DOCSLIB.ORG

Introduction to R and Statistical Data Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to R and Statistical Data Analysis Microarray Center IntroductionIntroduction toto RR andand StatisticalStatistical DataData AnalysisAnalysis PARTPART IIII Petr Nazarov petr.nazarov@crp -sante.lu 22-11-2010 LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu OUTLINE PART II Descriptive statistics in R (8) sum, mean, median, sd, var, cor, etc . Principle component analysis and clustering (9) PCA, k-means clustering, hierarchical clustering Random numbers (10) random number generators, distributions Statistical tests (11) t-test, Wilcoxon test, multiple test correction. ANOVA and Linear regression (12) ANOVA, linear regression LookLook for for corresponding corresponding scripts scripts at at http://edu.sablab.net/r2010/scriptshttp://edu.sablab.net/r2010/scripts LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 8. DESCRIPTIVE STATISTICS IN R 8.1-8.3. Center, Variation, Dependency LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.1. Iris Data from R.A.Fisher TheThe IrisIris flowerflower datadata setset oror Fisher'sFisher's IrisIris datadata setset isis aa multivariatemultivariate datadata setset introducedintroduced byby SirSir RonaldRonald AylmerAylmer FisherFisher (1936)(1936) asas anan exampleexample ofof discridiscriminantminant analysis.analysis. ItIt isis sometimessometimes calledcalled Anderson'sAnderson's IrisIris datadata setset becausebecause EdgarEdgar AndersonAnderson colcollectedlected thethe datadata toto quantifyquantify thethe geographicgeographic variationvariation ofof IrisIris flowersflowers inin thethe GaspéGaspé Pe Peninsula.ninsula. TheThe datasetdataset consistsconsists ofof 5050 samplessamples fromfrom eacheach ofof thrthreeee speciesspecies ofof Iris Iris flowersflowers (Iris(Iris setosasetosa ,, IrisIris virginicavirginica andand IrisIris versicolorversicolor ).). FourFour featuresfeatures werewere measuredmeasured fromfrom eacheach sample,sample, ththeyey areare thethe lengthlength andand thethe widthwidth ofof sepalsepal andand petal,petal, inin cencentimeters.timeters. BasedBased onon thethe combinationcombination ofof thethe fourfour features,features, FisherFisher developeddeveloped aa linearlinear discriminadiscriminantnt modelmodel toto distinguishdistinguish thethe speciesspecies fromfrom eacheach other.other. Iris setosa Iris versicolor Iris virginica LuciLinx: Luxembourg Bioinformatics Network http://wikipedia.prgwww.lucilinx.lu 9. PCA AND CLUSTERING 9.1. Data Presentation iris str (iris) 2.0 3.0 4.0 0.5 1.5 2.5 ## plot iris data x11 () Sepal.Length plot (iris[,-5]) plot (iris[,-5], 4.5 6.0 7.5 col = iris[,5]) Sepal.Width 2.0 3.0 4.0 Petal.Length 1 3 5 7 Petal.Width 0.5 1.5 2.5 4.5 5.5 6.5 7.5 1 2 3 4 5 6 7 http://urbanext.illinois.edu/gpe/case4/c4facts1a.html LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.2 Principle Component Analysis (PCA) PrincipalPrincipal component component analysis analysis (PCA) (PCA) isis a a vector vector space space transform transform used used to to reduce reduce multidim multidimensionalensional 20000 genes → 2 dimensions datadata sets sets to to lower lower dimensions dimensions for for analysis. analysis. It It sele selectscts the the 2 dimensions coordinatescoordinates along along which which the the variation variation of of the the data data i sis bigger bigger.. For the simplicity let us consider 2 parametric situation both i n terms of data and resulting PCA. Scatter plot in Scatter plot in PC “natural ” coordinates Variable 2 Variable Variable 2 Variable Second component Second Second component Second Variable 1 First component Instead of using 2 “natural ” parameters for the classification, we can use the first compone nt! LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.2. Data Transformation for PCA Data = as.matrix (iris[,-5]) row.names (Data) = as.character (iris[,5]) classes = as.integer (iris[,5]) ## plot data in 3d Iris library (scatterplot3d) x11 () scatterplot3d (iris[,1],iris[,2],iris[,3], pch=19,color=classes, main = "Iris", setosa xlab = names (iris)[1], versicolor ylab = names (iris)[2], virginica zlab = names (iris)[3]) legend (4,7, levels (iris$Species), col=c(1,2,3),pch=19) Petal.Length 4.5 