Classification and K-Nearest Neighbors
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1 Classification and K-Nearest Neighbors
2 Administrivia o Reminder: Homework 1 is due by 5pm Friday on Moodle o Reading Quiz associated with today s lecture. Due before class Wednesday. NOTETAKER 2
3 Regression vs Classification So far, we ve taken a vector of features (x 1,x 2,...,x p ) and predicted a numerical response ŷ = ˆ0 + ˆ1x 1 + ˆ2x 2 + ˆpx p $360, 000 = (2000 sq-ft) + 2 (3 bath rooms) Regression deals with quantitative predictions But what if we re interested in making qualitative predictions? 3
4 Classification Examples Text Classification: Spam or Not Spam 4
5 Classification Examples Text Classification: Fake News or Real News 5
6 Classification Examples Text Classification: Sentiment Analysis (Tweets, Amazon Reviews) 6
7 Classification Examples Image Classification: Handwritten Digits 7
8 Classification Examples Image Classification: Stop Sign or No Stop Sign (Autonomous Vehicles) 8
9 Classification Examples Image Classification: Hot Dog or Not Hot Dog (No real application whatsoever) 9
10 Classification Examples Image Classification: Labradoodle or Fried Chicken (Important for not eating your dog) 10
11 Classification with Probability In classification problems our outputs are discrete classes Goal: Given features predict which class belongs to x =(x 1,x 2,...,x p ) Y = k, x 11
12 Classification with Probability In classification problems our outputs are discrete classes Goal: Given features predict which class belongs to x =(x 1,x 2,...,x p ) Y = k, x Conditional probability gives a nice framework for many classification models Suppose our data can belong to one of k =1, 2,...,K classes We model the conditional probability p(y = k x) for each k =1, 2,...,K and we make the prediction ŷ = argmax k p(y = k x) 12
13 Classification with Probability In classification problems our outputs are discrete classes Goal: Given features x =(x 1,x 2,...,x p ) predict which class Y = k, x belongs to Example: In Spam Classification, the features are the words in the The classes are SPAM and HAM We estimate. r p(spam ) and p(ham ) 0.75= o. zs y ' = SPAM 13
14 K-Nearest Neighbors Our first classifier is a very simple one Assume we have a labeled training set Features can be anything: words, pixel intensities, numerical quantities Instead of continuous response, now have discrete labels (encoded as 0, 1, 2, etc) All Labeled Data Training Set Validation Set 14
15 K-Nearest Neighbors General Idea: Suppose we have an example x that we want to classify o Look in training set for K examples that are nearest to x o Predict label for Questions: x using majority vote of labels for K nearest neighbors o What does nearest mean? o What if there s a tie in the vote? 15
16 K-Nearest Neighbors KNN: o Find N K (x) : the set of K training examples nearest to o Predict - ŷ to be majority label in N K (x) x Question: o What does nearest mean? Answer: o Measure closeness using Euclidean Distance: g kx i xk 2 test Example Tnthimng vector 16
17 K-Nearest Neighbors Classes = { RED, Blue 3 Euclidean Distance: kx i xk 2 o 1-NN predicts: o 2-NN predicts: o 5-NN predicts: 17
18 K-Nearest Neighbors Euclidean Distance: kx i xk 2 o 1-NN predicts: BLUE o 2-NN predicts: o 5-NN predicts: 18
19 K-Nearest Neighbors Euclidean Distance: kx i xk 2 o 1-NN predicts: blue unclear o 2-NN predicts: o 5-NN predicts: 19
20 K-Nearest Neighbors Euclidean Distance: kx i xk 2 o 1-NN predicts: blue o 2-NN predicts: unclear RED o 5-NN predicts: 20
21 K-Nearest Neighbors Euclidean Distance: kx i xk 2 o 1-NN predicts: blue o 2-NN predicts: unclear o 5-NN predicts: red 21
22 What About Probability? INDICATOR Function ˆp(Y = k x) = 1 X : I(y i = k) K - i2n = K o 5-NN: p ( Y Blue / x ) }= - = ftp.redk/)=3@ I (b) = { 1 it b is true 0 IF b is FADE 22
23 What About Ties? Lots of reasonable strategies Example: If there is a tie: o reduce K by 1 and try again. o repeat until the tie is broken. 23
24 K-Nearest Neighbors Predict ŷ with K =1 Closest Points: N, 1 1 = { Coin } Prediction: ya = RED 24
25 K-Nearest Neighbors Predict ŷ with K =2 Closest Points: [ Nzlx ) > 310,4/11,4 ) } t µ, (1) a { ( 0,2 ) } Prediction: RED =p 25
26 K-Nearest Neighbors Predict ŷ with K =3 Closest Points: N } ( x ) = { ( 0,2 ), 11,4 ) ( 2) 4)} Prediction: ya = BLUE 26
