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A practical guide to averaging functions

Author: Gleb Beliakov; Humberto Bustince Sola; Tomasa Calvo Sánchez
Publisher: Cham : Springer, 2016.
Series: Studies in fuzziness and soft computing, 329.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Additional Physical Format: Printed edition:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Gleb Beliakov; Humberto Bustince Sola; Tomasa Calvo Sánchez
ISBN: 9783319247533 3319247530
OCLC Number: 932169510
Description: 1 online resource (xix, 352 pages) : color illustrations.
Contents: Machine generated contents note: 1. Review of Aggregation Functions --
1.1. Aggregation Functions --
1.2. Applications of Aggregation Functions --
1.3. Classification and General Properties --
1.3.1. Main Classes --
1.3.2. Main Properties --
1.3.3. Duality --
1.3.4.Comparability --
1.3.5. Continuity and Stability --
1.4. Main Families and Prototypical Examples --
1.4.1. Min and Max --
1.4.2. Means --
1.4.3. Medians --
1.4.4. Ordered Weighted Averaging --
1.4.5. Choquet and Sugeno Integrals --
1.4.6. Conjunctive and Disjunctive Functions --
1.4.7. Mixed Aggregation --
1.5.Composition and Transformation of Aggregation Functions --
1.6. How to Choose an Aggregation Function --
1.7. Supplementary Material: Some Methods for Approximation and Optimization --
1.7.1. Univariate Approximation and Smoothing --
1.7.2. Approximation with Constraints --
1.7.3. Multivariate Approximation --
1.7.4. Convex and Non-convex Optimization --
1.7.5. Main Tools and Libraries --
References. Note continued: 2. Classical Averaging Functions --
2.1. Semantics --
2.1.1. Measure of Orness --
2.2. Classical Means --
2.2.1. Arithmetic Mean --
2.3. Weighted Quasi-arithmetic Means --
2.3.1. Definitions --
2.3.2. Main Properties --
2.3.3. Examples --
2.3.4. Calculation --
2.3.5. Weighting Triangles --
2.3.6. Weights Dispersion --
2.3.7. How to Choose Weights --
2.4. Other Means --
2.4.1. Gini Means --
2.4.2. Bonferroni Means --
2.4.3. Heronian Mean --
2.4.4. Generalized Logarithmic Means --
2.4.5. Cauchy and Lagrangean Means --
2.4.6. Mean of Bajraktarevic --
2.4.7. Mixture Functions --
2.4.8.Compound Means --
2.4.9. Extending Bivariate Means to More Than Two Arguments --
References --
3. Ordered Weighted Averaging --
3.1. Definitions --
3.2. Main Properties --
3.2.1. Orness Measure --
3.2.2. Entropy --
3.3. Other Types of OWA Functions --
3.3.1. Neat OWA --
3.3.2. Generalized OWA --
3.3.3. Weighted OWA --
3.4. How to Choose Weights in OWA --
3.4.1. Methods Based on Data. Note continued: 3.4.2. Methods Based on a Measure of Dispersion --
3.4.3. Methods Based on Weight Generating Functions --
3.4.4. Fitting Weight Generating Functions --
3.4.5. Choosing Parameters of Generalized OWA --
3.5. Induced OWA --
3.5.1. Definition --
3.5.2. Properties --
3.5.3. Induced Generalized OWA --
3.5.4. Choices for the Inducing Variable --
3.6. Medians and Order Statistics --
3.6.1. Median --
3.6.2. Order Statistics --
References --
4. Fuzzy Integrals --
4.1. Choquet Integral --
4.1.1. Semantics --
4.1.2. Definitions and Properties --
4.1.3. Types of Fuzzy Measures --
4.1.4. Interaction, Importance and Other Indices --
4.1.5. Special Cases of the Choquet Integral --
4.1.6. Fitting Fuzzy Measures --
4.1.7. Generalized Choquet Integral --
4.2. Sugeno Integral --
4.2.1. Definition and Properties --
4.2.2. Special Cases --
4.3. Induced Fuzzy Integrals --
References --
5. Penalty Based Averages --
5.1. Motivation and Definitions --
5.2. Types of Penalty Functions. Note continued: 5.2.1. Faithful Penalty Functions --
5.2.2. Restricted Dissimilarity Functions --
5.2.3. Minkowski Gauge Based Penalties --
5.3. Examples --
5.3.1. Quasi-arithmetic Means, OWA and Choquet Integral --
5.3.2. Deviation Means --
5.3.3. Entropic Means --
5.3.4. Bregman Loss Functions --
5.4. New Penalty Based Aggregation Functions --
5.5. Relation to the Maximum Likelihood Principle --
5.6. Representation of Averages --
References --
6. More Types of Averaging and Construction Methods --
