Application of genetic algorithms and Kohonen networks to cluster analysis
|
|
- Sharyl McCarthy
- 5 years ago
- Views:
Transcription
1 Application of genetic algorithms and Kohonen networks to cluster analysis Marian B. Gorza lczany and Filip Rudziński Department of Electrical and Computer Engineering Kielce University of Technology Al lecia P.P. 7, Kielce, Poland Abstract. The paper presents two methods offering flexible solutions to cluster-analysis problems. The first one employs a genetic-algorithmbased Travelling-Salesman-Problem-solution, and the second one - selforganizing Kohonen networks. The operation of both techniques has been illustrated with the use of synthetic data set and then they have been tested and compared by means of real-life multidimensional Mushrooms Database (8124 records) available from the FTP server of the University of California at Irvine (ftp.ics.uci.edu). 1 Introduction Cluster analysis is an important topic in various problems of classification, pattern recognition, association, intelligent decision support, etc. [2]. The performance of the majority of existing clustering methods is influenced by several more or less subjective factors such as the choice of the number of clusters (usually unknown in advance), the choice of the similarity measure, the initial cluster centers, different cluster-validity criteria, and so on. This paper outlines two methods offering more flexible solutions to clusteranalysis problems. These methods aim at providing the user with an image (in two-dimensional space) of the cluster distribution in a given data set. The first approach employs a genetic-algorithm-based solution of the Travelling Salesman Problem, and the second one - a self-organizing Kohonen network. First, both techniques have been presented by means of an exemplary, synthetic data set and then they have been tested using a real-life, multidimensional data set (the so-called Mushrooms Database) [7]. The considerations presented in this paper are continued in paper [1] included in this volume. 2 Cluster analysis by means of genetic algorithms Travelling Salesman Problem (TSP for short), conceptually, can be formulated as follows: the travelling salesman must visit every city in the area of his activity exactly once and then return to the starting point. There is also a variant of TSP without the demand of returning to the starting point - it of the interest
2 in the present paper. Given the cost of travel between all cities, the salesman must plan his itinerary so as to achieve the minimal total cost of the entire tour [5]; in our approach it is equivalent to designing the minimal (shortest) route connecting all cities). Application of a genetic-algorithm-based TSP-solution method to cluster analysis consists of two main stages. In the first stage, a minimal route connecting all the points of the considered data set should be determined. An analysis of distances between neighbouring points along the route obtained in such a way provides the user with an image of the internal structure of the data set. It also gives the basis for the inference regarding the number and position of clusters within a given data set, which is an essence of cluster analysis. Therefore, in the second stage of the proposed methodology, a histogram of distances between neighbouring points along the route is determined and then the corresponding histogram of nearness is calculated. The distance d i,j between two points A i, A j R n (A i = (a i1, a i2,..., a in ), A j = (a j1, a j2,..., a jn )) can be determined, e.g., by means of the Euclidean norm as follows d i,j = d(a i, A j ) = A i A j = n (a ik a jk ) 2 (1) The histogram of distances Hi dist between two neighbouring points A i, A i+1 along the route (i = 1, 2,..., r 1, r - number of points along the route) is defined by Hi dist = d i,i+1 and the corresponding histogram of nearness Hi near - by the following k=1 H near i = ( max j=1,2,...,r 1 Hdist j ) H dist i = ( max d j,j+1) d i,i+1, i = 1, 2,..., r 1 j=1,2,...,r 1 (2) The higher the bars of the nearness histogram are, the closer the corresponding data points are situated (they belong to a more compact clusters). Fig. 1a presents a synthetic data set (250 points on the plane) for the purpose of illustration of the proposed technique. Using genetic algorithm, a minimal route connecting all the points has been determined as shown in Fig. 1b. In turn, in Fig. 2, the histogram of nearness - calculated according to (2) - has been presented. Although the histogram shows the existence of 3 clusters in the data set, sometimes (including the present case) it is worth to subject the histogram a filtering or smoothing action. Fig. 3 shows an approximation of nearness histogram of Fig. 2 by means of a set of second-order polynomials [4]. An analysis of either original histogram or the approximated one gives an image of the cluster distribution in the given data set. A simple criterion for determining the number and locations of clusters may be a cut-off of the histogram at some level of nearness (see Fig. 3) associated with the degree of compactness of the clusters obtained. The higher the histogram cut-off level, the more compact (characterized by small and close-to-each-other distances between data points) clusters in data set are generated. On the other hand, the lower the histogram
3 cut-off level, the more fuzzy clusters are produced. This technique also allows us to define membership functions for fuzzy relations formally representing particular clusters. a) b) Fig. 1. Exemplary, synthetic data set (a) and a minimal route in it determined by a genetic algorithm (b) +LVWRJUDPRIQHDUQHVV Fig. 2. Histogram of nearness between neighbouring points along the route of Fig. 1b $SSUR[LPDWLRQRIQHDUQHVVKLVWRJUDP ([HPSODU\OHYHORIFXWRIIRIKLVWRJUDP Fig. 3. Approximation of the nearness histogram of Fig. 2 There is one drawback of the proposed technique (although not affecting - in a significant way - the results obtained): not always remote points along the route generated by genetic algorithm in a given data set are remote in the original data set. In order to alleviate this problem, the afore-presented technique can be treated, conceptually, as a point of departure to other approach. Instead of looking for the route crossing all the original data points, it is worth considering to find shorter, simplified, averaged route coming through low-level, locally defined (by the method itself) clusters in the original data space. An excellent tool for performing this task is offered by self-organizing Kohonen networks [3].
