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Recentes Avanços em Sistemas Fuzzy
Editores:
Marcos Eduardo Ribeiro do Valle Mesquita
Graçaliz Pereira Dimuro
Regivan Hugo Nunes Santiago
Estevão Esmi Laureano
ISBN: 978-85-8215-079-5
Campinas - SP, Novembro de 2016.
Sociedade Brasileira de Matemática Aplicada e Computacional
Recent Trends on Fuzzy Systems
Editors:
Marcos Eduardo Ribeiro do Valle Mesquita
Graçaliz Pereira Dimuro
Regivan Hugo Nunes Santiago
Estevão Esmi Laureano
ISBN: 978-85-8215-079-5
Campinas - Brazil, November 2016.
Sociedade Brasileira de Matemática Aplicada e Computacional
Prefácio
O termo sistema fuzzy usado nessa obra compreende sistemas computacionais ou sistemas teóricos basea-
dos na lógica fuzzy ou conjuntos fuzzy. Tal como a lógica fuzzy, conjuntos fuzzy são usados para descrever
conceitos vagos ou incertos comuns na linguagem natural.
Este livro biĺıngue contém alguns dos recentes avanços na área de sistemas fuzzy, uma área de pesquisa
ativa e crescente no cenário mundial. Especificamente, essa obra contém 45 contribuições escritas ou em
português ou em ĺıngua inglesa. Todas as contribuições foram avaliadas por um comitê cient́ıfico que
atestou a relevância das mesmas. Além disso, elas foram selecioadas para serem apresentadas na forma
oral durante o Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF), realizado em Novembro de
2016, na cidade de Campinas – São Paulo, Brasil.
O IV CBSF reuniu 121 participantes, incluindo palestrantes e membros da comissão organizadora,
divididos entre professores, estudantes e profissionais interessados em sistemas fuzzy. O congresso contou
com 6 palestras, 4 mini-cursos e diversas sessões técnicas nos quais os trabalhos aprovados pelo comitê
cient́ıfico foram apresentados na forma oral ou pôster. Durante a chamada de trabalhos do IV CBSF, foram
submetidas 102 contribuições distribúıdas entre 52 trabalhos completos e 50 resumos estendidos. Cada
trabalho completo foi avaliado por pelo menos dois revisores anônimos que, de um modo geral, atribúıram
uma nota no conjunto {−3,−2,−1, 0,+1,+2,+3}, no qual -3 representa “rejeita enfaticamente” e +3
corresponde à “aceita enfaticamente”. Foram aceitos somente os trabalhos completos que receberam notas
positivas de todos os revisores. Com isso, foram selecionados 45 contribuições, que corresponde à uma
taxa de rejeição de 13%. Os 45 trabalhos completos foram organizados em 12 temas tal como esse livro.
Aproveitamos a oportunidade para agradecer a todos envolvidos no IV CBSF: os palestrantes que
dedicaram seu tempo compartilhando seus conhecimentos conosco, os responsáveis pelos mini-cursos que
divulgaram de forma brilhante temas relacionados a sistemas fuzzy, os autores que submeteram suas con-
tribuições, os membros do comitê cient́ıfico que avaliaram os trabalhos e a todos os membros da comissão
organizadora. Agradecemos também à Universidade Estadual de Campinas (Unicamp) e ao Instituto de
Matemática, Estat́ıstica e Computação Cient́ıfica (IMECC) pela realização do IV CBSF, à Coordenação
de Aperfeiçoamento de Pessoal de Nı́vel Superior (CAPES) e à Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP) pelo suporte financeiro e ao apoio das sociedades SBMAC, SBIC, SBA,
NAFIPS, IFSA e EUSFLAT.
Campinas, Novembro de 2016.
Marcos Eduardo Ribeiro do Valle Mesquita
Coordenador Geral do IV CBSF
Foreword
The term fuzzy system used in this book comprises computational systems or theoretical systems based
on fuzzy logic or fuzzy sets. Like fuzzy logic, fuzzy sets are used to describe vague or uncertain concepts
common in natural language.
This bilingual book contains some of the recent trends in the area of fuzzy systems, an active and
growing research area. Specifically, this book contains 45 contributions written in either Portuguese or
English. All the contributions were evaluated by a scientific committee that proved their relevance. In
addition, they were selected to be presented orally during the Fourth Brazilian Congress of Fuzzy Systems
(IV CBSF), held in November 2016, in the city of Campinas – São Paulo, Brazil.
The IV CBSF brought together 121 participants, including speakers and members of the organizing
committee, divided among teachers, students, and professionals interested in fuzzy systems. The congress
counted on 6 lectures, 4 tutorials, and several technical sessions in which the works approved by the
scientific committee were presented either orally or by poster. For the call for papers of the IV CBSF,
102 contributions distributed as 52 full papers and 50 extended abstracts were submitted. Each full
paper was evaluated by at least two anonymous referees who generally assigned a grade in the set
{−3,−2,−1, 0,+1,+2,+3}, in which -3 represents “strong reject” and +3 corresponds to “strong accept”.
Only the full papers that received positive grade from all the reviewers were accepted. As a consequence,
45 contributions were selected, corresponding to a rate of rejection of 13%. The 45 selected full papers
were organized in 12 subjects such as this book.
We took the opportunity to thank everyone involved in the IV CBSF: the speakers who dedicated
their time sharing their knowledge with us, those responsible for the tutorials that brilliantly spread
topics related to fuzzy systems, the authors who submitted their contributions, the scientific committee
who evaluated the work and all members of the organizing committee. We also thank the University of
Campinas (Unicamp) and the Institute of Mathematics, Statistics and Scientific Computation (IMECC)
for the accomplishment of the IV CBSF, CAPES and FAPESP for the sponsorship, and the support of
the societies SBMAC, SBIC, SBA, NAFIPS, IFSA and EUSFLAT.
Campinas, November 2016.
Marcos Eduardo Ribeiro do Valle Mesquita
General Chair of IV CBSF
Organização / Organization
Comissão Organizadora / Steering Committee
Marcos Eduardo Valle (Unicamp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coordenador Geral / General Chair.
Estevão Esmi (Unicamp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Vice-coordenador Geral / General Co-chair.
Graçaliz Dimuro (UFRG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coordenadora Cient́ıfica / Program Chair.
Regivan Santiago (UFRN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vice-coordenador Cient́ıfico / Program co-chair.
Organização Geral e Colaboradores / General Coordination and
Collaborators
Adrião Doria Neto, Universidade Federal do Rio Grande do Norte.
Benjamin Bedregal, Universidade Federal do Rio Grande do Norte.
Eduardo Palmeira, Universidade Estadual de Santa Cruz.
Estevão Esmi, Universidade Estadual de Campinas.
Fernando Gomide, Universidade Estadual de Campinas.
Graçaliz Dimuro, Universidade Federal do Rio Grande.
José Arnaldo Roveda, Universidade Estadual Paulista.
Laecio Barros, Universidade Estadual de Campinas.
Luciana Gomes, Universidade Federal de São Carlos.
Magda Peixoto, Universidade Federal de São Carlos.
Marcos Eduardo Valle, Universidade Estadual de Campinas.
Regivan Santiago, Universidade Federal do Rio Grande do Norte.
Ricardo Coelho, Universidade Federal do Ceará.
Ricardo Tanscheit, Pontif́ıcia Universidade Católica do Rio de Janeiro.
Ronei Moraes, Universidade Federal da Paráıba.
Rosana Jafelice, Universidade Federal de Uberlândia.
Sandra Sandri, Instituto Nacional de Pesquisas Espaciais.
Sérgio Meriga, Pontificia Universidad Católica de Chile. Santiago - Chile.
Weldon Lodwick, University of Colorado, Denver - Estados Unidos da América.
Avaliadores / Referees
Adrião Duarte, Universidade Federal do Rio Grande do Norte.
Alexsandra Andrade, Universidade Estadual do Sudoeste da Bahia.
André Lemos, Universidade Federal de Minas Gerais.
Antônio Diego Silva Farias, Universidade Federal Rural do Semi-Árido.
Benjamin Bedregal, Universidade Federal do Rio Grande do Norte.
David Correa Martins Jr, Universidade Federal do ABC.
EmmanuellyMonteiro, Universidade Federal do Rio Grande do Norte.
Estevão Esmi Laureano, Universidade Estadual de Campinas.
Fagner Santana, Universidade Federal do Rio Grande do Norte.
Fernando Gomide, Universidade Estadual de Campinas.
Flaulles Bergamaschi, Universidade Estadual do Sudoeste da Bahia.
Francisco de Carvalho, Universidade Federal do Pernambuco.
Graçaliz Dimuro, Universidade Federal do Rio Grande.
Guilherme Barreto, Universidade Federal do Ceará.
Helida Santos,Universidade Federal do Rio Grande do Norte.
Heloisa Camargo, Universidade Federal de São Carlos.
Humberto Bustince, Universidad Pública de Navarra - Espanha.
Ing Ren Tsang, Universidade Federal do Pernambuco.
Ivan Mezzomo, Universidade Federal Rural do Semi-Árido.
Javier Fernandez, Universidad Pública de Navarra - Espanha.
João Alcantara, Universidade Federal do Ceará.
Jose Alfredo Ferreira Costa, Universidade Federal do Rio Grande do Norte.
Jose Arnaldo Roveda, Universidade Estadual Paulista.
Laécio C. Barros, Universidade Estadual de Campinas.
Liliane Silva, Universidade Federal do Rio Grande do Norte.
Magda Peixoto, Universidade Federal de São Carlos.
Marcos Eduardo R. Valle Mesquita, Universidade Estadual de Campinas.
Marilton Sanchotene de Aguiar, Universidade Federal de Pelotas.
Marina Tuyako Mizukoshi, Universidade Federal de Goiás.
Mario Benevides, Universidade Federal do Rio de Janeiro.
Márjory Da Costa-Abreu, Universidade Federal do Rio Grande do Norte.
Marley Vellasco, Pontif́ıcia Universidade Católica do Rio de Janeiro.
Michal Baczynski, University of Silesia - Polônia.
Myriam Delgado, Universidade Tecnológica Federal do Paraná.
Paulo Almeida, Centro Federal de Educação Tecnológica de Minas Gerais.
Peter Sussner, Universidade Estadual de Campinas.
Regivan Santiago, Universidade Federal do Rio Grande do Norte.
Renata Reiser, Universidade Federal de Pelotas.
Ricardo Tanscheit, Pontif́ıcia Universidade Católica do Rio de Janeiro.
Ronei Moraes, Universidade Federal do Paráıba.
Sandra Regina Monteiro Masalskiene Roveda, Universidade Estadual Paulista.
Sandra Sandri, Instituto Nacional de Pesquisas Espaciais.
Vilem Novak, University of Ostrava - República Checa.
Weldon Lodwick, University of Colorado - Estados Unidos da América.
Realização / Realization
Universidade Estadual de Campinas (Unicamp)
Instituto de Matemática, Estat́ıstica e Computação Cient́ıfica (IMECC)
Suporte Financeiro / Sponsors
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior (CAPES)
Universidade Estadual de Campinas (Unicamp)
Apoio / Support
Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC)
Sociedade Brasileira de Inteligência Computacional (SBIC)
Sociedade Brasileira de Automática (SBA)
North American Fuzzy Information Processing Society (NAFIPS)
European Society for Fuzzy Logic and Technology (EUSFLAT)
International Fuzzy Systems Association (IFSA)
Sumário / Table of Contents
I Matemática Intervalar e Extensões de Conjuntos Fuzzy / Interval-valued Mathematics and Extensions
of Fuzzy Sets
Prediction of the Economically Active Population Index Using Interval-Valued Fuzzy Associative
Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Tiago Schuster and Peter Sussner
