Contents Lecture 4. Lecture 4 Linear Discriminant Analysis. Summary of Lecture 3 (II/II) Summary of Lecture 3 (I/II)
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1 Contents Lecture Lecture Linear Discriminant Analysis Fredrik Lindsten Division of Systems and Control Department of Information Technology Uppsala University Summary of lecture 3 Using Bayes theorem for classification 3 Linear Discriminant Analysis A nonparametric classifier k-nearest Neighbors fredriklindsten@ituuse / fredriklindsten@ituuse Summary of Lecture 3 I/II) Summary of Lecture 3 II/II) The classification problem amounts to modeling the relationship between the input X and a qualitative output Y, ie, the output belongs to one out of K distinct classes A classifier is a prediction model Ŷ = ĜX) that maps any input X into a predicted class Ŷ {,, K} The Bayes classifier predicts each input as belonging to the most likely class, according to the conditional probabilities PrY = k X) for k {,, K} The Bayes classifier is optimal wrt misclassification error For binary two-class) classification, Y {, }, the logistic regression model is px) := PrY = X) = eβt X + e βt X The model parameters β are found by maximum likelihood by numerically) maximizing the log-likelihood function, lβ) = N i= )) y i β T x i log + e βt x i Approximating the Bayes classifier, we get the prediction model ĜX) = I ˆpX) > ) = I ˆβT X > ) / fredriklindsten@ituuse 3 / fredriklindsten@ituuse
2 Bayes theorem Bayes theorem for classification Figure: Possibly) Thomas Bayes, c 7 76 Bayes theorem describes a fundamental relationship between conditional and marginal probabilities If A and B are two events with PrB) >, then PrA B) = PrB A) PrA) PrB) Recall the Bayes classifier, based on the conditional probabilities PrY = k X = x) Let π k := PrY = k) denote the prior marginal) probability of class k {,, K} f k x) := px Y = k) denote the probability density of X for an observation that comes from the kth class The Bayes classifier can then be expressed using Bayes theorem: PrY = k X = x) = f k x)π k Kj= f j x)π j / fredriklindsten@ituuse / fredriklindsten@ituuse Multivariate Gaussian density Multivariate Gaussian density The p-dimensional Gaussian normal) probability density function with mean vector µ and covariance matrix Σ is, N x µ, Σ) = π) p/ exp ) Σ / x µ)t Σ x µ), where µ : p vector and Σ : p p positive definite matrix Let X =,, X p ) T N µ, Σ), µ j is the mean of X j, Σ jj is the variance of X j, Σ ij i j) is the covariance between X i and X j Value of PDF x - - ) x Value of PDF 3 x - - x ) 6 / fredriklindsten@ituuse 7 / fredriklindsten@ituuse
3 Multivariate Gaussian density LDA, summary 3 3 The LDA classifier assigns a test input X = x to class k for which, ˆδ k x) = x T ˆΣ ˆµ k ˆµT k ˆΣ ˆµ k + log ˆπ k x x 6 ) x x ) is largest, where ˆπ k = N k /N k =,, K, ˆµ k = N k x i, k =,, K, ˆ i:y i =k K N K k= i:y i =k x i ˆµ k )x i ˆµ k ) T ˆδ k x) is a linear function in x and LDA is therefore a linear classifier its decision boundaries are linear) 8 / fredriklindsten@ituuse 9 / fredriklindsten@ituuse ex) LDA decision boundary Illustration of LDA decision boundary the level curves of two Gaussian PDFs with the same covariance matrix intersect along a straight line, ˆδ x) = ˆδ x) x ex) Simple spam filter We will use LDA to construct a simple spam filter: Output: Y {spam, ham} Input: X = 7-dimensional vector of features extracted from the Frequencies of 8 predefined words make, address, all, ) Frequencies of 6 predefined characters ;,, [,!, $, #) Average length of uninterrupted sequences of capital letters Length of longest uninterrupted sequence of capital letters Total number of capital letters in the Dataset consists of, 6 s classified as either spam or ham split into 7 % training and % testing) UCI Machine Learning Repository Spambase Dataset x / fredriklindsten@ituuse / fredriklindsten@ituuse