4 3 Cluster analysis with the use of Kohonen networks In this paper, a self-organizing Kohonen network with one-dimensional neighbourhood is used. The network has an input layer with n units x 1, x 2,..., x n and an output layer with m neurons arranged in a chain; their outputs are y 1, y 2,..., y m, where y j = n i=1 w jix i, j = 1, 2,..., m and w ji are weights connecting the output of j-th neuron with i-th input. Using vector notation (x = (x 1, x 2,..., x n ), w j = (w j1, w j2,..., w jn )), y j = w j x. In the learning process, the network aims - through the competition of neurons - to such an arrangement of neurons (that is, the selection of their weight vectors) to minimize a global error E of approximation of particular input vectors x l (l = 1, 2,..., L) by the weight vectors w wl of neurons w l winning in the competition when x l is presented. Applying the Euclidean norm (1), global error E can be expressed as follows: E = 1 L L l=1 x l w wl. Assuming the normalization of input vector and using the Euclidean norm, the winning neuron for the input vector is selected such that x l w wl = min j=1,2,...,m x l w j. In practice, Winner-Takes-Most (WTM) learning algorithms are applied, in which not only the winning neuron but also neurons from its specific neighbourhood update their weights. However, the further a given neuron in the neighbourhood of the winning one is located, the lesser update of its weights takes place. The learning rule can be formulated as follows: w l (k +1) = w j (k)+ η j (k)n(j, wx, k)[x (k) w j (k)], where η j (k) is the learning coefficient, N(j, wx, k) is the neighbourhood function (both shrink while the learning progresses) and k is the iteration number. There are various WTM learning algorithms with different neighbourhood functions N(j, wx, k). In this paper, a linear neighbourhood function has been used: N(j, wx, k) = { 1 j w x η j (k), for j wx λ, 0, for j wx > λ, (3) where λ is the number of neurons belonging to the neighbourhood of the winning neuron and, obviously, j wx 1 is always fulfilled for j wx λ. Fig. 4 shows the route determined by the self-organizing network with 50 neurons arranged in the chain for the data set of Fig. 1. Fig. 5 presents a polynomial approximation of the nearness histogram for the route of Fig. 4. An analysis of the histogram gives a clear image of the cluster distribution in the original data set. Introducing - as in the previous section of the paper - a cut-off of the histogram at some level of nearness (see Fig. 5) allows us to determine either more compact or more fuzzy clusters. 4 Application of both techniques to multidimensional cluster-analysis problems Both techniques that have been presented in previous sections of the paper, now will be tested with the use of a real-life, multidimensional data set (the so-called
5 Mushrooms Database) [7]. It is worth mentioning that the number of classes (equal to 4) and the class assignments are known here; it allows us for direct verification of the results obtained. Fig. 4. Minimal route determined by 50-neuron self-organizing network for data set of Fig. 1 Fig. 6. Sammon s planar mapping of multidimensional attribute space of Mushrooms Database $SSUR[LPDWLRQRIQHDUQHVVKLVWRJUDP ([HPSODU\OHYHORIFXWRIIRIKLVWRJUDP Fig. 5. Approximation of the nearness histogram between neighbouring points along the route of Fig. 4 The Mushrooms Database is the data set that contains as many as 8124 records (mushroom descriptions); each record is described by 22 nominal attributes. Due to high dimensionality of the attribute space, an important issue is graphical presentation of the distribution of these data in a way the human being is able to comprehend. For this purpose, a well-known Sammon s mapping [6] for the presentation of multi-dimensional data on the plane has been used. Fig. 6 presents the Sammon s planar mapping of multidimensional attribute space of Mushrooms Database. Fig. 7 shows the envelope of the polynomial approximation of the nearness histogram for the minimal route - determined by means of genetic algorithm - connecting all the data points. Three local minima of the plot of Fig. 7 indicate the boundaries between particular four clusters (classes). The percentage of correct decisions regarding the assignment of data points to particular classes is very high (91.6%). In turn, the self-organizing network with 100 neurons arranged in a chain has been applied to determine a simplified, averaged route through the attribute space of Mushrooms Database. Fig. 8 presents the nearness histogram for this route. Without performing a polynomial approximation, four bar clusters - corresponding to four clusters (classes) in original data space - are clearly visible in Fig. 8. The percentage of correct decisions is also very high (93.2 %).