4.0 3.5 Sepal.Width 3.0 2.5 1 2 3 4 5 6 7 2.0 4 5 6 7 8 Sepal.Length LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.2. PCA LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.3. k-Means Clustering kk-Means-Means Clustering Clustering kk-means-means clustering clustering is is a a method method of of cluster cluster analysis analysis wh whichich aims aims to to p partitionartition n n observations observations intointo k k clusters clusters in in which which each each observation observation belongs belongs t oto the the cluster clusterwith with the the nearest nearest mean. mean. 1) k initial "means" (in 4) Steps 2 and 3 are this case k=3) are repeated until 2) k clusters are created randomly selected from convergence has been by associating every the data set (shown in reached. color). observation with the 3) The centroid of each nearest mean. of the k clusters becomes the new means. http://wikipedia.org LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.3. k-Means Clustering clusters = kmeans (x=Data,centers=3,nstart=10)$cluster x11 () plot (PC$x[,1],PC$x[,2],col = classes,pch=clusters) legend (2,1.4, levels (iris$Species),col=c(1,2,3),pch=19) legend (-2.5,1.4,c("c1","c2","c3"),col=4,pch=c(1,2,3)) c1 setosa c2 versicolor c3 virginica PC$x[, 2] PC$x[, -1.0 -0.5 0.0 0.5 1.0 -3 -2 -1 0 1 2 3 4 PC$x[, 1] LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.4. Hierarchical Clustering HierarchicalHierarchical Clustering Clustering HierarchicalHierarchical clustering clustering creates creates a a hierarchy hierarchy of of clus clustersters which which ma mayy be be represented represented in in a a tree tree structurestructure called called a a dendrogram. dendrogram. TheThe root root of of the the tree tree consists consists of of a a single single cluster cluster c containingontaining allall observations, observations, and and the the leaves leaves correspond correspond to to indi individualvidual observ observations.ations. AlgorithmsAlgorithms for for hierarchical hierarchical clustering clustering are are generall generallyy either either agglomerative agglomerative , ,in in which which one one startsstarts at at the the leaves leaves and and successively successively merges merges cluste clustersrs together; together; or or divisive divisive , ,in in which which one one startsstarts at at the the root root and and recursively recursively splits splits the the clust clusters.ers. Elements Divisive Agglomerative Dendrogram Distance : Euclidean http://wikipedia.org LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.4. Hierarchical Clustering ## use heatmap heatmap (data) ## use heatmap with colors color = character (length (classes)) color[classes == 1] = "black" color[classes == 2] = "red" color[classes == 3] = "green" heatmap (data,RowSideColors=color) Iris setosa Iris versicolor Iris virginica LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 9. PCA AND CLUSTERING 9.5. Example: Task 8a http://edu.sablab.net/data/txt/all_data.txthttp://edu.sablab.net/data/txt/all_data.txt Acute lymphoblastic leukemia (ALL), is a form of leukemia, or ca ncer of the white blood cells characterized by excess lymphoblasts. all_data.xls contains the results of full -trancript profiling for ALL patients and healthy donors using Affymetrix microarrays. The data were downloaded from ArrayExpress repository and normalized. The expression values in the table are in log 2 scale. LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 10. RANDOM NUMBERS AND DISTRIBUTIONS See Source Code LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 11. STATISTICAL TESTS See Source Code LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 12. ANOVA and LINEAR REGRESSION 12.1. Why ANOVA? If we would use pairwise comparisons, MeansMeans for for more more than than 2 2 populations populations what will be the probability of getting error? WeWe have have measurements measurements for for 5 5 !5 Number of comparisons: C 5 = = 10 conditions.conditions. Are Are the the means means for for these these 2 !3!2 conditionsconditions equal? equal? Probability of an error : 1 –(0.95) 10 = 0.4 ValidationValidation of of the the effects effects WeWe assume assume that that we we have have several several factorsfactors affecting affecting