6.1. Some Construction Methods --
6.1.1. Idempotization --
6.1.2. Means Defined by Using Graduation Curves --
6.1.3. Aggregation Functions with Flying Parameter --
6.1.4. Construction of Shift-Invariant Functions --
6.1.5. Interpolatory Constructions --
6.2. Other Types of Aggregation and Properties --
6.2.1. Bi-capacities --
6.2.2. Linguistic Aggregation Functions --
6.2.3. Multistage Aggregation --
6.2.4. Migrativity --
6.3. Overlap and Grouping Functions. Note continued: 6.3.1. Definition of Overlap Functions and Basic Properties --
6.3.2. Characterization of Overlap Functions --
6.3.3. Homogeneous Overlap Functions --
6.3.4.k-Lipschitz Overlap Functions --
6.3.5.n-dimensional Overlap Functions --
6.3.6. Grouping Functions --
6.4. Generalized Bonferroni Mean --
6.4.1. Main Definitions --
6.4.2. Properties of the Generalized Bonferroni Mean --
6.4.3. Replacing the Outer Mean --
6.4.4. Replacing the Inner Mean --
6.4.5. Replacing the Product Operation --
6.4.6. Extensions to BKM --
6.4.7. Boundedness of the Generalized Bonferroni Mean --
6.4.8.k-intolerance Boundedness --
6.4.9. Generated t-norm and Generated Quasi-arithmetic Means as Components of BM --
6.5. Consistency and Stability --
6.5.1. Motivation --
6.5.2. Strictly Stable Families --
6.5.3.R-strict Stability --
6.5.4. Learning Consistent Weights --
6.5.5. Consistency and Global Monotonicity --
References --
7. Non-monotone Averages --
7.1. Motivation. Note continued: 7.2. Weakly Monotone Functions --
7.2.1. Basic Properties of Weakly Monotone Functions --
7.3. Robust Estimators of Location --
7.3.1. Mode --
7.3.2. Shorth --
7.3.3. Least Median of Squares (LMS) --
7.3.4. Least Trimmed Squares (LTS) --
7.3.5. Least Trimmed Absolute Deviations (LTA) --
7.3.6. The Least Winsorized Squares Estimator --
7.3.7. OWA Penalty Functions --
7.4. Lehmer and Gini Means --
7.4.1. Lehmer Means --
7.4.2. Gini Means --
7.5. Mixture Functions --
7.5.1. Some Special Cases of Weighting Functions --
7.5.2. Affine Weighting Functions --
7.5.3. Linear Combinations of Weighting Functions --
7.5.4. The Duals of Lehmer Mean and Other Mixture Functions --
7.6. Density Based Means and Medians --
7.6.1. Density Based Means --
7.6.2. Density Based Medians and ML Estimators --
7.6.3. Modified Weighting Functions --
7.7. Mode-Like Averages --
7.8. Spatial-Tonal Filters --
7.9. Transforms --
7.10. Cone Monotone Functions --
7.10.1. Formal Definitions and Properties. Note continued: 7.10.2. Verification of Cone Monotonicity --
7.10.3. Construction of Cone Monotone Lipschitz Functions --
7.11. Monotonicity with Respect to Coalitions --
7.11.1. Simple Majority --
7.11.2. Majority and Preferential Inputs --
7.11.3. Coalitions --
7.12. Directional Monotonicity --
7.12.1. Properties of r-Monotone functions --
7.12.2. The Set of Directions of Increasingness --
7.13. Pre-aggregation Functions --
7.13.1. Definitions and Properties --
7.13.2. Construction of Pre-aggregation Functions by Composition --
7.13.3. Choquet-Like Construction Method of Pre-aggregation Functions --
7.13.4. Sugeno-Like Construction Method of Pre-aggregation Functions --
References --
8. Averages on Lattices --
8.1. Aggregation of Intervals and Intuitionistic Fuzzy Values --
8.1.1. Preliminary Definitions --
8.1.2. Aggregation on Product Lattices --
8.1.3. Arithmetic Means and OWA for AIFV --
8.1.4. Alternative Definitions of Aggregation Functions on AIFV. Note continued: 8.1.5. Consistency with Operations on Ordinary Fuzzy Sets --
8.1.6. Medians for AIFV --
8.1.7. Bonferroni Means --
8.2. Medians on Lattices --
8.2.1. Medians as Penalty Based Functions --
8.2.2. Median Graphs and Distributive Lattices --
8.2.3. Medians on Infinite Lattices and Fermat Points --
8.2.4. Medians Based on Distances Between Intervals --
8.2.5. Numerical Comparison --
8.3. Penalty Functions on Cartesian Products of Lattices --
References.
Series Title: Studies in fuzziness and soft computing, 329.
Responsibility: by Gleb Beliakov, Humberto Bustince Sola, Tomasa Calvo Sánchez.

