Fuzzy lattice reasoning (FLR) classifier and its application for ambient ozone estimation

The fuzzy lattice reasoning (FLR) classifier is presented for inducing descriptive, decision-making knowledge (rules) in a mathematical lattice data domain including space R^N. Tunable generalization is possible based on non-linear (sigmoid) positive valuation functions; moreover, the FLR classifier can deal with missing data. Learning is carried out both incrementally and fast by computing disjunctions of join-lattice interval conjunctions, where a join-lattice interval conjunction corresponds to a hyperbox in R^N. Our testbed in this work concerns the problem of estimating ambient ozone concentration from both meteorological and air-pollutant measurements. The results compare favorably with results obtained by C4.5 decision trees, fuzzy-ART as well as back-propagation neural networks. Novelties and advantages of classifier FLR are detailed extensively and in comparison with related work from the literature.

[1]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[2]  Thomas G. Dietterich,et al.  Solving the Multiple Instance Problem with Axis-Parallel Rectangles , 1997, Artif. Intell..

[3]  Michael R. Berthold,et al.  Constructing fuzzy graphs from examples , 1999, Intell. Data Anal..

[4]  Henk J. A. M. Heijmans,et al.  Nonlinear multiresolution signal decomposition schemes. I. Morphological pyramids , 2000, IEEE Trans. Image Process..

[5]  Athanasios Kehagias,et al.  L-fuzzy valued inclusion measure, L-fuzzy similarity and L-fuzzy distance , 2001, Fuzzy Sets Syst..

[6]  Vassilios Petridis,et al.  Fuzzy Lattice Neurocomputing (FLN) models , 2000, Neural Networks.

[7]  José Miguel Mantas,et al.  Extraction of similarity based fuzzy rules from artificial neural networks , 2006, Int. J. Approx. Reason..

[8]  Yoav Shoham,et al.  An overview of agent-oriented programming , 1997 .

[9]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, CACM.

[10]  Vassilios Petridis,et al.  Learning in the framework of fuzzy lattices , 1999, IEEE Trans. Fuzzy Syst..

[11]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[12]  Heiko Timm,et al.  Different approaches to fuzzy clustering of incomplete datasets , 2004, Int. J. Approx. Reason..

[13]  Athanasios Kehagias,et al.  Novel Fuzzy Inference System (FIS) Analysis and Design Based on Lattice Theory , 2007, IEEE Transactions on Fuzzy Systems.

[14]  Petros Maragos,et al.  Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems , 2005, Journal of Mathematical Imaging and Vision.

[15]  E. H. Mamdani,et al.  An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller , 1999, Int. J. Man Mach. Stud..

[16]  Richard L. Smith,et al.  Meteorologically‐dependent trends in urban ozone , 1999 .

[17]  Didier Dubois,et al.  A New Perspective on Reasoning with Fuzzy Rules , 2002, AFSS.

[18]  J. Ross Quinlan,et al.  Decision trees and decision-making , 1990, IEEE Trans. Syst. Man Cybern..

[19]  Vassilis G. Kaburlasos.  Adaptive resonance theory with supervised learning and large database applications , 1992 .

[20]  Jerry M. Mendel,et al.  Type-2 fuzzy logic systems , 1999, IEEE Trans. Fuzzy Syst..

[21]  Plamen Angelov,et al.  Evolving Rule-Based Models: A Tool For Design Of Flexible Adaptive Systems , 2002 .

[22]  Athanasios Kehagias,et al.  Novel fuzzy inference system (FIS) analysis and design based on lattice theory. Part I: Working principles , 2006, Int. J. Gen. Syst..

[23]  Kevin H. Knuth,et al.  Lattice duality: The origin of probability and entropy , 2013, Neurocomputing.

[24]  John F. Sowa,et al.  Knowledge representation: logical, philosophical, and computational foundations , 2000 .

[25]  Chris Cornelis,et al.  Efficient Approximate Reasoning with Positive and Negative Information , 2004, KES.

[26]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[27]  Michael R. Genesereth,et al.  Software agents , 1994, CACM.

