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- The Course DS OLAP DS DP DW DM Association DS ClassificationDS = Data source ClusteringDW = Data warehouseDM = Data MiningDP = Staging Database
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Chapter Objectives Learn basic techniques for data classification and prediction. Realize the difference between the following classifications of data: – supervised classification – prediction – unsupervised classification
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Chapter Outline What is classification and prediction of data? How do we classify data by decision tree induction? What are neural networks and how can they classify? What is Bayesian classification? Are there other classification techniques? How do we predict continuous values?
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What is Classification? The goal of data classification is to organize and categorize data in distinct classes. – A model is first created based on the data distribution. – The model is then used to classify new data. – Given the model, a class can be predicted for new data. Classification = prediction for discrete and nominal values
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What is Prediction? The goal of prediction is to forecast or deduce the value of an attribute based on values of other attributes. – A model is first created based on the data distribution. – The model is then used to predict future or unknown values In Data Mining – If forecasting discrete value Classification – If forecasting continuous value Prediction
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Supervised and Unsupervised Supervised Classification = Classification – We know the class labels and the number of classes Unsupervised Classification = Clustering – We do not know the class labels and may not know the number of classes
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Preparing Data Before Classification Data transformation: – Discretization of continuous data – Normalization to [-1..1] or [0..1] Data Cleaning: – Smoothing to reduce noise Relevance Analysis: – Feature selection to eliminate irrelevant attributes
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Application Credit approval Target marketing Medical diagnosis Defective parts identification in manufacturing Crime zoning Treatment effectiveness analysis Etc
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Classification is a 3-step process 1. Model construction (Learning): • Each tuple is assumed to belong to a predefined class, as determined by one of the attributes, called the class label. • The set of all tuples used for construction of the model is called training set. – The model is represented in the following forms: • Classification rules, (IF-THEN statements), • Decision tree • Mathematical formulae
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1. Classification Process (Learning)Name Income Age CreditSamir Low <30 rating Classification Method badAhmed Medium [30...40 ] goodSalah High <30 goodAli Medium >40 good Classification ModelSami Low [30..40] goodEmad Medium <30 bad IF Income = ‘High’ Training Data class OR Age > 30 THEN Class = ‘Good OR Decision Tree OR Mathematical For
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Classification is a 3-step process2. Model Evaluation (Accuracy): – Estimate accuracy rate of the model based on a test set. – The known label of test sample is compared with the classified result from the model. – Accuracy rate is the percentage of test set samples that are correctly classified by the model. – Test set is independent of training set otherwise over-fitting will occur
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2. Classification Process (Accuracy Evaluation) Classification ModelName Income Age Credit rating ModelNaser Low <30 Bad Bad AccuracyLutfi Medium <30 Bad good 75%Adel High >40 good goodFahd Medium [30..40] good good class
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Classification is a three-step process3. Model Use (Classification): – The model is used to classify unseen objects. • Give a class label to a new tuple • Predict the value of an actual attribute
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3. Classification Process (Use) Classification Model Name Income Age Credit rating Adham Low <30 ?
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Classification Methods Classification Method Decision Tree Induction Neural Networks Bayesian Classification Association-Based Classification K-Nearest Neighbour Case-Based Reasoning Genetic Algorithms Rough Set Theory Fuzzy Sets Etc.
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Evaluating Classification Methods Predictive accuracy – Ability of the model to correctly predict the class label Speed and scalability – Time to construct the model – Time to use the model Robustness – Handling noise and missing values Scalability – Efficiency in large databases (not memory resident data) Interpretability: – The level of understanding and insight provided by the model
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Chapter Outline What is classification and prediction of data? How do we classify data by decision tree induction ? What are neural networks and how can they classify? What is Bayesian classification? Are there other classification techniques? How do we predict continuous values?
