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© Vipin Kumar CSci 8980 Fall 2002 1 CSci 8980: Data Mining (Fall 2002) Vipin Kumar Army High Performance Computing Research Center Department of Computer Science University of Minnesota http://www.cs.umn.edu/~kumar

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© Vipin Kumar CSci 8980 Fall 2002 2 Instance-Based Classifiers Store the training instances Use the training instances to predict the class label of unseen cases

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© Vipin Kumar CSci 8980 Fall 2002 3 Nearest-Neighbor Classifiers l For data set with continuous attributes l Requires three things –The set of stored instances –Distance Metric to compute distance between instances. –The value of k, the number of nearest neighbors to retrieve l For classification : –Retrieve the k nearest neighbors –Use class labels of nearest neighbors to determine the class label of unseen instance (e.g., by taking majority vote)

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© Vipin Kumar CSci 8980 Fall 2002 4 Definition of Nearest Neighbor K-nearest neighbors of an instance x are data points that have the k smallest distance to x

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© Vipin Kumar CSci 8980 Fall 2002 5 1 nearest-neighbor Voronoi Diagram

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© Vipin Kumar CSci 8980 Fall 2002 6 Nearest neighbor classification l Compute distance between two points: –Euclidean distance take the majority vote of class labels among the k-nearest neighbors –Weighted distance weight factor, w = 1/d 2 weigh the vote according to distance

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© Vipin Kumar CSci 8980 Fall 2002 7 Nearest Neighbor classification… l Choosing the value of k: –If k too small, sensitive to noise points –If k too large, it can be computationally expensive neighborhood may include points from other classes

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© Vipin Kumar CSci 8980 Fall 2002 8 Nearest Neighbor Classification… l Problem with Euclidean measure: –High dimensional data curse of dimensionality –Can produce counter-intuitive results –Normalization 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 vs

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© Vipin Kumar CSci 8980 Fall 2002 9 Nearest neighbor classification… l Other issues: –k-NN classifiers are lazy learners it does not build models explicitly unlike eager learners such as decision tree induction and rule-based systems classifying instances are relatively expensive –Scaling of attributes Person: (height in meters, weight in lbs, Class) –Height may vary from 1.5m to 1.85m –Weight may vary from 90lbs to 250lbs –distance measure could be dominated by difference in weights

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© Vipin Kumar CSci 8980 Fall 2002 10 Bayes Classifier l A probabilistic (Bayesian) framework to solve the classification problem l Conditional Probability: l Bayes theorem:

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© Vipin Kumar CSci 8980 Fall 2002 11 Example of Bayes Theorem l Given: –A doctor knows that meningitis causes stiff neck 50% of the time –Prior probability of any patient having meningitis is 1/50,000 –Prior probability of any patient having stiff neck is 1/20 l If a patient has stiff neck, what’s the probability he/she has meningitis?

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© Vipin Kumar CSci 8980 Fall 2002 12 Bayesian Classifiers l Consider each attribute and class label as random variables. l Given a record of attributes (A 1, A 2,…,A n ) –Goal is to predict class C –Specifically, we want to find value of C that maximizes P(C| A 1, A 2,…,A n ) l Can we estimate P(C| A 1, A 2,…,A n ) directly from data?

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© Vipin Kumar CSci 8980 Fall 2002 13 Bayesian Classifiers l Approach: –compute the posterior probability P(C | A 1, A 2, …, A n ) for all values of C using the Bayes theorem –Choose value of C that maximizes P(C | A 1, A 2, …, A n ) –Equivalent to choosing value of C that maximizes P(A 1, A 2, …, A n |C) P(C) l How to estimate P(A 1, A 2, …, A n | C )?

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© Vipin Kumar CSci 8980 Fall 2002 14 Naïve Bayes Classifier l Assume independence among attributes A i when class is given: –P(A 1, A 2, …, A n |C) = P(A 1 | C j ) P(A 2 | C j )… P(A n | C j ) –Can estimate P(A i | C j ) for all A i and C j. –New point is classified to C j if P(C j ) P(A i | C j ) is maximal.

