24 Nov In chapter 8 we focus on cluster analysis.? Therefore, after reading the chapter answer the following questions: What are the characteristics of data? Compare the difference in each of the f
In chapter 8 we focus on cluster analysis. Therefore, after reading the chapter answer the following questions:
- What are the characteristics of data?
- Compare the difference in each of the following clustering types: prototype-based, density-based, graph-based.
- What is a scalable clustering algorithm?
- How do you choose the right algorithm?
- Students must not copy and post from sources. When referencing sources, students must rephrase all work from author’s and include in-text citations and references in APA format.
INTRODUCTION TO DATA MINING
INTRODUCTION TO DATA MINING SECOND EDITION
PANG-NING TAN Michigan State Universit MICHAEL STEINBACH University of Minnesota ANUJ KARPATNE University of Minnesota VIPIN KUMAR University of Minnesota
330 Hudson Street, NY NY 10013
Director, Portfolio Management: Engineering, Computer Science & Global Editions: Julian Partridge
Specialist, Higher Ed Portfolio Management: Matt Goldstein
Portfolio Management Assistant: Meghan Jacoby
Managing Content Producer: Scott Disanno
Content Producer: Carole Snyder
Web Developer: Steve Wright
Rights and Permissions Manager: Ben Ferrini
Manufacturing Buyer, Higher Ed, Lake Side Communications Inc (LSC): Maura Zaldivar-Garcia
Inventory Manager: Ann Lam
Product Marketing Manager: Yvonne Vannatta
Field Marketing Manager: Demetrius Hall
Marketing Assistant: Jon Bryant
Cover Designer: Joyce Wells, jWellsDesign
Full-Service Project Management: Chandrasekar Subramanian, SPi Global
Copyright ©2019 Pearson Education, Inc. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsonhighed.com/permissions/.
Many of the designations by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data on File
Names: Tan, Pang-Ning, author. | Steinbach, Michael, author. | Karpatne, Anuj, author. | Kumar, Vipin, 1956- author.
Title: Introduction to Data Mining / Pang-Ning Tan, Michigan State University, Michael Steinbach, University of Minnesota, Anuj Karpatne, University of Minnesota, Vipin Kumar, University of Minnesota.
Description: Second edition. | New York, NY : Pearson Education,  | Includes bibliographical references and index.
Identifiers: LCCN 2017048641 | ISBN 9780133128901 | ISBN 0133128903
Subjects: LCSH: Data mining.
Classification: LCC QA76.9.D343 T35 2019 | DDC 006.3/12–dc23 LC record available at https://lccn.loc.gov/2017048641
To our families …
Preface to the Second Edition Since the first edition, roughly 12 years ago, much has changed in the field of data analysis. The volume and variety of data being collected continues to increase, as has the rate (velocity) at which it is being collected and used to make decisions. Indeed, the term, Big Data, has been used to refer to the massive and diverse data sets now available. In addition, the term data science has been coined to describe an emerging area that applies tools and techniques from various fields, such as data mining, machine learning, statistics, and many others, to extract actionable insights from data, often big data.
The growth in data has created numerous opportunities for all areas of data analysis. The most dramatic developments have been in the area of predictive modeling, across a wide range of application domains. For instance, recent advances in neural networks, known as deep learning, have shown impressive results in a number of challenging areas, such as image classification, speech recognition, as well as text categorization and understanding. While not as dramatic, other areas, e.g., clustering, association analysis, and anomaly detection have also continued to advance. This new edition is in response to those advances.
As with the first edition, the second edition of the book provides a comprehensive introduction to data mining and is designed to be accessible and useful to students, instructors, researchers, and professionals. Areas
covered include data preprocessing, predictive modeling, association analysis, cluster analysis, anomaly detection, and avoiding false discoveries. The goal is to present fundamental concepts and algorithms for each topic, thus providing the reader with the necessary background for the application of data mining to real problems. As before, classification, association analysis and cluster analysis, are each covered in a pair of chapters. The introductory chapter covers basic concepts, representative algorithms, and evaluation techniques, while the more following chapter discusses advanced concepts and algorithms. As before, our objective is to provide the reader with a sound understanding of the foundations of data mining, while still covering many important advanced topics. Because of this approach, the book is useful both as a learning tool and as a reference.