Transformações de Matrizes Intervalares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Guilherme Andrade, Gildson de Jesus and Eduardo Palmeira
Produto de Kronecker para Matrizes Intervalares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Rudhero Dos Santos, Gildson de Jesus and Eduardo Palmeira
Atanassov’s Intuitionistic Fuzzy Entropy: Conjugation and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Lidiane Costa, Alice Finger, Mateus Nascimento, Monica Matzenauer, Rosana Zanotelli,
Renata Hax Sander Reiser, Adenauer Yamin and Mauŕıcio Pilla
II Reconhecimento de Padrões e Otimização Flex́ıvel / Pattern Recognition and Flexible Optimization
Pattern recognition based on soft boundaries: a proposal applied to tree species identification
from texture in trunk images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Adriano Bressane, José Arnaldo Frutuoso Roveda, Antonio Cesar Germano Martins and
Sandra Regina Monteiro Masalskiene Roveda
Assessment of Poisson Naive Bayes Classifier with Fuzzy Parameters Using Data from Different
Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Elaine Anita de Melo Gomes Soares and Ronei Marcos de Moraes
O Problema da Árvore Geradora Mı́nima Fuzzy: um algoritmo para o caso envolvendo incertezas
nos pesos das arestas e na estrutura da rede . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Fábio Hernandes and Cassiano Sampaio
III Lógica Fuzzy e Matemática Fuzzy / Fuzzy Logic and Fuzzy Mathematics
New results about De Morgan triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Ivan Mezzomo and Benjamı́n Bedregal
A theorem to construct fuzzy subsethood measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Helida Santos, Benjamin Bedregal, Regivan Santiago, Humberto Bustinceand Edurne
Barrenechea
Diferentes Abordagens à Teoria de Possibilidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Estevão Esmi and Laécio C. Barros
Notas sobre Fuzzy x Probabilidades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Laécio C. Barros and Estevão Esmi
IV Tomada de Decisão / Decision Making
Lógica Fuzzy Aplicada na Avaliação da Satisfação de Usuários de Sistemas de Gestão . . . . . . . . . . . 133
Alisson Marques Silva, Edilson Hélio Santana and Marco Antônio Pinheiro Silveira
Fuzzy system for indicating the fuel quantity in the sintering process of products ceramic . . . . . . . . 145
Antônio Eudson Costa Cabó, Rommel Wladimir de Lima, Danniel Cavalcante Lopes,
Emerson de Andrade Lima
Prinćıpio de Extensão Multivariável de Zadeh para a Variável de Decisão de um Plano de
Radioterapia de Intensidade Modulada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Ana Maria Bertone and Rosana Sueli Da Motta Jafelice
Tomadas de Decisões Fuzzy Através da Inferência de Larsen para o Diagnóstico dos Indicadores
de QEE: DTHV, DTHI, FP, VTRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Diego Nolasco and Eduardo Palmeira
V Sistemas Dinâmicos e Análise Fuzzy / Fuzzy Dynamic Systems and Fuzzy Analysis
Uma Análise de Bifurcação de Hopf Fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Marina T. Mizukoshi and Moiseis Dos S. Cecconello
Um estudo sobre derivada lateral interativa fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Francielle Santo Pedro, Estevão Esmi and Laécio C. Barros
Autômatos Lineares Fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Valdigleis Costa and Benjamı́n Bedregal
Ensaios do Método de Elementos Finitos com Integral Fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Daniel Ibanez, Luana Bassani, Laecio Barros and Estevão Laureano
VI Sistemas Evolucionários e Suporte a Decisão / Evolutionary Systems and Decision Making
Algoritmo evolucionário com inspiração quântica para śıntese de sistemasde classificação fuzzy . . . 229
Waldir Nunes, Marley Vellasco and Ricardo Tanscheit
Sistema de Inferência Fuzzy para Diagnóstico de Desempenho de Turbinas a Gás Aeronáuticas . . . 242
Tairo Teixeira, Ricardo Tanscheit and Marley Vellasco
Um sistema inteligente para análise de condição de máquinas rotativas baseado em indicadores
de vibração . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Thiago Felicio, Marilza Lemos, Marcio Marques and Alexandre Simões
VII Processamento de Sinais e Imagens / Image and Signal Processing
Type-1 Fuzzy Logic System Applied to Classification of Rail Head Defects . . . . . . . . . . . . . . . . . . . . . 269
Eduardo Pestana de Aguiar, Lucas Pereira Verneck Da Silva, Adler Ferreira Moreira,
Leonardo Goliatt Da Fonseca, Fernando Marques de Almeida Nogueira, Marley Maria
Bernardes Rebuzzi Vellasco, Moises Vidal Ribeiro
WOWA image filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Leonardo Torres, José Carlos Becceneri, Corina Freitas, Sidnei Santanna and Sandra Sandri
Peak detection using fuzzy inference systems in gamma-ray spectrometry . . . . . . . . . . . . . . . . . . . . . . 293
Bruno Arine, José Arnaldo Roveda, Sandra Roveda and Antonio Martins
Adaptive Median and Wiener Filters as Reference Functions for Morphological Associative
Memories in Complete Inf-Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Majid Ali and Peter Sussner
VIII Neuro-fuzzy, Robótica e Controle / Neuro-fuzzy, Robotics, and Control
Modelagem em Tempo Real do TRMS Usando Rede Neuro-Fuzzy Evolutiva . . . . . . . . . . . . . . . . . . . . 317
Alisson Marques Silva, Walmir Matos Caminhas, André Paim Lemos and Fernando Gomide
Dispositivo Robótico para Assistência à Locomoção de Pessoas Idosas em Ambientes Urbanos . . . . 329
Daniel de Sousa Leite, Karla Figueiredo and Marley Vellasco
Seleção de Terrenos para Edificações Comerciais na Cidade do Rio de Janeiro: Aplicação da
Lógica Fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Nilson Brandalise, Amanda Sexto Alexandre Pereira and Luiz Carlos Brasil de Brito Mello
Controle Fuzzy de um Sistema Auxiliar de Navegação de um Robô Ambiental Hı́brido . . . . . . . . . . . 353
Cristhian Gomez, Marley Vellasco and Ricardo Tanscheit
IX Biomatemática Fuzzy / Fuzzy Biomathematics
Utilizando Lógica Fuzzy para Modelagem Computacional de Qualidade da Água do Rio
Cachoeira, Região Sul da Bahia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Valdex Santos, Francisco B. S. Oliveira and Eduardo S. Palmeira
Avaliação da cobertura do solo no entorno de rodovias usando uma abordagem fuzzy baseada
no método de inferência de Mamdani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
Felipe Goulart Moraes, André Bairros Peres and Antonio Cesar Germano Martins
Matriz de Relações Fuzzy: Aplicações em Questões Ambientais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Lucirene França, José Roveda and Sandra Roveda
Estimativa de Risco e Perigo de Incêndios Florestais Utilizando Subconjuntos Fuzzy, k-NN
Fuzzy e Subtractive Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Isaac Silva, Diego Gomes, Marcos Valle, Laécio Barros and João Meyer
X Medidas Fuzzy e Agregações / Fuzzy Measures and Aggregations
Pre-aggregation functions constructed by CO-integrals applied in classication problems . . . . . . . . . . 413
Giancarlo Lucca, Graçaliz Dimuro, José Antonio Sanz, Benjamı́n Callejas Bedregal and
Humberto Bustince
Funções Mistura Generalizada Construdas via Funções Mistura Generalizada Limitada . . . . . . . . . . 424
Antonio D. Silva Farias, Valdigleis S. Costa, Regivan H. N. Santiago and Benjamı́n Bedregal
Aplicação do Defuzzificador para Eventos Fuzzy no Estudo da Evolução da Doença de Chagas . . . . 436
Ricardo Augusto Watanabe, Estevão Esmi Laureano and Laécio Carvalho Barros
XI Epidemiologia e Diagnóstico Médico / Epidemiology and Medical Diagnosis
Análise da Malária no Estado do Amazonas através de Sistema de Base de Regras Fuzzy . . . . . . . . . 451
Lee Ketlen Costa Farias R. Dos Santos and Roberto Antonio Cordeiro Prata
Diagnósticos de Risco da Incidência de Doenças Cardiovasculares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Luana Tais Bassani, Augusto Terranova Rocha and Rodney Carlos Bassanezi
Modelo de suporte à decisão baseado em regras fuzzy para casos do dengue na Paráıba entre os
anos de 2010 e 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
José Carlos Da Silva Melo and Ronei Marcos de Moraes
Uma proposta de diagnóstico médico por meio de relações fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
Lázaro R. Marins and Magda S. Peixoto
XII Inteligência Computacional e Aprendizado de Máquinas / Computational Intelligence and Machine
Learning
Uma Introdução as Memórias Autoassoaciativas Fuzzy de Projeçõoes Max-C . . . . . . . . . . . . . . . . . . . 491
Alex S. Dos Santos and Marcos Eduardo Valle
Memória Associativa Bidirecional Exponencial Fuzzy Generalizada Aplicada ao Reconhecimento
de Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Aline Cristina Souza and Marcos Eduardo Valle
Metodologia para Processamento de Dados para Previsão de Energia e Curva de Carga em
Edificações . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Leandro Gomes and Patŕıcia Jota
Modelagem nebulosa c-regressão para combinação de previsões de séries temporais . . . . . . . . . . . . . . 527
Fernanda Cristina Janoele, Leandro Maciel, Rosangela Ballini and Fernando Gomide
I
Matemática Intervalar e Extensões
de Conjuntos Fuzzy
Interval-valued Mathematics and
Extensions of Fuzzy Sets
Prediction of the Economically Active
Population Index Using Interval-Valued Fuzzy
Morphological Associative Memories
Tiago Schuster and Peter Sussner
Department of Applied Mathematics, University of Campinas,
13081-970, Campinas, SP, Brazil
{ra073785,sussner}@ime.unicamp.br
http://www.ime.unicamp.br/
Abstract. The last few decades have witnessed rapid progress in the
field of type-2 fuzzy systems and in particular interval type-2 fuzzy sys-
tems. Encouraged by this progress, we present in this paper some theoret-
ical foundations and applications of interval-valued fuzzy morphological
associative memories (IV-FMAMs) as a rule-based system. We perform
simulations concerning the application of IV-FMAMs to the prediction
of the monthly rate of participation of certain age groups in the work
force of the metropolitan area of São Paulo. The performance in terms
of mean squared prediction errors of the IV-FMAM approach is then
compared with the one of a conventional type-2 fuzzy inference system.
1 Introduction
Type-2 fuzzy sets, and in particular interval type-2 fuzzy sets have found a
wide variety of applications in engineering, control, computing with words, and
approximate reasoning [8,16,17]. Although interval-valued fuzzy sets (IV fuzzy
sets) are related to interval type-2 fuzzy sets [2], these concepts are different.
In fact, interval-valued fuzzy sets correspond to closed interval type-2 fuzzy sets
whose secondary membership functionsare characteristic functions of closed
intervals [14]. To our knowledge, all interval type-2 fuzzy systems that have
appeared in the literature and are used in practice represent closed interval
type-2 fuzzy systems.
Like general type-2 fuzzy sets and (closed) interval type-2 fuzzy sets, interval-
