Data Mining: Concepts and Techniques (3rd ed.) — Chapter

Data Mining: Concepts and Techniques (3rd ed.) — Chapter 9 — Classification: Advanced Methods Jiawei Han, Micheline Kamber, and Jian Pei University of Illinois at Urbana-Champaign & Simon Fraser University 2011 Han, Kamber & Pei. All rights reserved. 1

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 2

Bayesian Belief Networks Bayesian belief networks (also known as Bayesian networks, probabilistic networks): allow class conditional independencies between subsets of variables A (directed acyclic) graphical model of causal relationships Represents dependency among the variables Gives a specification of joint probability distribution Nodes: random variables Links: dependency Y X Z X and Y are the parents of Z, and P Y is the parent of P No dependency between Z and P Has no loops/cycles 3

Bayesian Belief Network: An Example Family History (FH) Smoker (S) CPT: Conditional Probability Table for variable LungCancer: (FH, S) (FH, S) ( FH, S) ( FH, S) LungCancer (LC) PositiveXRay Emphysema Dyspnea LC 0.8 0.5 0.7 0.1 LC 0.2 0.5 0.3 0.9 shows the conditional probability for each possible combination of its parents Derivation of the probability of a particular combination of values of X, from CPT: Bayesian Belief Network P ( x ,., x 1 n n ) P ( x i Parents (Y i )) i 1 4

Training Bayesian Networks: Several Scenarios Scenario 1: Given both the network structure and all variables observable: compute only the CPT entries Scenario 2: Network structure known, some variables hidden: gradient descent (greedy hill-climbing) method, i.e., search for a solution along the steepest descent of a criterion function Weights are initialized to random probability values At each iteration, it moves towards what appears to be the best solution at the moment, w.o. backtracking Weights are updated at each iteration & converge to local optimum Scenario 3: Network structure unknown, all variables observable: search through the model space to reconstruct network topology Scenario 4: Unknown structure, all hidden variables: No good algorithms known for this purpose D. Heckerman. A Tutorial on Learning with Bayesian Networks. In Learning in Graphical Models, M. Jordan, ed. MIT Press, 1999. 5

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 6

Classification by Backpropagation Backpropagation: A neural network learning algorithm Started by psychologists and neurobiologists to develop and test computational analogues of neurons A neural network: A set of connected input/output units where each connection has a weight associated with it During the learning phase, the network learns by adjusting the weights so as to be able to predict the correct class label of the input tuples Also referred to as connectionist learning due to the connections between units 7

Neural Network as a Classifier Weakness Long training time Require a number of parameters typically best determined empirically, e.g., the network topology or “structure.” Poor interpretability: Difficult to interpret the symbolic meaning behind the learned weights and of “hidden units” in the network Strength High tolerance to noisy data Ability to classify untrained patterns Well-suited for continuous-valued inputs and outputs Successful on an array of real-world data, e.g., hand-written letters Algorithms are inherently parallel Techniques have recently been developed for the extraction of rules from trained neural networks 8

A Multi-Layer Feed-Forward Neural Network Output vector w(jk 1) w(jk ) ( yi yˆ i( k ) ) xij Output layer Hidden layer wij Input layer Input vector: X 9

How A Multi-Layer Neural Network Works The inputs to the network correspond to the attributes measured for each training tuple Inputs are fed simultaneously into the units making up the input layer They are then weighted and fed simultaneously to a hidden layer The number of hidden layers is arbitrary, although usually only one The weighted outputs of the last hidden layer are input to units making up the output layer, which emits the network's prediction The network is feed-forward: None of the weights cycles back to an input unit or to an output unit of a previous layer From a statistical point of view, networks perform nonlinear regression: Given enough hidden units and enough training samples, they can closely approximate any function 10

Defining a Network Topology Decide the network topology: Specify # of units in the input layer, # of hidden layers (if 1), # of units in each hidden layer, and # of units in the output layer Normalize the input values for each attribute measured in the training tuples to [0.0—1.0] One input unit per domain value, each initialized to 0 Output, if for classification and more than two classes, one output unit per class is used Once a network has been trained and its accuracy is unacceptable, repeat the training process with a different network topology or a different set of initial weights 11