To help readers better understand the concepts that have been presented, we provide an extensive set of examples, figures, and exercises. The solutions to the original exercises, which are already circulating on the web, will be made public. The exercises are mostly unchanged from the last edition, with the exception of new exercises in the chapter on avoiding false discoveries. New exercises for the other chapters and their solutions will be available to instructors via the web. Bibliographic notes are included at the end of each chapter for readers who are interested in more advanced topics, historically important papers, and recent trends. These have also been significantly updated. The book also contains a comprehensive subject and author index.
What is New in the Second Edition?
Some of the most significant improvements in the text have been in the two chapters on classification. The introductory chapter uses the decision tree classifier for illustration, but the discussion on many topics—those that apply
across all classification approaches—has been greatly expanded and clarified, including topics such as overfitting, underfitting, the impact of training size, model complexity, model selection, and common pitfalls in model evaluation. Almost every section of the advanced classification chapter has been significantly updated. The material on Bayesian networks, support vector machines, and artificial neural networks has been significantly expanded. We have added a separate section on deep networks to address the current developments in this area. The discussion of evaluation, which occurs in the section on imbalanced classes, has also been updated and improved.
The changes in association analysis are more localized. We have completely reworked the section on the evaluation of association patterns (introductory chapter), as well as the sections on sequence and graph mining (advanced chapter). Changes to cluster analysis are also localized. The introductory chapter added the K-means initialization technique and an updated the discussion of cluster evaluation. The advanced clustering chapter adds a new section on spectral graph clustering. Anomaly detection has been greatly revised and expanded. Existing approaches—statistical, nearest neighbor/density-based, and clustering based—have been retained and updated, while new approaches have been added: reconstruction-based, one- class classification, and information-theoretic. The reconstruction-based approach is illustrated using autoencoder networks that are part of the deep learning paradigm. The data chapter has been updated to include discussions of mutual information and kernel-based techniques.
The last chapter, which discusses how to avoid false discoveries and produce valid results, is completely new, and is novel among other contemporary textbooks on data mining. It supplements the discussions in the other chapters with a discussion of the statistical concepts (statistical significance, p-values, false discovery rate, permutation testing, etc.) relevant to avoiding spurious results, and then illustrates these concepts in the context of data
mining techniques. This chapter addresses the increasing concern over the validity and reproducibility of results obtained from data analysis. The addition of this last chapter is a recognition of the importance of this topic and an acknowledgment that a deeper understanding of this area is needed for those analyzing data.
The data exploration chapter has been deleted, as have the appendices, from the print edition of the book, but will remain available on the web. A new appendix provides a brief discussion of scalability in the context of big data.
To the Instructor
As a textbook, this book is suitable for a wide range of students at the advanced undergraduate or graduate level. Since students come to this subject with diverse backgrounds that may not include extensive knowledge of statistics or databases, our book requires minimal prerequisites. No database knowledge is needed, and we assume only a modest background in statistics or mathematics, although such a background will make for easier going in some sections. As before, the book, and more specifically, the chapters covering major data mining topics, are designed to be as self-contained as possible. Thus, the order in which topics can be covered is quite flexible. The core material is covered in chapters 2 (data), 3 (classification), 5 (association analysis), 7 (clustering), and 9 (anomaly detection). We recommend at least a cursory coverage of Chapter 10 (Avoiding False Discoveries) to instill in students some caution when interpreting the results of their data analysis. Although the introductory data chapter (2) should be covered first, the basic classification (3), association analysis (5), and clustering chapters (7), can be covered in any order. Because of the relationship of anomaly detection (9) to classification (3) and clustering (7), these chapters should precede Chapter 9.
Various topics can be selected from the advanced classification, association analysis, and clustering chapters (4, 6, and 8, respectively) to fit the schedule and interests of the instructor and students. We also advise that the lectures be augmented by projects or practical exercises in data mining. Although they are time consuming, such hands-on assignments greatly enhance the value of the course.
Support materials available to all readers of this book are available at http://www-users.cs.umn.edu/~kumar/dmbook.
PowerPoint lecture slides Suggestions for student projects Data mining resources, such as algorithms and data sets Online tutorials that give step-by-step examples for selected data mining techniques described in the book using actual data sets and data analysis software
Additional support materials, including solutions to exercises, are available only to instructors adopting this textbook for classroom use. The book’s resources will be mirrored at www.pearsonhighered.com/cs-resources. Comments and suggestions, as well as reports of errors, can be sent to the authors through [email protected]
Many people contributed to the first and second editions of the book. We begin by acknowledging our families to whom this book is dedicated. Without their patience and support, this project would have been impossible.