our our data. data. Which Which factorsfactors are are most most significant? significant? Which Which cancan be be neglected? neglected? ANOVA example from Partek™ http://easylink.playstream.com/affymetrix/ambsymposium/partek_08.wvx LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 12. ANOVA and LINEAR REGRESSION 12.1. Meaning of ANOVA H : µµµ = µµµ = µµµ H00: µµµ11= µµµ22= µµµ33 H : not all 3 means are equal Haa: not all 3 means are equal 14 12 10 m2 8 m3 6 m1 Depression level Depression 4 2 0 FL FL FL FL FL FL FL NY NY NY NY NY NY NY NC NC NC NC NC NC Measures LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 12. ANOVA and LINEAR REGRESSION 12.1. ANOVA in R: Fast and Simple salaries.txtsalaries.txt LuciLinx: Luxembourg Bioinformatics Network www.lucilinx.lu 12. ANOVA and LINEAR REGRESSION 12.2. Regression Model and Regression Line RegressionRegression model model
Load more

Recommended publications
  • Crop Monitoring System Using Image Processing and Environmental Data
    Crop Monitoring System Using Image Processing and Environmental Data Shadman Rabbi - 18341015 Ahnaf Shabik - 13201013 Supervised by: Dr. Md. Ashraful Alam Department of Computer Science and Engineering BRAC University Declaration We, hereby declare that this thesis is based on the results found by ourselves. Materials of work found by other researcher are mentioned by reference. This thesis, neither in whole or in part, has been previously submitted for any degree. ______________________ Signature of Supervisor (Dr. Md. Ashraful Alam) ___________________ Signature of Author (Shadman Rabbi) ___________________ Signature of Author (Ahnaf Shabik) i Acknowledgement Firstly we would like to thank the almighty for enabling us to initiate our research, to put our best efforts and successfully conclude it. Secondly, we offer our genuine and heartiest appreciation to our regarded Supervisor Dr. Md. Ashraful Alam for his contribution, direction and support in leading the research and preparation of the report. His involvement, inclusion and supervision have inspired us and acted as a huge incentive all through our research. Last but not the least, we are grateful to the resources, seniors, companions who have been indirectly but actively helpful with the research. Especially Monirul Islam Pavel, Ekhwan Islam, Touhidul Islam and Tahmidul Haq has propelled and motivated us through this journey. We would also like to acknowledge the help we got from various assets over the internet; particularly from fellow researchers’ work. ii Table of Contents Declaration
    [Show full text]
  • Scikit-Learn
    Scikit-Learn i Scikit-Learn About the Tutorial Scikit-learn (Sklearn) is the most useful and robust library for machine learning in Python. It provides a selection of efficient tools for machine learning and statistical modeling including classification, regression, clustering and dimensionality reduction via a consistence interface in Python. This library, which is largely written in Python, is built upon NumPy, SciPy and Matplotlib. Audience This tutorial will be useful for graduates, postgraduates, and research students who either have an interest in this Machine Learning subject or have this subject as a part of their curriculum. The reader can be a beginner or an advanced learner. Prerequisites The reader must have basic knowledge about Machine Learning. He/she should also be aware about Python, NumPy, Scipy, Matplotlib. If you are new to any of these concepts, we recommend you take up tutorials concerning these topics, before you dig further into this tutorial. Copyright & Disclaimer Copyright 2019 by Tutorials Point (I) Pvt. Ltd. All the content and graphics published in this e-book are the property of Tutorials Point (I) Pvt. Ltd. The user of this e-book is prohibited to reuse, retain, copy, distribute or republish any contents or a part of contents of this e-book in any manner without written consent of the publisher. We strive to update the contents of our website and tutorials as timely and as precisely as possible, however, the contents may contain inaccuracies or errors. Tutorials Point (I) Pvt. Ltd. provides no guarantee regarding the accuracy, timeliness or completeness of our website or its contents including this tutorial.