6 $SSUR[LPDWLRQRIQHDUQHVVKLVWRJUDP Fig. 7. Envelope of the approximation of the nearness histogram for minimal route in attribute space of Mushrooms Database determined by genetic algorithm +LVWRJUDPRIQHDUQHVV Fig. 8. Histogram of nearness for minimal route in attribute space of Mushroom Database determined by 100-neuron self-organizing network 5 Conclusions Two methods offering flexible solutions to cluster-analysis problems have been presented in this paper. The first one employs the genetic-algorithm-based TSPsolution, and the second one - the self-organizing Kohonen networks. The operation of both techniques has been illustrated with the use of synthetic data set and then they have been tested by means of real-life, multidimensional Mushrooms Database (8124 records) [7]. The considerations presented in this paper are continued in paper [1] included in this volume. References 1. Gorza lczany M.B., Rudziński F.: Generalized Kohonen networks for complex cluster-analysis problems, in Proc. of Seventh Int. Conference on Artificial Intelligence and Soft Computing IEEE ICAISC 2004, Zakopane, 2004, in this volume. 2. Höppner F., Klawon F., Kruse R., Runkler T.: Fuzzy Cluster Analysis. J.Wiley&Sons, Chichester, Kohonen T.: Self-organizing Maps. Springer-Verlag, Berlin, Mathcad 2001 Professional (Polynomial regression). MathSoft, Inc., Michalewicz Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, Berlin, Heidelberg, Sammon J.: A nonlinear mapping for data structure analysis. IEEE Trans. on Computers 18, 1969, pp Machine Learning Database Repository, University of California at Irvine (ftp.ics.uci.edu).
Generalized Tree-Like Self-Organizing Neural Networks with Dynamically Defined Neighborhood for Cluster Analysis
Generalized Tree-Like Self-Organizing Neural Networks with Dynamically Defined Neighborhood for Cluster Analysis Marian B. Gorza lczany, Jakub Piekoszewski, and Filip Rudziński Department of Electrical
More informationGeneralized SOMs with Splitting-Merging Tree-Like Structures for WWW-Document Clustering
1th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT) Generalized SOMs with Splitting-Merging Tree-Like
More informationRepresentation of 2D objects with a topology preserving network
Representation of 2D objects with a topology preserving network Francisco Flórez, Juan Manuel García, José García, Antonio Hernández, Departamento de Tecnología Informática y Computación. Universidad de
More informationFunction approximation using RBF network. 10 basis functions and 25 data points.
1 Function approximation using RBF network F (x j ) = m 1 w i ϕ( x j t i ) i=1 j = 1... N, m 1 = 10, N = 25 10 basis functions and 25 data points. Basis function centers are plotted with circles and data
More informationFuzzy Inspired Hybrid Genetic Approach to Optimize Travelling Salesman Problem
Fuzzy Inspired Hybrid Genetic Approach to Optimize Travelling Salesman Problem Bindu Student, JMIT Radaur binduaahuja@gmail.com Mrs. Pinki Tanwar Asstt. Prof, CSE, JMIT Radaur pinki.tanwar@gmail.com Abstract
More informationEfficient Object Extraction Using Fuzzy Cardinality Based Thresholding and Hopfield Network
Efficient Object Extraction Using Fuzzy Cardinality Based Thresholding and Hopfield Network S. Bhattacharyya U. Maulik S. Bandyopadhyay Dept. of Information Technology Dept. of Comp. Sc. and Tech. Machine
More informationArtificial Neural Networks Unsupervised learning: SOM
Artificial Neural Networks Unsupervised learning: SOM 01001110 01100101 01110101 01110010 01101111 01101110 01101111 01110110 01100001 00100000 01110011 01101011 01110101 01110000 01101001 01101110 01100001
More informationto the Traveling Salesman Problem 1 Susanne Timsj Applied Optimization and Modeling Group (TOM) Department of Mathematics and Physics
An Application of Lagrangian Relaxation to the Traveling Salesman Problem 1 Susanne Timsj Applied Optimization and Modeling Group (TOM) Department of Mathematics and Physics M lardalen University SE-721
More informationExploratory Data Analysis using Self-Organizing Maps. Madhumanti Ray
Exploratory Data Analysis using Self-Organizing Maps Madhumanti Ray Content Introduction Data Analysis methods Self-Organizing Maps Conclusion Visualization of high-dimensional data items Exploratory data
More informationA Study on Clustering Method by Self-Organizing Map and Information Criteria
A Study on Clustering Method by Self-Organizing Map and Information Criteria Satoru Kato, Tadashi Horiuchi,andYoshioItoh Matsue College of Technology, 4-4 Nishi-ikuma, Matsue, Shimane 90-88, JAPAN, kato@matsue-ct.ac.jp
More informationDESIGN OF KOHONEN SELF-ORGANIZING MAP WITH REDUCED STRUCTURE
DESIGN OF KOHONEN SELF-ORGANIZING MAP WITH REDUCED STRUCTURE S. Kajan, M. Lajtman Institute of Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak University
More informationIntroduction to Approximation Algorithms
Introduction to Approximation Algorithms Dr. Gautam K. Das Departmet of Mathematics Indian Institute of Technology Guwahati, India gkd@iitg.ernet.in February 19, 2016 Outline of the lecture Background
More informationHARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION
HARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION 1 M.S.Rekha, 2 S.G.Nawaz 1 PG SCALOR, CSE, SRI KRISHNADEVARAYA ENGINEERING COLLEGE, GOOTY 2 ASSOCIATE PROFESSOR, SRI KRISHNADEVARAYA
More informationA fuzzy k-modes algorithm for clustering categorical data. Citation IEEE Transactions on Fuzzy Systems, 1999, v. 7 n. 4, p.