We would like to thank the current and former students of our data mining groups at the University of Minnesota and Michigan State for their contributions. Eui-Hong (Sam) Han and Mahesh Joshi helped with the initial data mining classes. Some of the exercises and presentation slides that they created can be found in the book and its accompanying slides. Students in our data mining groups who provided comments on drafts of the book or who contributed in other ways include Shyam Boriah, Haibin Cheng, Varun Chandola, Eric Eilertson, Levent Ertöz, Jing Gao, Rohit Gupta, Sridhar Iyer, Jung-Eun Lee, Benjamin Mayer, Aysel Ozgur, Uygar Oztekin, Gaurav Pandey, Kashif Riaz, Jerry Scripps, Gyorgy Simon, Hui Xiong, Jieping Ye, and Pusheng Zhang. We would also like to thank the students of our data mining classes at the University of Minnesota and Michigan State University who worked with early drafts of the book and provided invaluable feedback. We specifically note the helpful suggestions of Bernardo Craemer, Arifin Ruslim, Jamshid Vayghan, and Yu Wei.
Joydeep Ghosh (University of Texas) and Sanjay Ranka (University of Florida) class tested early versions of the book. We also received many useful suggestions directly from the following UT students: Pankaj Adhikari, Rajiv Bhatia, Frederic Bosche, Arindam Chakraborty, Meghana Deodhar, Chris Everson, David Gardner, Saad Godil, Todd Hay, Clint Jones, Ajay Joshi, Joonsoo Lee, Yue Luo, Anuj Nanavati, Tyler Olsen, Sunyoung Park, Aashish Phansalkar, Geoff Prewett, Michael Ryoo, Daryl Shannon, and Mei Yang.
Ronald Kostoff (ONR) read an early version of the clustering chapter and offered numerous suggestions. George Karypis provided invaluable LATEX assistance in creating an author index. Irene Moulitsas also provided
assistance with LATEX and reviewed some of the appendices. Musetta Steinbach was very helpful in finding errors in the figures.
We would like to acknowledge our colleagues at the University of Minnesota and Michigan State who have helped create a positive environment for data mining research. They include Arindam Banerjee, Dan Boley, Joyce Chai, Anil Jain, Ravi Janardan, Rong Jin, George Karypis, Claudia Neuhauser, Haesun Park, William F. Punch, György Simon, Shashi Shekhar, and Jaideep Srivastava. The collaborators on our many data mining projects, who also have our gratitude, include Ramesh Agrawal, Maneesh Bhargava, Steve Cannon, Alok Choudhary, Imme Ebert-Uphoff, Auroop Ganguly, Piet C. de Groen, Fran Hill, Yongdae Kim, Steve Klooster, Kerry Long, Nihar Mahapatra, Rama Nemani, Nikunj Oza, Chris Potter, Lisiane Pruinelli, Nagiza Samatova, Jonathan Shapiro, Kevin Silverstein, Brian Van Ness, Bonnie Westra, Nevin Young, and Zhi-Li Zhang.
The departments of Computer Science and Engineering at the University of Minnesota and Michigan State University provided computing resources and a supportive environment for this project. ARDA, ARL, ARO, DOE, NASA, NOAA, and NSF provided research support for Pang-Ning Tan, Michael Stein- bach, Anuj Karpatne, and Vipin Kumar. In particular, Kamal Abdali, Mitra Basu, Dick Brackney, Jagdish Chandra, Joe Coughlan, Michael Coyle, Stephen Davis, Frederica Darema, Richard Hirsch, Chandrika Kamath, Tsengdar Lee, Raju Namburu, N. Radhakrishnan, James Sidoran, Sylvia Spengler, Bhavani Thuraisingham, Walt Tiernin, Maria Zemankova, Aidong Zhang, and Xiaodong Zhang have been supportive of our research in data mining and high-performance computing.
It was a pleasure working with the helpful staff at Pearson Education. In particular, we would like to thank Matt Goldstein, Kathy Smith, Carole Snyder,
and Joyce Wells. We would also like to thank George Nichols, who helped with the art work and Paul Anagnostopoulos, who provided LATEX support.