valued fuzzy sets can be employed to model the inherent uncertainties regarding
fuzzy set membership functions. An approach for dealing with interval-valued
fuzzy data is given by IV-FMAMs [24], a recent extension of fuzzy morphological
associative memories. The latter can be used to implement fuzzy rule-based
systems, in particular for applications in time-series prediction [22,27]. In this
paper, we apply the proposed IV-FMAM model to the problem of predicting
the monthly percentages of participation of certain age groups in the work force
of the metropolitan area of São Paulo [4]. Solving this prediction problem may
offer decision support for government, industry and trade unions.
Fourth Brazilian Conference on Fuzzy Systems (IV CBSF)
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The paper is organized as follows. Section 2 provides a brief review of per-
tinent notions on lattice theory, L-fuzzy logical operators and L-fuzzy mathe-
matical morphology. In Section 3, we present the interval-fuzzy morphological
associative memories (IV-FMAMs), that yield associations between IV fuzzy
sets. Section 4 describes an application of the IV-FMAMs as rule-based systems
to a social index time-series prediction. The results are compared with the ones
obtained using a Mamdani(-Assilian) type-2 fuzzy system [13], followed by some
concluding remarks.
2 Theoretical background
2.1 Some Relevant Concepts of Lattice Theory and Mathematical
Morphology
A complete lattice L is a partially ordered set such that every subset X ∈ L
has an infimum
∧
X and a supremum
∨
X in L. Recall that a partial order
is a reflexive, antisymmetric, and transitive binary relation “≤” [7]. The unit
interval [0, 1] with the usual (total) ordering yields an example of a complete
lattice. Another example is given by I = {u = [u, u] ⊆ [0, 1]} if we consider the
following partial order:
u ≤ v ⇔ u ≤ v and u ≤ v . (1)
A component-wise partial order can also be defined on Ln by setting
(a1, . . . , an) ≤ (b1, . . . , bn)⇔ ai ≤ bi, i = 1, . . . , n. (2)
If L is a (complete) lattice, then Ln is also a (complete) lattice. Similarly, we
have that if L is a (complete) lattice then the class of functions X → L, LX , is
also a (complete) lattice. Here, the partial order on LX is given as follows for all
f, g ∈ LX :
f ≤ g ⇔ f(x) ≤ g(x) ∀x ∈ X. (3)
Let L be a complete lattice and X 6= ∅. The partial order on LX of Equation
3 induces a partial order on FL(X), the class of L-fuzzy sets over the universe X.
Recall that an L-fuzzy set A consists of a universe X together with a membership
function µA : X → L [6]. For L-fuzzy sets A and B, we have A ≤ B if and only
if µA ≤ µB . The class of fuzzy sets over the universe X,F(X), and the class of
interval-valued fuzzy sets over the universe X,FI(X), represent particular classes
of L-fuzzy sets for particular choices of L.
It is well established that complete lattices form an appropriate framework for
mathematical morphology (MM) [19,26]. Although MM was originally conceived
as a theory for image and signal processing based on geometrical and topological
concepts, its operators naturally adhere to the lattice theory as their algebraic
framework [10,20].
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In this framework, the basic operators of MM are the (algebraic) erosion,
ε : L → M and the (algebraic) dilation, δ : L → M, that satisfies, respectively,
the equations below for all M ⊆ L:
ε(
∧
M) =
∧
ε(M) and δ(
∨
M) =
∨
δ(M). (4)
The concept of an adjunction, that is also known as a monotone or isotone
Galois connection [1] plays a very prominent role in MM [10].
Definition 1. A pair (ε, δ) consisting of mappings ε : M→ L and δ : L→M is
called an adjunction between M and L if and only if for all x ∈ L and all y ∈M:
δ(x) ≤ y ⇔ x ≤ ε(y). (5)
If the pair (ε, δ) is an adjunction then ε is an erosion and δ a dilation [11].
2.2 L-Fuzzy Operators
The purpose of this section is to recall the definitions of some morphological
operators and matrix products on the complete lattice FL(X) [24]. Let us begin
by the definition of L-fuzzy conjunctions and implications [5]:
Definition 2. Let L be a complete lattice.
– A conjunction on L or L-fuzzy conjunction is defined as an increasing map-
ping C : L × L → L that satisfies C(0L, 0L) = C(0L, 1L) = C(1L, 0L) = 0L
and C(1L, 1L) = 1L. In particular, a commutative and associative L-fuzzy
conjunction T : L × L → L that satisfies T (x, 1L) = x for every x ∈ L is
called triangular norm or simply t-norm on L.
– An operator I : L × L → L that is decreasing in the first argument and
that is increasing in the second argument is called an implication on L or
L-fuzzy implication if the equations I(0L, 0L) = I(0L, 1L) = I(1L, 1L) = 1L
and I(1L, 0L) = 0L are satisfied.
– In the special case where L = I, we speak of interval-valued fuzzy (IV fuzzy)
operators, in particular of IV fuzzy conjunctions, t-norms, and implications.
Several types of matrix products arise from L-fuzzy operators including the
following ones [24]:
Definition 3. Given an L-fuzzy conjunction C and an L-fuzzy implication I,
the sup-C product of A ∈ Lm×k and B ∈ Lk×n, denoted by E = A ◦C B,
and the inf-I product, denoted by G = A ~ B are defined, respectively, for all
i = 1, . . . ,m and j = 1, . . . , n, as follows:
eij =
k∨
ξ=1
C(aiξ, bξj), and gij =
k∧
ξ=1
I(bξj , aiξ) . (6)
The following proposition holds for every complete lattice L:
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Proposition 1 The operator δW : Ln → Lm that are given by
δW (x) = W ◦ x, ∀x ∈ Ln. (7)
represents a dilation for every W ∈ Lm×n from the complete lattice Ln to the
complete lattice Lm if and only if C(w, ·) : L→ L represents a dilation for every
w ∈ L.
As we shall point out in the next section for the interval-valued case, the
output of an interval-valued fuzzy associative memory can be modeled by means
of Equation 7. Furthermore, the model given by Equation 7 will be called mor-
phological. The next section focuses on interval-valued fuzzy morphological as-
sociative memories.
3 Interval-Valued FMAMs
In this section we will present an associative memory model for interval-valued
fuzzy sets. Consider initiallyX and Y arbitrary universes and
{(
pξ,qξ
)
: ξ ∈ K
}
⊆
FI(X)×FI(Y ) a set of pairs or associations, the set
{(
pξ,qξ
)
: ξ ∈ K
}
is called
the fundamental memory set and its elements are called fundamental memories.
An IV fuzzy associative memory (IV -FAM) is an input-output system given
by a mapping W : FI(X) → FI(Y ) that should ideally satisfy the following
conditions [9]:
1. W(pξ) = qξ for all ξ ∈ K;
2. W(p̃ξ) = qξ for p̃ξ ≈ pξ.
Since our objective is to employ IV-FAMs in order to implement rule-based
systems, it would be more adequate to replace items 1 and 2 with the following
condition:
p̃ξ ≈ pξ ⇒W(p̃ξ) ≈ qξ . (8)
For simplicity, we concentrate on the case where X, Y , and K are finite.
Let |X| = n, |Y | = m, and |K| = k. Thus, we view an IV-FAM as a mapping
W : In → Im. We say that W represents a sup-C IV-FAM if W(x) = δW (x) =
W ◦C x, ∀x ∈ In, for some W ∈ Im×n.
In this paper we focus on the construction of IV-FMAMs based on repre-
sentable IV fuzzy conjunctions and their adjoint implications [5,23]. We will
now present a recipe for constructing IV fuzzy conjunctions and their adjoint
implications, beginning by the definition of a representable IV fuzzy conjunction
[5]:
Proposition 2 If C is a fuzzy conjunction, then the operator CrC , that is defined
as follows for all u = [u, u], v = [v, v] ∈ I, yields an IV fuzzy conjunction.
CrC(u, v) = [C(u, v), C(u, v)].(9)
Definition 4. The IV fuzzy conjunction CrC of Equation 9 is referred to as the
representable conjunction with representative C.
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Let CrC be a representable interval-valued fuzzy conjunction with representa-
tive C and I the adjoint fuzzy implication of C. The adjoint IV fuzzy implication
of CrC can be obtained by means of Equation 10, as follows [23]:
InI (u, v) = [I(u, v) ∧ I(u, v), I(u, v)]. (10)
An example of dilative fuzzy conjunction is the cross-ratio uninorm [30],
denoted using the symbol CF . The formulas for CF and its adjoint implication
IF can be found below:
CF (x, y) =
{
1, if (x, y) = (0, 1) or (1, 0)
xy
(1−y)(1−x)+xy) , otherwise.
(11)
IF (x, y) =
{
1, if (x, y) = (0, 0) or (1, 1)
(1−x)y
y(1−x)+x(1−y) , otherwise.
(12)
Observe that the representable IV fuzzy conjunction CrCF and its adjoint
implication InIF , given by means of Equation 10, represents a pair consisting of
an erosion and a dilation. Hence, an IV-FAM based on CrCF , that will be denoted
here by WrF , will be morphological.
In this research paper, we will employ sup-CrC matrix products in the re-
call phases of IV-FAMs, in particular sup-CrC matrix products that give rise to
dilations and, therefore, IV-FMAMs.
Let P = [p1, ...,pk] ∈ In×k, Q = [q1, ...,qk] ∈ Im×k be matrices which
columns are formed by the pairs (pj ,qj) of fundamental memories and let CrC
be an IV fuzzy conjunction. Consider the problem of determining the weight
matrix W of a sup-CrC IV-FMAM given by the dilation δW . As an extension of
the fuzzy learning by adjunction [22] for sup-C FMAMs to IV fuzzy learning by
adjunction (IV-FLA) for sup-CrC FMAMs, we propose to construct its weight
matrix W ∈ In×k as follows:
W = Q~n,I P t, (13)
where the IV fuzzy implication used in the Inf-I product is the adjoint pair of
CrC . Finally, upon presentation of an input pattern x ∈ In, the output pattern
y ∈ Im of the corresponding IV-FMAM can be calculated by the sup-CrC product
of W and x. In the following section, we employ the IV-FMAM WrF in order to
implement a IV fuzzy inference system for a time series prediction problem.
4 An IV-FMAM Approach Towards Time Series
Prediction
4.1 Experimental Setup for the IV-FMAM Approach
In this section, we present the experimental setup regarding the use of sup-CrC IV-
FMAMs in a time-series forecasting problem, namely the Economically Active
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Population Index (PEA), a socio-economic index used by Brazilian governmental
and non-governmental sectors for strategic decision making [4]. Specifically, a
sup-CrF IV-FMAM was employed to model a rule-based system consisting of k
rules, each one having a IV fuzzy antecedents and b IV fuzzy consequents. To
this end, the crisp training data, that are contained along with the testing data
in a finite universe U × V ⊆ Ra × Rb, were clustered by means of the IV fuzzy
c-means clustering technique (IV-FCM) [12]. More precisely, given a number k
of clusters and a “fuzzifier parameter m = [m1,m2], IV fuzzy c-means clustering
produces the cluster centers cγp ∈ Ra and cξq ∈ Rb with respective component-
wise standard deviations σp ∈ Ra and σq ∈ Rb of IV fuzzy Gaussian membership
functions pξ and qξ for ξ = 1, . . . , k.
The antecedents pξ and the consequents qξ are respectively contained in
FI(U) and FI(V), where U = {u1, . . . ,um} and V = {v1, . . . ,vn}. Their compo-
nents can be calculated as follows:
pξj = exp