    [Show full text]
  • Implementation of Multivariate Data Set by Cart Algorithm
    International Journal of Information Technology and Knowledge Management July-December 2010, Volume 2, No. 2, pp. 455-459 IMPLEMENTATION OF MULTIVARIATE DATA SET BY CART ALGORITHM Sneha Soni Data mining deals with various applications such as the discovery of hidden knowledge, unexpected patterns and new rules from large Databases that guide to make decisions about enterprise to products and services competitive. Basically, data mining is concerned with the analysis of data and the use of software techniques for finding patterns and regularities in sets of data. Data Mining, which is known as knowledge discovery in databases has been defined as the nontrivial extraction of implicit, previous unknown and potentially useful information from data. In this paper CART Algorithm is presented which is well known for classification task of the datamining.CART is one of the best known methods for machine learning and computer statistical representation. In CART Result is represented in the form of Decision tree diagram or by flow chart. This paper shows results of multivariate dataset Classification by CART Algorithm. Multivariate dataset Encompassing the Simultaneous observation and analysis of more than one statistical variable. Keywords: Data Mining, Decision Tree, Multivariate Dataset, CART Algorithm 1. INTRODUCTION 2. CART ALGORITHM In data mining and machine learning different classifier are CART stands for Classification and Regression Trees a used for classifying different dataset for finding optimal classical statistical and machine learning method introduced classification. In this paper Classification and Regression by Leo Breiman, Jerome Friedman, Richard Olsen and Tree or CART Algorithm is implanted on multivariate Charles Stone in 1984.it is a data mining procedure to present dataset.
    [Show full text]
  • B. Gnana Priya Assistant Professor Department of Computer Science and Engineering Annamalai University
    A COMPARISON OF VARIOUS MACHINE LEARNING ALGORITHMS FOR IRIS DATASET B. Gnana Priya Assistant Professor Department of Computer Science and Engineering Annamalai University ABSTRACT Machine learning techniques are applications includes video surveillance, e- used for classification to predict group mail spam filtering, online fraud detection, membership of the data given. In this virtual personal assistance like Alexa, paper different machine learning automatic traffic prediction using GPS and algorithms are used for Iris plant many more. classification. Plant attributes like petal and sepal size of the Iris plant are taken Machine learning algorithms are and predictions are made from analyzing broadly classified into supervised and unsupervised algorithms. Supervised the pattern to find the class of Iris plant. Algorithms such as K-nearest neigbour, learning is a method in which we train the Support vector machine, Decision Trees, machine using data which are well labelled Random Forest and Naive Bayes classifier or the problem for which answers are well were applied to the Iris flower dataset and known. Then the machine is provided with were compared. new set of examples and the supervised learning algorithm analyses the training KEYWORDS: Machine learning, KNN, data and comes out with predictions and SVM, Naive Bayes, Decision Trees, produces an correct outcome from labelled Random Forest data. Unsupervised learning is the training of machine using information that is 1. INTRODUCTION neither classified nor labelled and allowing Machine learning is employed in the algorithm to act on that information almost every field nowadays. Starting without guidance. Reinforcement learning from the recommendations based on our approach is based on observation.