4 ex) Simple spam filter ex) Simple spam filter A linear classifier uses a linear decision boundary for separating the two classes as much as possible > ldafit = lday,data=dataset,subset=-test) LDA confusion matrix test data): Ŷ = Ŷ = Y = Y = 38 Table: Threshold r = Ŷ = Ŷ = Y = Y = 37 3 Table: Threshold r = 9 In R p this is a p )-dimensional hyperplane How can we visualize the classifier? One way: project the labeled) test inputs onto the normal of the decision boundary Y= - Projection on normal / fredriklindsten@ituuse 3 / fredriklindsten@ituuse ex) Simple spam filter Gmail s spam filter Feature Count Input X) make = 8 will 6 = 66 people 3 = 8 report = addresses = free 6 = 8 business 7 = 3 8 = 83 you 7 9 = 9 #Capitals X 7 = Official Gmail blog, July 9, the spam filter now uses an artificial neural network to detect and block the especially sneaky spam the kind that could actually pass for wanted mail Sri Harsha Somanchi, Product Manager Prham X) = 9% according to LDA model) / fredriklindsten@ituuse / fredriklindsten@ituuse
5 Quadratic discriminant analysis Do we have to assume a common covariance matrix? No! Estimating a separate covariance matrix for each class leads to an alternative method, Quadratic Discriminant Analysis QDA) Whether we should choose LDA or QDA has to do with the bias-variance trade-off! Parametric and nonparametric models So far we have looked at a few parametric models, linear regression, logistic regression, LDA and QDA, all of which are parametrized by a fixed-dimensional parameter Non-parametric models instead allow the flexibility of the model to grow with the amount of available data QDA: more flexible more parameters higher variance NB QDA is not a linear classifier can be very flexible = low bias) can suffer from high variance can be compute and memory intensive As always, the bias-variance trade-off is key also when working with non-parametric models 6 / fredriklindsten@ituuse 7 / fredriklindsten@ituuse k-nn The k-nearest neighbors k-nn) classifier is a simple non-parametric method We illustrate the k-nn classifier on a synthetic example where the Bayes classifier is known colored regions) Given training data T = {x i, y i )} N i=, for a test input X = x, Identify the k training inputs x i nearest to x, represented by the set N = {i : x i is in the k-neighborhood of x} Classify x according to a majority vote within the neighborhood N, ie assign x to class j for which i N Iy i = j) X - Y= is largest ties are often handled by a coin flip) -6-8 / fredriklindsten@ituuse 9 / fredriklindsten@ituuse
6 The decision boundaries for the k-nn classifier are shown as black lines The decision boundaries for the k-nn classifier are shown as black lines k= k= Y= Y= X X / fredriklindsten@ituuse / fredriklindsten@ituuse The decision boundaries for the k-nn classifier are shown as black lines k= k= k= Y= k= X - Y= X - Y= X - Y= X - The choice of the tuning parameter k controls the model flexibility: Small k small bias, large variance Large k large bias, small variance -6 - / fredriklindsten@ituuse 3 / fredriklindsten@ituuse
7 Error A few concepts to summarize lecture 3 Training error Test error Bayes error /k Bayes theorem: A formula relating conditional and marginal probabilities of two events Linear discriminant analysis LDA): A classifier based on the assumption that the distribution of the input X is multivariate Gaussian for each class, with different means but the same covariance LDA is a linear classifier Quadratic discriminant analysis QDA): Same as LDA, but where a different covariance matrix is used for each class QDA is not a linear classifier Non-parametric model: A model where the flexibility is allowed to grow with the amount of available training data k-nn classifier: A simple non-parametric classifier based on classifying a test input X = x according to the class labels of the k training samples nearest to x / fredriklindsten@ituuse / fredriklindsten@ituuse
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