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What is a Decision Tree? A decision tree is a flow-chart-like tree structure. – Internal node denotes a test on an attribute – Branch represents an outcome of the test • All tuples in branch have the same value for the tested attribute. Leaf node represents class label or class label distribution
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Sample Decision Tree Excellent customers Fair customers 80 Income < 6K >= 6KAge 50 No YES 20 2000 6000 10000 Income
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Sample Decision Tree 80 Income <6k >=6k NO AgeAge 50 >=50 <50 NO Yes 20 2000 6000 10000 Income
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Sample Decision TreeOutlook Temp Humidity Windy Play?sunny hot high FALSE Nosunny hot high TRUE Noovercast hot high FALSE Yesrainy mild high FALSE Yesrainy cool normal FALSE Yesrainy cool Normal TRUE Noovercast cool Normal TRUE Yessunny mild High FALSE Nosunny cool Normal FALSE Yesrainy mild Normal FALSE Yessunny mild normal TRUE Yesovercast mild High TRUE Yesovercast hot Normal FALSE Yesrainy mild high TRUE No http://www-lmmb.ncifcrf.gov/~toms/paper/primer/latex/index.html http://directory.google.com/Top/Science/Math/Applications/Information_Theory/Papers/
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Decision-Tree Classification Methods The basic top-down decision tree generation approach usually consists of two phases: 1. Tree construction • At the start, all the training examples are at the root. • Partition examples are recursively based on selected attributes. 2. Tree pruning • Aiming at removing tree branches that may reflect noise in the training data and lead to errors when classifying test data improve classification accuracy
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How to Specify Test Condition? Depends on attribute types – Nominal – Ordinal – Continuous Depends on number of ways to split – 2-way split – Multi-way split
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Splitting Based on Nominal Attributes Multi-way split: Use as many partitions as distinct values. CarType Family Luxury Sports Binary split: Divides values into two subsets. Need to find optimal partitioning. CarType CarType {Sports, OR {Family, Luxury} {Family} Luxury} {Sports}
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Splitting Based on Ordinal Attributes Multi-way split: Use as many partitions as distinct values. Size Small Large Medium Binary split: Divides values into two subsets. Need to find optimal partitioning. Size Size {Medium,{Small, {Large} OR Large} {Small}Medium} Size {Small, What about this split? Large} {Medium}
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Splitting Based on Continuous Attributes Different ways of handling – Discretization to form an ordinal categorical attribute • Static – discretize once at the beginning • Dynamic – ranges can be found by equal interval bucketing, equal frequency bucketing (percentiles), or clustering. – Binary Decision: (A < v) or (A ≥ v) • consider all possible splits and finds the best cut • can be more compute intensive
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Tree Induction Greedy strategy. – Split the records based on an attribute test that optimizes certain criterion. Issues – Determine how to split the records • How to specify the attribute test condition? • How to determine the best split? – Determine when to stop splitting
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How to determine the Best Split Good customers fair customers Customers Income Age <10k >=10k young old
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How to determine the Best Split Greedy approach: – Nodes with homogeneous class distribution are preferred Need a measure of node impurity: High degree Low degree pure of impurity of impurity 50% red 75% red 100% red 50% green 25% green 0% green
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Measures of Node Impurity Information gain – Uses Entropy Gain Ratio – Uses Information Gain and Splitinfo Gini Index – Used only for binary splits
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Algorithm for Decision Tree Induction Basic algorithm (a greedy algorithm) – Tree is constructed in a top-down recursive divide-and-conquer manner – At start, all the training examples are at the root – Attributes are categorical (if continuous-valued, they are discretized in advance) – Examples are partitioned recursively based on selected attributes – Test attributes are selected on the basis of a heuristic or statistical measure (e.g., information gain) Conditions for stopping partitioning – All samples for a given node belong to the same class – There are no remaining attributes for further partitioning – majority voting is employed for classifying the leaf – There are no samples left
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Classification Algorithms ID3 – Uses information gain C4.5 – Uses Gain Ratio CART – Uses Gini
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Entropy: Used by ID3 Entropy(S) = - p log2 p - q log2 q Entropy measures the impurity of S S is a set of examples p is the proportion of positive examples q is the proportion of negative examples