    [Show full text]
  • The Iris Dataset Revisited... Informatica 44 (2020) 35–44 37
    https://doi.org/10.31449/inf.v44i1.2715 Informatica 44 (2020) 35–44 35 The Iris Dataset Revisited – a Partial Ordering Study Lars Carlsen Awareness Center, Linkøpingvej 35, Trekroner, DK-4000 Roskilde4, Denmark E-mail: [email protected] Rainer Bruggemann Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Department: Ecohydrology D-92421 Schwandorf, Oskar - Kösters-Str. 11, Germany E-mail: [email protected] Keywords: IRIS data set, partial ordering, separability, dominance, classification Received: March 11, 2019 The well-known Iris data set has been studied applying partial ordering methodology. Previous studies, e.g., applying supervision learning such as neural networks (NN) and support-vector machines (SVM) perfectly distinguish between the three Iris subgroups, i.e., Iris Setosa, Iris Versicolour and Iris Virginica, respectively, in contrast to, e.g., K-means clustering that only separates the full Iris data set in two clusters. In the present study applying partial ordering methodology further discloses the difference between the different classification methods. The partial ordering results appears to be in perfect agreement with the results of the K-means clustering, which means that the clear separation in the three Iris subsets applying NN and SVM is neither recognized by clustering nor by partial ordering methodology. Povzetek: Analizirana je znana baza učnih domen Iris s poudarkom na nekaterih metodah, recimo gručenju. 1 Introduction One of the most often applied datasets in machine discipline with the methodological components of learning studies test cases is the Iris dataset [1, 2]. This combinatorics, algebra and graph theory, can be dataset includes 150 entries comprising 3 x 50 entries for attributed to the work of Birkhoff [5] and Hasse [6].
    [Show full text]
  • Identification of Iris Flower Species Using Machine Learning
    IPASJ International Journal of Computer Science (IIJCS) Web Site: http://www.ipasj.org/IIJCS/IIJCS.htm A Publisher for Research Motivation ........ Email:[email protected] Volume 5, Issue 8, August 2017 ISSN 2321-5992 Identification Of Iris Flower Species Using Machine Learning Shashidhar T Halakatti1, Shambulinga T Halakatti2 1Department. of Computer Science Engineering, Rural Engineering College ,Hulkoti – 582205 Dist : Gadag State : Karnataka Country : India 2Department of Electronics & Communication Engineering BVVS Polytechnic, Bagalkot – 587101 Dist : Bagalkot State : Karnataka Country: India ABSTRACT In Machine Learning, we are using semi-automated extraction of knowledge of data for identifying IRIS flower species. Classification is a supervised learning in which the response is categorical that is its values are in finite unordered set. To simply the problem of classification, scikit learn tools has been used. This paper focuses on IRIS flower classification using Machine Learning with scikit tools. Here the problem concerns the identification of IRIS flower species on the basis of flowers attribute measurements. Classification of IRIS data set would be discovering patterns from examining petal and sepal size of the IRIS flower and how the prediction was made from analyzing the pattern to from the class of IRIS flower. In this paper we train the machine learning model with data and when unseen data is discovered the predictive model predicts the species using what it has been learnt from the trained data. Keywords: Classification, Logistic Regression, K Nearest Neighbour, Machine Learning. 1. INTRODUCTION The Machine Learning is the subfield of computer science, according to Arthur Samuel in 1959 told “computers are having the ability to learn without being explicitly programmed”.
    [Show full text]
  • Machine-Learning