−1
2
a∑
l=1
∣∣∣∣∣
(
uj
)
l
−
(
cξp
)
l
(σp)l
∣∣∣∣∣
2
m2−1
,−1
2
a∑
l=1
∣∣∣∣∣
(
uj
)
l
−
(
cξp
)
l
(σp)l
∣∣∣∣∣
2
m1−1

 , (14)
qξi = exp

−1
2
b∑
l=1
∣∣∣∣∣
(
vi
)
l
−
(
cξq
)
l
(σq)l
∣∣∣∣∣
2
m2−1
,−1
2
b∑
l=1
∣∣∣∣∣
(
vi
)
l
−
(
cξq
)
l
(σq)l
∣∣∣∣∣
2
m1−1

 . (15)
In principle, the pairs (pξ,qξ) corresponding to the training data can be used
to compute the weight matrix W ∈ Im×n by means of Equation 13.
Training:
1. Apply the IV-FCM clustering technique with k centers and fuzzifier param-
eter m to the training data in U ×V ⊆ Ra ×Rb in order to obtain k centers
cγp ∈ Ra and cξq ∈ Rb with respective component-wise standard deviations
σp ∈ Ra and σq ∈ Rb;
2. Create the discrete intervalar gaussian antecedents pξ in FI(U) and conse-
quents qξ in FI(V) using Equations 14 and 15;
3. Compute the weight matrix W ∈ Im×n by means of the inf-I product W =
(qξ)t ~ pξ.
The test is performed as follows. Given the weight matrix W , for each p in
the test data set, do:
1. Compute the sup-C product q = W ◦ p to obtain an interval-valued fuzzy
vector;
2. Average q component-wise to type-reduce it (Nie-Tan type-reduction method
[18]);
3. Defuzzify the type-reduced vector using the centroid defuzzification method
to obtain a real valued output.
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However, an explicit construction of W is not required since the represen-
tation as an IV fuzzy set of the antecedent part of a crisp test datum ud in U
yields an input pattern p ∈ FI(U) of the form p = [0I, . . . , 0I, eI, 0I, . . . , 0I]T .
More precisely, we have pj =
{
eI, if j = d
0I, otherwise.
. Here, the symbol eI denotes the
identity element [0.5, 0.5] of the IV fuzzy conjunction CrF . For p ∈ FI(U) ' Im
of this form, q = W ◦r,F p has the following components for i = 1, . . . ,m:
qi = W ◦r,F pi =
n∨
j=1
CrF (wij , pj) = wid =
k∧
ξ=1
InF (pξj , qξi ) . (16)
We obtain a final, crisp prediction value from q after an application of type-
reduction and defuzzification. For computational reasons, we employed the Nie-
Tan method, which is nearly as fast as a conventional fuzzy centroid defuzzifica-
tion [28]. The Nie-Tan method consists of a type-reduction procedure followed
by a centroid defuzzification of the resulting type-1 fuzzy set.
4.2 Prediction of a Brazilian Socioeconomic Index
The Economically Active Population Index (PEA) refers to the percentage of
economically active persons, i.e., people who are currently employed or actively
looking for a job, within certain age groups. The data for the computation of the
PEA index are collected by DIEESE, the “Inter-Union Department of Statistics
and Socio-Economic Studies”, a creation of the Brazilian trade union movement.
DIEESE was founded in 1955 to develop research on which workers’ demands
could be based [4].
The PEA index, since its conception, has provided guidance - not only to
some governmental sectors such as the social security office but also to trade
unions and private corporations since this index yields valuable information con-
cerning the current state of the workforce. This information can also be used as
a decision making tool for politicians to avoid future social and economical prob-
lems. The PEA index is computed monthly as part of some other more compre-
hensive indices, such as the Monthly Employment Survey Index (PME/IBGE)
and the Employment and Unemployment ResearchIndex (PED/DIEESE), in the
metropolitan area of São Paulo and other major Brazilian metropolitan areas.
Here, we employed the methodology described in order to forecast the PEA
index of the metropolitan area of São Paulo from January 1985 to December
2012. The population under consideration comprises approximately 17 million
inhabitants and is divided into the following age brackets: 10-15, 16-24, 25-39,
40-49, 50-59 and 60+. The PEA index values are given by the percentage of the
working age population that includes every person older than nine.
Note that the PEA index includes 10 to 15 year old economically active
children that are part, albeit illegally, of Brazil’s working population. Thus, the
PEA index serves to aide governmental and non-governmental organizations to
understand and remedy the child labor problem. Since the insertion of the 10-15
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9
years age group in the PEA index, initiatives such as the creation of the Statute
of the Child and the Adolescent (“Estatuto da Criança e do Adolescente”) and
several non-governmental campaigns helped to decrease the amount of child
labor from more than 20% in 1985 to 4.5% at the end of 2012.
We chose to use one predictor model for each month and for age-group, in
view of seasonal differences in the PEA values. We standardized the original
data to lie within the range [−5, 5] by subtracting the mean and dividing by the
standard deviation. Let sγ ∈ R be samples of the seasonal time series. The goal
is to estimate the value of sq from a subset of the past values {s1, s2, ..., sq−1}. In
particular, we simply used the last three monthly PEA indices {sγ−3, sγ−2, sγ−1}
to predict the next index sγ .
Given a fuzzifier parameter m and a number of clusters k, IV-FCM clustering
produced cluster centers cξp ∈ R3 and cξq ∈ R with respective component-wise
standard deviations σp ∈ R3 and σq ∈ R. We considered finite universes of
discourse U = {u1, . . . ,um} and V = {v1, . . . ,vn} comprising m = 503 and
n = 50 equally spaced points in [−5, 5]3 and [−5, 5], respectively. The values
cξp, σp, c
ξ
q, and σq can be used to compute the entries p
ξ
j = [p
ξ
j , p
ξ
j ] and q
ξ
i =
[qξi , q
ξ
i ] of the finite Gaussian IV fuzzy sets p
ξ ∈ FI(U) ' In and qξ ∈ FI(V) ' Im
via Equations 14 to 15.
The performance of the IV-FMAM approach suggested in this paper depends
on the choices of the number of clusters and a fuzzifier parameter m = [m1,m2]
as inputs to the IV-FCM clustering algorithm. We employed a fixed number of
clusters k = 10 for all monthly models. We considered three different options of
m, namely [2.0 − α, 2.0 − α] for α = 0, 0.1, 0.2. Note that m = [2.0, 2.0] yields
fuzzy Gaussian membership functions pξ ∈ F(U) and qξ ∈ F(V) and therefore
WrC corresponds to WC for every commutative and dilative fuzzy conjunction
C. Since WF achieved the best validation performance in previous experiments
concerning time-series prediction [27], we only performed simulations usingWrF .
The fuzzifier parameter m was selected by means of leave-one-out cross-
validation on the data from January 1985 to December 2002, according to the
Table 1 for each age bracket. Here, the performance was measured in terms of
the mean absolute error (MAE), the root mean squared error (RMSE), the mean
percentage error (MPE), and the correlation coefficient ρ (the higher the values
of ρ, the better the performance).
We compared the prediction results produced by the sup-CrF IV-FMAM
model with m = [1.9, 2.1] with the ones produced by an interval type-2 fuzzy
inference system (IT-2 FIS) with the same fuzzifier parameter m = [1.9, 2.1]
using the test data from January 2003 to December 2012. Note that the IT2-FIS
with m = [2.0, 2.0] corresponds to a conventional type-1 fuzzy inference system
[15]. Tables 2 and 3 display the testing errors and the correlation coefficients pro-
duced by the WrF and the interval type-2 fuzzy inference system, respectively.
Note that the IV-FMAMWrF evidently outperformed the IT-2 FIS in these sim-
ulations. Figures 1 and 2 illustrate the prediction results obtained by WrF and a
conventional interval type-2 fuzzy system, respectively, in comparison with the
real data for the testing period.
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Table 1. Leave-one-out cross-validation errors from 1985 to 2003.
Age Bracket m MAE RMSE MRE(%) ρ
[2.0, 2.0] 1.70 2.77 11.10 0.798
10-15 [1.9, 2.1] 1.70 2.75 10.97 0.802
[1.8,2.2] 1.68 2.75 10.91 0.803
[2.0, 2.0] 3.17 8.25 4.79 0.231
16-24 [1.9,2.1] 3.04 8.10 4.74 0.233
[1.8, 2.2] 3.11 8.32 4.87 0.230
[2.0, 2.0] 3.03 7.80 4.40 0.278
25-39 [1.9,2.1] 3.00 7.76 4.38 0.283
[1.8, 2.2] 3.04 7.96 4.46 0.2801
[2.0, 2.0] 2.45 6.00 3.69 0.334
40-49 [1.9,2.1] 2.45 5.99 3.69 0.335
[1.8, 2.2] 2.54 6.30 3.89 0.330
[2.0, 2.0] 2.44 5.35 4.95 0.418
50-59 [1.9,2.1] 2.45 5.34 4.95 0.425
[1.8, 2.2] 2.48 5.45 5.05 0.417
[2.0, 2.0] 1.32 2.53 6.87 0.144
60+ [1.9,2.1] 1.32 2.51 6.86 0.144
[1.8, 2.2] 1.34 2.54 6.92 0.148
Table 2. Testing errors produced by the WrF for the each age group of the PEA index.
Age Bracket MAE RMSE MRE(%) ρ
10-15 0.90 0.60 8.66 0.870
16-24 0.62 0.50 0.66 0.861
25-39 0.50 0.35 0.41 0.809
40-49 0.62 0.46 0.58 0.876
50-59 0.91 0.70 1.13 0.920
60+ 0.83 0.66 3.04 0.624
Table 3. Testing errors produced by the IT2 Mamdani inference system for the each
age group of the PEA index.
Age Bracket MAE RMSE MRE(%) ρ