[28]  Andreja Tepavcevic,et al.  L-fuzzy lattices: an introduction , 2001, Fuzzy Sets Syst..

[29]  Anil K. Jain,et al.  Data clustering: a review , 1999, CSUR.

[30]  Philip M. Long,et al.  PAC Learning Axis-Aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples , 1996, COLT.

[31]  Donald E. Knuth,et al.  Sorting and Searching , 1973 .

[32]  Leslie G. Valiant,et al.  Learning Boolean formulas , 1994, JACM.

[33]  David Haussler,et al.  Learning Conjunctive Concepts in Structural Domains , 1989, Machine Learning.

[34]  Weixin Xie,et al.  Subsethood measure: new definitions , 1999, Fuzzy Sets Syst..

[35]  Ah-Hwee Tan,et al.  Cascade ARTMAP: integrating neural computation and symbolic knowledge processing , 1997, IEEE Trans. Neural Networks.

[36]  V. Petridis,et al.  Learning and decision-making in the framework of fuzzy lattices , 2002 .

[37]  Thomas P. Caudell,et al.  Acquiring rule sets as a product of learning in a logical neural architecture , 1997, IEEE Trans. Neural Networks.

[38]  Ernest Edmonds.  Lattice Fuzzy Logics , 1980, Int. J. Man Mach. Stud..

[39]  C. J. Mantas,et al.  Extraction of OR fuzzy rules from Artificial Neural Networks , 2006 .

[40]  Aiko M. Hormann,et al.  Programs for Machine Learning. Part I , 1962, Inf. Control..

[41]  Pericles A. Mitkas,et al.  Applying Machine Learning Techniques on Air Quality Data for Real-Time Decision Support , 2003 .

[42]  E. Dougherty,et al.  Fuzzification of set inclusion: theory and applications , 1993 .

[43]  Edward R. Dougherty,et al.  Computational Gray-scale Mathematical Morphology on Lattices (A Comparator-based Image Algebra) Part II: Image Operators , 1995, Real Time Imaging.

[44]  W. Pedrycz,et al.  Fuzzy computing for data mining , 1999, Proc. IEEE.

[45]  J. Goguen.  L-fuzzy sets , 1967 .

[46]  Georgios C. Anagnostopoulos,et al.  Category regions as new geometrical concepts in Fuzzy-ART and Fuzzy-ARTMAP , 2002, Neural Networks.

[47]  Chuen-Tsai Sun,et al.  Neuro-fuzzy modeling and control , 1995, Proc. IEEE.

[48]  V. Prybutok,et al.  A neural network model forecasting for prediction of daily maximum ozone concentration in an industrialized urban area. , 1996, Environmental pollution.

[49]  Edward R. Dougherty,et al.  A general axiomatic theory of intrinsically fuzzy mathematical morphologies , 1995, IEEE Trans. Fuzzy Syst..

[50]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[51]  David Haussler,et al.  Learnability and the Vapnik-Chervonenkis dimension , 1989, JACM.

[52]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[53]  Virginia R. Young,et al.  Fuzzy subsethood , 1996, Fuzzy Sets Syst..

[54]  Vassilis G. Kaburlasos.  FINs: lattice theoretic tools for improving prediction of sugar production from populations of measurements , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[55]  Leo Breiman,et al.  Bagging Predictors , 1996, Machine Learning.

[56]  Andrzej Bargiela,et al.  General fuzzy min-max neural network for clustering and classification , 2000, IEEE Trans. Neural Networks Learn. Syst..

[57]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[58]  Thomas G. Dietterich.  What is machine learning? , 2020, Archives of Disease in Childhood.

[59]  Zhaohao Sun,et al.  Case base building with similarity relations , 2004, Inf. Sci..

[60]  Stuart C. Shapiro.  Review of Knowledge representation: logical, philosophical, and computational foundations by John F. Sowa. Brooks/Cole 2000. , 2001 .

[61]  Przemyslaw Grzegorzewski,et al.  Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric , 2004, Fuzzy Sets Syst..