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ID3outlook temperature humidity windy play playsunny hot high FALSE nosunny hot high TRUE no don’t playovercast hot high FALSE yesrainy mild high FALSE yes pno = 5/14rainy cool normal FALSE yesrainy cool normal TRUE noovercast cool normal TRUE yessunny mild high FALSE nosunny cool normal FALSE yesrainy mild normal FALSE yessunny mild normal TRUE yes pyes = 9/14overcast mild high TRUE yesovercast hot normal FALSE yesrainy mild high TRUE no Impurity = - pyes log2 pyes - pno log2 pno = - 9/14 log2 9/14 - 5/14 log2 5/14 = 0.94 bits
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ID3 0.94 bits play don’t play al play xim tion 2 dont play play dont play play dont play play dont play ma ma sunny 3 high 3 4 hot 2 2 FALSE 6 2 or overcast 4 0 mild 4 2 infrainy ain 3 g 2 normal 6 1 cool 3 1 TRUE 3 3 outlook humidity temperature windy sunny overcast rainy high normal hot mild cool false true amount of information required to specify class of an example given that it reaches node0.97 bits 0.0 bits 0.97 bits 0.98 bits 0.59 bits 1.0 bits 0.92 bits 0.81 bits 0.81 bits 1.0 bits* 5/14 * 4/14 * 5/14 * 7/14 * 7/14 * 4/14 * 6/14 * 4/14 * 8/14 * 6/14 + + + + = 0.69 bits = 0.79 bits = 0.91 bits = 0.89 bits gain: 0.25 bits gain: 0.15 bits gain: 0.03 bits gain: 0.05 bits
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ID3 outlook play don’t play sunny overcast rainy 0.97 bits outlook sunny temperature hot humidity high windy FALSE play no sunny hot high TRUE no sunny mild high FALSE no sunny cool normal FALSE yes al xim tion sunny mild normal TRUE yes ma ma humidity or inf gain temperature windy high normal hot mild cool false true 0.0 bits 0.0 bits 0.0 bits 1.0 bits 0.0 bits 0.92 bits 1.0 bits * 3/5 * 2/5 * 2/5 * 2/5 * 1/5 * 3/5 * 2/5 + + + = 0.0 bits = 0.40 bits = 0.95 bits gain: 0.97 bits gain: 0.57 bits gain: 0.02 bits
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ID3 outlook play don’t play outlook temperature humidity windy play sunny overcast rainy rainy mild high FALSE yes rainy cool normal FALSE yes 0.97 bits rainy rainy cool mild normal normal TRUE FALSE no yes rainy mild high TRUE nohumidity humidity temperature windyhigh normal high normal hot mild cool false true ∅ 1.0 bits 0.92 bits 0.92 bits 1.0 bits 0.0 bits 0.0 bits *2/5 * 3/5 * 3/5 * 2/5 * 3/5 * 2/5 + + + = 0.95 bits = 0.95 bits = 0.0 bits gain: 0.02 bits gain: 0.02 bits gain: 0.97 bits
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ID3outlook temperature humidity windy playsunny hot high FALSE nosunny hot high TRUE noovercast hot high FALSE yesrainy mild high FALSE yesrainy cool normal FALSE yesrainy cool normal TRUE noovercast cool normal TRUE yessunny mild high FALSE nosunny cool normal FALSE yesrainy mild normal FALSE yes playsunny mild normal TRUE yesovercastovercast mild hot high normal TRUE FALSE yes yes outlook don’t playrainy mild high TRUE no sunny overcast rainy Yes humidity windy high normal false true No Yes Yes No
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C4.5 Information gain measure is biased towards attributes with a large number of values C4.5 (a successor of ID3) uses gain ratio to overcome the problem (normalization to information gain) – GainRatio(A) = Gain(A)/SplitInfo(A) v | Dj | | Dj | SplitInfo A ( D ) = −∑ × log 2 ( ) j =1 |D| |D| Ex. 5 5 4 4 5 5 SplitInfo A ( D) = − ×log 2 ( ) − ×log 2 ( ) − ×log 2 ( ) = 0.926 14 14 14 14 14 14 – gain_ratio(income) = 0.029/0.926 = 0.031 The attribute with the maximum gain ratio is selected as the splitting attribute
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CART If a data set D contains examples from n classes, gini index, gini(D) is defined as n 2 gini( D) =1− ∑ p j j =1 where pj is the relative frequency of class j in D If a data set D is split on A into two subsets D1 and D2, the gini index gini(D) is defined as |D1| |D | gini A ( D) = gini( D1) + 2 gini( D 2) |D| |D| Reduction in Impurity: ∆gini( A) = gini( D) − giniA ( D) The attribute provides the smallest ginisplit(D) (or the largest reduction in impurity) is chosen to split the node (need to enumerate all the possible splitting points for each attribute)
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CART Ex. D has 9 tuples in buys_computer = “yes” and 5 in “no” 2 2 9 5 gini ( D) = 1 − − = 0.459 14 14 Suppose the attribute income partitions D into 10 in D1: {low, medium} and 4 in D2 10 4 giniincome∈{low,medium} ( D ) = Gini ( D1 ) + Gini ( D1 ) 14 14 but gini{medium,high} is 0.30 and thus the best since it is the lowest All attributes are assumed continuous-valued May need other tools, e.g., clustering, to get the possible split values Can be modified for categorical attributes
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Comparing Attribute Selection Measures The three measures, in general, return good results but – Information gain: • biased towards multivalued attributes – Gain ratio: • tends to prefer unbalanced splits in which one partition is much smaller than the others – Gini index: • biased to multivalued attributes • has difficulty when # of classes is large • tends to favor tests that result in equal-sized partitions and purity in both partitions