Backpropagation Iteratively process a set of training tuples & compare the network's prediction with the actual known target value For each training tuple, the weights are modified to minimize the mean squared error between the network's prediction and the actual target value Modifications are made in the “backwards” direction: from the output layer, through each hidden layer down to the first hidden layer, hence “backpropagation” Steps Initialize weights to small random numbers, associated with biases Propagate the inputs forward (by applying activation function) Backpropagate the error (by updating weights and biases) Terminating condition (when error is very small, etc.) 12

Neuron: A Hidden/Output Layer Unit bias x0 w0 x1 w1 xn k f wn output y For Example n Input weight vector x vector w weighted sum Activation function y sign( wi xi k ) i 0 An n-dimensional input vector x is mapped into variable y by means of the scalar product and a nonlinear function mapping The inputs to unit are outputs from the previous layer. They are multiplied by their corresponding weights to form a weighted sum, which is added to the bias associated with unit. Then a nonlinear activation function is applied to it. 13

Efficiency and Interpretability Efficiency of backpropagation: Each epoch (one iteration through the training set) takes O( D * w), with D tuples and w weights, but # of epochs can be exponential to n, the number of inputs, in worst case For easier comprehension: Rule extraction by network pruning Simplify the network structure by removing weighted links that have the least effect on the trained network Then perform link, unit, or activation value clustering The set of input and activation values are studied to derive rules describing the relationship between the input and hidden unit layers Sensitivity analysis: assess the impact that a given input variable has on a network output. The knowledge gained from this analysis can be represented in rules 14

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 15

Classification: A Mathematical Mapping Classification: predicts categorical class labels E.g., Personal homepage classification xi (x1, x2, x3, ), yi 1 or –1 x1 : # of word “homepage” x2 : # of word “welcome” x x x x x x x x x Mathematically, x X n, y Y { 1, –1}, o We want to derive a function f: X Y o o o x Linear Classification ooo Binary Classification problem o o o o o o Data above the red line belongs to class ‘x’ Data below red line belongs to class ‘o’ Examples: SVM, Perceptron, Probabilistic Classifiers 16

Discriminative Classifiers Advantages Prediction accuracy is generally high As compared to Bayesian methods – in general Robust, works when training examples contain errors Fast evaluation of the learned target function Bayesian networks are normally slow Criticism Long training time Difficult to understand the learned function (weights) Bayesian networks can be used easily for pattern discovery Not easy to incorporate domain knowledge Easy in the form of priors on the data or distributions 17

SVM—Support Vector Machines A relatively new classification method for both linear and nonlinear data It uses a nonlinear mapping to transform the original training data into a higher dimension With the new dimension, it searches for the linear optimal separating hyperplane (i.e., “decision boundary”) With an appropriate nonlinear mapping to a sufficiently high dimension, data from two classes can always be separated by a hyperplane SVM finds this hyperplane using support vectors (“essential” training tuples) and margins (defined by the support vectors) 18

SVM—History and Applications Vapnik and colleagues (1992)—groundwork from Vapnik & Chervonenkis’ statistical learning theory in 1960s Features: training can be slow but accuracy is high owing to their ability to model complex nonlinear decision boundaries (margin maximization) Used for: classification and numeric prediction Applications: handwritten digit recognition, object recognition, speaker identification, benchmarking time-series prediction tests 19

SVM—General Philosophy Small Margin Large Margin Support Vectors 20

SVM—Margins and Support Vectors July 2, 2024 Data Mining: Concepts and Techniques 21

SVM—When Data Is Linearly Separable m Let data D be (X1, y1), , (X D , y D ), where Xi is the set of training tuples associated with the class labels yi There are infinite lines (hyperplanes) separating the two classes but we want to find the best one (the one that minimizes classification error on unseen data) SVM searches for the hyperplane with the largest margin, i.e., maximum marginal hyperplane (MMH) 22

