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Supervised Learning Regression, Classification Linear regression, k-NN classification Debapriyo Majumdar Data Mining – Fall 2014 Indian Statistical Institute Kolkata August 11, 2014

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An Example: Size of Engine vs Power 2 Engine displacement (cc) Power (bhp) An unknown car has an engine of size 1800cc. What is likely to be the power of the engine?

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An Example: Size of Engine vs Power 3 Engine displacement (cc) Power (bhp) Intuitively, the two variables have a relation Learn the relation from the given data Predict the target variable after learning Target Variable

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Exercise: on a simpler set of data points Predict y for x = y x xy ?

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Linear Regression 5 Engine displacement (cc) Power (bhp) Assume: the relation is linear Then for a given x (=1800), predict the value of y Training set

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Linear Regression 6 Engine displacement (cc) Power (bhp) Linear regression Assume y = a. x + b Try to find suitable a and b Optional exercise Engine (cc) Power (bhp)

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Exercise: using Linear Regression Define a regression line of your choice Predict y for x = y x xy ?

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Choosing the parameters right The data points: (x 1, y 1 ), (x 2, y 2 ), …, (x m, y m ) The regression line: f(x) = y = a. x + b Least-square cost function: J = Σ i ( f(x i ) – y i ) 2 Goal: minimize J over choices of a and b 8 x y Goal: minimizing the deviation from the actual data points

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How to Minimize the Cost Function? Goal: minimize J for all values of a and b Start from some a = a 0 and b = b 0 Compute: J(a 0,b 0 ) Simultaneously change a and b towards the negative gradient and eventually hope to arrive an optimal Question: Can there be more than one optimal? 9 a b Δ

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Another example: Given that a person’s age is 24, predict if (s)he has high blood sugar Discrete values of the target variable (Y / N) Many ways of approaching this problem 10 High blood sugar N Y Age Training set

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Classification problem One approach: what other data points are nearest to the new point? Other approaches? 11 High blood sugar N Y Age ? ? 24

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Classification Algorithms The k-nearest neighbor classification Naïve Bayes classification Decision Tree Linear Discriminant Analysis Logistics Regression Support Vector Machine 12

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Classification or Regression? Given data about some cars: engine size, number of seats, petrol / diesel, has airbag or not, price Problem 1: Given engine size of a new car, what is likely to be the price? Problem 2: Given the engine size of a new car, is it likely that the car is run by petrol? Problem 3: Given the engine size, is it likely that the car has airbags? 13

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Classification

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Example: Age, Income and Owning a flat 15 Monthly income (thousand rupees) Age Training set Owns a flat Does not own a flat Given a new person’s age and income, predict – does (s)he own a flat?

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Example: Age, Income and Owning a flat 16 Monthly income (thousand rupees) Age Nearest neighbor approach Find nearest neighbors among the known data points and check their labels Training set Owns a flat Does not own a flat

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Example: Age, Income and Owning a flat 17 Monthly income (thousand rupees) Age The 1-Nearest Neighbor (1-NN) Algorithm: – Find the closest point in the training set – Output the label of the nearest neighbor Training set Owns a flat Does not own a flat

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The k-Nearest Neighbor Algorithm 18 Monthly income (thousand rupees) Age The k-Nearest Neighbor (k-NN) Algorithm: – Find the closest k point in the training set – Majority vote among the labels of the k points Training set Owns a flat Does not own a flat

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Distance measures How to measure distance to find closest points? Euclidean: Distance between vectors x = (x 1, …, x k ) and y = (y 1, …, y k ) 19 Manhattan distance: Generalized squared interpoint distance: S is the covariance matrix The Maholanobis distance (1936)

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Classification setup 20 Training data / set: set of input data points and given answers for the data points Labels: the list of possible answers Test data / set: inputs to the classification algorithm for finding labels – Used for evaluating the algorithm in case the answers are known (but known to the algorithm) Classification task: Determining labels of the data points for which the label is not known or not passed to the algorithm Features: attributes that represent the data

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Evaluation Test set accuracy: the correct performance measure Accuracy = #of correct answer / #of all answers Need to know the true test labels – Option: use training set itself – Parameter selection (for k-NN) by accuracy on training set Overfitting: a classifier performs too good on training set compared to new (unlabeled) test data 21

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Better validation methods Leave one out: – For each training data point x of training set D – Construct training set D – x, test set {x} – Train on D – x, test on x – Overall accuracy = average over all such cases – Expensive to compute Hold out set: – Randomly choose x% (say 25-30%) of the training data, set aside as test set – Train on the rest of training data, test on the test set – Easy to compute, but tends to have higher variance 22

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The k-fold Cross Validation Method Randomly divide the training data into k partitions D 1,…, D k : possibly equal division For each fold D i – Train a classifier with training data = D – D i – Test and validate with D i Overall accuracy: average accuracy over all cases 23

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References Lecture videos by Prof. Andrew Ng, Stanford University Available on Coursera (Course: Machine Learning) Data Mining Map: 24

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