Application of genetic algorithms and Kohonen networks to cluster analysis

Size: px
Start display at page:

Download "Application of genetic algorithms and Kohonen networks to cluster analysis"

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 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 information

Generalized SOMs with Splitting-Merging Tree-Like Structures for WWW-Document Clustering

Generalized 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 information

Representation of 2D objects with a topology preserving network

Representation 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 information

Function approximation using RBF network. 10 basis functions and 25 data points.

Function 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 information

Fuzzy Inspired Hybrid Genetic Approach to Optimize Travelling Salesman Problem

Fuzzy 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 information

Efficient Object Extraction Using Fuzzy Cardinality Based Thresholding and Hopfield Network

Efficient 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 information

Artificial Neural Networks Unsupervised learning: SOM

Artificial 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 information

to the Traveling Salesman Problem 1 Susanne Timsj Applied Optimization and Modeling Group (TOM) Department of Mathematics and Physics

to 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 information

Exploratory Data Analysis using Self-Organizing Maps. Madhumanti Ray

Exploratory 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 information

A Study on Clustering Method by Self-Organizing Map and Information Criteria

A 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 information

DESIGN OF KOHONEN SELF-ORGANIZING MAP WITH REDUCED STRUCTURE

DESIGN 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 information

Introduction to Approximation Algorithms

Introduction 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 information

HARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION

HARD, 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 information

A fuzzy k-modes algorithm for clustering categorical data. Citation IEEE Transactions on Fuzzy Systems, 1999, v. 7 n. 4, p.

A 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 information

COMBINED METHOD TO VISUALISE AND REDUCE DIMENSIONALITY OF THE FINANCIAL DATA SETS

COMBINED 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 information

Reducing topological defects in self-organizing maps using multiple scale neighborhood functions

Reducing 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 information

Methods for Intelligent Systems

Methods 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 information

APPLICATION 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 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 information

Controlling the spread of dynamic self-organising maps

Controlling 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 information

Finding Sets of Non-Dominated Solutions with High Spread and Well-Balanced Distribution using Generalized Strength Pareto Evolutionary Algorithm

Finding 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 information

Machine Learning for Signal Processing Clustering. Bhiksha Raj Class Oct 2016

Machine 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 information

Chapter 7: Competitive learning, clustering, and self-organizing maps

Chapter 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 information

MURDOCH RESEARCH REPOSITORY

MURDOCH 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 information

Cluster analysis of 3D seismic data for oil and gas exploration

Cluster 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

^ 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 information

Figure (5) Kohonen Self-Organized Map

Figure (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 information

CHAPTER FOUR NEURAL NETWORK SELF- ORGANIZING MAP

CHAPTER 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 information

A Topography-Preserving Latent Variable Model with Learning Metrics

A 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 information

Flexible-Hybrid Sequential Floating Search in Statistical Feature Selection

Flexible-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 information

INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN ICED 05 MELBOURNE, AUGUST 15-18, 2005

INTERNATIONAL 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 information

Two-step Modified SOM for Parallel Calculation

Two-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 information

The Modified IWO Algorithm for Optimization of Numerical Functions

The 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 information

Seismic regionalization based on an artificial neural network

Seismic 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 information

Travelling salesman problem using reduced algorithmic Branch and bound approach P. Ranjana Hindustan Institute of Technology and Science

Travelling 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 information

Cluster 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 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 information

11/14/2010 Intelligent Systems and Soft Computing 1

11/14/2010 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 information

Machine Learning for Software Engineering

Machine 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 information

Outline of the module

Outline 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 information

SYDE Winter 2011 Introduction to Pattern Recognition. Clustering

SYDE 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 information

Nonlinear dimensionality reduction of large datasets for data exploration

Nonlinear 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 information

Unsupervised Learning

Unsupervised 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 information

What 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? 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 information

Method and Algorithm for solving the Bicriterion Network Problem

Method 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 information

Unsupervised learning

Unsupervised 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 information

FEATURE EXTRACTION USING FUZZY RULE BASED SYSTEM

FEATURE 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 information

Assignment 3b: The traveling salesman problem

Assignment 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 information

Solving the Traveling Salesman Problem using Reinforced Ant Colony Optimization techniques

Solving 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 information

An algorithmic method to extend TOPSIS for decision-making problems with interval data

