth......................................................................257
7.3.3 Least Median of Squares (LMS)................................258
7.3.4 Least Trimmed Squares (LTS)....................................258
7.3.5 Least Trimmed Absolute Deviations (LTA)..................259
7.3.6 The Least Winsorized Squares Estimator......................260
7.3.7 OWA Penalty Functions............................................260
Contents xv
7.4 Lehmer and Gini Means........................................................261
7.4.1 Lehmer Means..........................................................261
7.4.2 Gini Means..............................................................263
7.5 Mixture Functions................................................................265
7.5.1 Some Special Cases of Weighting Functions................266
7.5.2 Affine Weighting Functions........................................267
7.5.3 Linear Combinations of Weighting Functions..............269
7.5.4 The Duals of Lehmer Mean and Other Mixture
Functions................................................................270
7.6 Density Based Means and Medians........................................271
7.6.1 Density Based Means................................................271
7.6.2 Density Based Medians and ML Estimators..................272
7.6.3 Modified Weighting Functions....................................273
7.7 Mode-Like Averages............................................................275
7.8 Spatial-Tonal Filters..............................................................277
7.9 Transforms..........................................................................278
7.10 Cone Monotone Functions......................................................281
7.10.1 Formal Definitions and Properties..............................281
7.10.2 Verification of Cone Monotonicity..............................285
7.10.3 Construction of Cone Monotone Lipschitz
Functions................................................................286
7.11 Monotonicity with Respect to Coalitions..................................288
7.11.1 Simple Majority........................................................288
7.11.2 Majority and Preferential Inputs..................................290
7.11.3 Coalitions................................................................291
7.12 Directional Monotonicity........................................................293
7.12.1 Properties of r-Monotone functions............................294
7.12.2 The Set of Directions of Increasingness........................295
7.13 Pre-aggregation Functions......................................................296
7.13.1 Definitions and Properties..........................................296
7.13.2 Construction of Pre-aggregation Functions
by Composition........................................................297
7.13.3 Choquet-Like Construction Method
of Pre-aggregation Functions......................................299
7.13.4 Sugeno-Like Construction Method
of Pre-aggregation Functions......................................300
References....................................................................................302
8 Averages on Lattices....................................................................305
8.1 Aggregation of Intervals and Intuitionistic Fuzzy Values............305
8.1.1 Preliminary Definitions..............................................306
8.1.2 Aggregation on Product Lattices................................306
8.1.3 Arithmetic Means and OWA for AIFV........................309
8.1.4 Alternative Definitions of Aggregation Functions
on AIFV..................................................................313
xvi Contents
8.1.5 Consistency with Operations on Ordinary
Fuzzy Sets ..............................................................316
8.1.6 Medians for AIFV....................................................319
8.1.7 Bonferroni Means....................................................320
8.2 Medians on Lattices..............................................................327
8.2.1 Medians as Penalty Based Functions............................327
8.2.2 Median Graphs and Distributive Lattices......................328
8.2.3 Medians on Infinite Lattices and Fermât Points............330
8.2.4 Medians Based on Distances Between Intervals............331
8.2.5 Numerical Comparison..............................................333
8.3 Penalty Functions on Cartesian Products of Lattices..................336
References....................................................................................342
Index................................................................................................347
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| id | DE-604.BV043211911 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-01T12:12:10Z |
| institution | BVB |
| isbn | 9783319247533 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028635050 |
| oclc_num | 930004057 |
| open_access_boolean | |
| owner | DE-1043 DE-Aug4 DE-573 DE-859 DE-863 DE-BY-FWS DE-92 DE-862 DE-BY-FWS DE-898 DE-BY-UBR DE-634 DE-1046 DE-861 DE-706 |
| owner_facet | DE-1043 DE-Aug4 DE-573 DE-859 DE-863 DE-BY-FWS DE-92 DE-862 DE-BY-FWS DE-898 DE-BY-UBR DE-634 DE-1046 DE-861 DE-706 |
| physical | 1 Online Ressource (xix, 352 p. 44 illus. in color) |
| psigel | ZDB-2-ENG ZDB-2-ENG_2016 |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Springer International Publishing |
| record_format | marc |
| series | Studies in fuzziness and soft computing |
| series2 | Studies in fuzziness and soft computing |
| spellingShingle | Beliakov, Gleb A practical guide to averaging functions Studies in fuzziness and soft computing Engineering Artificial intelligence Operations research Management science Computational intelligence Control engineering Robotics Mechatronics Computational Intelligence Control, Robotics, Mechatronics Artificial Intelligence (incl. Robotics) Operations Research, Management Science Ingenieurwissenschaften Künstliche Intelligenz |
| title | A practical guide to averaging functions |
| title_auth | A practical guide to averaging functions |
| title_exact_search | A practical guide to averaging functions |
| title_full | A practical guide to averaging functions Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo Sánchez |
| title_fullStr | A practical guide to averaging functions Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo Sánchez |
| title_full_unstemmed | A practical guide to averaging functions Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo Sánchez |
| title_short | A practical guide to averaging functions |
| title_sort | a practical guide to averaging functions |
| topic | Engineering Artificial intelligence Operations research Management science Computational intelligence Control engineering Robotics Mechatronics Computational Intelligence Control, Robotics, Mechatronics Artificial Intelligence (incl. Robotics) Operations Research, Management Science Ingenieurwissenschaften Künstliche Intelligenz |
| topic_facet | Engineering Artificial intelligence Operations research Management science Computational intelligence Control engineering Robotics Mechatronics Computational Intelligence Control, Robotics, Mechatronics Artificial Intelligence (incl. Robotics) Operations Research, Management Science Ingenieurwissenschaften Künstliche Intelligenz |
| url | https://doi.org/10.1007/978-3-319-24753-3 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028635050&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028635050&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV020843944 |
| work_keys_str_mv | AT beliakovgleb apracticalguidetoaveragingfunctions AT bustincehumberto apracticalguidetoaveragingfunctions AT calvosancheztomasa apracticalguidetoaveragingfunctions |