[62]  Vassilis G. Kaburlasos,et al.  Granular self-organizing map (grSOM) for structure identification , 2006, Neural Networks.

[63]  Vassilios Petridis,et al.  Fuzzy lattice neural network (FLNN): a hybrid model for learning , 1998, IEEE Trans. Neural Networks.

[64]  Gerhard X. Ritter,et al.  Lattice algebra approach to single-neuron computation , 2003, IEEE Trans. Neural Networks.

[65]  Andrzej Bargiela,et al.  Granular clustering: a granular signature of data , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[66]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[67]  J. Ross Quinlan,et al.  C4.5: Programs for Machine Learning , 1992 .

[68]  Jerry M. Mendel,et al.  Interval type-2 fuzzy logic systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[69]  Ian H. Witten,et al.  Data mining: practical machine learning tools and techniques with Java implementations , 2002, SGMD.

[70]  P. K. Simpson.  Fuzzy Min-Max Neural Networks-Part 1 : Classification , 1992 .

[71]  Ioannis B. Theocharis,et al.  A GA-based fuzzy modeling approach for generating TSK models , 2002, Fuzzy Sets Syst..

[72]  Donald E. Knuth,et al.  The Art of Computer Programming, Volumes 1-3 Boxed Set , 1998 .

[73]  Shigeo Abe DrEng.  Pattern Classification , 2001, Springer London.

[74]  Kankana Chakrabarty.  On Fuzzy Lattice , 2000, Rough Sets and Current Trends in Computing.

[75]  M. Jenkin,et al.  Analysis of the relationship between ambient levels of O3, NO2 and NO as a function of NOx in the UK , 2001 .

[76]  Edward R. Dougherty,et al.  Computational Gray-scale Mathematical Morphology on Lattices (A Comparator-based Image Algebra) Part 1: Architecture , 1995, Real Time Imaging.

[77]  Christos H. Papadimitriou,et al.  On the analysis of indexing schemes , 1997, PODS '97.

[78]  Chris Cornelis,et al.  Sinha-Dougherty approach to the fuzzification of set inclusion revisited , 2003, Fuzzy Sets Syst..

[79]  Brian R. Gaines,et al.  Fuzzy and Probability Uncertainty Logics , 1993 .

[80]  S. Salzberg.  A nearest hyperrectangle learning method , 2004, Machine Learning.

[81]  Yvon Savaria,et al.  Generalization, discrimination, and multiple categorization using adaptive resonance theory , 1999, IEEE Trans. Neural Networks.

[82]  Uta Priss,et al.  Lattice-based information retrieval , 2000 .

[83]  Detlef D. Nauck.  Special issue on hybrid methods for adaptive systems , 2004, Fuzzy Sets Syst..

[84]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[85]  Jacek M. Zurada,et al.  Extraction of rules from artificial neural networks for nonlinear regression , 2002, IEEE Trans. Neural Networks.

[86]  Pedro M. Domingos.  Two-way induction , 1995, Proceedings of 7th IEEE International Conference on Tools with Artificial Intelligence.

[87]  Peter N. Brett,et al.  Estimation of the stapes-bone thickness in the stapedotomy surgical procedure using a machine-learning technique , 1999, IEEE Transactions on Information Technology in Biomedicine.

[88]  Stanislaw Heilpern,et al.  Representation and application of fuzzy numbers , 1997, Fuzzy Sets Syst..

[89]  Thomas G. Dietterich,et al.  An experimental comparison of the nearest-neighbor and nearest-hyperrectangle algorithms , 1995, Machine Learning.

[90]  Rudolf Kruse,et al.  Neuro-fuzzy systems for function approximation , 1999, Fuzzy Sets Syst..

[91]  Vassilios Petridis,et al.  Clustering and Classification in Structured Data Domains Using Fuzzy Lattice Neurocomputing (FLN) , 2001, IEEE Trans. Knowl. Data Eng..

[92]  Nikolaos Avouris,et al.  Air Quality Management Using a Multi‐Agent System , 2002 .