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Other Attribute Selection Measures CHAID: a popular decision tree algorithm, measure based on χ2 test for independence C-SEP: performs better than info. gain and gini index in certain cases G-statistics: has a close approximation to χ2 distribution MDL (Minimal Description Length) principle (i.e., the simplest solution is preferred): – The best tree as the one that requires the fewest # of bits to both (1) encode the tree, and (2) encode the exceptions to the tree Multivariate splits (partition based on multiple variable combinations) – CART: finds multivariate splits based on a linear comb. of attrs. Which attribute selection measure is the best? – Most give good results, none is significantly superior than others
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Underfitting and Overfitting OverfittingUnderfitting: when model is too simple, both training andtest errors are large
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Overfitting due to NoiseDecision boundary is distorted by noise point
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Underfitting due to Insufficient ExamplesLack of data points in the lower half of the diagram makes it difficultto predict correctly the class labels of that region- Insufficient number of training records in the region causes thedecision tree to predict the test examples using other trainingrecords that are irrelevant to the classification task
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Two approaches to avoid Overfitting Prepruning: – Halt tree construction early—do not split a node if this would result in the goodness measure falling below a threshold – Difficult to choose an appropriate threshold Postpruning: – Remove branches from a “fully grown” tree—get a sequence of progressively pruned trees – Use a set of data different from the training data to decide which is the “best pruned tree”
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Scalable Decision Tree Induction Methods ID3, C4.5, and CART are not efficient when the training set doesn’t fit the available memory. Instead the following algorithms are used – SLIQ • Builds an index for each attribute and only class list and the current attribute list reside in memory – SPRINT • Constructs an attribute list data structure – RainForest • Builds an AVC-list (attribute, value, class label) – BOAT • Uses bootstrapping to create several small samples
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BOAT BOAT (Bootstrapped Optimistic Algorithm for Tree Construction) – Use a statistical technique called bootstrapping to create several smaller samples (subsets), each fits in memory – Each subset is used to create a tree, resulting in several trees – These trees are examined and used to construct a new tree T’ • It turns out that T’ is very close to the tree that would be generated using the whole data set together – Adv: requires only two scans of DB, an incremental alg.
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Why decision tree induction in data mining? Relatively faster learning speed (than other classification methods) Convertible to simple and easy to understand classification rules Comparable classification accuracy with other methods
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Converting Tree to Rules Outlook Sunny Overcast Rain Humidity Yes WindHigh Normal Strong WeakNo Yes No Yes R1: IF (Outlook=Sunny) AND (Humidity=High) THEN Play=No R2: IF (Outlook=Sunny) AND (Humidity=Normal) THEN Play=Yes R3: IF (Outlook=Overcast) THEN Play=Yes R4: IF (Outlook=Rain) AND (Wind=Strong) THEN Play=No R5: IF (Outlook=Rain) AND (Wind=Weak) THEN Play=Yes
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Decision trees: The Weka tool@relation weather.symbolic@attribute outlook {sunny, overcast, rainy}@attribute temperature {hot, mild, cool}@attribute humidity {high, normal}@attribute windy {TRUE, FALSE}@attribute play {yes, no}@datasunny,hot,high,FALSE,nosunny,hot,high,TRUE,noovercast,hot,high,FALSE,yesrainy,mild,high,FALSE,yesrainy,cool,normal,FALSE,yesrainy,cool,normal,TRUE,noovercast,cool,normal,TRUE,yessunny,mild,high,FALSE,nosunny,cool,normal,FALSE,yesrainy,mild,normal,FALSE,yessunny,mild,normal,TRUE,yesovercast,mild,high,TRUE,yesovercast,hot,normal,FALSE,yesrainy,mild,high,TRUE,nohttp://www.cs.waikato.ac.nz/ml/weka/
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Basic StatisticsAssume• D = All students• X = ICS students• C = SWE students 74 D X 6 4 C 16 |X| = 10 P(X) = 10/100 P(X|C) = P(X,C)/P(C) = 4/20 |C| = 20 P(C) = 20/100 P(C|X) = P(X,C)/P(X) = 4/10 |D| = 100 P(X,C) = 4/100 P(X,C) = P(C|X)*P(X) = P(X|C)*P(C)
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Bayesian Classifier – Basic Equation P(X,C) = P(C|X)*P(X) = P(X|C)*P(C) Class Prior Probability Descriptor Posterior Probability P( C ) P( X | C ) P( C | X ) = P( X ) Class Posterior Probability Descriptor Prior Probability
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Naive Bayesian Classifier P ( C | X ) = P( C ) P( X | C ) P( X ) P (C1 )P( C1 | X ) = P( x1 | C1 ) P( x2 | C1 ) P( x3 | C1 ) .... P( xn | C1 ) P(X) P(C2 )P( C2 | X ) = P( x1 | C2 ) P( x2 | C2 ) P( x3 | C2 ) .... P( xn | C2 ) P( X) P(Cm )P( Cm | X ) = P( x1 | Cm ) P( x2 | Cm ) P( x3 | Cm ) .... P( xn | Cm ) P( X) Independence assumption about descriptors
