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Regression “A new perspective on freedom” TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A A AAA A A

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Classification

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? CatDog

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Cleanliness Size

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? $$$$$$$$$$

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Regression

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$ $$ $$$ $$$$ Price Top speed x y

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Regression Data Goal: given, predict i.e. find a prediction function

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Nearest neighbor -50510152025 -10 -5 0 5 10 15

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Nearest neighbor To predict x –Find the data point x i closest to x –Choose y = y i + No training – Finding closest point can be expensive – Overfitting

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Kernel Regression To predict X –Give data point x i weight –Normalize weights –Let e.g.

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Kernel Regression -50510152025 -10 -5 0 5 10 15 [matlab demo]

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Kernel Regression + No training + Smooth prediction – Slower than nearest neighbor – Must choose width of

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Linear regression

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0 10 20 30 40 0 10 20 30 20 22 24 26 Temperature [start Matlab demo lecture2.m] Given examples Predict given a new point 0 10 20 30 40 0 10 20 30 20 22 24 26 Temperature

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0 10 20 30 40 0 10 20 30 20 22 24 26 Temperature Linear regression Prediction

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Linear Regression Error or “residual” Prediction Observation Sum squared error

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Linear Regression n d Solve the system (it’s better not to invert the matrix)

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Minimize the sum squared error Sum squared error Linear equation Linear system

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LMS Algorithm (Least Mean Squares) where Online algorithm

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Beyond lines and planes everything is the same with still linear in 01020 0 40

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Linear Regression [summary] n d Let For example Let Minimize by solving Given examples Predict

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Probabilistic interpretation Likelihood

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Overfitting 02468101214161820 -15 -10 -5 0 5 10 15 20 25 30 [Matlab demo] Degree 15 polynomial

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Ridge Regression (Regularization) 02468101214161820 -10 -5 0 5 10 15 Effect of regularization (degree 19) with “small” Minimize Solve Let

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Probabilistic interpretation Likelihood Prior Posterior

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Locally Linear Regression

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[source: http://www.cru.uea.ac.uk/cru/data/temperature] 1840186018801900192019401960198020002020 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Global temperature increase

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Locally Linear Regression To predict X –Give data point x i weight –Let e.g.

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Locally Linear Regression + Good even at the boundary (more important in high dimension) – Solve linear system for each new prediction – Must choose width of To minimize Solve Predict where

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[source: http://www.cru.uea.ac.uk/cru/data/temperature] Locally Linear Regression Gaussian kernel 180

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[source: http://www.cru.uea.ac.uk/cru/data/temperature] Locally Linear Regression Laplacian kernel 180

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L1 Regression

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Sensitivity to outliers High weight given to outliers Influence function

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L 1 Regression Linear program Influence function

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Spline Regression Regression on each interval 5200540056005800 50 60 70

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Spline Regression With equality constraints 5200540056005800 50 60 70

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Spline Regression With L 1 cost 5200540056005800 50 60 70

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To learn more The Elements of Statistical Learning, Hastie, Tibshirani, Friedman, Springer

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1 1 Slide The Simple Linear Regression Model n Simple Linear Regression Model y = 0 + 1 x + n Simple Linear Regression Equation E( y ) = 0 +

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