10-15 1.15 0.91 13.16 0.791
16-24 0.78 0.63 0.84 0.754
25-39 0.60 0.45 0.53 0.710
40-49 0.82 0.64 0.81 0.804
50-59 1.40 1.17 1.87 0.849
60+ 0.97 0.76 3.49 0.447
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Fig. 1. Predictions obtained by WrF (top row) and the IT2-FIS (bottom row) for the
age groups: 10-15, 16-24 and 25-39, from left to right.
Fig. 2. Predictions obtained by WrF (top row) and the IT2-FIS (bottom row) for the
age groups: 40-49, 50-59 and 60+, from left to right.
5 Concluding Remarks
Type-2 fuzzy systems have been successfully employed in a variety of applications
for their capability of handling uncertainties that are intrinsic in real-world data
better then traditional (type-1) fuzzy systems [3]. In particular, interval-valued
FSs have found far more applications then full type-2 FS due to their simplicity
and to the merely linear increase in computational complexity in comparison to
type-1 fuzzy systems [13,25].
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We applied in this paper our alternative approach towards an IV fuzzy sys-
tem, namely the sup-CrC IV-FMAM, to the problem of forecasting the monthly
rates of participation of given age groups in the work force of the metropolitan
area of São Paulo. To this purpose, we generated an IV fuzzy inference system
based on the aforementioned IV-FMAMs. The same methodology was employed
in conjunction with an Mamdani(-Assilian) IT-2 FIS. In our simulations the sup-
CrC IV-FMAM approach exhibited significantly better results than the interval
type-2 fuzzy system with respect to four different performance measures.
Note that the IV-FMAM approach presented in this paper depends on the
complete lattice structure of I and in particular on the fact that elements of
I were partially ordered in terms of Equation 1. In the future, we intend to
investigate the suitability of other partial orders in conjunction with IV-FAMs.
Furthermore, we plan to develop full type-2 fuzzy associative memories (T2-
FAMs) as particular cases of L-fuzzy associative memories and use them to
build full type-2 fuzzy inference systems.
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Fourth Brazilian Conference on Fuzzy Systems (IV CBSF)
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Transformações de Matrizes Intervalares
Guilherme P. Andrade, Gildson Q. de Jesus, and Eduardo S. Palmeira
Programa de Pós-Graduação em Modelagem Computacional em Ciência e Tecnologia
Departamento de Ciências Exatas e Tecnológicas - DCET
Universidade Estadual de Santa Cruz - UESC
gildsonj@gmail.com,
{gpandrade,espalmeira}@uesc.br
Resumo Este trabalho consiste em demonstrar as propriedades de trans-
formações elementares de matrizes, aplicadas à matrizes cujo os elemen-
tos são definidos como intervalos, denominadas matrizes intervalares.
Nele pôde-se definir as transformações básicas e suas respectivas re-
versões, além de demonstrar aplicações de matrizes elementares.
Keywords: Matrizes Intervalares, Transformações Elementares, Matri-
zes Elementares
1 Introdução
O estudo da matemática intervalar tem uma grande relevância no campo da mo-
delagem computacional, pois tenta resolver o problema da propagação de erros
cometidos no processamento de dados com arredondamento dada a dificuldade
de se representar determinados números reais, além de ser uma boa maneira de
fazer um controle e estimativa desses erros através da utilizando um intervalo
ao redor do um valor que se pretende representar, possibilitando uma melhor
análise dos valores em torno do resultado esperado.
Assim para que se tenha uma matemática intervalar bem fundamentada é
necessário investir esforços de pesquisa na direção de verificar e comprovar quais
propriedades aritméticas e algébricas dos números reais são válidas no escopo
intervalar. Por exemplo, a utilização de matrizes e suas propriedades na modela-
gem computacional é imprescind́ıvel, pois constitui uma ferramenta para resolver
e manipular sistema de equações, além de ser altamente aplicável em qualquer
linguagem de programação [3]. Por esse motivo, transformar os elementos de uma
matriz de números reais em intervalos, expandir as propriedades da matemática
intervalar para transformação de matrizes e aplicar as operações intervalares em
matrizes, pode determinar uma melhora na análise de resultados.
Outro aspecto importante da teoria intervalar se dá na sua relação direta
com os números fuzzy, pois esses números tem sua representação clássica (CRIP)
dada em termos de intervalos (i.e. o suporte do número) e portanto, a aritmética
intervalar serve de suporte para a computação na aritmética dos números fuzzy.
As operações aritméticas para intervalos foram definidas por [2], denomina-
das operações clássicas. Utilizando destas definições existentes, pode-se definir
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as operações de matrizes com elementos intervalares, denominadas matrizes in-
tervalares, sendo necessário definir mudança em algumas operações para se obter
o resultado pretendido.
Os resultados objetivados foram, provar e demonstrar que as propriedades
de transformação de matrizes com elementos reais se aplicam a matrizes inter-
valares, aplicando os teoremas e proposições existentes para transformações de
matrizes à matrizes intervalares e verificando se o resultado obtido constitui uma
operação válida para matrizes intervalares.
2 Matrizes Intervalares
Nesta seção serão apresentados alguns resultados importantes para o desenvol-
vimento da teoria do produto de Kronecker para matrizes intervalares.
O conjunto M = I(R) ∪ I(R), sendo
I(R) = {[a, a] ; a ≤ a, a, a ∈ R} e I(R) = {[a, a] ; [a, a] ∈ I (R)} . (1)
Em [1] foi provado que o conjunto M munido com as seguintes operações de
soma e multiplicação
[a1, a2] + [b1, b2] = [a1 + b1, a2 + b2] e [a1,a2] · [b1, b2] = [a1b1, a2b2] . (2)
é um corpo, este fato foi provado em [5]. Na proposição a seguir será mostrado
que o conjunto das matrizes intervalares Mm×n(M), ou seja, o conjunto das
matrizes cuja as entradas são elementos de M é um espaço vetorial sobre M .
Proposição 21 Seja Mm×n(M) o conjunto das matrizes de ordem m× n com
entradas em M . Dizemos que Mm×n(M) é um espaço vetorial sobre M se,
∀A,B,C ∈Mm×n(M) e k, x ∈M , estiverem definidas as seguintes operações
(A+B)ij =
[
[aij , aij ]
]
+
[
[bij , bij ]
]
=
[
[aij , aij ] + [bij , bij ]
]
=
[
[aij + bij , aij + bij ]
]
,
(k ·A)ij =
[
k, k
] [
[aij , aij ]
]
=
[
[k, k][aij , aij ]
]
=
[
[kaij , kaij ]
]
, (3)
com 0 < i ≤ m e 0 < j ≤ n e satisfazerem as seguintes propriedades:
(A1) A+B = B +A, ∀A,B ∈Mm×n(M);
(A2) (A+B) + C = A+ (B + C), ∀A,B,C ∈Mm×n(M);
(A3) Existe um elemento 0 =
[
[0ij , 0ij ]
]
∈ Mm×n(M) tal que A + 0 = A,
∀A ∈Mm×n(M);
(A4) Existe um elemento −A =
[
[−aij ,−aij ]
]
∈Mm×n(M) tal que A+ (−A) =
0;
(M1) (k · x)A = k(x ·A), ∀k, x ∈M e ∀A ∈Mm×n(M);
(M2) Existe um elemento 1 =
[
1, 1
]
∈M tal que A · 1 = A, ∀A ∈Mm×n(M);
(D1) k(A+B) = kA+ kB, ∀k ∈M e ∀A,B ∈Mm×n(M);
(D2) (k + x)A = kA+ xA, ∀k, x ∈M , ∀A ∈Mm×n(M);
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Prova: Visto que, tem-se pouco espaço serão provadas apenas as propriedades
(A1) e (M1), as outras propriedades seguem de maneira análoga. Sejam A =[
[aij , aij ]
]
, B =
[
[bij , bij ]
]
, C =
[
[cij , cij ]
]
pertencentes a Mm×n(M), e sabendo
que M é corpo a prova da propriedade (A1) é dada a seguir
A+B =
[
[aij , aij ]
]
+
[
[bij , bij ]
]
=
[
[aij + bij , aij + bij ]
]
=
[
[bij + aij , bij + aij ]
]
=
[
[bij , bij ]
]
+
[
[aij , aij ]
]
= B +A.
Já a propriedade (M1) é provada como segue
(kx)A = (
[
k, k
]
[x, x])
[
[aij , aij ]
]
=
[
([k, k][x, x])[aij , aij ]
]
(4)
=
[
[k, k]([x, x][aij , aij ])
]
=
[
k, k
] [
([x, x][aij , aij ])
]
=
[
k, k
]
([x, x]
[
[aij , aij ]
]
) = k(xA).
A seguir, algumas definições sobre a teoria de matrizes intervalares.
Definição 21 [4] (Produto de Matrizes) Dadas duas matrizes A ∈ Mm×n(M)
e B ∈Mn×t(M), a matriz produto (A ·B)ij (0 < i ≤ m e 0 < j ≤ t) é dada por
(A ·B)ij =
[
[aij , aij ]
] [
[bij , bij ]
]
=
[
[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ]
]
.
Definição 22 [4] (Matriz Identidade) Seja I ∈ Mn(M). A matriz intervalar
In é denominada matriz intervalar identidade de ordem n se os seus elementos
são dados da seguinte forma (I)ij =
[
0, 0
]
se i 6= j e (I)ij =
[
1, 1
]
se i = j.
Definição 23 [4] (Matriz transposta) Seja A ∈ Mm×n(M). Dizemos que a
transposta de A, e denotamos por AT , é uma matriz n ×m obtida a partir de
trocas das linhas por colunas de A, isto é, se (A)ij =
[
[aij , aij ]
]
então (AT )ji =[
[aji, aji]
]
.
Definição 24 [4] (Matriz Inversa) Sejam A,B ∈ Mn(M). Sendo B = A−1 a
matriz inversa de A, deve então satisfazer AB = I. Assim,
{
(A ·B)ij = [1, 1], para i = j
(A ·B)ij = [0, 0], para i 6= j (5)
=⇒