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Training DataOutlook Temp Humidity Windy Play?sunny hot high FALSE Nosunny hot high TRUE Noovercast hot high FALSE Yesrainy mild high FALSE Yesrainy cool normal FALSE Yesrainy cool Normal TRUE Noovercast cool Normal TRUE Yessunny mild High FALSE Nosunny cool Normal FALSE Yesrainy mild Normal FALSE Yessunny mild normal TRUE Yesovercast mild High TRUE Yesovercast hot Normal FALSE Yesrainy mild high TRUE No P(yes) = 9/14 P(no) = 5/14
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Bayesian Classifier – Probabilities for the weather data Frequency Tables Outlook | No Yes Temp. | No Yes Humidity | No Yes Windy | No Yes ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- Sunny | 3 2 Hot | 2 2 High | 4 3 False | 2 6 ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- Overcast | 0 4 Mild | 2 4 Normal | 1 6 True | 3 3 ---------------------------------- ---------------------------------- Rainy | 2 3 Cool | 1 3 Outlook | No Yes Temp. | No Yes Humidity | No Yes Windy | No Yes ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- Sunny | 3/5 2/9 Hot | 2/5 2/9 High | 4/5 3/9 False | 2/5 6/9 ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- Overcast | 0/5 4/9 Mild | 2/5 4/9 Normal | 1/5 6/9 True | 3/5 3/9 ---------------------------------- ---------------------------------- Rainy | 2/5 3/9 Cool | 1/5 3/9 Likelihood Tables
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Bayesian Classifier – Predicting a new day Outlook Temp. Humidity Windy PlayX sunny cool high true ? Class? P(yes|X) = p(sunny|yes) x p(cool|yes) x p(high|yes) x p(true|yes) x p(yes) = 2/9 x 3/9 x 3/9 x 3/9 x 9/14 = 0.0053 => 0.0053/(0.0053+0.0206) = 0.205 P(no|X) = p(sunny|no) x p(cool|no) x p(high|no) x p(true|no) x p(no) = 3/5 x 1/5 x 4/5 x 3/5 x 5/14 = 0.0206=0.0206/(0.0053+0.0206) = 0.795
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Bayesian Classifier – zero frequency problem What if a descriptor value doesn’t occur with every class value P(outlook=overcast|No)=0 Remedy: add 1 to the count for every descriptor-class combination (Laplace Estimator)Outlook | No Yes Temp. | No Yes Humidity | No Yes Windy | No Yes---------------------------------- ---------------------------------- ---------------------------------- ----------------------------------Sunny | 3+1 2+1 Hot | 2+1 2+1 High | 4+1 3+1 False | 2+1 6+1---------------------------------- ---------------------------------- ---------------------------------- ----------------------------------Overcast | 0+1 4+1 Mild | 2+1 4+1 Normal | 1+1 6+1 True | 3+1 3+1---------------------------------- ----------------------------------Rainy | 2+1 3+1 Cool | 1+1 3+1
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Bayesian Classifier – General Equation P ( X | Ck ) P( Ck ) P ( Ck | X ) = P( X )Likelihood: P ( X | Ck ) 1 ( x − µ )2 Continues variable: P ( x | C ) = exp− (2πσ ) 2 1/ 2 2σ 2
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Bayesian Classifier – Dealing with numeric attributes
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Bayesian Classifier – Dealing with numeric attributes
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Naïve Bayesian Classifier: Comments Advantages – Easy to implement – Good results obtained in most of the cases Disadvantages – Assumption: class conditional independence, therefore loss of accuracy – Practically, dependencies exist among variables • E.g., hospitals: patients: Profile: age, family history, etc. Symptoms: fever, cough etc., Disease: lung cancer, diabetes, etc. • Dependencies among these cannot be modeled by Naïve Bayesian Classifier How to deal with these dependencies? – Bayesian Belief Networks
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Bayesian Belief Networks Bayesian belief network allows a subset of the variables conditionally independent A graphical model of causal relationships – Represents dependency among the variables – Gives a specification of joint probability distribution Nodes: random variables Links: dependency X Y X and Y are the parents of Z, and Y is the parent of P Z No dependency between Z and P P Has no loops or cycles
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Bayesian Belief Network: An Example The conditional probability table Family (CPT) for variable LungCancer: Smoker History (FH, S) (FH, ~S) (~FH, S) (~FH, ~S) LC 0.8 0.5 0.7 0.1 ~LC 0.2 0.5 0.3 0.9LungCancer Emphysema CPT shows the conditional probability for each possible combination of its parentsPositiveXRay Dyspnea Derivation of the probability of a particular combination of values of X, from CPT: n Bayesian Belief Networks P ( x1 ,..., xn ) = ∏ P ( x i | Parents (Y i )) i =1
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Training Bayesian Networks Several scenarios: – Given both the network structure and all variables observable: learn only the CPTs – Network structure known, some hidden variables: gradient descent (greedy hill-climbing) method, analogous to neural network learning – Network structure unknown, all variables observable: search through the model space to reconstruct network topology – Unknown structure, all hidden variables: No good algorithms known for this purpose.