    machine-learning #machine- learning Table of Contents About 1 Chapter 1: Getting started with machine-learning 2 Remarks 2 Examples 2 Installation or Setup using Python 2 Installation or Setup using R Language 5 Chapter 2: An introduction to Classificiation: Generating several models using Weka 7 Introduction 7 Examples 7 Getting started: Loading a dataset from file 7 Train the first classifier: Setting a baseline with ZeroR 8 Getting a feel for the data. Training Naive Bayes and kNN 9 Putting it together: Training a tree 11 Chapter 3: Deep Learning 13 Introduction 13 Examples 13 Short brief of Deep learning 13 Chapter 4: Evaluation Metrics 18 Examples 18 Area Under the Curve of the Receiver Operating Characteristic (AUROC) 18 Overview – Abbreviations 18 Interpreting the AUROC 18 Computing the AUROC 19 Confusion Matrix 21 ROC curves 22 Chapter 5: Getting started with Machine Learning using Apache spark MLib 24 Introduction 24 Remarks 24 Examples 24 Write your first classification problem using Logistic Regression model 24 Chapter 6: Machine learning and it's classification 28 Examples 28 What is machine learning ? 28 What is supervised learning ? 28 What is unsupervised learning ? 29 Chapter 7: Machine Learning Using Java 30 Examples 30 tools list 30 Chapter 8: Natural Language Processing 33 Introduction 33 Examples 33 Text Matching or Similarity 33 Chapter 9: Neural Networks 34 Examples 34 Getting Started: A Simple ANN with Python 34 Backpropagation - The Heart of Neural Networks 37 1. Weights Initialisation 37 2. Forward Pass 38 3. Backward
    [Show full text]
  • A Study of Pattern Recognition of Iris Flower Based on Machine Learning 2
    Bachelor’s Thesis (UAS) Degree Program: Information Technology Specialization: Internet Technology 2013 Yu Yang A study of pattern recognition of Iris flower based on Machine Learning 2 BACHELOR’S THESIS | ABSTRACT TURKU UNIVERSITY OF APPLIED SCIENCES Degree Program: Information Technology | Specialization: Internet Technology 2013 | 43 Instructor: Patric Granholm Yu Yang A study of pattern recognition of Iris flower based on Machine Learning As we all know from the nature, most of creatures have the ability to recognize the objects in order to identify food or danger. Human beings can also recognize the types and application of objects. An interesting phenomenon could be that machines could recognize objects just like us someday in the future. This thesis mainly focuses on machine learning in pattern recognition applications. Machine learning is the core of Artificial Intelligence (AI) and pattern recognition is also an important branch of AI. In this thesis, the conception of machine learning and machine learning algorithms are introduced. Moreover, a typical and simple machine learning algorithm called K-means is introduced. A case study about Iris classification is introduced to show how the K-means works in pattern recognition. The aim of the case study is to design and implement a system of pattern recognition for the Iris flower based on Machine Learning. This project shows the workflow of pattern recognition and how to use machine learning approach to achieve this goal. The data set was collected from an open source website of
    [Show full text]
  • Robust Fuzzy Cluster Ensemble on Cancer Gene Expression Data
    University of Nevada Reno Robust Fuzzy Cluster Ensemble on Cancer Gene Expression Data A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering by Yan Yan Dr. Frederick C. Harris, Jr./Dissertation Advisor May, 2019 Copyright by Yan Yan 2018 All Rights Reserved THE GRADUATE SCHOOL We recommend that the dissertation prepared under our supervision by YAN YAN Entitled Robust Improved Fuzzy Cluster Ensemble On Cancer Gene Expression Data be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Dr. Frederick C. Harris, Jr., Advisor Dr. Sergiu Dascalu, Committee Member Dr. Dwight Egbert, Committee Member Dr. Ania Panorska, Committee Member Dr. Tin Nguyen, Committee Member Dr. Yantao Shen, Graduate School Representative David W. Zeh, Ph. D., Dean, Graduate School May, 2019 i Abstract In the past few decades, there has been tremendous growth in the scale and complexity of biological data generated by emerging high-throughput biotechnologies, including gene expression data generated by microarray technology. High-throughput gene expression data may contain gene expression measurements of thousands or millions of genes in a single data set, and provide us opportunities to explore the cell on a genome wide scale. Finding patterns in genomic data is a very important task in bioinformatics research and biomedical applications. Many clustering algorithms have been applied to gene expression data to find patterns. Nonetheless, there are still a number of challenges for clustering gene expression data because of the specific characteristics of such data and the special requirements from the domain of biology.