An 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 information

Self-Organizing Maps for cyclic and unbounded graphs

Self-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 information

ENHANCED BEE COLONY ALGORITHM FOR SOLVING TRAVELLING SALESPERSON PROBLEM

ENHANCED 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 information

A novel firing rule for training Kohonen selforganising

A 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 information

Comparison Study of Multiple Traveling Salesmen Problem using Genetic Algorithm

Comparison 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 information

Finding Clusters 1 / 60

Finding 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 information

Unsupervised Learning

Unsupervised 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 information

Neural 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 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 information

ESANN'2001 proceedings - European Symposium on Artificial Neural Networks Bruges (Belgium), April 2001, D-Facto public., ISBN ,

ESANN'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 information

The role of Fisher information in primary data space for neighbourhood mapping

The 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 information

USING 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 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 information

Collaborative Rough Clustering

Collaborative 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 information

Supervised vs.unsupervised Learning

Supervised 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 information

Table of Contents. Recognition of Facial Gestures... 1 Attila Fazekas

Table 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 information

CHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS

CHAPTER 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 information

SOLVING TRAVELING SALESMAN PROBLEMS USING HEURISTIC LEARNING APPROACH

SOLVING 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 information

Cluster Analysis. Angela Montanari and Laura Anderlucci

Cluster 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 information

The Heuristic Strategy Implementation to the Hopfield -Tank TSP Neural Algorithm

The 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 information

Computational Complexity CSC Professor: Tom Altman. Capacitated Problem

Computational 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 information

Problems 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 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 information

Encoding Techniques in Genetic Algorithms

Encoding 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 information

Semi-Supervised Clustering with Partial Background Information

Semi-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 information

A Modular Reduction Method for k-nn Algorithm with Self-recombination Learning

A 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 information

NCC 2009, January 16-18, IIT Guwahati 267

NCC 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 information

BRACE: A Paradigm For the Discretization of Continuously Valued Data

BRACE: 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 information

Revision of a Floating-Point Genetic Algorithm GENOCOP V for Nonlinear Programming Problems

Revision 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 information

Graph Based Image Segmentation

Graph 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 information

Notes for Lecture 24

Notes 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 information

A 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 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 information

Ant Colony Optimization for dynamic Traveling Salesman Problems

Ant 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 information

ENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition

ENG 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 information

Modelling Data Segmentation for Image Retrieval Systems

Modelling 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 information

Intuitionistic Fuzzy Estimations of the Ant Colony Optimization

Intuitionistic 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 information

Generalized Tree-Based Wavelet Transform and Applications to Patch-Based Image Processing

Generalized 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 information

COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION. Ivan P. Stanimirović. 1. Introduction

COMPENDIOUS 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 information

SOMSN: An Effective Self Organizing Map for Clustering of Social Networks

SOMSN: 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 information

A NEW ALGORITHM FOR OPTIMIZING THE SELF- ORGANIZING MAP

A 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 information

Cluster Tendency Assessment for Fuzzy Clustering of Incomplete Data

Cluster 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 information

Cluster Analysis using Spherical SOM

Cluster 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 information

Improvement heuristics for the Sparse Travelling Salesman Problem

Improvement 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 information

Constrained Optimization of the Stress Function for Multidimensional Scaling

Constrained 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 information

Neuro-Fuzzy Comp. Ch. 8 May 12, 2005

Neuro-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 information

Fuzzy Segmentation. Chapter Introduction. 4.2 Unsupervised Clustering.

Fuzzy 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 information

A Web-Based Evolutionary Algorithm Demonstration using the Traveling Salesman Problem

A 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 information

A Method for the Identification of Inaccuracies in Pupil Segmentation

A 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 information

Metaheuristic Optimization with Evolver, Genocop and OptQuest

Metaheuristic 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 information

A Parallel Evolutionary Algorithm for Discovery of Decision Rules

A 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 information

9.1. K-means Clustering

9.1. 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 information

A Hybrid Intelligent System for Fault Detection in Power Systems

A 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 information

Modification of the Growing Neural Gas Algorithm for Cluster Analysis

Modification 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 information

4. Feedforward neural networks. 4.1 Feedforward neural network structure

4. 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 information

Analytical model A structure and process for analyzing a dataset. For example, a decision tree is a model for the classification of a dataset.

Analytical 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 information

Applications of Mathematics to Real-World Problems

Applications 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