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Support Vector Machines w• x + b = 0 w • x + b = +1 w • x + b = −1 1 if w • x + b ≥ 1 2f ( x) = Margin = 2 −1 if w • x + b ≤ −1 || w ||
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Finding the Decision Boundary Let {x1, ..., xn} be our data set and let yi ∈ {1,-1} be the class label of xi The decision boundary should classify all points correctly ⇒ The decision boundary can be found by solving the following constrained optimization problem This is a constrained optimization problem. Solving it is beyond our course
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Support Vector Machines 2 We want to maximize: Margin = 2 || w || 2 || w || – Which is equivalent to minimizing: L( w) = 2 – But subjected to the following constraints: 1 if w • x i + b ≥ 1 f ( xi ) = −1 if w • x i + b ≤ −1 • This is a constrained optimization problem – Numerical approaches to solve it (e.g., quadratic programming)
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Classifying new Tuples The decision boundary is determined only by the support vectors Let tj (j=1, ..., s) be the indices of the s support vectors. For testing with a new data z – Compute and classify z as class 1 if the sum is positive, and class 2 otherwise
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Support Vector Machines What if the training set is not linearly separable? Slack variables ξi can be added to allow misclassification of difficult or noisy examples, resulting margin called soft. ξi ξi
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Support Vector Machines What if the problem is not linearly separable? – Introduce slack variables • Need to minimize: 2 || w || N k L( w) = + C ∑ ξi 2 i =1 • Subject to: 1 if w • x i + b ≥ 1 - ξi f ( xi ) = −1 if w • x i + b ≤ −1 + ξi
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Nonlinear Support Vector Machines What if decision boundary is not linear?
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Non-linear SVMs Datasets that are linearly separable with some noise work out great: 0 x But what are we going to do if the dataset is just too hard? 0 x How about… mapping data to a higher-dimensional space: x2 0 x
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Non-linear SVMs: Feature spaces General idea: the original feature space can always be mapped to some higher-dimensional feature space where the training set is separable: Φ: x → φ(x)
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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 predictor variables and a response variable
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PredictionResponse Training data Attribute (Y) Attribute (X) Predictor
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Types of CorrelationPositive correlation Negative correlation No correlation
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Regression Analysis Simple Linear regression multiple regression Non-linear regression Other regression methods: – generalized linear model, – Poisson regression, – log-linear models, – regression trees
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Simple Linear Regressiondescribes the linear relationship between a predictor variable,plotted on the x-axis, and a response variable, plotted on they-axis Y X
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Simple Linear Regression Y = βo + β X 1 β1 Y 1.0 βo X
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Simple Linear RegressionFitting data to a linear modelYi = β o + β1 X i + ε i intercept slope residuals
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Simple Linear RegressionHow to fit data to a linear model? Least Square Method
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Least Squares Regression ˆ Model line: Y = β 0 + β1 X Residual (ε) = Y − Yˆ Sum of squares of residuals = ∑ ˆ (Y − Y ) 2 we must find values of β o and β1 that minimise ∑ ˆ (Y − Y ) 2
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Linear Regression A model line: y = w0 + w1 x acquired by using Method of least squares to estimates the best-fitting straight line has: w = y−w x 0 1 | D| ∑( x − x )( yi − y ) w = i i=1 1 ∑( x | D| i − x )2 i=1
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Multiple Linear Regression Multiple linear regression: involves more than one predictor variable The linear model with a single predictor variable X can easily be extended to two or more predictor variables Y = β o + β1 X 1 + β 2 X 2 + ... + β p X p + ε – Solvable by extension of least square method or using SAS, S-Plus
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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 – possible to obtain least square estimates through extensive calculation on more complex formulae
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What is a ANN? ANN is a data structure that supposedly simulates the behavior of neurons in a biological brain. ANN is composed of layers of units interconnected. Messages are passed along the connections from one unit to the other. Messages can change based on the weight of the connection and the value in the node
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General Structure of ANNx0 w0 - µkx1 w1 ∑ f xn wn
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ANNOutput Y is 1 if at least two of the three inputs are equal to 1.