27 o KNN is an instance-based method A Subtle Distinction o Given an unseen query point, look directly at training data and then predict o Compare to regression where we use training data to learn parameters o When we learn parameters we call the model a parametric. model o Instance-based methods like KNN are called non-parametric models Question: When would parametric vs non-parametric make a big difference? 27
28 Naïve Implementation How much does KNN cost? klxi - x 113 Suppose your training set has n examples, each with p features Now you classify a single example x Naïve Method: Loop over each training example, compute distances, cache min indices ocpn ) o The naïve method is in time olpn ) o The naïve method is in space 28
29 Tree-Based Implementation Build a KD- or Ball-tree on the training data 29
30 Tree-Based Implementation Build a KD- or Ball-tree on the training data Higher up-front cost in time and storage to build data structure ocpnlogn) Ocpu ) Up-front cost is in time and in space Now you classify a single example x ocplogn ) Classification time for single query is Good strategy if lots of data and need to do lots of queries 30
31 Performance Metrics and Choosing K Question: How do we evaluate our KNN classifier? Answer: Perform prediction on labeled data, count how many we got wrong Question: How do we choose K? Error Rate = 1 n nx I(y i 6=ŷ i ) i=1 [ # wrong PRED - # predictions 31
32 Performance Metrics and Choosing K Like all models, KNN can overfit and underfit, depending on choice of K K=1 K = 15 DECISION ay Boundary 32
33 Performance Metrics and Choosing K Choose optimal K by varying K and computing error rate on train and validation set o Note: Plot vs 1/K o Lower K leads to more flexibility o Choose K to minimize validation error Error Rate Training Errors Test. Errors VALIDATION /K 33
34 Feature Standardization o Important to normalize/standardize features to similar sizes o Consider doing 2-NN on unscaled and scaled data O 34
35 Feature Standardization o Important to normalize/standardize features to similar sizes o Consider doing 2-NN on unscaled and scaled data " y = BLUE ya = RED 0 o 35
36 Generalization Error and Analysis Bias-Variance Trade-Off: o Pretty much analogous to our experience with polynomial regression o As 1/K! 1, variance increases and bias decreases Reducible and Irreducible Errors: o This is more interesting, and more complicated o What follows applies to ALL classification methods, not just KNN 36
37 The Optimal Bayes Classifier Suppose we knew the true probabilistic distribution of our data: Example: p(y = k X) QI o Equal prob a 2D point is in Atta Q1 or Q3 o If point in Q1, 0.9 probability it s red. o If point in Q2, 0.9 probability it s blue. o If classifier uses true model to predict o will be wrong 04 of time on average o Bayes Error Rate = : 37
38 The Optimal Bayes Classifier The Bayes Error Rate is the theoretical measure of Irreducible Error _ o In long run, can never do better than the Bayes Error Rate on validation data o Of course, just like Var( ) in the regression setting, we don t actually know what this error is. We just know it s there! Error Rate t Training Errors Test Errors /K 38
39 K-Nearest Neighbors Wrap-Up o Talked about the general classification setting o Learned how to do simple KNN o Looked at possible implementations and their complexity Next Time: o Look at our first parametric classifier: The Perceptron o Kinda a weak model, but forms the basis for Neural Networks 39
40 If-Time Bonus: Weighted KNN Question: What should 5-NN predict in the following picture? 40
41 If-Time Bonus: Weighted KNN Question: What should 5-NN predict in the following picture? 41
42 If-Time Bonus: Weighted KNN Weighted-KNN: o Find N K (x) : the set of K training examples nearest to x o Predict ŷ to be weighted-majority label in N K (x), weighted by inverse-distance Improvements over KNN: o Gives more weight to examples that are very close to query point o Less tie-breaking required 42
43 If-Time Bonus: Weighted KNN Question: What should 5-NN predict in the following picture? Red Distances: Blue Distances: Red Weighted-Majority Vote: Blue Weighted-Majority Vote: Prediction: 43
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