Title A fuzzy k-modes algorithm for clustering categorical data Author(s) Huang, Z; Ng, MKP Citation IEEE Transactions on Fuzzy Systems, 1999, v. 7 n. 4, p. 446-452 Issued Date 1999 URL http://hdl.handle.net/10722/42992
More informationCOMBINED METHOD TO VISUALISE AND REDUCE DIMENSIONALITY OF THE FINANCIAL DATA SETS
COMBINED METHOD TO VISUALISE AND REDUCE DIMENSIONALITY OF THE FINANCIAL DATA SETS Toomas Kirt Supervisor: Leo Võhandu Tallinn Technical University Toomas.Kirt@mail.ee Abstract: Key words: For the visualisation
More informationReducing topological defects in self-organizing maps using multiple scale neighborhood functions
Reducing topological defects in self-organizing maps using multiple scale neighborhood functions Kazushi Murakoshi,YuichiSato Department of Knowledge-based Information Engineering, Toyohashi University
More informationMethods for Intelligent Systems
Methods for Intelligent Systems Lecture Notes on Clustering (II) Davide Eynard eynard@elet.polimi.it Department of Electronics and Information Politecnico di Milano Davide Eynard - Lecture Notes on Clustering
More informationAPPLICATION OF THE FUZZY MIN-MAX NEURAL NETWORK CLASSIFIER TO PROBLEMS WITH CONTINUOUS AND DISCRETE ATTRIBUTES
APPLICATION OF THE FUZZY MIN-MAX NEURAL NETWORK CLASSIFIER TO PROBLEMS WITH CONTINUOUS AND DISCRETE ATTRIBUTES A. Likas, K. Blekas and A. Stafylopatis National Technical University of Athens Department
More informationControlling the spread of dynamic self-organising maps
Neural Comput & Applic (2004) 13: 168 174 DOI 10.1007/s00521-004-0419-y ORIGINAL ARTICLE L. D. Alahakoon Controlling the spread of dynamic self-organising maps Received: 7 April 2004 / Accepted: 20 April
More informationFinding Sets of Non-Dominated Solutions with High Spread and Well-Balanced Distribution using Generalized Strength Pareto Evolutionary Algorithm
16th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT) Finding Sets of Non-Dominated Solutions with High
More informationMachine Learning for Signal Processing Clustering. Bhiksha Raj Class Oct 2016
Machine Learning for Signal Processing Clustering Bhiksha Raj Class 11. 13 Oct 2016 1 Statistical Modelling and Latent Structure Much of statistical modelling attempts to identify latent structure in the
More informationChapter 7: Competitive learning, clustering, and self-organizing maps
Chapter 7: Competitive learning, clustering, and self-organizing maps António R. C. Paiva EEL 6814 Spring 2008 Outline Competitive learning Clustering Self-Organizing Maps What is competition in neural
More informationMURDOCH RESEARCH REPOSITORY
MURDOCH RESEARCH REPOSITORY http://dx.doi.org/10.1109/19.668276 Fung, C.C., Wong, K.W. and Eren, H. (1997) Modular artificial neural network for prediction of petrophysical properties from well log data.
More informationCluster analysis of 3D seismic data for oil and gas exploration
Data Mining VII: Data, Text and Web Mining and their Business Applications 63 Cluster analysis of 3D seismic data for oil and gas exploration D. R. S. Moraes, R. P. Espíndola, A. G. Evsukoff & N. F. F.
More information^ Springer. Computational Intelligence. A Methodological Introduction. Rudolf Kruse Christian Borgelt. Matthias Steinbrecher Pascal Held
Rudolf Kruse Christian Borgelt Frank Klawonn Christian Moewes Matthias Steinbrecher Pascal Held Computational Intelligence A Methodological Introduction ^ Springer Contents 1 Introduction 1 1.1 Intelligent
More informationFigure (5) Kohonen Self-Organized Map
2- KOHONEN SELF-ORGANIZING MAPS (SOM) - The self-organizing neural networks assume a topological structure among the cluster units. - There are m cluster units, arranged in a one- or two-dimensional array;
More informationCHAPTER FOUR NEURAL NETWORK SELF- ORGANIZING MAP
96 CHAPTER FOUR NEURAL NETWORK SELF- ORGANIZING MAP 97 4.1 INTRODUCTION Neural networks have been successfully applied by many authors in solving pattern recognition problems. Unsupervised classification
More informationA Topography-Preserving Latent Variable Model with Learning Metrics
A Topography-Preserving Latent Variable Model with Learning Metrics Samuel Kaski and Janne Sinkkonen Helsinki University of Technology Neural Networks Research Centre P.O. Box 5400, FIN-02015 HUT, Finland
More informationFlexible-Hybrid Sequential Floating Search in Statistical Feature Selection
Flexible-Hybrid Sequential Floating Search in Statistical Feature Selection Petr Somol 1,2, Jana Novovičová 1,2, and Pavel Pudil 2,1 1 Dept. of Pattern Recognition, Institute of Information Theory and
More informationINTERNATIONAL CONFERENCE ON ENGINEERING DESIGN ICED 05 MELBOURNE, AUGUST 15-18, 2005
INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN ICED MELBOURNE, AUGUST -, METHOD USING A SELF-ORGANISING MAP FOR DRIVER CLASSIFI- CATION AS A PRECONDITION FOR CUSTOMER ORIENTED DESIGN Albert Albers and
More informationTwo-step Modified SOM for Parallel Calculation