We are grateful to the following Pearson reviewers: Leman Akoglu (Carnegie Mellon University), Chien-Chung Chan (University of Akron), Zhengxin Chen (University of Nebraska at Omaha), Chris Clifton (Purdue University), Joy- deep Ghosh (University of Texas, Austin), Nazli Goharian (Illinois Institute of Technology), J. Michael Hardin (University of Alabama), Jingrui He (Arizona State University), James Hearne (Western Washington University), Hillol Kargupta (University of Maryland, Baltimore County and Agnik, LLC), Eamonn Keogh (University of California-Riverside), Bing Liu (University of Illinois at Chicago), Mariofanna Milanova (University of Arkansas at Little Rock), Srinivasan Parthasarathy (Ohio State University), Zbigniew W. Ras (University of North Carolina at Charlotte), Xintao Wu (University of North Carolina at Charlotte), and Mohammed J. Zaki (Rensselaer Polytechnic Institute).
Over the years since the first edition, we have also received numerous comments from readers and students who have pointed out typos and various other issues. We are unable to mention these individuals by name, but their input is much appreciated and has been taken into account for the second edition.
Contents Preface to the Second Edition v
1 Introduction 1 1.1 What Is Data Mining? 4
1.2 Motivating Challenges 5
1.3 The Origins of Data Mining 7
1.4 Data Mining Tasks 9
1.5 Scope and Organization of the Book 13
1.6 Bibliographic Notes 15
1.7 Exercises 21
2 Data 23 2.1 Types of Data 26
2.1.1 Attributes and Measurement 27
2.1.2 Types of Data Sets 34
2.2 Data Quality 42 2.2.1 Measurement and Data Collection Issues 42
2.2.2 Issues Related to Applications 49
2.3 Data Preprocessing 50 2.3.1 Aggregation 51
2.3.2 Sampling 52
2.3.3 Dimensionality Reduction 56
2.3.4 Feature Subset Selection 58
2.3.5 Feature Creation 61
2.3.6 Discretization and Binarization 63
2.3.7 Variable Transformation 69
2.4 Measures of Similarity and Dissimilarity 71 2.4.1 Basics 72
2.4.2 Similarity and Dissimilarity between Simple Attributes 74
2.4.3 Dissimilarities between Data Objects 76
2.4.4 Similarities between Data Objects 78
2.4.5 Examples of Proximity Measures 79
2.4.6 Mutual Information 88
2.4.7 Kernel Functions* 90
2.4.8 Bregman Divergence* 94
2.4.9 Issues in Proximity Calculation 96
2.4.10 Selecting the Right Proximity Measure 98
2.5 Bibliographic Notes 100
2.6 Exercises 105
3 Classification: Basic Concepts and Techniques 113
3.1 Basic Concepts 114
3.2 General Framework for Classification 117
3.3 Decision Tree Classifier 119 3.3.1 A Basic Algorithm to Build a Decision Tree 121
3.3.2 Methods for Expressing Attribute Test Conditions 124
3.3.3 Measures for Selecting an Attribute Test Condition 127
3.3.4 Algorithm for Decision Tree Induction 136
3.3.5 Example Application: Web Robot Detection 138
3.3.6 Characteristics of Decision Tree Classifiers 140
3.4 Model Overfitting 147 3.4.1 Reasons for Model Overfitting 149
3.5 Model Selection 156 3.5.1 Using a Validation Set 156
3.5.2 Incorporating Model Complexity 157
3.5.3 Estimating Statistical Bounds 162
3.5.4 Model Selection for Decision Trees 162
3.6 Model Evaluation 164 3.6.1 Holdout Method 165
3.6.2 Cross-Validation 165
3.7 Presence of Hyper-parameters 168 3.7.1 Hyper-parameter Selection 168
3.7.2 Nested Cross-Validation 170
3.8 Pitfalls of Model Selection and Evaluation 172 3.8.1 Overlap between Training and Test Sets 172
3.8.2 Use of Validation Error as Generalization Error 172
3.9 Model Comparison 173 3.9.1 Estimating the Confidence Interval for Accuracy 174
3.9.2 Comparing the Performance of Two Models 175
3.10 Bibliographic Notes 176
3.11 Exercises 185
4 Classification: Alternative Techniques 193 4.1 Types of Classifiers 193