[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ] = [1, 1], para i = j,
[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ] = [0, 0], para i 6= j
(6)
=⇒



n∑
t=1
aitbtj =
n∑
t=1
aitbtj = 1, para i = j,
n∑
t=1
aitbtj =
n∑
t=1
aitbtj = 0, para i 6= j
(7)
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3 Transformações Elementares de Matrizes Intervalares
Definição 31 (Transformações Elementares) Seja A ∈ Mmxn(M), para cada
Li com 1 ≤ i ≤ m, a i-ésima linha de A. Definimos as transformações elemen-
tares nas linha da matriz A como
(i) Permutação das linhas: Li ↔ Lj;
(ii) Multiplicação da linha Li por um intervalo constante [c, c] ∈M com [c, c] 6=
[0, 0]: Li → [c, c]Li;
(iii) Substituição da linha Li pela adição desta mesma linha com [c, c] vezes uma
outra linha Lj, onde [c, c] ∈M com [c, c] 6= [0, 0]: Li → Li + [c, c]Lj.
A matriz obtida de A aplicando a transformação elementar ‘e”é denotada por
e(A).
Ilustração 31 As transformações elemenares (i), (ii) e (iii) podem ser aplica-
das em uma matriz
A =


[a11, a11] [a12, a12] . . . [a1n, a1n]
[a21, a21] [a22, a22] . . . [a2n, a2n]
...
...
. . .
...
[am1, am1] [am2, am2] . . . [amn, amn]

 (8)
da seguinte forma:
L1 ↔ Lm em A, tem-se:
e(A) =


[am1, am1] [am2, am2] . . . [amn, amn]
[a21, a21] [a22, a22] . . . [a2n, a2n]
...
...
. . .
...
[a11, a11] [a12, a12] . . . [a1n, a1n] .

 (9)
L2 → [c, c]L2 em A, tem-se:
e(A) =


[a11, a11] [a12, a12] . . . [a1n, a1n]
[ca21, ca21] [ca22, ca22] . . . [ca2n, ca2n]
...
...
. . .
...
[am1, am1] [am2, am2] . . . [amn, amn] .

 (10)
L2 → L2 + [c, c]L2 em A, tem-se:
e(A) =


[
a11, a11
] [
a12, a12
]
. . .
[
a1n, a1n
]
[
a21 + ca11, a21 + ca11
] [
a22 + ca12, a22 + ca12
]
. . .
[
a2n + ca1n, a2n + ca1n
]
.
.
.
.
.
.
. . .
.
.
.[
am1, am1
] [
am2, am2
]
. . .
[
amn, amn
]
.

(11)
Definição 32 (Matrizes Equivalentes) Sejam A e B matrizes de ordem m× n.
A matriz A é dita equivalente por linhas a matriz B, se B pode ser obtida de A
pela aplicação sucessiva de transformações elementares sobre linhas.
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Proposição 31 Toda transformação elementar “e”nas linhas de matrizes de
ordem m × n é reverśıvel, no sentido que existe uma transformação “e′”, tal
que, e′(e(A)) = A e e(e′(A)) = A para todo A ∈ Am×n(M).
Demonstração: Considere uma matriz B = e(A) uma matriz obtida por trans-
formações elementares aplicadas na matriz A, sendo B,A ∈ Mm×n(M). Defi-
nindo as transformações elementares reversas
(i) Permutação das linhas: Lj ↔ Li;
(ii) Multiplicação da linha Lj pelo intervalo constante
[
1
c ,
1
c
]
∈ M com [c, c] 6=
[0, 0]: Lj →
[
1
c ,
1
c
]
Lj ;
(iii) Substituição da linha Lj pela subtração desta mesma linha com
[
1
c ,
1
c
]
vezes
uma outra linha Li, onde
[
1
c ,
1
c
]
∈ M com [c, c] 6= [0, 0]: Lj → Lj −
[
1
c ,
1
c
]
Li.
Claramente verifica-se que aplicando as transformações reversas (i), (ii) e (iii)
nas linhas de B obtém-se a matriz A.
Observação 31 Se A é equivalente por linhas a B, então B é equivalente por
linhas a A, visto que as transformações elementares sobre linhas são reverśıveis.
Definição 33 (Matrizes Elementares) Uma matriz elementar intervalar de or-
dem n é uma matriz obtida a partir de transformações elementares aplicadas na
matriz identidade intervalar In, isto é, trata-se de uma matriz da forma
E = e(In). (12)
Teorema 31 Seja “e”uma transformação elementar sobre matrizes intervalares
de ordem n. Considere a matriz elementar intervalar E = e(In), então:
e(A) = EA, A ∈Mn(M). (13)
Demonstração: Aplicando as transformações elementares nas linhas da matriz
identidade e calculando, temos (i) L1 ↔ Ln :
EA = e(In)A =


[0, 0] [0, 0] . . . [1, 1]
[0, 0] [1, 1] . . . [0, 0]
...
...
. . .
...
[1, 1] [0, 0] . . . [0, 0]




[a11, a11] [a12, a12] . . . [a1n, a1n]
[a21, a21] [a22, a22] . . . [a2n, a2n]
...
...
. . .
...
[an1, an1] [an2, an2] . . . [ann, ann]


=


[an1, an1] [an2, an2] . . . [ann, ann]
[a21, a21] [a22, a22] . . . [a2n, a2n]
...
...
. . .
...
[a11, a11] [a12, a12] . . . [a1n, a1n]

 = e(A). (14)
(ii) L2 → [c, c]L2.
EA = e(In)A =


[1, 1] [0, 0] . . . [0, 0]
[0, 0] [c, c] . . . [0, 0]
...
...
. . .
...
[0, 0] [0, 0] . . . [1, 1]




[a11, a11] [a12, a12] . . . [a1n, a1n]
[a21, a21] [a22, a22] . . . [a2n, a2n]
...
...
. . .
...
[an1, an1] [an2, an2] . . . [ann, ann]


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=


[a11, a11] [a12, a12] . . . [a1n, a1n]
[ca21, ca21] [ca22, ca22] . . . [ca2n, ca2n]
...
...
. . .
...
[an1, an1] [an2, an2] . . . [ann, ann]
 = e(A). (15)
(iii) L2 → L2 + [c, c]L1
EA = e(In)A =


[1, 1] [0, 0] . . . [0, 0]
[c, c] [1, 1] . . . [0, 0]
.
.
.
.
.
.
.
.
.
.
.
.
[0, 0] [0, 0] . . . [1, 1]