Abstract:

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  Read more...

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    schema:description "Note continued: 8.1.5. Consistency with Operations on Ordinary Fuzzy Sets -- 8.1.6. Medians for AIFV -- 8.1.7. Bonferroni Means -- 8.2. Medians on Lattices -- 8.2.1. Medians as Penalty Based Functions -- 8.2.2. Median Graphs and Distributive Lattices -- 8.2.3. Medians on Infinite Lattices and Fermat Points -- 8.2.4. Medians Based on Distances Between Intervals -- 8.2.5. Numerical Comparison -- 8.3. Penalty Functions on Cartesian Products of Lattices -- References."@en ;
    schema:description "Note continued: 7.2. Weakly Monotone Functions -- 7.2.1. Basic Properties of Weakly Monotone Functions -- 7.3. Robust Estimators of Location -- 7.3.1. Mode -- 7.3.2. Shorth -- 7.3.3. Least Median of Squares (LMS) -- 7.3.4. Least Trimmed Squares (LTS) -- 7.3.5. Least Trimmed Absolute Deviations (LTA) -- 7.3.6. The Least Winsorized Squares Estimator -- 7.3.7. OWA Penalty Functions -- 7.4. Lehmer and Gini Means -- 7.4.1. Lehmer Means -- 7.4.2. Gini Means -- 7.5. Mixture Functions -- 7.5.1. Some Special Cases of Weighting Functions -- 7.5.2. Affine Weighting Functions -- 7.5.3. Linear Combinations of Weighting Functions -- 7.5.4. The Duals of Lehmer Mean and Other Mixture Functions -- 7.6. Density Based Means and Medians -- 7.6.1. Density Based Means -- 7.6.2. Density Based Medians and ML Estimators -- 7.6.3. Modified Weighting Functions -- 7.7. Mode-Like Averages -- 7.8. Spatial-Tonal Filters -- 7.9. Transforms -- 7.10. Cone Monotone Functions -- 7.10.1. Formal Definitions and Properties."@en ;
    schema:description "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."@en ;
    schema:description "Note continued: 7.10.2. Verification of Cone Monotonicity -- 7.10.3. Construction of Cone Monotone Lipschitz Functions -- 7.11. Monotonicity with Respect to Coalitions -- 7.11.1. Simple Majority -- 7.11.2. Majority and Preferential Inputs -- 7.11.3. Coalitions -- 7.12. Directional Monotonicity -- 7.12.1. Properties of r-Monotone functions -- 7.12.2. The Set of Directions of Increasingness -- 7.13. Pre-aggregation Functions -- 7.13.1. Definitions and Properties -- 7.13.2. Construction of Pre-aggregation Functions by Composition -- 7.13.3. Choquet-Like Construction Method of Pre-aggregation Functions -- 7.13.4. Sugeno-Like Construction Method of Pre-aggregation Functions -- References -- 8. Averages on Lattices -- 8.1. Aggregation of Intervals and Intuitionistic Fuzzy Values -- 8.1.1. Preliminary Definitions -- 8.1.2. Aggregation on Product Lattices -- 8.1.3. Arithmetic Means and OWA for AIFV -- 8.1.4. Alternative Definitions of Aggregation Functions on AIFV."@en ;
    schema:description "Note continued: 5.2.1. Faithful Penalty Functions -- 5.2.2. Restricted Dissimilarity Functions -- 5.2.3. Minkowski Gauge Based Penalties -- 5.3. Examples -- 5.3.1. Quasi-arithmetic Means, OWA and Choquet Integral -- 5.3.2. Deviation Means -- 5.3.3. Entropic Means -- 5.3.4. Bregman Loss Functions -- 5.4. New Penalty Based Aggregation Functions -- 5.5. Relation to the Maximum Likelihood Principle -- 5.6. Representation of Averages -- References -- 6. More Types of Averaging and Construction Methods -- 6.1. Some Construction Methods -- 6.1.1. Idempotization -- 6.1.2. Means Defined by Using Graduation Curves -- 6.1.3. Aggregation Functions with Flying Parameter -- 6.1.4. Construction of Shift-Invariant Functions -- 6.1.5. Interpolatory Constructions -- 6.2. Other Types of Aggregation and Properties -- 6.2.1. Bi-capacities -- 6.2.2. Linguistic Aggregation Functions -- 6.2.3. Multistage Aggregation -- 6.2.4. Migrativity -- 6.3. Overlap and Grouping Functions."@en ;