4.2 Rule-Based Classifier 195 4.2.1 How a Rule-Based Classifier Works 197
4.2.2 Properties of a Rule Set 198
4.2.3 Direct Methods for Rule Extraction 199
4.2.4 Indirect Methods for Rule Extraction 204
4.2.5 Characteristics of Rule-Based Classifiers 206
4.3 Nearest Neighbor Classifiers 208 4.3.1 Algorithm 209
4.3.2 Characteristics of Nearest Neighbor Classifiers 210
4.4 Naïve Bayes Classifier 212 4.4.1 Basics of Probability Theory 213
4.4.2 Naïve Bayes Assumption 218
4.5 Bayesian Networks 227 4.5.1 Graphical Representation 227
4.5.2 Inference and Learning 233
4.5.3 Characteristics of Bayesian Networks 242
4.6 Logistic Regression 243 4.6.1 Logistic Regression as a Generalized Linear Model 244
4.6.2 Learning Model Parameters 245
4.6.3 Characteristics of Logistic Regression 248
4.7 Artificial Neural Network (ANN) 249 4.7.1 Perceptron 250
4.7.2 Multi-layer Neural Network 254
4.7.3 Characteristics of ANN 261
4.8 Deep Learning 262 4.8.1 Using Synergistic Loss Functions 263
4.8.2 Using Responsive Activation Functions 266
4.8.3 Regularization 268
4.8.4 Initialization of Model Parameters 271
4.8.5 Characteristics of Deep Learning 275
4.9 Support Vector Machine (SVM) 276 4.9.1 Margin of a Separating Hyperplane 276
4.9.2 Linear SVM 278
4.9.3 Soft-margin SVM 284
4.9.4 Nonlinear SVM 290
4.9.5 Characteristics of SVM 294
4.10 Ensemble Methods 296 4.10.1 Rationale for Ensemble Method 297
4.10.2 Methods for Constructing an Ensemble Classifier 297
4.10.3 Bias-Variance Decomposition 300
4.10.4 Bagging 302
4.10.5 Boosting 305
4.10.6 Random Forests 310
4.10.7 Empirical Comparison among Ensemble Methods 312
4.11 Class Imbalance Problem 313 4.11.1 Building Classifiers with Class Imbalance 314
4.11.2 Evaluating Performance with Class Imbalance 318
4.11.3 Finding an Optimal Score Threshold 322
4.11.4 Aggregate Evaluation of Performance 323
4.12 Multiclass Problem 330
4.13 Bibliographic Notes 333
4.14 Exercises 345
5 Association Analysis: Basic Concepts and Algorithms 357 5.1 Preliminaries 358
5.2 Frequent Itemset Generation 362 5.2.1 The Apriori Principle 363
5.2.2 Frequent Itemset Generation in the Apriori Algorithm 364
5.2.3 Candidate Generation and Pruning 368
5.2.4 Support Counting 373
5.2.5 Computational Complexity 377
5.3 Rule Generation 380 5.3.1 Confidence-Based Pruning 380
5.3.2 Rule Generation in Apriori Algorithm 381
5.3.3 An Example: Congressional Voting Records 382
5.4 Compact Representation of Frequent Itemsets 384 5.4.1 Maximal Frequent Itemsets 384
5.4.2 Closed Itemsets 386
5.5 Alternative Methods for Generating Frequent Itemsets* 389
5.6 FP-Growth Algorithm* 393 5.6.1 FP-Tree Representation 394
5.6.2 Frequent Itemset Generation in FP-Growth Algorithm 397
5.7 Evaluation of Association Patterns 401
5.7.1 Objective Measures of Interestingness 402
5.7.2 Measures beyond Pairs of Binary Variables 414
5.7.3 Simpson’s Paradox 416
5.8 Effect of Skewed Support Distribution 418
5.9 Bibliographic Notes 424
5.10 Exercises 438
6 Association Analysis: Advanced Concepts 451 6.1 Handling Categorical Attributes 451
6.2 Handling Continuous Attributes 454 6.2.1 Discretization-Based Methods 454
6.2.2 Statistics-Based Methods 458
6.2.3 Non-discretization Methods 460
6.3 Handling a Concept Hierarchy 462
6.4 Sequential Patterns 464 6.4.1 Preliminaries 465
6.4.2 Sequential Pattern Discovery 468
6.4.3 Timing Constraints 473
6.4.4 Alternative Counting Schemes 477
6.5 Subgraph Patterns 479 6.5.1 Preliminaries 480
6.5.2 Frequent Subgraph Mining 483
6.5.3 Candidate Generation 487
6.5.4 Candidate Pruning 493
6.5.5 Support Counting 493
6.6 Infrequent Patterns 493 6.6.1 Negative Patterns 494
6.6.2 Negatively Correlated Patterns 495