Two-step Modified SOM for Parallel Calculation Two-step Modified SOM for Parallel Calculation Petr Gajdoš and Pavel Moravec Petr Gajdoš and Pavel Moravec Department of Computer Science, FEECS, VŠB Technical
More informationThe Modified IWO Algorithm for Optimization of Numerical Functions
The Modified IWO Algorithm for Optimization of Numerical Functions Daniel Kostrzewa and Henryk Josiński Silesian University of Technology, Akademicka 16 PL-44-100 Gliwice, Poland {Daniel.Kostrzewa,Henryk.Josinski}@polsl.pl
More informationSeismic regionalization based on an artificial neural network
Seismic regionalization based on an artificial neural network *Jaime García-Pérez 1) and René Riaño 2) 1), 2) Instituto de Ingeniería, UNAM, CU, Coyoacán, México D.F., 014510, Mexico 1) jgap@pumas.ii.unam.mx
More informationTravelling salesman problem using reduced algorithmic Branch and bound approach P. Ranjana Hindustan Institute of Technology and Science
Volume 118 No. 20 2018, 419-424 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Travelling salesman problem using reduced algorithmic Branch and bound approach P. Ranjana Hindustan
More informationCluster Analysis. Mu-Chun Su. Department of Computer Science and Information Engineering National Central University 2003/3/11 1
Cluster Analysis Mu-Chun Su Department of Computer Science and Information Engineering National Central University 2003/3/11 1 Introduction Cluster analysis is the formal study of algorithms and methods
More information11/14/2010 Intelligent Systems and Soft Computing 1
Lecture 8 Artificial neural networks: Unsupervised learning Introduction Hebbian learning Generalised Hebbian learning algorithm Competitive learning Self-organising computational map: Kohonen network
More informationMachine Learning for Software Engineering
Machine Learning for Software Engineering Introduction and Motivation Prof. Dr.-Ing. Norbert Siegmund Intelligent Software Systems 1 2 Organizational Stuff Lectures: Tuesday 11:00 12:30 in room SR015 Cover
More informationOutline of the module
Evolutionary and Heuristic Optimisation (ITNPD8) Lecture 2: Heuristics and Metaheuristics Gabriela Ochoa http://www.cs.stir.ac.uk/~goc/ Computing Science and Mathematics, School of Natural Sciences University
More informationSYDE Winter 2011 Introduction to Pattern Recognition. Clustering
SYDE 372 - Winter 2011 Introduction to Pattern Recognition Clustering Alexander Wong Department of Systems Design Engineering University of Waterloo Outline 1 2 3 4 5 All the approaches we have learned
More informationNonlinear dimensionality reduction of large datasets for data exploration
Data Mining VII: Data, Text and Web Mining and their Business Applications 3 Nonlinear dimensionality reduction of large datasets for data exploration V. Tomenko & V. Popov Wessex Institute of Technology,
More informationUnsupervised Learning
Unsupervised Learning Learning without a teacher No targets for the outputs Networks which discover patterns, correlations, etc. in the input data This is a self organisation Self organising networks An
More informationWhat is a receptive field? Why a sensory neuron has such particular RF How a RF was developed?
What is a receptive field? Why a sensory neuron has such particular RF How a RF was developed? x 1 x 2 x 3 y f w 1 w 2 w 3 T x y = f (wx i i T ) i y x 1 x 2 x 3 = = E (y y) (y f( wx T)) 2 2 o o i i i
More informationMethod and Algorithm for solving the Bicriterion Network Problem
Proceedings of the 00 International Conference on Industrial Engineering and Operations Management Dhaka, Bangladesh, anuary 9 0, 00 Method and Algorithm for solving the Bicriterion Network Problem Hossain
More informationUnsupervised learning
Unsupervised learning Enrique Muñoz Ballester Dipartimento di Informatica via Bramante 65, 26013 Crema (CR), Italy enrique.munoz@unimi.it Enrique Muñoz Ballester 2017 1 Download slides data and scripts:
More informationFEATURE EXTRACTION USING FUZZY RULE BASED SYSTEM
International Journal of Computer Science and Applications, Vol. 5, No. 3, pp 1-8 Technomathematics Research Foundation FEATURE EXTRACTION USING FUZZY RULE BASED SYSTEM NARENDRA S. CHAUDHARI and AVISHEK
More informationAssignment 3b: The traveling salesman problem
Chalmers University of Technology MVE165 University of Gothenburg MMG631 Mathematical Sciences Linear and integer optimization Optimization with applications Emil Gustavsson Assignment information Ann-Brith
More informationSolving the Traveling Salesman Problem using Reinforced Ant Colony Optimization techniques
Solving the Traveling Salesman Problem using Reinforced Ant Colony Optimization techniques N.N.Poddar 1, D. Kaur 2 1 Electrical Engineering and Computer Science, University of Toledo, Toledo, OH, USA 2
More informationAn algorithmic method to extend TOPSIS for decision-making problems with interval data
Applied Mathematics and Computation 175 (2006) 1375 1384 www.elsevier.com/locate/amc An algorithmic method to extend TOPSIS for decision-making problems with interval data G.R. Jahanshahloo, F. Hosseinzadeh