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ANN Y = I (0.3 X 1 + 0.3 X 2 + 0.3 X 3 − 0.4 > 0) 1 if z is true where I ( z ) = 0 otherwise
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Artificial Neural Networks Model is an assembly of inter-connected nodes and weighted links Output node sums up each of its input value according to the weights of its links Perceptron Model Compare output node Y = I ( ∑wi X i − t ) or against some threshold t i Y = sign( ∑ wi X i − t ) i
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Neural Networks Advantages – prediction accuracy is generally high. – robust, works when training examples contain errors. – output may be discrete, real-valued, or a vector of several discrete or real-valued attributes. – fast evaluation of the learned target function. Criticism – long training time. – difficult to understand the learned function (weights). – not easy to incorporate domain knowledge.
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Learning Algorithms Back propagation for classification Kohonen feature maps for clustering Recurrent back propagation for classification Radial basis function for classification Adaptive resonance theory Probabilistic neural networks
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Major Steps for Back Propagation Network Constructing a network – input data representation – selection of number of layers, number of nodes in each layer. Training the network using training data Pruning the network Interpret the results
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A Multi-Layer Feed-Forward Neural Network wij I j = ∑ wij Oi + θ j i 1 Oj = −I j 1+ e
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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 networks prediction The network is feed-forward in that 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
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Defining a Network Topology First decide the network topology: # of units in the input layer, # of hidden layers (if > 1), # of units in each hidden layer, and # of units in the output layer Normalizing the input values for each attribute measured in the training tuples to [0.0—1.0] One input unit per domain value 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
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Backpropagation Iteratively process a set of training tuples & compare the networks prediction with the actual known target value For each training tuple, the weights are modified to minimize the mean squared error between the networks 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 #s) and biases in the network – Propagate the inputs forward (by applying activation function) – Backpropagate the error (by updating weights and biases) – Terminating condition (when error is very small, etc.)
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Backpropagation Err j = O j (1 − O j )∑ Errk w jk k wij = wij + (l ) Err j Oi θ j = θ j + (l) Err j Err j = O j (1 − O j )(T j − O j )Generated value Correct value
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Network Pruning Fully connected network will be hard to articulate n input nodes, h hidden nodes and m output nodes lead to h(m+n) links (weights) Pruning: Remove some of the links without affecting classification accuracy of the network.
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Other Classification Methods Associative classification : Association rule based condSet class Genetic algorithm : Initial population of encoded rules are changed by mutation and cross-over based on survival of accurate once (survival). K-nearest neighbor classifier : Learning by analogy. Case-based reasoning : Similarity with other cases. Rough set theory : Approximation to equivalence classes. Fuzzy sets: Based on fuzzy logic (truth values between 0..1).
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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 set, constructs a classification model before receiving new (e.g., test) data to classify Lazy: less time in training but more time in predicting
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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. – Case-based reasoning • Uses symbolic representations and knowledge- based inference
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Nearest Neighbor Classifiers Basic idea: – If it walks like a duck, quacks like a duck, then it’s probably a duck Compute Distance Test Record Choose k of the “nearest” recordsTrainingrecords
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Instance-Based Classifiers • Store the training records • Use training records to predict the class label of unseen cases
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Definition of Nearest Neighbor X X X(a) 1-nearest neighbor (b) 2-nearest neighbor (c) 3-nearest neighbor K-nearest neighbors of a record x are data points that have the k smallest distance to x
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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 real- valued 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 _ + .
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Nearest-Neighbor ClassifiersRequires three things – The set of stored records – Distance Metric to compute distance between records – The value of k, the number of nearest neighbors to retrieveTo classify an unknown record: – Compute distance to other training records – Identify k nearest neighbors – Use class labels of nearest neighbors to determine the class label of unknown record (e.g., by taking majority vote)
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Nearest Neighbor Classification Compute distance between two points: – Euclidean distance d ( p, q ) = ∑( p i i −q ) i 2 Determine the class from nearest neighbor list – take the majority vote of class labels among the k- nearest neighbors – Weigh the vote according to distance • weight factor, w = 1/d2
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Nearest Neighbor Classification… Scaling issues – Attributes may have to be scaled to prevent distance measures from being dominated by one of the attributes – Example: • height of a person may vary from 1.5m to 1.8m • weight of a person may vary from 90lb to 300lb • income of a person may vary from $10K to $1M