6.6.3 Comparisons among Infrequent Patterns, Negative Patterns, and Negatively Correlated Patterns 496
6.6.4 Techniques for Mining Interesting Infrequent Patterns 498
6.6.5 Techniques Based on Mining Negative Patterns 499
6.6.6 Techniques Based on Support Expectation 501
6.7 Bibliographic Notes 505
6.8 Exercises 510
7 Cluster Analysis: Basic Concepts and Algorithms 525 7.1 Overview 528
7.1.1 What Is Cluster Analysis? 528
7.1.2 Different Types of Clusterings 529
7.1.3 Different Types of Clusters 531
7.2 K-means 534 7.2.1 The Basic K-means Algorithm 535
7.2.2 K-means: Additional Issues 544
7.2.3 Bisecting K-means 547
7.2.4 K-means and Different Types of Clusters 548
7.2.5 Strengths and Weaknesses 549
7.2.6 K-means as an Optimization Problem 549
7.3 Agglomerative Hierarchical Clustering 554 7.3.1 Basic Agglomerative Hierarchical Clustering Algorithm 555
7.3.2 Specific Techniques 557
7.3.3 The Lance-Williams Formula for Cluster Proximity 562
7.3.4 Key Issues in Hierarchical Clustering 563
7.3.5 Outliers 564
7.3.6 Strengths and Weaknesses 565
7.4 DBSCAN 565 7.4.1 Traditional Density: Center-Based Approach 565
7.4.2 The DBSCAN Algorithm 567
7.4.3 Strengths and Weaknesses 569
7.5 Cluster Evaluation 571 7.5.1 Overview 571
7.5.2 Unsupervised Cluster Evaluation Using Cohesion and Separation 574
7.5.3 Unsupervised Cluster Evaluation Using the Proximity Matrix 582
7.5.4 Unsupervised Evaluation of Hierarchical Clustering 585
7.5.5 Determining the Correct Number of Clusters 587
7.5.6 Clustering Tendency 588
7.5.7 Supervised Measures of Cluster Validity 589
7.5.8 Assessing the Significance of Cluster Validity Measures 594
7.5.9 Choosing a Cluster Validity Measure 596
7.6 Bibliographic Notes 597
7.7 Exercises 603
8 Cluster Analysis: Additional Issues and Algorithms 613 8.1 Characteristics of Data, Clusters, and Clustering Algorithms 614
8.1.1 Example: Comparing K-means and DBSCAN 614
8.1.2 Data Characteristics 615
8.1.3 Cluster Characteristics 617
8.1.4 General Characteristics of Clustering Algorithms 619
8.2 Prototype-Based Clustering 621 8.2.1 Fuzzy Clustering 621
8.2.2 Clustering Using Mixture Models 627
8.2.3 Self-Organizing Maps (SOM) 637
8.3 Density-Based Clustering 644 8.3.1 Grid-Based Clustering 644
8.3.2 Subspace Clustering 648
8.3.3 DENCLUE: A Kernel-Based Scheme for Density-Based Clustering 652
8.4 Graph-Based Clustering 656 8.4.1 Sparsification 657
8.4.2 Minimum Spanning Tree (MST) Clustering 658
8.4.3 OPOSSUM: Optimal Partitioning of Sparse Similarities Using METIS 659
8.4.4 Chameleon: Hierarchical Clustering with Dynamic Modeling 660
8.4.5 Spectral Cl
HOW OUR WEBSITE WORKS
Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of
HIGH QUALITY & PLAGIARISM FREE.
To make an Order you only need to click ORDER NOW and we will direct you to our Order Page at WriteDen. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.
Deadline range from 6 hours to 30 days.
Once done with writing your paper we will upload it to your account on our website and also forward a copy to your email.
Upon receiving your paper, review it and if any changes are needed contact us immediately. We offer unlimited revisions at no extra cost.
Is it Safe to use our services?
We never resell papers on this site. Meaning after your purchase you will get an original copy of your assignment and you have all the rights to use the paper.
Our price ranges from $8-$14 per page. If you are short of Budget, contact our Live Support for a Discount Code. All new clients are eligible for 20% off in their first Order. Our payment method is safe and secure.
Please note we do not have prewritten answers. We need some time to prepare a perfect essay for you.