More informationSelf-Organizing Maps for cyclic and unbounded graphs
Self-Organizing Maps for cyclic and unbounded graphs M. Hagenbuchner 1, A. Sperduti 2, A.C. Tsoi 3 1- University of Wollongong, Wollongong, Australia. 2- University of Padova, Padova, Italy. 3- Hong Kong
More informationENHANCED BEE COLONY ALGORITHM FOR SOLVING TRAVELLING SALESPERSON PROBLEM
ENHANCED BEE COLONY ALGORITHM FOR SOLVING TRAVELLING SALESPERSON PROBLEM Prateek Agrawal 1, Harjeet Kaur 2, and Deepa Bhardwaj 3 123 Department of Computer Engineering, Lovely Professional University (
More informationA novel firing rule for training Kohonen selforganising
A novel firing rule for training Kohonen selforganising maps D. T. Pham & A. B. Chan Manufacturing Engineering Centre, School of Engineering, University of Wales Cardiff, P.O. Box 688, Queen's Buildings,
More informationComparison Study of Multiple Traveling Salesmen Problem using Genetic Algorithm
IOSR Journal of Computer Engineering (IOSR-JCE) e-issn: 2278-661, p- ISSN: 2278-8727Volume 13, Issue 3 (Jul. - Aug. 213), PP 17-22 Comparison Study of Multiple Traveling Salesmen Problem using Genetic
More informationFinding Clusters 1 / 60
Finding Clusters Types of Clustering Approaches: Linkage Based, e.g. Hierarchical Clustering Clustering by Partitioning, e.g. k-means Density Based Clustering, e.g. DBScan Grid Based Clustering 1 / 60
More informationUnsupervised Learning
Networks for Pattern Recognition, 2014 Networks for Single Linkage K-Means Soft DBSCAN PCA Networks for Kohonen Maps Linear Vector Quantization Networks for Problems/Approaches in Machine Learning Supervised
More informationNeural Network and SVM classification via Decision Trees, Vector Quantization and Simulated Annealing
Neural Network and SVM classification via Decision Trees, Vector Quantization and Simulated Annealing JOHN TSILIGARIDIS Mathematics and Computer Science Department Heritage University Toppenish, WA USA
More informationESANN'2001 proceedings - European Symposium on Artificial Neural Networks Bruges (Belgium), April 2001, D-Facto public., ISBN ,
Interpretation and Comparison of Multidimensional Data Partitions Esa Alhoniemi and Olli Simula Neural Networks Research Centre Helsinki University of Technology P. O.Box 5400 FIN-02015 HUT, Finland esa.alhoniemi@hut.fi
More informationThe role of Fisher information in primary data space for neighbourhood mapping
The role of Fisher information in primary data space for neighbourhood mapping H. Ruiz 1, I. H. Jarman 2, J. D. Martín 3, P. J. Lisboa 1 1 - School of Computing and Mathematical Sciences - Department of
More informationUSING OF THE K NEAREST NEIGHBOURS ALGORITHM (k-nns) IN THE DATA CLASSIFICATION
USING OF THE K NEAREST NEIGHBOURS ALGORITHM (k-nns) IN THE DATA CLASSIFICATION Gîlcă Natalia, Roșia de Amaradia Technological High School, Gorj, ROMANIA Gîlcă Gheorghe, Constantin Brîncuși University from
More informationCollaborative Rough Clustering
Collaborative Rough Clustering Sushmita Mitra, Haider Banka, and Witold Pedrycz Machine Intelligence Unit, Indian Statistical Institute, Kolkata, India {sushmita, hbanka r}@isical.ac.in Dept. of Electrical
More informationSupervised vs.unsupervised Learning
Supervised vs.unsupervised Learning In supervised learning we train algorithms with predefined concepts and functions based on labeled data D = { ( x, y ) x X, y {yes,no}. In unsupervised learning we are
More informationTable of Contents. Recognition of Facial Gestures... 1 Attila Fazekas
Table of Contents Recognition of Facial Gestures...................................... 1 Attila Fazekas II Recognition of Facial Gestures Attila Fazekas University of Debrecen, Institute of Informatics
More informationCHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS
CHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS 4.1. INTRODUCTION This chapter includes implementation and testing of the student s academic performance evaluation to achieve the objective(s)
More informationSOLVING TRAVELING SALESMAN PROBLEMS USING HEURISTIC LEARNING APPROACH
SOLVING TRAVELING SALESMAN PROBLEMS USING HEURISTIC LEARNING APPROACH Sim Kim Lau 1), Li-Yen Shue 2) 1) Department of Information Systems, University of Wollongong, Australia (simlau@uow.edu.au) 2) Department
More informationCluster Analysis. Angela Montanari and Laura Anderlucci
Cluster Analysis Angela Montanari and Laura Anderlucci 1 Introduction Clustering a set of n objects into k groups is usually moved by the aim of identifying internally homogenous groups according to a
More informationThe Heuristic Strategy Implementation to the Hopfield -Tank TSP Neural Algorithm
The Heuristic Strategy Implementation to the Hopfield -Tank TSP Neural Algorithm N. Kovač, S. Bauk Faculty of Maritime Studies, University of Montenegro Dobrota 36, 85 330 Kotor, Serbia and Montenegro
More informationComputational Complexity CSC Professor: Tom Altman. Capacitated Problem
