Machine Learning Fisher’s Criteria & Linear Discriminant Analysis

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Presentation transcript:

Machine Learning Fisher’s Criteria & Linear Discriminant Analysis Supervised Learning Classification and Regression K-Nearest Neighbor Classification Fisher’s Criteria & Linear Discriminant Analysis Perceptron: Linearly Separable Multilayer Perceptron & EBP & Deep Learning, RBF Network Support Vector Machine Ensemble Learning: Voting, Boosting(Adaboost) Unsupervised Learning Principle Component Analysis Independent Component Analysis Clustering: K-means Semi-supervised Learning & Reinforcement Learning

Linear Discriminant Analysis (LDA) Problem: Given a set of data points, each of which is labelled with a class, such as the classes 0 to 9 in the digits data, find the best set of basis vectors for projecting the data points such that classification is improved. Idea: Form the projection such that the variability across the different classes is maximized, while the variability within each class is minimized.

LDA: Supervised Linear Projections Suppose we have data points that are labelled either class 1 or class 2. N1 data points for class 1 and N2 data points for class 2, And we wish to find a linear projection And, for classification, for a new points , if its projection is close to class 1 projected data, we expect x to belong to class 1.

LDA: Maximize Difference of Means Considering the simple problem of projecting onto one dimension The two classes should be well separated in this single dimension A simple idea is to maximize the difference of the means in the projected space What is a problem with this solution?

LDA: A Better Idea Fisher's idea: Fisher's linear discriminant Maximize a function that will give a large separation between the projected class means While also giving a small variance within each class, thereby minimizing the class overlap

LDA: Fisher Linear Discriminant Find w that maximizes Between-class scatter (numerator): Within-class scatter (denominator):

LDA: Fisher Linear Discriminant Find w that maximizes Solution: Parametric solution:

More than 2 Classes (K > 2) Projection from d-dim space to (c-1)-dim space:

More than 2 Classes (K > 2)

LDA on Vowels Data 11 vowels from words spoken by fteen speakers. (Source: David Deterding, Mahesan Niranjan, Tony Robinson, see Hastie book website for data)

LDA on Vowels Data: Decision Boundaries