[
a11, a11
] [
a12, a12
]
. . .
[
a1n, a1n
]
[
a21, a21
] [
a22, a22
]
. . .
[
a2n, a2n
]
.
.
.
.
.
.
.
.
.
.
.
.[
an1, an1
] [
an2, an2
]
. . .
[
ann, ann
]

 =


[
a11, a11
] [
a12, a12
]
. . .
[
a1n, a1n
]
[
a21 + ca11, a21 + ca11
] [
a22 + ca12, a22 + ca12
]
. . .
[
a2n + ca1n, a2n + ca1n
]
.
.
.
.
.
.
.
.
.
.
.
.[
am1, am1
] [
am2, am2
]
. . .
[
amn, amn
]

 = e(A). (16)
O resultado é verificado para operações em qualquer linha da matriz.
Corolário 31 Sejam A,B ∈ Mn(M). Então, A é equivalente a B por linhas
se, e somente se, existem matrizes elementares E1, E2, ..., Es de ordem n, tais
que, Es(...(E2(E1(A))...) = B.
Demonstração: Por definição, A é equivalente a B por linhas quando existem
transformações elementares e1, e2, ..., es, tais que
(es...(e2(e1(
[
[aij , aij ]
]
))...) =
[
[bij , bij ]
]
. (17)
Mas, pelo Teorema 31, sabe-se que (17) é equivalente a
(Es...(E2(E1(
[
[aij , aij ]
]
))...) =
[
[bij , bij ]
]
. (18)
Corolário 32 Toda matriz elementar intervalar é inverśıvel e sua inversa também
é uma matriz elementar intervalar.
Demonstração: Considere as matrizes elementares intervalares E = e(
[
[Iij , Iij ]
]
) =[
[Eij , Eij ]
]
e E′ = e′(
[
[Iij , Iij ]
]
) =
[
[E′ij , E′ij ]
]
, onde “e′”é a transformação ele-
mentar inversa de “e”, pelo Teorema 31, segue que:
[
[Iij , Iij ]
]
= e′(e(
[
[Iij , Iij ])
]
) = e′(
[
[Eij , Eij ]
]
) = e′(
[
[Iij , Iij ]
]
)
[
[Eij , Eij ]
]
=
[
[E′ij , E′ij ]
]
·
[
[Eij , Eij ]
]
, (19)
e
[
[Iij , Iij ]
]
= e(e′(
[
[Iij , Iij ]
]
)) = e(
[
[E′ij , E′ij ]
]
) = e(
[
[Iij , Iij ]
]
)
[
[E′ij , E′ij ]
]
=
[
[Eij , Eij ]
]
·
[
[E′ij , E′ij ]
]
. (20)
Logo,
[
[Eij , Eij ]
]
é inverśıvel e
[
[Eij , Eij ]
]−1
=
[
[E′ij , E′ij ]
]
. (21)
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Proposição 32 Seja A ∈Mn(M) uma matriz intervalar inverśıvel e e1, e2, ..., es
uma sequência de transformações elementares, tais que,
(es...(e2(e1
[
[aij , aij ]
]
))...) =
[
[Iij , Iij ]
]
, (22)
onde In =
[
[Iij , Iij ]
]
é a matriz identidade intervalar. Então, essa mesma
sequência de transformações elementares aplicadas a In, produzem A
−1, isto
é,
(es...(e2(e1(
[
[Iij , Iij ]
]
))...) =
[
[a−1ij , a
−1
ij ]
]
= A−1. (23)
Observação 32 Para transformações elementares de matrizes intervalares, ob-
servou-se que a operação de multiplicação de intervalos usual [2]
[a, a] · [b, b] = [min
{
ab, ab, ab, ab
}
,max
{
ab, ab, ab, ab
}
] (24)
não se aplica, pois para uma transformação elementar ser válida ela deverá ter
uma transformação reversa como descrito na Proposição 31, o que não é posśıvel,
como pode ser visto no exemplo abaixo. Sejam X=[-3,4] e Y=[2,6], pela multi-
plicação (24) tem-se:
X · Y = [min {−6,−18, 8, 24} ,max {−6,−18, 8, 24}] = [−18, 24] = Z. (25)
A reversão direta de (25) é a divisão de Z por X para obter Y e de Z por Y para
obter X, porém quando isto é feito o resultado não é alcançado, como se pode
ver:
Z · 1/X = [min {6,−9/2,−8, 6} ,max {6,−9/2,−8, 6}] = [−8, 6] 6= Y. (26)
Z · 1/Y = [min {−9, 12,−3, 4} ,max {−9, 12,−3, 4}] = [−9, 12] 6= X. (27)
4 Conclusões
Pode-se concluir que as transformações elementares para matrizes se aplicam
a matrizes intervalares, uma vez que sejam utilizadas matrizes intervalares no
espaço vetorial Mmxn e a multiplicação de extremos de intervalos demonstrada
neste trabalho. Além disto, pôde-se definir aplicações fundamentais para trans-
formação de matrizes, como matrizes elementares e a matriz inversa intervalar.
Agradecimentos
A FAPESB e a UESC pela oportunidade concedida em realizar esta pesquisa.
Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF)
16–18 de Novemebro de 2016, Campinas – SP, Brasil.
21
Referências
1. Costa, T. M. and Chalco-Cano, Y. and Lowick,W. A., Generalized interval vector
spaces and interval optimization, Information Sciences, 311, 74-85, 2015.
2. Sunaga, T., Theory of an Interval Algebra and its Application to Numerical Analy-
sis, RAAG Memoirs,1958.
3. Hefez, A. and Fernandes, C. S., Introdução á Álgebra Linear, Coleção PROFMAT,
SBM, 2014.
4. Moore, R. E. and Kearfott, R. B. and Colud, M. J., Introduction to Interval Analysis,
SIAM, 2009.
5. Rufino, V. N. and Palmeira, E. S. and de Jesus, G. Q., Influência de Uma Variável
Fuzzy no Problema dos Mı́nimos Quadrados, Anais do Encontro Nacional de Mo-
delagem Computacional (XVIII ENMC), Salvador, Bahia, Brasil, 2015.
Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF)
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22
Produto de Kronecker para Matrizes
Intervalares
Rudhero M. dos Santos, Gildson Q. de Jesus, and Eduardo S. Palmeira
Programa de Pós-Graduação em Modelagem Computacional em Ciência e Tecnologia
Departamento de Ciências Exatas e Tecnológicas - DCET
Universidade Estadual de Santa Cruz - UESC
espalmeira@uesc.br,
{rudhero,gildsonj}@gmail.com
Resumo Nas últimas três décadas o lugar dos intervalos compactos
como objetos independentes tem crescido continuamente na análise nu-
mérica, na verificação ou determinação de soluções de vários problemas
matemáticos ou na prova de que tais problemas não possuem solução
em um domı́nio particular. Observamos aplicações em diversas áreas tais
como problemas em engenharias, robótica, controle etc. Em especial, no
presente artigo, usamos noções da matemática intervalar para provar um
lema com algumas propriedades do produto de Kronecker, sendo que este
é basicamente aplicado na resolução de equações matriciais a exemplo da
equação de Lyapunov P = FPA + Q.
Keywords: Matrizes Intervalares, Produto de Kronecker, Modelagem
Computacional
1 Introdução
A matemática intervalar é uma área relativamente nova que vem se desenvol-
vendo ao longo dos anos. Os primeiros textos que discorrem sobre tal tema datam
da década de 50 tais como [8]. Na aritmética intervalar introduzida por [8] e [3]
o conjunto dos intervalos reais, denotado por I(R), provê as operações binárias
de adição, subtração, multiplicação e divisão, bem como operações unárias. En-
tretanto essa aritmética não é, em muitos casos, razoável visto que I(R) munido
desses operadores não possui estrutura de corpo, fazendo com que muitas pro-
priedades interessantes não funcionem nesse conjunto.
Nesse artigo não utilizaremos as definições para subtração e multiplicação tal
como na literatura, mas sim uma extensão dos intervalos como o descrito em [2],
visto que, com elas, não conseguimos uma estrutura algébrica consistente como
acontece com o conjunto dos números reais.
Pensando nisso, tomamos o conjunto M como sendo a união entre os inter-
valos próprios (I(R)) e impróprios (I(R)) de forma que todos os intervalos em
M tenham um inverso aditivo bem como um inverso multiplicativo, os quais na
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literatura são denominados de Pseudo Inverso Aditivo e Pseudo Inverso multipli-
cativo. Dessa forma, definimos a multiplicação intervalar como sendo o produto
entre os extremos correspondentes e, com isso, conseguimos mostrar que M é
um corpo e o conjunto das matrizes de ordem qualquer com entradas em M é
um espaço vetorial sobre M . As definições de corpo e espaço vetorial podem ser
consultadas em [7].
No que se refere a aplicação, especialmente do ponto de vista matemático,
pode-se citar problemas intervalares associados à solução de sistemas lineares
ou não lineares, otimização (restrita ou global), determinação de valores e ve-
tores próprios, solução de problemasde contorno e de equações diferenciais,
entre outros. Isto foi posśıvel através da compreensão de intervalos como ex-
tensões de números reais ou complexos, da introdução de funções intervalares e
de aritméticas intervalares. Além disso, temos aplicações em áreas como enge-
nharia mecânica, robótica, controle, etc.
Ainda, como é bem conhecido na literatura, a teoria intervalar tem forte
ligação com a teoria dos conjuntos fuzzy e em especial com os números fuzzy,
através dos alfa cortes. Nesse trabalho buscamos formalizar determinados ope-
radores de matrizes intervalares para servir de base para definirmos operadores
num espaço vetorial fuzzy.
Na seção (2) evidenciamos algumas definições e proposições preliminares a
cerca da matemática intervalar a exemplo de operações de soma e multiplicação
de intervalos e operações entre matrizes intervalares. Tendo em vista a estrutura
algébrica constrúıda, na seção (3) provamos algumas propriedades do produto
de Kronecker para matrizes intervalares condensadas em um lema e verificamos
que, com a multiplicação usual, não se obtém os mesmos resultados. Vale ressal-
tar que o produto de Kronecker é aplicado na resolução de equações matriciais
a exemplo da equação de Lyapunov P = FPA+Q, como pode ser visto em [9].
2 Matrizes Intervalares
Nesta seção serão apresentados alguns resultados importantes para o desenvol-
vimento da teoria do produto de Kronecker para matrizes intervalares.
O conjunto M = I(R) ∪ I(R), sendo
I(R) = {[a, a] ; a ≤ a, a, a ∈ R} e I(R) = {[a, a] ; [a, a] ∈ I (R)} . (1)
Em [1] foi provado que o conjunto M munido com as seguintes operações de
soma e multiplicação
[a1, a2] + [b1, b2] = [a1 + b1, a2 + b2] e [a1, a2] · [b1, b2] = [a1b1, a2b2] . (2)
é um corpo, este fato foi provado em [10]. Na proposição a seguir será mostrado
que o conjunto das matrizes intervalares Mm×n(M), ou seja, o conjunto das
matrizes cuja as entradas são elementos de M é um espaço vetorial sobre M .
Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF)
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Proposição 21 Seja Mm×n(M) o conjunto das matrizes de ordem m× n com
entradas em M . Então,Mm×n(M) é um espaço vetorial sobre M se, ∀A,B,C ∈
Mm×n(M) e k, x ∈M , estiverem definidas as seguintes operações
(A+B)ij =
[
[aij , aij ]
]
+
[
[bij , bij ]
]
=
[
[aij , aij ] + [bij , bij ]
]
=
[
[aij + bij , aij + bij ]
]
,
(k ·A)ij =
[
k, k
] [
[aij , aij ]
]
=
[
[k, k][aij , aij ]
]
=
[
[kaij , kaij ]
]
, (3)
com 0 < i ≤ m e 0 < j ≤ n e satisfazerem as seguintes propriedades:
(A1) A+B = B +A, ∀A,B ∈Mm×n(M);
(A2) (A+B) + C = A+ (B + C), ∀A,B,C ∈Mm×n(M);
(A3) Existe um elemento 0 =
[
[0ij , 0ij ]
]
∈ Mm×n(M) tal que A + 0 = A,
∀A ∈Mm×n(M);
(A4) Existe um elemento −A =
[
[−aij ,−aij ]
]
∈Mm×n(M) tal que A+ (−A) =
0;
(M1) (k · x)A = k(x ·A), ∀k, x ∈M e ∀A ∈Mm×n(M);
(M2) Existe um elemento 1 =
[
1, 1
]
∈M tal que A · 1 = A, ∀A ∈Mm×n(M);
(D1) k(A+B) = kA+ kB, ∀k ∈M e ∀A,B ∈Mm×n(M);
(D2) (k + x)A = kA+ xA, ∀k, x ∈M , ∀A ∈Mm×n(M);
Prova: Visto que, tem-se pouco espaço serão provadas apenas as propriedades
(A1) e (M1), as outras propriedades seguem de maneira análoga. Sejam A =[
[aij , aij ]
]
, B =
[
[bij , bij ]
]
, C =
[
[cij , cij ]
]
pertencentes a Mm×n(M), e sabendo
que M é corpo a prova da propriedade (A1) é dada a seguir
A+B =
[
[aij , aij ]
]
+
[
[bij , bij ]
]
=
[
[aij + bij , aij + bij ]
]
=
[
[bij + aij , bij + aij ]
]
=
[
[bij , bij ]
]
+
[
[aij , aij ]
]
= B +A.
Já a propriedade (M1) é provada como segue
(kx)A = (
[
k, k
]
[x, x])
[
[aij , aij ]
]
=
[
(kx, kx)
] [
[aij , aij ]
]
=
[
[(kx), (kx)][aij , aij ]
]
=
[
([k, k][x, x])[aij , aij ]
]
=
[
[k, k]([x, x][aij , aij ])
]
=
[
k, k
] [
([x, x][aij , aij ])
]
=
[
k, k
]
([x, x]
[
[aij , aij ]
]
) = k(xA).
A seguir, algumas definições sobre a teoria de matrizes intervalares.
Definição 21 [8] (Produto de Matrizes) Dadas duas matrizes A ∈ Mm×n(M)
e B ∈Mn×t(M), a matriz produto (A ·B)ij (0 < i ≤ m e 0 < j ≤ t) é dada por
(A ·B)ij =
[
[aij , aij ]
] [
[bij , bij ]
]
=
[
[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ]
]
.
Definição 22 [8] (Matriz Identidade) Seja I ∈ Mn(M). A matriz intervalar
In é denominada matriz intervalar identidade de ordem n se os seus elementos
são dados da seguinte forma (I)ij =
[
0, 0
]
se i 6= j e (I)ij =
[
1, 1
]
se i = j.
Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF)
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Definição 23 [8] (Matriz transposta) Seja A ∈ Mm×n(M). Dizemos que a
transposta de A, e denotamos por AT , é uma matriz n ×m obtida a partir de
trocas das linhas por colunas de A, isto é, se (A)ij =
[
[aij , aij ]
]
então (AT )ij =[
[aji, aji]
]
.
Definição 24 [8] (Matriz Inversa) Sejam A,B ∈ Mn(M). Sendo B = A−1 a
matriz inversa de A, deve então satisfazer AB = I. Assim,
{
(A ·B)ij = [1, 1], para i = j
(A ·B)ij = [0, 0], para i 6= j (4)
=⇒