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Nearest Neighbor Classification… Choosing the value of k: – If k is too small, sensitive to noise points – If k is too large, neighborhood may include points from other classes
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Metrics for Performance Evaluation Focus on the predictive capability of a model – Rather than how fast it takes to classify or build models, scalability, etc. Confusion Matrix: PREDICTED CLASS a: TP (true positive) Class=Yes Class=No b: FN (false negative) c: FP (false positive) Class=Yes a b ACTUAL d: TN (true negative) CLASS Class=No c d
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Metrics for Performance Evaluation… PREDICTED CLASS Class=Yes Class=No ACTUAL Class=Yes a b CLASS (TP) (FN) Class=No c d (FP) (TN) Most widely-used metric: a+d TP + TN Accuracy = = a + b + c + d TP + TN + FP + FN Error Rate = 1 - Accuracy
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Limitation of Accuracy Consider a 2-class problem – Number of Class 0 examples = 9990 – Number of Class 1 examples = 10 If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 % – Accuracy is misleading because model does not detect any class 1 example
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Predictor Error Measures Test error (generalization error): the average loss over the test set d – Mean absolute error: ∑| yi − yi | i =1 d d – Mean squared error: ∑(y i =1 i − yi ) 2 d d ∑y | i −yi | – Relative absolute error: i= d 1 ∑y | i=1 i −y | d ∑(y i =1 i − yi ) 2 – Relative squared error: d ∑(y i =1 i − y)2 – The mean squared-error exaggerates the presence of outliers Popularly use (square) root mean-square error, similarly, root relative squared error
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Evaluating Accuracy Holdout method – Given data is randomly partitioned into two independent sets • Training set (e.g., 2/3) for model construction • Test set (e.g., 1/3) for accuracy estimation – Random sampling: a variation of holdout • Repeat holdout k times, accuracy = avg. of the accuracies obtained Cross-validation (k-fold, where k = 10 is most popular) – Randomly partition the data into k mutually exclusive subsets, each approximately equal size – At i-th iteration, use Di as test set and others as training set
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Evaluating Accuracy Bootstrap – Works well with small data sets – Samples the given training tuples uniformly with replacement Several boostrap methods, and a common one is .632 boostrap – Suppose we are given a data set of d tuples. The data set is sampled d times, with replacement, resulting in a training set of d samples. The data tuples that did not make it into the training set end up forming the test set. About 63.2% of the original data will end up in the bootstrap, and the remaining 36.8% will form the test set (since (1 – 1/d)d ≈ e-1 = 0.368) – Repeat the sampling procedure k times, overall accuracy of the model: k acc( M ) = ∑ (0.632 × acc( M i ) test _ set +0.368 × acc( M i ) train _ set ) i =1
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Ensemble Methods Construct a set of classifiers from the training data Predict class label of previously unseen records by aggregating predictions made by multiple classifiers – Use a combination of models to increase accuracy – Combine a series of k learned models, M1, M2, …, Mk, with the aim of creating an improved model M* Popular ensemble methods – Bagging • averaging the prediction over a collection of classifiers – Boosting • weighted vote with a collection of classifiers
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Bagging: Boostrap Aggregation Analogy: Diagnosis based on multiple doctors’ majority vote Training – Given a set D of d tuples, at each iteration i, a training set Di of d tuples is sampled with replacement from D (i.e., boostrap) – A classifier model Mi is learned for each training set Di Classification: classify an unknown sample X – Each classifier Mi returns its class prediction – The bagged classifier M* counts the votes and assigns the class with the most votes to X Prediction: can be applied to the prediction of continuous values by taking the average value of each prediction for a given test tuple
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Bagging: Boostrap Aggregation Accuracy – Often significant better than a single classifier derived from D – For noise data: not considerably worse, more robust – Proved improved accuracy in prediction
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Boosting Analogy: Consult several doctors, based on a combination of weighted diagnoses—weight assigned based on the previous diagnosis accuracy How boosting works? – Weights are assigned to each training tuple – A series of k classifiers is iteratively learned – After a classifier Mi is learned, the weights are updated to allow the subsequent classifier, Mi+1, to pay more attention to the training tuples that were misclassified by Mi – The final M* combines the votes of each individual classifier, where the weight of each classifiers vote is a function of its accuracy
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Boosting The boosting algorithm can be extended for the prediction of continuous values Comparing with bagging: boosting tends to achieve greater accuracy, but it also risks overfitting the model to misclassified data
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Boosting: Adaboost Given a set of d class-labeled tuples, (X1, y1), …, (Xd, yd) Initially, all the weights of tuples are set the same (1/d) Generate k classifiers in k rounds. At round i, – Tuples from D are sampled (with replacement) to form a training set Di of the same size – Each tuple’s chance of being selected is based on its weight – A classification model Mi is derived from Di – Its error rate is calculated using Di as a test set – If a tuple is misclassified, its weight is increased, otherwise it is decreased Error rate: err(Xj) is the misclassification error of tuple Xj. Classifier Mi error rate is the sum of the weights of the misclassified tuples: d error ( M i ) = ∑ j ×err ( X j ) w j 1 − error ( M i ) log error ( M i ) The weight of classifier Mi’s vote is
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