Lecture outline• Classification• Naïve Bayes classifier• Nearest-neighbor classifierEager vs Lazy learners• Eager learners: learn the model as soon as the training data becomes available• Lazy learners: delay model-building until testing data needs to be classified– Rote classifier: memorizes the entire training datak-nearest neighbor classifiersXXX(a) 1-nearest neighbor (b) 2-nearest neighbor (c) 3-nearest neighbork-nearest neighbors of a record x are data points that have the k smallest distance to xk-nearest neighbor classification• Given a data record x find its k closest points– Closeness: Euclidean, Hamming, Jaccard distance • Determine the class of x based on the classes in the neighbor list– Majority vote– Weigh the vote according to distance• e.g., weight factor, w = 1/d2– Probabilistic votingCharacteristics of nearest-neighbor classifiers• Instance of instance-based learning• No model building (lazy learners)– Lazy learners: computational time in classification– Eager learners: computational time in model building• Decision trees try to find global models, k-NN take into account local information• K-NN classifiers depend a lot on the choice of proximity measureBayes Theorem• X, Y random variables• Joint probability: Pr(X=x,Y=y)• Conditional probability: Pr(Y=y | X=x)• Relationship between joint and conditional probability distributions• Bayes Theorem:)Pr()|Pr()Pr()|Pr(),Pr( XXYYYXYX)Pr()Pr()|Pr()|Pr(XYYXXYBayes Theorem for Classification• X: attribute set• Y: class variable• Y depends on X in a non-determininstic way• We can capture this dependence using Pr(Y|X) : Posterior probabilityvsPr(Y): Prior probabilityBuilding the Classifier• Training phase:– Learning the posterior probabilities Pr(Y|X) for every combination of X and Y based on training data• Test phase:– For test record X’, compute the class Y’ that maximizes the posterior probabilityPr(Y’|X’)Bayes Classification: ExampleX’=(Home Owner=No, Marital Status=Married, AnnualIncome=120K)Compute: Pr(Yes|X’), Pr(No|X’) pick No or Yes with max Prob.How can we compute these probabilities??Computing posterior probabilities• Bayes Theorem• P(X) is constant and can be ignored• P(Y): estimated from training data; compute the fraction of training records in each class• P(X|Y)?)Pr()Pr()|Pr()|Pr(XYYXXYNaïve Bayes Classifier• Attribute set X = {X1,…,Xd} consists of dattributes• Conditional independence:– X conditionally independent of Y, given X: Pr(X|Y,Z) = Pr(X|Z)– Pr(X,Y|Z) = Pr(X|Z)xPr(Y|Z)diiyYXyYX1)|Pr()|Pr(Naïve Bayes Classifier• Attribute set X = {X1,…,Xd} consists of dattributesdiiyYXyYX1)|Pr()|Pr()Pr()|Pr()Pr()|Pr(1XYXYXYdiiConditional probabilities for categorical attributes• Categorical attribute Xi• Pr(Xi = xi|Y=y): fraction of training instances in class y that take value xion the i-th attributePr(homeOwner = yes|No) = 3/7Pr(MaritalStatus = Single| Yes) = 2/3Estimating conditional probabilities for continuous attributes?• Discretization?• How can we discretize?Naïve Bayes Classifier: Example• X’ = (HomeOwner = No, MaritalStatus = Married, Income=120K)• Need to compute Pr(Y|X’) or Pr(Y)xPr(X’|Y)• But Pr(X’|Y) is– Y = No:• Pr(HO=No|No)xPr(MS=Married|No)xPr(Inc=120K|No) = 4/7x4/7x0.0072 = 0.0024– Y=Yes:• Pr(HO=No|Yes)xPr(MS=Married|Yes)xPr(Inc=120K|Yes) = 1x0x1.2x10-9= 0Naïve Bayes Classifier: Example• X’ = (HomeOwner = No, MaritalStatus = Married, Income=120K)• Need to compute Pr(Y|X’) or Pr(Y)xPr(X’|Y)• But Pr(X’|Y = Yes) is 0?• Correction process:mnmpnyYxXcjii)|Pr(nc: number of training examples from class yjthat take value xin: total number of instances from class yjm: equivalent sample size (balance between prior and posterior)p: user-specified parameter (prior probability)Characteristics of Naïve BayesClassifier• Robust to isolated noise points – noise points are averaged out• Handles missing values – Ignoring missing-value examples• Robust to irrelevant attributes– If Xiis irrelevant, P(Xi|Y) becomes almost uniform• Correlated attributes degrade the performance of NB
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