SVM—Linearly Separable A separating hyperplane can be written as W X b 0 where W {w1, w2, , wn} is a weight vector and b a scalar (bias) For 2-D it can be written as w0 w1 x1 w2 x2 0 The hyperplane defining the sides of the margin: H 1: w 0 w 1 x 1 w 2 x 2 1 for yi 1, and H2: w0 w1 x1 w2 x2 – 1 for yi –1 Any training tuples that fall on hyperplanes H1 or H2 (i.e., the sides defining the margin) are support vectors This becomes a constrained (convex) quadratic optimization problem: Quadratic objective function and linear constraints Quadratic Programming (QP) 23

Why Is SVM Effective on High Dimensional Data? The complexity of trained classifier is characterized by the # of support vectors rather than the dimensionality of the data The support vectors are the essential or critical training examples —they lie closest to the decision boundary (MMH) If all other training examples are removed and the training is repeated, the same separating hyperplane would be found The number of support vectors found can be used to compute an (upper) bound on the expected error rate of the SVM classifier, which is independent of the data dimensionality Thus, an SVM with a small number of support vectors can have good generalization, even when the dimensionality of 24

SVM—Linearly Inseparable A 2 Transform the original input data into a higher dimensional space A Search for a linear separating hyperplane in the new space 25 1

SVM: Different Kernel functions Instead of computing the dot product on the transformed data, it is math. equivalent to applying a kernel function K(Xi, Xj) to the original data, i.e., K(Xi, Xj) Φ(Xi) Φ(Xj) Typical Kernel Functions SVM can also be used for classifying multiple ( 2) classes and for regression analysis (with additional 26

Scaling SVM by Hierarchical MicroClustering SVM is not scalable to the number of data objects in terms of training time and memory usage H. Yu, J. Yang, and J. Han, “ Classifying Large Data Sets Using SVM with Hierarchical Clusters ”, KDD'03) CB-SVM (Clustering-Based SVM) Given limited amount of system resources (e.g., memory), maximize the SVM performance in terms of accuracy and the training speed Use micro-clustering to effectively reduce the number of points to be considered At deriving support vectors, de-cluster micro-clusters near “candidate vector” to ensure high classification accuracy 27

CF-Tree: Hierarchical Microcluster Read the data set once, construct a statistical summary of the data (i.e., hierarchical clusters) given a limited amount of memory Micro-clustering: Hierarchical indexing structure provide finer samples closer to the boundary and coarser samples farther from the boundary 28

Selective Declustering: Ensure High Accuracy CF tree is a suitable base structure for selective declustering De-cluster only the cluster Ei such that Di – Ri Ds, where Di is the distance from the boundary to the center point of Ei and Ri is the radius of Ei Decluster only the cluster whose subclusters have possibilities to be the support cluster of the boundary “Support cluster”: The cluster whose centroid is a support vector 29

CB-SVM Algorithm: Outline Construct two CF-trees from positive and negative data sets independently Need one scan of the data set Train an SVM from the centroids of the root entries De-cluster the entries near the boundary into the next level The children entries de-clustered from the parent entries are accumulated into the training set with the non-declustered parent entries Train an SVM again from the centroids of the entries in the training set Repeat until nothing is accumulated 30

Accuracy and Scalability on Synthetic Dataset Experiments on large synthetic data sets shows better accuracy than random sampling approaches and far more scalable than the original SVM algorithm 31

SVM vs. Neural Network SVM Deterministic algorithm Nice generalization properties Hard to learn – learned in batch mode using quadratic programming techniques Using kernels can learn very complex functions Neural Network Nondeterministic algorithm Generalizes well but doesn’t have strong mathematical foundation Can easily be learned in incremental fashion To learn complex functions—use multilayer perceptron (nontrivial) 32

SVM Related Links SVM Website: http://www.kernel-machines.org/ Representative implementations LIBSVM: an efficient implementation of SVM, multiclass classifications, nu-SVM, one-class SVM, including also various interfaces with java, python, etc. SVM-light: simpler but performance is not better than LIBSVM, support only binary classification and only in C SVM-torch: another recent implementation also written in C 33

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 34

Associative Classification Associative classification: Major steps Mine data to find strong associations between frequent patterns (conjunctions of attribute-value pairs) and class labels Association rules are generated in the form of P1 p2 pl “Aclass C” (conf, sup) Organize the rules to form a rule-based classifier Why effective? It explores highly confident associations among multiple attributes and may overcome some constraints introduced by decision-tree induction, which considers only one attribute at a time Associative classification has been found to be often more accurate than some traditional classification methods, such as C4.5 35

