Supervised Learning Regression Classification Linear regression kNN classification

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

Power (bhp) An Example: Size of Engine vs Power 200 180 160 140 120 100 80 60 40 20 0 0 500 1000 1500 Engine displacement (cc) 2000 2500 § An unknown car has an engine of size 1800 cc. What is likely to be the power of the engine? 2

Power (bhp) An Example: Size of Engine vs Power Target Variable 200 180 160 140 120 100 80 60 40 20 0 0 500 1000 1500 Engine displacement (cc) 2000 2500 § Intuitively, the two variables have a relation § Learn the relation from the given data § Predict the target variable after learning 3

Exercise: on a simpler set of data points 12 x 1 2 3 4 2. 5 10 y 8 6 4 2 y 1 3 7 10 ? 0 0 1 2 x 3 4 5 § Predict y for x = 2. 5 4

Power (bhp) Linear Regression 200 180 160 140 120 100 80 60 40 20 0 Training set 0 500 1000 1500 Engine displacement (cc) 2000 2500 § Assume: the relation is linear § Then for a given x (=1800), predict the value of y 5

Power (bhp) Linear Regression 200 180 160 140 120 100 80 60 40 20 0 0 500 1000 1500 Engine displacement (cc) § Linear regression § Assume y = a. x + b § Try to find suitable a and b 2000 2500 Engine Power (cc) (bhp) 800 60 1000 90 1200 80 1200 75 1400 90 1500 120 1800 160 2000 140 2000 170 2400 180 Optional exercise 6

Exercise: using Linear Regression 12 x 1 2 3 4 2. 5 10 y 8 6 4 2 y 1 3 7 10 ? 0 0 1 2 x 3 4 5 § Define a regression line of your choice § Predict y for x = 2. 5 7

Choosing the parameters right 200 Goal: minimizing the deviation from the actual data points 150 y 100 50 0 0 § § 500 1000 x 1500 2000 2500 The data points: (x 1, y 1), (x 2, y 2), … , (xm, ym) The regression line: f(x) = y = a. x + b Least-square cost function: J = Σi ( f(xi) – yi )2 Goal: minimize J over choices of a and b 8

How to Minimize the Cost Function? b a § § 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? Δ 9

Another example: High blood sugar Y Training set N 0 20 40 60 80 Age § 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

Classification problem High blood sugar Y N 0 20 ? 24 40 60 80 Age § One approach: what other data points are nearest to the new point? § Other approaches? 11

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

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

Classification

Example: Age, Income and Owning a flat Monthly income (thousand rupees) 250 Training set • Owns a flat 200 150 • Does not own a flat 100 50 0 0 10 20 30 40 50 60 70 Age § Given a new person’s age and income, predict – does (s)he own a flat? 15

Example: Age, Income and Owning a flat Monthly income (thousand rupees) 250 Training set • Owns a flat 200 150 • Does not own a flat 100 50 0 0 10 20 30 40 50 60 70 Age § Nearest neighbor approach § Find nearest neighbors among the known data points and check their labels 16

Example: Age, Income and Owning a flat Monthly income (thousand rupees) 250 Training set • Owns a flat 200 150 • Does not own a flat 100 50 0 0 10 20 30 40 50 60 70 Age § The 1 -Nearest Neighbor (1 -NN) Algorithm: – Find the closest point in the training set – Output the label of the nearest neighbor 17

The k-Nearest Neighbor Algorithm Monthly income (thousand rupees) 250 Training set • Owns a flat 200 150 • Does not own a flat 100 50 0 0 10 20 30 40 50 60 70 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 18

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

Classification setup § 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 not 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 20

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

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

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

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