Gaussian Classifiers CS498


 James Stone
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1 Gaussian Classifiers CS498
2 Today s lecture The Gaussian Gaussian classifiers A slightly more sophisticated classifier
3 Nearest Neighbors We can classify with nearest neighbors x m 1 m 2 Decision boundary Can we get a probability?
4 Nearest Neighbors Nearest neighbors offers an intuitive distance measure x d 1 d 2 m 1 m 2 d i (x 1 m i,1 ) 2 +(x 2 m i,2 ) 2 = x m
5 Making a Soft Decision What if I didn t want to classify What if I wanted a degree of belief How would you do that? x d 1 d 2 m 1 m 2
6 From a Distance to a Probability If the distance is 0 the probability is high If the distance is the probability is zero How do we make a function like that?
7 Here s a first crack at it Use exponentiation: 1 e x m
8 Adding an importance factor Let s try to tune the output by adding a factor denoting importance e x m c
9 One more problem Not all dimensions are equal
10 Adding variable importance to dimensions Somewhat more complicated now: e (x m)t C 1 (x m)
11 The Gaussian Distribution This is the idea behind the Gaussian Adding some normalization we get: P(x;m,C) = 1 2π k/2 e (x m) 1/2 C T C 1 (x m)
12 Gaussian models We can now describe data using Gaussians x How? That s very easy
13 Learn Gaussian parameters Estimate the mean: m = 1 x N i Estimate the covariance: C = 1 N 1 (x m)t (x m)
14 Now we can make classifiers We will use probabilities this time We ll compute a belief of class assignment
15 The Classification Process We provide examples of classes We make models of each class We assign all new input data to a class
16 Making an assignment decision Face classification example Having a probability for each face how do we make a decision?
17 Motivating example Face 1 is more likely x y P(y {face 1, face 2 }) Template face 1 Template face 2 Unknown face
18 How the decision is made In simple cases the answer is intuitive To get a complete picture we need to probe a bit deeper
19 Starting simple Two class case, ω 1 and ω 2
20 Starting simple Given a sample x, is it ω 1 or ω 2? i.e. P(ω i x) =?
21 Getting the answer The class posterior probability is: Likelihood P(ω i x) = P(x ω i )P(ω i ) P(x) Evidence Priors To find the answer we need to fill in the terms in the righthandside
22 Filling the unknowns Class priors How much of each class? P(ω 1 ) N 1 / N P(ω 2 ) N 2 / N Class likelihood: P(x ω i ) Requires that we know the distribution of ω i We ll assume it is the Gaussian
23 Filling the unknowns Evidence: P(x) = P(x ω 1 )P(ω 1 ) + P(x ω 2 )P(ω 2 ) We now have P(ω 1 x),p(ω 2 x)
24 Making the decision Bayes classification rule If P(ω 1 x) > P(ω 2 x) then x belongs to class ω 1 If P(ω 1 x) < P(ω 2 x) then x belongs to class ω 2 Easier version P(x ω 1 )P(ω 1 ) P(x ω 2 )P(ω 2 ) Equiprobable class version P(x ω 1 ) P(x ω 2 )
25 Visualizing the decision Assume Gaussian data P(x ω i ) = N (x µ i,σ i ) R 1 R 2 P(ω 1 x) = P(ω 2 x) x 0
26 Errors in classification We can t win all the time though Some inputs will be misclassified R 1 R 2 Probability of errors P error = 1 2 x 0 P(x ω 2 )dx x 0 x 0 P(x ω 1 )dx
27 Minimizing misclassifications The Bayes classification rule minimizes any potential misclassifications R 1 R 2 x 0 Can you do any better moving the line?
28 Minimizing risk Not all errors are equal! e.g. medical diagnoses R 1 R 2 x 0 Misclassification is often very tolerable depending on the assumed risks
29 Example Minimum classification error R 1 R 2 x 0 Minimum risk with λ 21 > λ 12 i.e. class 2 is more important R 1 R 2 x 0
30 True/False Positives/Negatives Naming the outcomes classifying for ω 1 x is ω 1 x is ω 2 x classified as ω 1 True positive False positive x classified as ω 2 False negative True negative False positive/false alarm/type I error False negative/miss/type II error
31 Receiver Operating Characteristic Visualize classification balance R.O.C. Curve R 1 R 2 x 0 True positive rate False positive rate
32 Classifying Gaussian data Remember that we need the class likelihood to make a decision For now we ll assume that: P(x ω i ) = N (x µ i,σ i ) i.e. that the input data is Gaussian distributed
33 Overall methodology Obtain training data Fit a Gaussian model to each class Perform parameter estimation for mean, variance and class priors Define decision regions based on models and any given constraints
34 1D example The decision boundary will always be a line separating the two class regions R 1 R 2 x 0
35 2D example
36 2D example fitted Gaussians
37 Gaussian decision boundaries The decision boundary is defined as: P(x ω 1 )P(ω 1 ) = P(x ω 2 )P(ω 2 ) We can substitute Gaussians and solve to find what the boundary looks like
38 Discriminant functions Define a function so that: classify x in ω i if g i (x) > g j (x), i j Decision boundaries are now defined as: g ij (x) ( g i (x) = g j (x))
39 Back to the data Σ i = σ 2 i I produces line boundaries Discriminant: g i (x) = w i T x + b w i = µ i / σ 2 b = µ i T µ i 2σ 2 + log P(ω i )
40 Quadratic boundaries Arbitrary covariance produces more elaborate patterns in the boundary g i (x) = x T W i x + w i T x + w i W i = 1 2 Σ 1 i w i = Σ i 1 µ i w = 1 2 µ T Σ 1 i i µ i 1 2 log Σ i + log P(ω i )
41 Quadratic boundaries Arbitrary covariance produces more elaborate patterns in the boundary
42 Quadratic boundaries Arbitrary covariance produces more elaborate patterns in the boundary
43 Example classification run Learning to recognize two handwritten digits Let s try this in MATLAB
44 Recap The Gaussian Bayesian Decision Theory Risk, decision regions Gaussian classifiers
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