Typical Associative Classification Methods CBA (Classification Based on Associations: Liu, Hsu & Ma, KDD’98) Mine possible association rules in the form of Build classifier: Organize rules according to decreasing precedence based on confidence and then support CMAR (Classification based on Multiple Association Rules: Li, Han, Pei, ICDM’01) Cond-set (a set of attribute-value pairs) class label Classification: Statistical analysis on multiple rules CPAR (Classification based on Predictive Association Rules: Yin & Han, SDM’03) Generation of predictive rules (FOIL-like analysis) but allow covered rules to retain with reduced weight Prediction using best k rules High efficiency, accuracy similar to CMAR 36

Frequent Pattern-Based Classification H. Cheng, X. Yan, J. Han, and C.-W. Hsu, “ Discriminative Frequent Pattern Analysis for Effective Clas sification ”, ICDE'07 Accuracy issue Increase the discriminative power Increase the expressive power of the feature space Scalability issue It is computationally infeasible to generate all feature combinations and filter them with an information gain threshold Efficient method (DDPMine: FPtree pruning): H. Cheng, X. Yan, J. Han, and P. S. Yu, " Direct Discriminative Pattern Mining for Effective Classi fication ", ICDE'08 37

Frequent Pattern vs. Single Feature The discriminative power of some frequent patterns is higher than that of single features. (a) Austral (b) Cleve (c) Sonar Fig. 1. Information Gain vs. Pattern Length 38

Empirical Results 1 InfoGain IG UpperBnd 0.9 0.8 Information Gain 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 100 200 300 400 500 600 700 Support (a) Austral (b) Breast (c) Sonar Fig. 2. Information Gain vs. Pattern Frequency 39

Feature Selection Given a set of frequent patterns, both nondiscriminative and redundant patterns exist, which can cause overfitting We want to single out the discriminative patterns and remove redundant ones The notion of Maximal Marginal Relevance (MMR) is borrowed A document has high marginal relevance if it is both relevant to the query and contains minimal marginal similarity to previously selected documents 40

Experimental Results 41 41

Scalability Tests 42

DDPMine: Branch-and-Bound Search sup(child ) sup( parent ) sup(b) sup(a) a b a: constant, a parent node b: variable, a descendent Association between information gain and frequency 43

DDPMine Efficiency: Runtime PatClass Harmon y PatClass: ICDE’07 Pattern Classification Alg. DDPMin e 44

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 45

Lazy vs. Eager Learning Lazy vs. eager learning Lazy learning (e.g., instance-based learning): Simply stores training data (or only minor processing) and waits until it is given a test tuple Eager learning (the above discussed methods): Given a set of training tuples, constructs a classification model before receiving new (e.g., test) data to classify Lazy: less time in training but more time in predicting Accuracy Lazy method effectively uses a richer hypothesis space since it uses many local linear functions to form an implicit global approximation to the target function Eager: must commit to a single hypothesis that covers the entire instance space 46

Lazy Learner: Instance-Based Methods Instance-based learning: Store training examples and delay the processing (“lazy evaluation”) until a new instance must be classified Typical approaches k-nearest neighbor approach Instances represented as points in a Euclidean space. Locally weighted regression Constructs local approximation Case-based reasoning Uses symbolic representations and knowledge-based inference 47

The k-Nearest Neighbor Algorithm All instances correspond to points in the n-D space The nearest neighbor are defined in terms of Euclidean distance, dist(X1, X2) Target function could be discrete- or realvalued For discrete-valued, k-NN returns the most common value among the k training examples nearest to xq Vonoroi diagram: the decision surface induced by 1-NN for a typical set of training examples . . xq . . . . 48

Discussion on the k-NN Algorithm k-NN for real-valued prediction for a given unknown tuple Returns the mean values of the k nearest neighbors Distance-weighted nearest neighbor algorithm Weight the contribution of each of the k neighbors according to their distance to the query xq 1 w Give greater weight to closer neighbors d ( xq , x )2 i Robust to noisy data by averaging k-nearest neighbors Curse of dimensionality: distance between neighbors could be dominated by irrelevant attributes To overcome it, axes stretch or elimination of the least relevant attributes 49

