1 k-nearest neighbor methods william cohen 10-601 april 2008
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K-nearest neighbor methods
William Cohen
10-601 April 2008
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But first….
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Number of Publications
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Onward: multivariate linear regression
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ACM Computing Surveys 2002
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Review of K-NN methods (so far)
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Kernel regression
• aka locally weighted regression, locally linear regression, LOESS, …
What does making the kernel wider do to bias and variance?
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BellCore’s MovieRecommender• Participants sent email to videos@bellcore.com• System replied with a list of 500 movies to rate on a
1-10 scale (250 random, 250 popular)– Only subset need to be rated
• New participant P sends in rated movies via email• System compares ratings for P to ratings of (a
random sample of) previous users• Most similar users are used to predict scores for
unrated movies (more later)• System returns recommendations in an email
message.
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Suggested Videos for: John A. Jamus.
Your must-see list with predicted ratings:
•7.0 "Alien (1979)"
•6.5 "Blade Runner"
•6.2 "Close Encounters Of The Third Kind (1977)"
Your video categories with average ratings:
•6.7 "Action/Adventure"
•6.5 "Science Fiction/Fantasy"
•6.3 "Children/Family"
•6.0 "Mystery/Suspense"
•5.9 "Comedy"
•5.8 "Drama"
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The viewing patterns of 243 viewers were consulted. Patterns of 7 viewers were found to be most similar. Correlation with target viewer:
•0.59 viewer-130 (unlisted@merl.com)
•0.55 bullert,jane r (bullert@cc.bellcore.com)
•0.51 jan_arst (jan_arst@khdld.decnet.philips.nl)
•0.46 Ken Cross (moose@denali.EE.CORNELL.EDU)
•0.42 rskt (rskt@cc.bellcore.com)
•0.41 kkgg (kkgg@Athena.MIT.EDU)
•0.41 bnn (bnn@cc.bellcore.com)
By category, their joint ratings recommend:
•Action/Adventure:
•"Excalibur" 8.0, 4 viewers
•"Apocalypse Now" 7.2, 4 viewers
•"Platoon" 8.3, 3 viewers
•Science Fiction/Fantasy:
•"Total Recall" 7.2, 5 viewers
•Children/Family:
•"Wizard Of Oz, The" 8.5, 4 viewers
•"Mary Poppins" 7.7, 3 viewers
Mystery/Suspense: •"Silence Of The Lambs, The" 9.3, 3 viewers
Comedy: •"National Lampoon's Animal House" 7.5, 4 viewers •"Driving Miss Daisy" 7.5, 4 viewers •"Hannah and Her Sisters" 8.0, 3 viewers
Drama: •"It's A Wonderful Life" 8.0, 5 viewers •"Dead Poets Society" 7.0, 5 viewers •"Rain Man" 7.5, 4 viewers
Correlation of predicted ratings with your actual ratings is: 0.64 This number measures ability to evaluate movies accurately for you. 0.15 means low ability. 0.85 means very good ability. 0.50
means fair ability.
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Algorithms for Collaborative Filtering 1: Memory-Based Algorithms (Breese et al, UAI98)
• vi,j= vote of user i on item j
• Ii = items for which user i has voted
• Mean vote for i is
• Predicted vote for “active user” a is weighted sum
weights of n similar usersnormalizer
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Basic k-nearest neighbor classification
• Training method:– Save the training examples
• At prediction time:– Find the k training examples (x1,y1),…(xk,yk) that
are closest to the test example x
– Predict the most frequent class among those yi’s.
• Example: http://cgm.cs.mcgill.ca/~soss/cs644/projects/simard/
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What is the decision boundary?Voronoi diagram
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Convergence of 1-NN
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Basic k-nearest neighbor classification
• Training method:– Save the training examples
• At prediction time:– Find the k training examples (x1,y1),…(xk,yk) that
are closest to the test example x– Predict the most frequent class among those yi’s.
• Improvements:– Weighting examples from the neighborhood– Measuring “closeness”– Finding “close” examples in a large training set
quickly
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K-NN and irrelevant features
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K-NN and irrelevant features
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K-NN and irrelevant features
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Ways of rescaling for KNN
Normalized L1 distance:
Scale by IG:
Modified value distance metric:
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Ways of rescaling for KNN
Dot product:
Cosine distance:
TFIDF weights for text: for doc j, feature i: xi=tfi,j * idfi :
#occur. of term i in
doc j
#docs in corpus
#docs in corpus that contain term i
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Combining distances to neighbors
Standard KNN:
Distance-weighted KNN:
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William W. Cohen & Haym Hirsh (1998): Joins that Generalize: Text Classification Using WHIRL in
KDD 1998: 169-173.
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M1
M2
Vitor Carvalho and William W. Cohen (2008): Ranking Users for Intelligent Message Addressing in
ECIR-2008, and current work with Vitor, me, and Ramnath Balasubramanyan
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Computing KNN: pros and cons
• Storage: all training examples are saved in memory– A decision tree or linear classifier is much smaller
• Time: to classify x, you need to loop over all training examples (x’,y’) to compute distance between x and x’.– However, you get predictions for every class y
• KNN is nice when there are many many classes
– Actually, there are some tricks to speed this up…especially when data is sparse (e.g., text)
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Efficiently implementing KNN (for text)
IDF is nice computationally
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Tricks with fast KNN
K-means using r-NN1. Pick k points c1=x1,….,ck=xk as centers
2. For each xi, find Di=Neighborhood(xi)
3. For each xi, let ci=mean(Di)
4. Go to step 2….
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Efficiently implementing KNN
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Selective classification: given a training set and test set, find the N test cases that you can most confidently classify
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Train once and select 100 test cases to classify
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