Report

5: Multivariate Regression CSC 4510 – Machine Learning Dr. Mary-Angela Papalaskari Department of Computing Sciences Villanova University Course website: www.csc.villanova.edu/~map/4510/ The slides in this presentation are adapted from: • Andrew Ng’s ML course http://www.ml-class.org/ CSC 4510 - M.A. Papalaskari - Villanova University 1 • • • • • Regression topics so far Introduction to linear regression Intuition – least squares approximation Intuition – gradient descent algorithm Hands on: Simple example using excel How to apply gradient descent to minimize the cost function for regression • linear algebra refresher CSC 4510 - M.A. Papalaskari - Villanova University 2 What’s next? • Multivariate regression • Gradient descent revisited – Feature scaling and normalization – Selecting a good value for α • Non-linear regression • Solving for analytically (Normal Equation) • Using Octave to solve regression problems CSC 4510 - M.A. Papalaskari - Villanova University 3 Multiple features (variables). Size (feet2) Number of bedrooms Number of floors 2104 1416 1534 852 … 5 3 3 2 … 1 2 2 1 … Age of home Price ($1000) (years) CSC 4510 - M.A. Papalaskari - Villanova University 45 40 30 36 … 460 232 315 178 … 5 Andrew Ng Multiple features (variables). Size (feet2) Number of bedrooms Number of floors Age of home (years) Price ($1000) 2104 1416 1534 852 … 5 3 3 2 … 1 2 2 1 … 45 40 30 36 … 460 232 315 178 … Notation: = number of features = input (features) of = value of feature in training example. training example. CSC 4510 - M.A. Papalaskari - Villanova University 6 Andrew Ng Multiple features (variables). Size (feet2) Price ($1000) 2104 1416 1534 852 … 460 232 315 178 … CSC 4510 - M.A. Papalaskari - Villanova University 7 Andrew Ng Hypothesis: Previously: Now: For convenience of notation, define . Multivariate linear regression CSC 4510 - M.A. Papalaskari - Villanova University 8 Hypothesis: Parameters: Cost function: Gradient descent: Repeat (simultaneously update for every CSC 4510 - M.A. Papalaskari - Villanova University ) 9 Gradient Descent Previously (n=1): Repeat (simultaneously update ) CSC 4510 - M.A. Papalaskari - Villanova University 10 New algorithm Repeat Gradient Descent : Previously (n=1): Repeat (simultaneously update ) (simultaneously update for ) CSC 4510 - M.A. Papalaskari - Villanova University 11 New algorithm Repeat Gradient Descent : Previously (n=1): Repeat (simultaneously update ) (simultaneously update for ) CSC 4510 - M.A. Papalaskari - Villanova University 12 Feature Scaling Idea: Make sure features are on a similar scale. E.g. = size (0-2000 feet2) = number of bedrooms (1-5) size (feet2) Get every feature into range number of bedrooms CSC 4510 - M.A. Papalaskari - Villanova University 13 Feature Scaling Idea: Make sure features are on a similar scale. E.g. = size (0-2000 feet2) = number of bedrooms (1-5) Mean normalization Replace with to make features have approximately zero mean (Do not apply to ). E.g. CSC 4510 - M.A. Papalaskari - Villanova University 14 Gradient descent - “Debugging”: How to make sure gradient descent is working correctly. - How to choose learning rate CSC 4510 - M.A. Papalaskari - Villanova University . 15 Making sure gradient descent is working correctly. - 0 For sufficiently small , should decrease on every iteration. But if is too small, gradient descent can be slow to converge. 100 200 300 400 Declare convergence if decreases by less than in one iteration? No. of iterations CSC 4510 - M.A. Papalaskari - Villanova University 16 Summary: Choosing - If is too small: slow convergence. - If is too large: may not decrease on every iteration; may not converge. To choose , try CSC 4510 - M.A. Papalaskari - Villanova University 17 Housing prices prediction CSC 4510 - M.A. Papalaskari - Villanova University 18 Andrew Ng Polynomial regression Price (y) Size (x) CSC 4510 - M.A. Papalaskari - Villanova University 19 Andrew Ng Choice of features Price (y) Size (x) CSC 4510 - M.A. Papalaskari - Villanova University 20 Andrew Ng Gradient Descent Normal equation: Method to solve for analytically. CSC 4510 - M.A. Papalaskari - Villanova University 21 Andrew Ng Intuition: If 1D (for every ) Solve for CSC 4510 - M.A. Papalaskari - Villanova University 22 Andrew Ng Examples: 1 1 1 1 Size (feet2) Number of bedrooms Number of floors Age of home (years) Price ($1000) 2104 1416 1534 852 5 3 3 2 1 2 2 1 45 40 30 36 460 232 315 178 CSC 4510 - M.A. Papalaskari - Villanova University 23 Andrew Ng examples ; features. E.g. If CSC 4510 - M.A. Papalaskari - Villanova University 25 Andrew Ng is inverse of matrix . Octave: pinv(X’*X)*X’*y CSC 4510 - M.A. Papalaskari - Villanova University 26 Andrew Ng training examples, Gradient Descent features. • Need to choose . • Needs many iterations. • Works well even when is large. Normal Equation • No need to choose . • Don’t need to iterate. • Need to compute • Slow if CSC 4510 - M.A. Papalaskari - Villanova University is very large. 27 Andrew Ng Notes on Supervised learning and Regression http://see.stanford.edu/materials/aimlcs229/cs229-notes1.pdf Octave http://www.gnu.org/software/octave/ Wiki: http://www.octave.org/wiki/index.php?title=Main_Page documentation: http://www.gnu.org/software/octave/doc/interpreter/ CSC 4510 - M.A. Papalaskari - Villanova University 28 Exercise For next class: 1. Download and install Octave (Alternative: if you have MATLAB, you can use it instead.) 2. Verify that it is working by typing in an Octave command window: x = [0 1 2 3] y = [0 2 4 6] plot(x,y) This example defines two vectors, x y and should display a plot showing a straight line (the line y=2x). If you get an error at this point, it may be that gnuplot is not installed or cannot access your display. If you are unable to get this to work, you can still do the rest of this exercise, because it does not involve any plotting (just restart Octave). You might refer to the Octave wiki for installation help but if you are stuck, you can get some help troubleshooting this on Friday afternoon 3-4pm in the software engineering lab (mendel 159). 3. Create a few matrices and vectors, eg: A = [1 2; 3 4; 5 6] V = [3 5 -1 0 7] 4. Try some of the elementary matrix and vector operations from our linear algebra slides (adding, multiplying between matrices, vectors and scalars) 5. Print out a log of your session CSC 4510 - M.A. Papalaskari - Villanova University 29