Case-Based Reasoning (CBR) CBR: Uses a database of problem solutions to solve new problems Store symbolic description (tuples or cases)—not points in a Euclidean space Applications: Customer-service (product-related diagnosis), legal ruling Methodology Instances represented by rich symbolic descriptions (e.g., function graphs) Search for similar cases, multiple retrieved cases may be combined Tight coupling between case retrieval, knowledge-based reasoning, and problem solving Challenges Find a good similarity metric Indexing based on syntactic similarity measure, and when failure, backtracking, and adapting to additional cases 50

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 51

Genetic Algorithms (GA) Genetic Algorithm: based on an analogy to biological evolution An initial population is created consisting of randomly generated rules Each rule is represented by a string of bits E.g., if A1 and A2 then C2 can be encoded as 100 If an attribute has k 2 values, k bits can be used Based on the notion of survival of the fittest, a new population is formed to consist of the fittest rules and their offspring The fitness of a rule is represented by its classification accuracy on a set of training examples Offspring are generated by crossover and mutation The process continues until a population P evolves when each rule in P satisfies a prespecified threshold Slow but easily parallelizable 52

Rough Set Approach Rough sets are used to approximately or “roughly” define equivalent classes A rough set for a given class C is approximated by two sets: a lower approximation (certain to be in C) and an upper approximation (cannot be described as not belonging to C) Finding the minimal subsets (reducts) of attributes for feature reduction is NP-hard but a discernibility matrix (which stores the differences between attribute values for each pair of data tuples) is used to reduce the computation intensity 53

Fuzzy Set Approaches Fuzzy logic uses truth values between 0.0 and 1.0 to represent the degree of membership (such as in a fuzzy membership graph) Attribute values are converted to fuzzy values. Ex.: Income, x, is assigned a fuzzy membership value to each of the discrete categories {low, medium, high}, e.g. 49K belongs to “medium income” with fuzzy value 0.15 but belongs to “high income” with fuzzy value 0.96 Fuzzy membership values do not have to sum to 1. Each applicable rule contributes a vote for membership in the categories Typically, the truth values for each predicted category are summed, and these sums are combined 54

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 55

Multiclass Classification Classification involving more than two classes (i.e., 2 Classes) Method 1. One-vs.-all (OVA): Learn a classifier one at a time Classifier j: treat tuples in class j as positive & all others as negative To classify a tuple X, the set of classifiers vote as an ensemble Method 2. All-vs.-all (AVA): Learn a classifier for each pair of classes Given m classes, construct m(m-1)/2 binary classifiers A classifier is trained using tuples of the two classes Given m classes, train m classifiers: one for each class To classify a tuple X, each classifier votes. X is assigned to the class with maximal vote Comparison All-vs.-all tends to be superior to one-vs.-all Problem: Binary classifier is sensitive to errors, and errors affect vote count 56

Error-Correcting Codes for Multiclass Classification Originally designed to correct errors during data transmission for communication tasks by exploring data redundancy Example A 7-bit codeword associated with classes 1-4 Clas s Error-Corr. Codeword C1 1 1 1 1 1 1 1 C2 0 0 0 0 1 1 1 C3 0 0 1 1 0 0 1 C4 0 1 0 1 0 1 0 Given a unknown tuple X, the 7-trained classifiers output: 0001010 Hamming distance: # of different bits between two codewords H(X, C ) 5, by checking # of bits between [1111111] & 1 [0001010] H(X, C ) 3, H(X, C ) 3, H(X, C ) 1, thus C as the label 2 3 4 4 for X Error-correcting codes can correct up to (h-1)/h 1-bit error, where h is the minimum Hamming distance between any two codewords 57

