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# L5_PR - ECEN 689 Statistical Computation in GSP...

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ECEN 689 Statistical Computation in GSP http://www.ece.tamu.edu/~ulisses/ECEN689/ Lecture 5: Review of Pattern Recognition Ulisses Braga Neto Genomic Signal Processing Laboratory Department of Electrical and Computer Engineering Texas A&M University

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Multivariate Classification Classification of expression profiles (vectors). Breast Cancer Data
Classifier Design Predictors (genes, epitopes) Target (disease, immunization, survivability) Probabilistic Relationship Classification Error: h (classifier)

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Error Estimation/Feature Selection Breast Cancer Data What predictors should I use and what is an estimate of the classification error based on these data?
Basic Pipeline

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Optimal Classifier Every problem has an optimal classifier, called the Bayes classifer . The corresponding classification error is called the Bayes error and is usually nonzero. To find the Bayes classifier and error one needs to know the joint distribution F XY This distribution is usually unknown or only known partially, so one must resort to design sub-optimal classifiers based on training data.
Gaussian Case

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Equal-Variance Case If we can assume that the covariance matrices are equal to each other then the optimal classifier is linear : where
Example Gene 1 Gene 2 μ 1 μ 0 0 + μ 1 ) 1 2 90 °

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Example - II From R. Duda, P. Hart and D. Stork, Pattern Classification, 2nd ed., John Wiley & Sons, 2001.
Optimal Classification Error The optimal classifier in the Gaussian equal- variance case is a hyperplane, and its error can be shown to be given by where Φ is the cdf of a standard Gaussian and δ is the Mahalanobis distance between classes:

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L5_PR - ECEN 689 Statistical Computation in GSP...

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