Computational Complexity CSC 5802 Professor: Tom Altman Capacitated Problem Agenda: Definition Example Solution Techniques Implementation Capacitated VRP (CPRV) CVRP is a Vehicle Routing Problem (VRP)
More informationProblems of Fuzzy c-means Clustering and Similar Algorithms with High Dimensional Data Sets
Problems of Fuzzy c-means Clustering and Similar Algorithms with High Dimensional Data Sets Roland Winkler, Frank Klawonn, and Rudolf Kruse Abstract Fuzzy c-means clustering and its derivatives are very
More informationEncoding Techniques in Genetic Algorithms
Encoding Techniques in Genetic Algorithms Debasis Samanta Indian Institute of Technology Kharagpur dsamanta@iitkgp.ac.in 01.03.2016 Debasis Samanta (IIT Kharagpur) Soft Computing Applications 01.03.2016
More informationSemi-Supervised Clustering with Partial Background Information
Semi-Supervised Clustering with Partial Background Information Jing Gao Pang-Ning Tan Haibin Cheng Abstract Incorporating background knowledge into unsupervised clustering algorithms has been the subject
More informationA Modular Reduction Method for k-nn Algorithm with Self-recombination Learning
A Modular Reduction Method for k-nn Algorithm with Self-recombination Learning Hai Zhao and Bao-Liang Lu Department of Computer Science and Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Rd.,
More informationNCC 2009, January 16-18, IIT Guwahati 267
NCC 2009, January 6-8, IIT Guwahati 267 Unsupervised texture segmentation based on Hadamard transform Tathagata Ray, Pranab Kumar Dutta Department Of Electrical Engineering Indian Institute of Technology
More informationBRACE: A Paradigm For the Discretization of Continuously Valued Data
Proceedings of the Seventh Florida Artificial Intelligence Research Symposium, pp. 7-2, 994 BRACE: A Paradigm For the Discretization of Continuously Valued Data Dan Ventura Tony R. Martinez Computer Science
More informationRevision of a Floating-Point Genetic Algorithm GENOCOP V for Nonlinear Programming Problems
4 The Open Cybernetics and Systemics Journal, 008,, 4-9 Revision of a Floating-Point Genetic Algorithm GENOCOP V for Nonlinear Programming Problems K. Kato *, M. Sakawa and H. Katagiri Department of Artificial
More informationGraph Based Image Segmentation
AUTOMATYKA 2011 Tom 15 Zeszyt 3 Anna Fabijañska* Graph Based Image Segmentation 1. Introduction Image segmentation is one of the fundamental problems in machine vision. In general it aims at extracting
More informationNotes for Lecture 24
U.C. Berkeley CS170: Intro to CS Theory Handout N24 Professor Luca Trevisan December 4, 2001 Notes for Lecture 24 1 Some NP-complete Numerical Problems 1.1 Subset Sum The Subset Sum problem is defined
More informationA COMPARATIVE STUDY OF FIVE PARALLEL GENETIC ALGORITHMS USING THE TRAVELING SALESMAN PROBLEM
A COMPARATIVE STUDY OF FIVE PARALLEL GENETIC ALGORITHMS USING THE TRAVELING SALESMAN PROBLEM Lee Wang, Anthony A. Maciejewski, Howard Jay Siegel, and Vwani P. Roychowdhury * Microsoft Corporation Parallel
More informationAnt Colony Optimization for dynamic Traveling Salesman Problems
Ant Colony Optimization for dynamic Traveling Salesman Problems Carlos A. Silva and Thomas A. Runkler Siemens AG, Corporate Technology Information and Communications, CT IC 4 81730 Munich - Germany thomas.runkler@siemens.com
More informationENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition
Memorial University of Newfoundland Pattern Recognition Lecture 15, June 29, 2006 http://www.engr.mun.ca/~charlesr Office Hours: Tuesdays & Thursdays 8:30-9:30 PM EN-3026 July 2006 Sunday Monday Tuesday
More informationModelling Data Segmentation for Image Retrieval Systems
Modelling Data Segmentation for Image Retrieval Systems Leticia Flores-Pulido 1,2, Oleg Starostenko 1, Gustavo Rodríguez-Gómez 3 and Vicente Alarcón-Aquino 1 1 Universidad de las Américas Puebla, Puebla,
More informationIntuitionistic Fuzzy Estimations of the Ant Colony Optimization
Intuitionistic Fuzzy Estimations of the Ant Colony Optimization Stefka Fidanova, Krasimir Atanasov and Pencho Marinov IPP BAS, Acad. G. Bonchev str. bl.25a, 1113 Sofia, Bulgaria {stefka,pencho}@parallel.bas.bg
More informationGeneralized Tree-Based Wavelet Transform and Applications to Patch-Based Image Processing
Generalized Tree-Based Wavelet Transform and * Michael Elad The Computer Science Department The Technion Israel Institute of technology Haifa 32000, Israel *Joint work with A Seminar in the Hebrew University
More informationCOMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION. Ivan P. Stanimirović. 1. Introduction
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 27, No 1 (2012), 55 66 COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION Ivan P. Stanimirović Abstract. A modification of the standard
More informationSOMSN: An Effective Self Organizing Map for Clustering of Social Networks
SOMSN: An Effective Self Organizing Map for Clustering of Social Networks Fatemeh Ghaemmaghami Research Scholar, CSE and IT Dept. Shiraz University, Shiraz, Iran Reza Manouchehri Sarhadi Research Scholar,