Semi-Supervised Classification Semi-supervised: Uses labeled and unlabeled data to build a classifier Self-training: Build a classifier using the labeled data Use it to label the unlabeled data, and those with the most confident label prediction are added to the set of labeled data Repeat the above process Adv: easy to understand; disadv: may reinforce errors Co-training: Use two or more classifiers to teach each other Each learner uses a mutually independent set of features of each tuple to train a good classifier, say f1 Then f1 and f2 are used to predict the class label for unlabeled data X Teach each other: The tuple having the most confident prediction from f1 is added to the set of labeled data for f2, & vice versa Other methods, e.g., joint probability distribution of features and labels 58

Active Learning Class labels are expensive to obtain Active learner: query human (oracle) for labels Pool-based approach: Uses a pool of unlabeled data L: a small subset of D is labeled, U: a pool of unlabeled data in D Use a query function to carefully select one or more tuples from U and request labels from an oracle (a human annotator) The newly labeled samples are added to L, and learn a model Goal: Achieve high accuracy using as few labeled data as possible Evaluated using learning curves: Accuracy as a function of the number of instances queried (# of tuples to be queried should be small) Research issue: How to choose the data tuples to be queried? Uncertainty sampling: choose the least certain ones Reduce version space, the subset of hypotheses consistent w. the training data Reduce expected entropy over U: Find the greatest reduction in the total number of incorrect predictions 59

Transfer Learning: Conceptual Framework Transfer learning: Extract knowledge from one or more source tasks and apply the knowledge to a target task Traditional learning: Build a new classifier for each new task Transfer learning: Build new classifier by applying existing knowledge learned from source tasks Different Tasks Learning System Learning System Traditional Learning Framework Source Tasks Learning System Knowledge Target Task Learning System Transfer Learning Framework 60

Transfer Learning: Methods and Applications Applications: Especially useful when data is outdated or distribution changes, e.g., Web document classification, e-mail spam filtering Instance-based transfer learning: Reweight some of the data from source tasks and use it to learn the target task TrAdaBoost (Transfer AdaBoost) Assume source and target data each described by the same set of attributes (features) & class labels, but rather diff. distributions Require only labeling a small amount of target data Use source data in training: When a source tuple is misclassified, reduce the weight of such tupels so that they will have less effect on the subsequent classifier Research issues Negative transfer: When it performs worse than no transfer at all Heterogeneous transfer learning: Transfer knowledge from different feature space or multiple source domains Large-scale transfer learning 61

Chapter 9. Classification: Advanced Methods Bayesian Belief Networks Classification by Backpropagation Support Vector Machines Classification by Using Frequent Patterns Lazy Learners (or Learning from Your Neighbors) Other Classification Methods Additional Topics Regarding Classification Summary 62

Summary Effective and advanced classification methods Bayesian belief network (probabilistic networks) Backpropagation (Neural networks) Support Vector Machine (SVM) Pattern-based classification Other classification methods: lazy learners (KNN, case-based reasoning), genetic algorithms, rough set and fuzzy set approaches Additional Topics on Classification Multiclass classification Semi-supervised classification Active learning Transfer learning 63

References Please see the references of Chapter 8 64

Surplus Slides

What Is Prediction? (Numerical) prediction is similar to classification construct a model use model to predict continuous or ordered value for a given input Prediction is different from classification Classification refers to predict categorical class label Prediction models continuous-valued functions Major method for prediction: regression model the relationship between one or more independent or predictor variables and a dependent or response variable Regression analysis Linear and multiple regression Non-linear regression Other regression methods: generalized linear model, Poisson regression, log-linear models, regression trees 66

Linear Regression Linear regression: involves a response variable y and a single predictor variable x y w0 w1 x where w0 (y-intercept) and w1 (slope) are regression coefficients Method of least squares: estimates the best-fitting straight line D (x i x )( yi y ) w 1 (x x) i 1 D 2 w y w x 0 1 i i 1 Multiple linear regression: involves more than one predictor variable Training data is of the form (X1, y1), (X2, y2), , (X D , y D ) Ex. For 2-D data, we may have: y w0 w1 x1 w2 x2 Solvable by extension of least square method or using SAS, SPlus Many nonlinear functions can be transformed into the above 67