[93]  Patrick K. Simpson,et al.  Fuzzy min-max neural networks. I. Classification , 1992, IEEE Trans. Neural Networks.

[94]  Zhi-Qiang Liu,et al.  Fuzzy neural network in case-based diagnostic system , 1997, IEEE Trans. Fuzzy Syst..

[95]  Janet L. Kolodner,et al.  Case-Based Reasoning , 1988, IJCAI 1989.

[96]  Naseem Ajmal,et al.  Fuzzy Lattices , 1994, Inf. Sci..

[97]  Àngela Nebot,et al.  Reasoning under uncertainty with FIR methodology , 2006, Int. J. Approx. Reason..

[98]  Peter Sussner,et al.  Associative morphological memories based on variations of the kernel and dual kernel methods , 2003, Neural Networks.

[99]  Patrick K. Simpson,et al.  Fuzzy min-max neural networks - Part 2: Clustering , 1993, IEEE Trans. Fuzzy Syst..

[100]  Claudio Carpineto,et al.  A lattice conceptual clustering system and its application to browsing retrieval , 2004, Machine Learning.

[101]  Athanasios Kehagias,et al.  A Comparison of Word- and Sense-Based Text Categorization Using Several Classification Algorithms , 2003, Journal of Intelligent Information Systems.

[102]  Umesh V. Vazirani,et al.  An Introduction to Computational Learning Theory , 1994 .

[103]  Jose Torres-Jimenez,et al.  Short-term ozone forecasting by artificial neural networks , 1995 .

[104]  Rudolf Kruse,et al.  Obtaining interpretable fuzzy classification rules from medical data , 1999, Artif. Intell. Medicine.

[105]  J. Ross Quinlan,et al.  Induction of Decision Trees , 1986, Machine Learning.

[106]  Pedro M. Domingos.  Unifying Instance-Based and Rule-Based Induction , 1996, Machine Learning.

[107]  Ian Witten,et al.  Data Mining , 2000 .

[108]  N. Schaumberger.  Generalization , 1989, Whitehead and Philosophy of Education.

[109]  Bart Kosko,et al.  Fuzzy Engineering , 1996 .

[110]  Elena Baralis,et al.  Compile-Time and Runtime Analysis of Active Behaviors , 1998, IEEE Trans. Knowl. Data Eng..

[111]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[112]  Hisao Ishibuchi,et al.  Performance evaluation of fuzzy classifier systems for multidimensional pattern classification problems , 1999, IEEE Trans. Syst. Man Cybern. Part B.

[113]  Ath. Kehagias,et al.  An example of L-fuzzy join space , 2002 .

[114]  Vassilios Petridis,et al.  FINkNN: A Fuzzy Interval Number k-Nearest Neighbor Classifier for Prediction of Sugar Production from Populations of Samples , 2003, J. Mach. Learn. Res..

[115]  Rudolf Kruse,et al.  A neuro-fuzzy method to learn fuzzy classification rules from data , 1997, Fuzzy Sets Syst..

[116]  Vassilis G. Kaburlasos.  Towards a Unified Modeling and Knowledge-Representation based on Lattice Theory: Computational Intelligence and Soft Computing Applications (Studies in Computational Intelligence) , 2006 .

[117]  Satish Kumar,et al.  Subsethood-product fuzzy neural inference system (SuPFuNIS) , 2002, IEEE Trans. Neural Networks.

[118]  David G. Stork,et al.  Pattern Classification , 1973 .

[119]  Yannis A. Dimitriadis,et al.  Study of distributed learning as a solution to category proliferation in Fuzzy ARTMAP based neural systems , 2003, Neural Networks.

[120]  Pericles A. Mitkas,et al.  An agent-based intelligent environmental monitoring system , 2004, ArXiv.

[121]  Ramón Fuentes-González,et al.  Inclusion grade and fuzzy implication operators , 2000, Fuzzy Sets Syst..

[122]  Vijay V. Raghavan,et al.  A new fuzzy clustering algorithm for optimally finding granular prototypes , 2005, Int. J. Approx. Reason..