    schema:description "Note continued: 2. Classical Averaging Functions -- 2.1. Semantics -- 2.1.1. Measure of Orness -- 2.2. Classical Means -- 2.2.1. Arithmetic Mean -- 2.3. Weighted Quasi-arithmetic Means -- 2.3.1. Definitions -- 2.3.2. Main Properties -- 2.3.3. Examples -- 2.3.4. Calculation -- 2.3.5. Weighting Triangles -- 2.3.6. Weights Dispersion -- 2.3.7. How to Choose Weights -- 2.4. Other Means -- 2.4.1. Gini Means -- 2.4.2. Bonferroni Means -- 2.4.3. Heronian Mean -- 2.4.4. Generalized Logarithmic Means -- 2.4.5. Cauchy and Lagrangean Means -- 2.4.6. Mean of Bajraktarevic -- 2.4.7. Mixture Functions -- 2.4.8.Compound Means -- 2.4.9. Extending Bivariate Means to More Than Two Arguments -- References -- 3. Ordered Weighted Averaging -- 3.1. Definitions -- 3.2. Main Properties -- 3.2.1. Orness Measure -- 3.2.2. Entropy -- 3.3. Other Types of OWA Functions -- 3.3.1. Neat OWA -- 3.3.2. Generalized OWA -- 3.3.3. Weighted OWA -- 3.4. How to Choose Weights in OWA -- 3.4.1. Methods Based on Data."@en ;
    schema:description "Note continued: 3.4.2. Methods Based on a Measure of Dispersion -- 3.4.3. Methods Based on Weight Generating Functions -- 3.4.4. Fitting Weight Generating Functions -- 3.4.5. Choosing Parameters of Generalized OWA -- 3.5. Induced OWA -- 3.5.1. Definition -- 3.5.2. Properties -- 3.5.3. Induced Generalized OWA -- 3.5.4. Choices for the Inducing Variable -- 3.6. Medians and Order Statistics -- 3.6.1. Median -- 3.6.2. Order Statistics -- References -- 4. Fuzzy Integrals -- 4.1. Choquet Integral -- 4.1.1. Semantics -- 4.1.2. Definitions and Properties -- 4.1.3. Types of Fuzzy Measures -- 4.1.4. Interaction, Importance and Other Indices -- 4.1.5. Special Cases of the Choquet Integral -- 4.1.6. Fitting Fuzzy Measures -- 4.1.7. Generalized Choquet Integral -- 4.2. Sugeno Integral -- 4.2.1. Definition and Properties -- 4.2.2. Special Cases -- 4.3. Induced Fuzzy Integrals -- References -- 5. Penalty Based Averages -- 5.1. Motivation and Definitions -- 5.2. Types of Penalty Functions."@en ;
    schema:description "Note continued: 6.3.1. Definition of Overlap Functions and Basic Properties -- 6.3.2. Characterization of Overlap Functions -- 6.3.3. Homogeneous Overlap Functions -- 6.3.4.k-Lipschitz Overlap Functions -- 6.3.5.n-dimensional Overlap Functions -- 6.3.6. Grouping Functions -- 6.4. Generalized Bonferroni Mean -- 6.4.1. Main Definitions -- 6.4.2. Properties of the Generalized Bonferroni Mean -- 6.4.3. Replacing the Outer Mean -- 6.4.4. Replacing the Inner Mean -- 6.4.5. Replacing the Product Operation -- 6.4.6. Extensions to BKM -- 6.4.7. Boundedness of the Generalized Bonferroni Mean -- 6.4.8.k-intolerance Boundedness -- 6.4.9. Generated t-norm and Generated Quasi-arithmetic Means as Components of BM -- 6.5. Consistency and Stability -- 6.5.1. Motivation -- 6.5.2. Strictly Stable Families -- 6.5.3.R-strict Stability -- 6.5.4. Learning Consistent Weights -- 6.5.5. Consistency and Global Monotonicity -- References -- 7. Non-monotone Averages -- 7.1. Motivation."@en ;
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<http://worldcat.org/entity/work/data/2817329063#CreativeWork/>
    a schema:CreativeWork ;
    schema:description "Printed edition:" ;
    schema:isSimilarTo <http://www.worldcat.org/oclc/932169510> ; # A practical guide to averaging functions
    .

<http://worldcat.org/isbn/9783319247533>
    a schema:ProductModel ;
    schema:isbn "3319247530" ;
    schema:isbn "9783319247533" ;
    .

<http://worldcat.org/issn/1434-9922> # Studies in fuzziness and soft computing ;
    a bgn:PublicationSeries ;
    schema:hasPart <http://www.worldcat.org/oclc/932169510> ; # A practical guide to averaging functions
    schema:issn "1434-9922" ;
    schema:name "Studies in fuzziness and soft computing ;" ;
    schema:name "Studies in Fuzziness and Soft Computing," ;
    .


Content-negotiable representations

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