Intuition Bayesian Classification
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1 Intuition Bayesian Classification More ockey fans in Canada tan in US Wic country is Tom, a ockey ball fan, from? Predicting Canada as a better cance to be rigt Prior probability P(Canadian=5%: reflect background knowledge 5% of total population is Canadians P(ockey fan Canadian=30%: te probability of a Canadian wo is also a ockey fan Posterior probability P(Canadian ockey fan: te probability of a ockey fan is from Canada Jian Pei: CMPT 741/459 Classification (4 1
2 Jian Pei: CMPT 741/459 Classification (4 2 Bayes Teorem Find te maximum a posteriori (MAP ypotesis Require background knowledge Computational cost ( ( ( ( D P P D P D P = ( ( max ( ( ( max ( max P D P D P P D P D P H H H MAP = =
3 Naïve Bayes Classifier Assumption: attributes are independent Given a tuple (a1, a2,, an, predict its class as C = = arg max f ( x : te value of x tat maximizes f(x Example: arg max P( a, a2, i i arg max P( C i arg max x {1,2, 3} x 1 2 j P( a = 3, a j n C C i i P( C i Jian Pei: CMPT 741/459 Classification (4 3
4 Example: Training Dataset Data sample X = (Outlook=sunny, Temp=mild, Humid=ig Wind=weak Will se play tennis? Yes P(Yes X = P(X Yes P(Yes = P(No X = P(X No P(No = Outlook Temp Humid Wind PlayTennis Sunny Hot Hig Weak No Sunny Hot Hig Strong No Overcast Hot Hig Weak Yes Rain Mild Hig Weak Yes Rain Cool Normal Weak Yes Rain Cool Normal Strong No Overcast Cool Normal Strong Yes Sunny Mild Hig Weak No Sunny Cool Normal Weak Yes Rain Mild Normal Weak Yes Sunny Mild Normal Strong Yes Overcast Mild Hig Strong Yes Overcast Hot Normal Weak Yes Rain Mild Hig Strong No Jian Pei: CMPT 741/459 Classification (4 4
5 Probability of Infrequent Values (outlook = Sunny, temp = ig, umid = low, wind = weak? P(umid = low = 0 Outlook Temp Humid Wind PlayTennis Sunny Hot Hig Weak No Sunny Hot Hig Strong No Overcast Hot Hig Weak Yes Rain Mild Hig Weak Yes Rain Cool Normal Weak Yes Rain Cool Normal Strong No Overcast Cool Normal Strong Yes Sunny Mild Hig Weak No Sunny Cool Normal Weak Yes Rain Mild Normal Weak Yes Sunny Mild Normal Strong Yes Overcast Mild Hig Strong Yes Overcast Hot Normal Weak Yes Rain Mild Hig Strong No Jian Pei: CMPT 741/459 Classification (4 5
6 Smooting Suppose an attribute as n different values: a 1,, a n Assume a small enoug value ε > 0 Let P i be te frequency of a i, P i = # tuples aving a i / total # of tuples Estimate n P (a i = + 1 n P i Jian Pei: CMPT 741/459 Classification (4 6
7 Handling Continuous Attributes Discretization Probability density estimation Jian Pei: CMPT 741/459 Classification (4 7
8 Density Estimation µ ij 2 ij Let and be te mean and variance of all samples of class C j, respectively P (X i = x i C j = 1 p 2 ij e (x i µ ij ij Jian Pei: CMPT 741/459 Classification (4 8
9 Caracteristics of Naïve Bayes Robust to isolated noise points Suc points are averaged out in probability computation Insensitive to missing values Robust to irrelevant attributes Distributions on suc attributes are almost uniform Correlated attributes degrade te performance Jian Pei: CMPT 741/459 Classification (4 9
10 Bayes Error Rate Te error rate of te ideal naïve Bayes classifier Err = Zˆx P (Crocodile XdX + Z 1 P (Alligator XdX 0 ˆx Jian Pei: CMPT 741/459 Classification (4 10
11 Pros and Cons Pros Easy to implement Good results obtained in many cases Cons A (too strong assumption: independent attributes How to andle dependent/correlated attributes? Bayesian belief networks Jian Pei: CMPT 741/459 Classification (4 11
12 Associative Classification Mine association possible rules (PR in form of condset à c Condset: a set of attribute-value pairs C: class label Build classifier Organize rules according to decreasing precedence based on confidence and support Classification Use te first matcing rule to classify an unknown case Jian Pei: CMPT 741/459 Classification (4 12