Nonlinear Regression Some nonlinear models can be modeled by a polynomial function A polynomial regression model can be transformed into linear regression model. For example, y w0 w1 x w2 x2 w3 x3 convertible to linear with new variables: x2 x2, x3 x3 y w0 w1 x w2 x2 w3 x3 Other functions, such as power function, can also be transformed to linear model Some models are intractable nonlinear (e.g., sum of exponential terms) possible to obtain least square estimates through extensive calculation on more complex formulae 68

Other Regression-Based Models Generalized linear model: Foundation on which linear regression can be applied to modeling categorical response variables Variance of y is a function of the mean value of y, not a constant Logistic regression: models the prob. of some event occurring as a linear function of a set of predictor variables Poisson regression: models the data that exhibit a Poisson distribution Log-linear models: (for categorical data) Approximate discrete multidimensional prob. distributions Also useful for data compression and smoothing Regression trees and model trees Trees to predict continuous values rather than class labels 69

Regression Trees and Model Trees Regression tree: proposed in CART system (Breiman et al. 1984) CART: Classification And Regression Trees Each leaf stores a continuous-valued prediction It is the average value of the predicted attribute for the training tuples that reach the leaf Model tree: proposed by Quinlan (1992) Each leaf holds a regression model—a multivariate linear equation for the predicted attribute A more general case than regression tree Regression and model trees tend to be more accurate than linear regression when the data are not represented well by a simple linear model 70

Predictive Modeling in Multidimensional Databases Predictive modeling: Predict data values or construct generalized linear models based on the database data One can only predict value ranges or category distributions Method outline: Minimal generalization Attribute relevance analysis Generalized linear model construction Prediction Determine the major factors which influence the prediction Data relevance analysis: uncertainty measurement, entropy analysis, expert judgement, etc. Multi-level prediction: drill-down and roll-up analysis 71

Prediction: Numerical Data 72

Prediction: Categorical Data 73

SVM—Introductory Literature “Statistical Learning Theory” by Vapnik: extremely hard to understand, containing many errors too. C. J. C. Burges. A Tutorial on Support Vector Machines for Pattern Recognition. Knowledge Discovery and Data Mining, 2(2), 1998. Better than the Vapnik’s book, but still written too hard for introduction, and the examples are so not-intuitive The book “An Introduction to Support Vector Machines” by N. Cristianini and J. Shawe-Taylor Also written hard for introduction, but the explanation about the mercer’s theorem is better than above literatures The neural network book by Haykins Contains one nice chapter of SVM introduction 74

Notes about SVM— Introductory Literature “Statistical Learning Theory” by Vapnik: difficult to understand, containing many errors. C. J. C. Burges. A Tutorial on Support Vector Machines for Pattern Recognition. Knowledge Discovery and Data Mining, 2(2), 1998. Easier than Vapnik’s book, but still not introductory level; the examples are not so intuitive The book An Introduction to Support Vector Machines by Cristianini and Shawe-Taylor Not introductory level, but the explanation about Mercer’s Theorem is better than above literatures Neural Networks and Learning Machines by Haykin Contains a nice chapter on SVM introduction 75

Associative Classification Can Achieve High Accuracy and Efficiency (Cong et al. SIGMOD05) 76

A Closer Look at CMAR CMAR (Classification based on Multiple Association Rules: Li, Han, Pei, ICDM’01) Efficiency: Uses an enhanced FP-tree that maintains the distribution of class labels among tuples satisfying each frequent itemset Rule pruning whenever a rule is inserted into the tree Given two rules, R and R , if the antecedent of R is more 1 2 1 general than that of R2 and conf(R1) conf(R2), then prune R2 Prunes rules for which the rule antecedent and class are not positively correlated, based on a χ2 test of statistical significance Classification based on generated/pruned rules If only one rule satisfies tuple X, assign the class label of the rule If a rule set S satisfies X, CMAR divides S into groups according to class labels uses a weighted χ2 measure to find the strongest group of rules, based on the statistical correlation of rules within a group assigns X the class label of the strongest group 77

Perceptron & Winnow Vector: x, w x2 Scalar: x, y, w Input: {(x1, y1), } Output: classification function f(x) f(xi) 0 for yi 1 f(xi) 0 for yi -1 f(x) wx b 0 or w1x1 w2update x2 b W 0 Perceptron: additively x1 Winnow: update W multiplicatively 78