More informationA NEW ALGORITHM FOR OPTIMIZING THE SELF- ORGANIZING MAP
A NEW ALGORITHM FOR OPTIMIZING THE SELF- ORGANIZING MAP BEN-HDECH Adil, GHANOU Youssef, EL QADI Abderrahim Team TIM, High School of Technology, Moulay Ismail University, Meknes, Morocco E-mail: adilbenhdech@gmail.com,
More informationCluster Tendency Assessment for Fuzzy Clustering of Incomplete Data
EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France Cluster Tendency Assessment for Fuzzy Clustering of Incomplete Data Ludmila Himmelspach 1 Daniel Hommers 1 Stefan Conrad 1 1 Institute of Computer Science,
More informationCluster Analysis using Spherical SOM
Cluster Analysis using Spherical SOM H. Tokutaka 1, P.K. Kihato 2, K. Fujimura 2 and M. Ohkita 2 1) SOM Japan Co-LTD, 2) Electrical and Electronic Department, Tottori University Email: {tokutaka@somj.com,
More informationImprovement heuristics for the Sparse Travelling Salesman Problem
Improvement heuristics for the Sparse Travelling Salesman Problem FREDRICK MTENZI Computer Science Department Dublin Institute of Technology School of Computing, DIT Kevin Street, Dublin 8 IRELAND http://www.comp.dit.ie/fmtenzi
More informationConstrained Optimization of the Stress Function for Multidimensional Scaling
Constrained Optimization of the Stress Function for Multidimensional Scaling Vydunas Saltenis Institute of Mathematics and Informatics Akademijos 4, LT-08663 Vilnius, Lithuania Saltenis@ktlmiilt Abstract
More informationNeuro-Fuzzy Comp. Ch. 8 May 12, 2005
Neuro-Fuzzy Comp. Ch. 8 May, 8 Self-Organizing Feature Maps Self-Organizing Feature Maps (SOFM or SOM) also known as Kohonen maps or topographic maps were first introduced by von der Malsburg (97) and
More informationFuzzy Segmentation. Chapter Introduction. 4.2 Unsupervised Clustering.
Chapter 4 Fuzzy Segmentation 4. Introduction. The segmentation of objects whose color-composition is not common represents a difficult task, due to the illumination and the appropriate threshold selection
More informationA Web-Based Evolutionary Algorithm Demonstration using the Traveling Salesman Problem
A Web-Based Evolutionary Algorithm Demonstration using the Traveling Salesman Problem Richard E. Mowe Department of Statistics St. Cloud State University mowe@stcloudstate.edu Bryant A. Julstrom Department
More informationA Method for the Identification of Inaccuracies in Pupil Segmentation
A Method for the Identification of Inaccuracies in Pupil Segmentation Hugo Proença and Luís A. Alexandre Dep. Informatics, IT - Networks and Multimedia Group Universidade da Beira Interior, Covilhã, Portugal
More informationMetaheuristic Optimization with Evolver, Genocop and OptQuest
Metaheuristic Optimization with Evolver, Genocop and OptQuest MANUEL LAGUNA Graduate School of Business Administration University of Colorado, Boulder, CO 80309-0419 Manuel.Laguna@Colorado.EDU Last revision:
More informationA Parallel Evolutionary Algorithm for Discovery of Decision Rules
A Parallel Evolutionary Algorithm for Discovery of Decision Rules Wojciech Kwedlo Faculty of Computer Science Technical University of Bia lystok Wiejska 45a, 15-351 Bia lystok, Poland wkwedlo@ii.pb.bialystok.pl
More information9.1. K-means Clustering
424 9. MIXTURE MODELS AND EM Section 9.2 Section 9.3 Section 9.4 view of mixture distributions in which the discrete latent variables can be interpreted as defining assignments of data points to specific
More informationA Hybrid Intelligent System for Fault Detection in Power Systems
A Hybrid Intelligent System for Fault Detection in Power Systems Hiroyuki Mori Hikaru Aoyama Dept. of Electrical and Electronics Eng. Meii University Tama-ku, Kawasaki 14-8571 Japan Toshiyuki Yamanaka
More informationModification of the Growing Neural Gas Algorithm for Cluster Analysis
Modification of the Growing Neural Gas Algorithm for Cluster Analysis Fernando Canales and Max Chacón Universidad de Santiago de Chile; Depto. de Ingeniería Informática, Avda. Ecuador No 3659 - PoBox 10233;
More information4. Feedforward neural networks. 4.1 Feedforward neural network structure
4. Feedforward neural networks 4.1 Feedforward neural network structure Feedforward neural network is one of the most common network architectures. Its structure and some basic preprocessing issues required
More informationAnalytical model A structure and process for analyzing a dataset. For example, a decision tree is a model for the classification of a dataset.
Glossary of data mining terms: Accuracy Accuracy is an important factor in assessing the success of data mining. When applied to data, accuracy refers to the rate of correct values in the data. When applied
More informationApplications of Mathematics to Real-World Problems
Applications of Mathematics to Real-World Problems Michelle Dunbar Maths Teachers Day @ UoW SMAS/SMART June 25, 2013 Michelle Dunbar, SMART, UoW Applications of Mathematics to Real-World Problems 1/28
More information