[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ] = [1, 1], para i = j,
[
n∑
t=1
aitbtj ,
n∑
t=1
aitbtj ] = [0, 0], para i 6= j
(5)
=⇒



n∑
t=1
aitbtj =
n∑
t=1
aitbtj = 1, para i = j,
n∑
t=1
aitbtj =
n∑
t=1
aitbtj = 0, para i 6= j
(6)
A proposição a seguir será muito útil para a prova das propriedades do produto
de Kronecker.
Proposição 22 Sejam U ∈ Mm×n(M) e V ∈ Mn×p(M) e sejam k =
[
k, k
]
,
x = [x, x] ∈M . Então, (kU)(xV ) = (kx)(UV ).
Prova: De fato, usando as definições e propriedades supracitadas, temos que
(kU)(xV ) = (
[
k, k
] [
[uij , uij ]
]
)([x, x]
[
[vij , vij ]
]
) =
[
[kuij , kuij ]
] [
[xvij , xvij ]
]
=
[
[
n∑
t=1
(kuit)(xvtj),
n∑
t=1
(kuit)(xvtj)]
]
=
[
kx, kx
] [
[
n∑
t=1
uitvtj ,
n∑
t=1
uitvtj ]
]
=
[
k, k
]
[[x, x]]
[
[uij , uij ]
] [
[vij , vij ]
]
= (kx)(UV ).
3 Produto de Kronecker para Matrizes Intervalares
Definição 31 O produto de Kronecker de A e B ∈ Mm×n(M), escrevemos
A⊗B, é uma matriz de ordem mn×mn, definida por:
A⊗B =


[a11, a11]B [a12, a12]B . . . [a1m, a1m]B
[a21, a21]B [a22, a22]B . . . [a2m, a2m]B
...
...
. . .
...
[am1, am1]B [am2, am2]B . . . [amm, amm]B

 . (7)
Quarto Congresso Brasileiro de Sistemas Fuzzy (IV CBSF)
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Lema 31 Dadas as matrizes A ∈ Mm(M), B ∈ Mn(M), C ∈ Mm×k(M) e
D ∈Mn×p(M), as seguintes afirmações são verdadeiras:
(i) (A⊗B)(C ⊗D) = AC ⊗BD.
(ii) Se A e B são inverśıveis, então (A⊗B)−1 = A−1 ⊗B−1.
(iii) Para todas as matrizes A,B,C e X ∈ Mm×n(M), se C = AXB, então
vec(C) = (BT ⊗A)vec(X).
Prova: (i) Calculando (A⊗C)(B⊗D), usando a Definição (31) e a Proposição
(22), obtém-se
(A⊗ C)(B ⊗D) =


[
m∑
t=1
a1tct1,
m∑
t=1
a1tct1
] [
[
n∑
t=1
bitdtj ,
n∑
t=1
bitdtj ]
]
. . .
[
m∑
t=1
a1tctk,
m∑
t=1
a1tctk
] [
[
n∑
t=1
bikdkj,
n∑
k=1
bitdtj ]
]
.
.
.
.
.
.
.
.
.[
m∑
t=1
amtct1,
m∑
t=1
a1tct1
] [
[
n∑
t=1
bitdtj ,
n∑
t=1
bitdtj ]
]
. . .
[
m∑
t=1
amtctk,
m∑
t=1
a1tctk
] [
[
n∑
t=1
bikdkj,
n∑
k=1
bitdtj ]
]

(8)
Por outro lado, AC e BD são dados por
AC =
[
[
m∑
t=1
aitctj ,
m∑
t=1
aitctj ]
]
e BD =
[
[
n∑
t=1
bitdtj ,
n∑
t=1
bitdtj ]
]
. (9)
Após alguns cálculos pode-se ver que o produto de Kronecker AC ⊗BD é dado
pela mesma representação obtida em (8). (ii) Considerando o item (i), seja
C = A−1 e D = B−1, então AC = I e BD = I. Assim,
(A⊗B)(C ⊗D) = AC ⊗BD = I ⊗ I =


[1, 1] I [0, 0] . . . [0, 0]
[0, 0] [1, 1] I . . . [0, 0]
...
...
. . .
...
[0, 0] [0, 0] . . . [1, 1] I

 . (10)
Logo, pode-se concluir que a matriz C⊗D é a matriz inversa de A⊗B e, portanto,
segue o resultado. (iii) Sejam A,B,C e X matrizes de ordem n. Calculando a
matriz C = AXB, tem-se
C = AXB =


[
n∑
l=1
(bl1
n∑
t=1
a1txtl),
n∑
l=1
(bl1
n∑
t=1
a1txtl)
]
. . .
[
n∑
l=1
(bln
n∑
t=1
a1txtl),
n∑
l=1
(bln
n∑
t=1
a1txtl)
]
.
.
.
.
.
.
.
.
.[
n∑
l=1
(bl1
n∑
t=1
antxtl),
n∑
l=1
(bl1
n∑
t=1