    [Show full text]
  • Spectral Clustering and Visualization: a Novel Clustering of Fisher’S Iris Data Set∗
    SPECTRAL CLUSTERING AND VISUALIZATION: A NOVEL CLUSTERING OF FISHER'S IRIS DATA SET∗ DAVID BENSON-PUTNINSy , MARGARET BONFARDINz , MEAGAN E. MAGNONIx , AND DANIEL MARTIN{ Advisors: Carl D. Meyer1 and Charles D. Wessell2 Abstract. Clustering is the act of partitioning a set of elements into subsets, or clusters, so that elements in the same cluster are, in some sense, similar. Determining an appropriate number of clusters in a particular data set is an important issue in data mining and cluster analysis. Another important issue is visualizing the strength, or connectivity, of clusters. We begin by creating a consensus matrix using multiple runs of the clustering algorithm k-means. This consensus matrix can be interpreted as a graph, which we cluster using two spectral clustering methods: the Fiedler Method and the MinMaxCut Method. To determine if increasing the number of clusters from k to k + 1 is appropriate, we check whether an existing cluster can be split. Finally, we visualize the strength of clusters by using the consensus matrix and the clustering obtained through one of the aforementioned spectral clustering techniques. Using these methods, we then investigate Fisher's Iris data set. Our methods support the exis- tence of four clusters, instead of the generally accepted three clusters in this data. Key words. cluster analysis, k-means, eigen decomposition, Laplacian matrix, data visualiza- tion, Fisher's Iris data set AMS subject classifications. 91C20, 15A18 1. Introduction. Clustering is the act of assigning a set of elements into subsets, or clusters, so that elements in the same cluster are, in some sense, similar.
    [Show full text]
  • 3D Data Visualization Techniques and Applications for Visual Multidimensional Data Mining
    UNIVERSITA` DEGLI STUDI DI SALERNO Dipartimento di Informatica Dottorato di Ricerca in Informatica XII Ciclo - Nuova Serie Tesi di Dottorato in 3D data visualization techniques and applications for visual multidimensional data mining Fabrizio Torre Ph.D. Program Chair Supervisors Ch.mo Prof. Giuseppe Persiano Ch.mo Prof. Gennaro Costagliola Dott. Vittorio Fuccella Anno Accademico 2013/2014 To my children: Sara and Luca in the hope of being able to lovingli preserve and grow... the little Angels who surely watch over me from the sky... Abstract Despite modern technology provide new tools to measure the world around us, we are quickly generating massive amounts of high-dimensional, spatial-temporal data. In this work, I deal with two types of datasets: one in which the spatial characteristics are relatively dynamic and the data are sampled at different periods of time, and the other where many dimensions prevail, although the spatial characteristics are relatively static. The first dataset refers to a peculiar aspect of uncertainty arising from contractual relationships that regulate a project execution: the dispute management. In recent years there has been a growth in size and complexity of the projects managed by public or private organizations. This leads to increased probability of project failures, frequently due to the difficulty and the ability to achieve the objectives such as on-time delivery, cost containment, expected quality achievement. In particular, one of the most common causes of project failure is the very high degree of uncertainty that affects the expected performance of the project, especially when different stakeholders with divergent aims and goals are involved in the project.
    [Show full text]
  • Comparative Evaluation of Various Techniques for Fisher's Iris
    ISSN (Online) : 2319 - 8753 ISSN (Print) : 2347 - 6710 International Journal of Innovative Research in Science, Engineering and Technology An ISO 3297: 2007 Certified Organization Volume 7, Special Issue 1, March 2018 6th National Conference on Frontiers in Communication and Signal Processing Systems (NCFCSPS '18) 13th-14th March 2018 Organized by Department of ECE, Adhiyamaan College of Engineering, Hosur, Tamilnadu, India Comparative Evaluation of Various Techniques for Fisher’s Iris Classification Mohan Prasath.M1, Paul Nathan.K2, Prakash.M3, Praveen Kumar.D4, Chidambaram.S5 UG Scholar, Department of ECE, Adhiyamaan College of Engineering, Hosur, TN, India1,2,3,4 Associate Professor, Department of ECE, Adhiyamaan College of Engineering, Hosur, TN, India5 ABSTRACT: Fisher’s Iris data base is perhaps the best known database to be found in the pattern recognition literature. The data set contains 3 classes of 50 instances each, where each class refers to a type of Iris plant. One class is linearly separable from the other two; the latter are not linearly separable from each other. Cluster analysis plays a vital role in various fields in order to group similar data from the available database. There are various clustering algorithms available in order to cluster the data but the entire algorithm are not suitable for all applications. Classification and clustering are both used to group data (observations) into multiple groups where those in the same group are in some way "similar" and those in different groups are not. In this project, we envisages distinct classification techniques and assessed the effectiveness of the algorithms on Fisher’s Iris dataset.
    [Show full text]