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© Vipin Kumar CSci 8980 Fall 2002 15 How to Estimate Probabilities from Data? l Class: P(C) = N c /N –e.g., P(No) = 7/10 l For discrete attributes: P(A i | C k ) = |A ik |/ N c –where |A ik | is number of instances having attribute A i and belongs to class C k –Examples: P(Married|No) = 4/7 P(Refund=Yes|Yes)=0 k

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© Vipin Kumar CSci 8980 Fall 2002 16 How to Estimate Probabilities from Data? l For continuous attributes: –Discretize the range into bins one ordinal attribute per bin violates independence assumption –Two-way split: (A v) choose only one of the two splits as new attribute –Assume attribute obeys certain probability distribution Typically, normal distribution is assumed Use data to estimate parameters of distribution (e.g., mean and standard deviation) Once probability distribution is known, can use it to estimate the conditional probability P(A i |c) k

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© Vipin Kumar CSci 8980 Fall 2002 17 How to Estimate Probabilities from Data? l Normal distribution: –One for each (A i,c i ) pair l For (Income, Class=No): –If Class=No sample mean = 110 sample variance = 2975

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© Vipin Kumar CSci 8980 Fall 2002 18 Example of Naïve Bayes Classifier l P(X|Class=No) = P(Refund=No|Class=No) P(Married| Class=No) P(Income=120K| Class=No) = 4/7 4/7 0.0072 = 0.0024 l P(X|Class=Yes) = P(Refund=No| Class=Yes) P(Married| Class=Yes) P(Income=120K| Class=Yes) = 1 0 1.2 10 -9 = 0 Since P(X|No)P(No) > P(X|Yes)P(Yes) Therefore P(No|X) > P(Yes|X) => Class = No Given a Test instance:

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© Vipin Kumar CSci 8980 Fall 2002 19 Naïve Bayes Classifier l If one of the conditional probability is zero, then the entire expression becomes zero –Independence multiplication of probabilities l Laplace correction (also known as m-estimate):

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© Vipin Kumar CSci 8980 Fall 2002 20 Example of Naïve Bayes Classifier A: attributes M: mammals N: non-mammals P(A|M)P(M) > P(A|N)P(N) => Mammals

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© Vipin Kumar CSci 8980 Fall 2002 21 Naïve Bayes (Summary) l Robust to isolated noise points l Handle missing values by ignoring the instance during probability estimate calculations l Robust to irrelevant attributes l Independence assumption may not hold for some attributes –Use other techniques such as Bayesian Belief Networks (BBN)

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© Vipin Kumar CSci 8980 Fall 2002 22 Model Evaluation l Evaluating the performance of a model (in terms of its predictive ability) l Confusion Matrix: PREDICTED CLASS ACTUAL CLASS Class=YesClass=No Class=Yesab Class=Nocd a: TP (true positive) b: FN (false negative) c: FP (false positive) d: TN (true negative)

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© Vipin Kumar CSci 8980 Fall 2002 23 Model Evaluation l What are the measures to estimate performance? PREDICTED CLASS ACTUAL CLASS Class=YesClass=No Class=Yesab Class=Nocd

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© Vipin Kumar CSci 8980 Fall 2002 24 Cost Matrix PREDICTED CLASS ACTUAL CLASS C(i|j) Class=YesClass=No Class=YesC(Yes|Yes)C(No|Yes) Class=NoC(Yes|No)C(No|No) C(i|j): Cost of misclassifying class j example as class i

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© Vipin Kumar CSci 8980 Fall 2002 25 Cost-Sensitive Measures l Precision is biased towards C(Yes|Yes) & C(Yes|No) l Recall is biased towards C(Yes|Yes) & C(No|Yes) l F-measure is biased towards all except C(No|No)

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© Vipin Kumar CSci 8980 Fall 2002 26 ROC (Receiver Operating Characteristic) l Developed in 1950s for signal detection theory to analyze noisy signals –Characterize the trade-off between positive hits and false alarms l ROC curve plots TP (on the y-axis) against FP (on the x-axis) l Performance of each classifier represented as a point in an ROC curve –changing the threshold of algorithm, sample distribution or cost matrix changes the location of the point

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© Vipin Kumar CSci 8980 Fall 2002 27 ROC Curve - 1-dimensional data set containing 2 classes (positive and negative) - any points located at x > t is classified as positive At threshold t: TP=0.5, FN=0.5, FP=0.12, FN=0.88

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© Vipin Kumar CSci 8980 Fall 2002 28 ROC Curve (TP,FP): l (0,0): everything is negative l (1,1): everything is positive l (1,0): ideal l Diagonal line: –Random guessing –Below diagonal line: prediction is opposite of the true class l Area under ROC curve –Another metric for comparison

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