13 Associative Classification Metods CBA (Classification By Association: Liu, Hsu & Ma, KDD 98 Mine association possible rules in te form of Cond-set (a set of attribute-value pairs à class label Build classifier: Organize rules according to decreasing precedence based on confidence and ten support CMAR (Classification based on Multiple Association Rules: Li, Han, Pei, ICDM 01 Classification: Statistical analysis on multiple rules Jian Pei: CMPT 741/459 Classification (4 13
14 CMAR Model Generation Classification based on Multiple Association Rules Efficiency: Uses an enanced FP-tree tat maintains te distribution of class labels among tuples satisfying eac frequent itemset Rule pruning wenever a rule is inserted into te tree Given two rules, R1 and R2, if te antecedent of R1 is more general tan tat of R2 and conf(r1 conf(r2, ten R2 is pruned Prune rules were te rule antecedent and class are not positively correlated, based on a χ 2 test of statistical significance Jian Pei: CMPT 741/459 Classification (4 14
15 CMAR Classification Classification based on generated/pruned rules If only one rule satisfies tuple X, assign te class label of te rule If a rule set S satisfies X, CMAR Divide S into groups according to class labels Use a weigted χ 2 measure to find te strongest group of rules, based on te statistical correlation of rules witin a group Assign X te class label of te strongest group Jian Pei: CMPT 741/459 Classification (4 15
16 Classification by Aggregating Emerging Patterns Emerging pattern (EP: A pattern frequent in one class of data but infrequent in oters Age<=30 is frequent in class buys_computer=yes and infrequent in class buys_computer=no Rule: age<=30 à buys computer G. Dong & J. Li. Efficient mining of emerging patterns: discovering trends and differences. In KDD 99 Jian Pei: CMPT 741/459 Classification (4 16
17 How to Mine Emerging Patterns? Border differential Max-patterns in D1 w.r.t. min_sup=90% Max-patterns in D2 w.r.t. min_sup=10% X is a pattern covered by a max-pattern in D1 but not by a max-pattern in D2 à X is an emerging pattern Metod Mine max-patterns in D1 and D2, respectively Compare te two sets of borders, find te maximal patterns tat are frequent in D1 and infrequent D2 Jian Pei: CMPT 741/459 Classification (4 17
18 Instance-based Metods Instance-based learning Store training examples and delay te processing until a new instance must be classified ( lazy evaluation Typical approaces K-nearest neigbor approac Instances represented as points in an Euclidean space Locally weigted regression Construct local approximation Case-based reasoning Use symbolic representations and knowledge-based inference Jian Pei: CMPT 741/459 Classification (4 18
19 Te K-Nearest Neigbor Metod Instances are points in an n-d space Te k-nearest neigbors (KNN in te Euclidean distance Return te most common value among te k training examples nearest to te query point Discrete-/real-valued target functions _ xq _ + Jian Pei: CMPT 741/459 Classification (4 19
20 KNN Metods For continuous-valued target functions, return te mean value of te k nearest neigbors Distance-weigted nearest neigbor algoritm Give greater weigts to closer neigbors Robust to noisy data by averaging k-nearest neigbors Curse of dimensionality w 1 d ( xq, x i 2 Distance could be dominated by irrelevant attributes Axes stretc or elimination of te least relevant attributes Jian Pei: CMPT 741/459 Classification (4 20
21 Lazy vs. Eager Learning Efficiency: lazy learning uses less training time but more predicting time Accuracy Lazy metod effectively uses a ricer ypotesis space Eager: must commit to a single ypotesis tat covers te entire instance space Jian Pei: CMPT 741/459 Classification (4 21
22 To-Do List Read Capters 8.3, 8.4, 9.4, and 9.5 Understand ow to use Naïve Bayes in Weka Jian Pei: CMPT 741/459 Classification (4 22
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