Lect15 Microarray analysis

An Introduction to Bioinformatics Algorithms (Computational Molecular Biology)

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L15:Microarray analysis (Classification)
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The Biological Problem Two conditions that need to be differentiated, (Have different treatments). EX: ALL (Acute Lymphocytic Leukemia) & AML (Acute Myelogenous Leukima) Possibly, the set of genes over-expressed are different in the two conditions
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Geometric formulation Each sample is a vector with dimension equal to the number of genes. We have two classes of vectors (AML, ALL), and would like to separate them, if possible, with a hyperplane.
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Basic geometry What is | | x | | 2 ? What is x /| | x | | Dot product? x=(x 1 ,x 2 ) y x T y = x 1 y 1 + x 2 y 2 = || x || || y ||cos q x cos q y + || x || || y ||sin( q x )sin( q y ) || x || || y ||cos( q x - q y )
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Dot Product Let β be a unit vector. | | β | | = 1 Recall that β T x = | | x| | cos θ What is β T x if x is orthogonal (perpendicular) to β ? θ x β β T x = | | x| | cos θ
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Hyperplane How can we define a hyperplane L? Find the unit vector that is perpendicular (normal to the hyperplane)
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Points on the hyperplane Consider a hyperplane L defined by unit vector β , and distance β 0 from the origin Notes; For all x L, x T β must be the same, x T β = β 0 For any two points x 1 , x 2 , (x 1 - x 2 ) T β =0 Therefore, given a vector β , and an offset β 0 , the hyperplane is the set of all points {x : x T β = β 0 } x 1 x 2
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Hyperplane properties Given an arbitrary point x, what is the distance from x to the plane L? D(x,L) = ( β T x - β 0 ) When are points x1 and x2 on different sides of the hyperplane? x β 0
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Hyperplane properties Given an arbitrary point x, what is the distance from x to the plane L? D(x,L) = ( β T x - β 0 ) When are points x 1 and x 2 on different sides of the hyperplane? Ans: If D(x 1 ,L)* D(x 2 ,L) < 0 x β 0
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Input: A training set of +ve & -ve examples Recall that a hyperplane is represented by {x:- β 0 + β 1 x 1 + β 2 x 2 =0} or (in higher dimensions) {x: β T x- β 0 =0} Goal: Find a hyperplane that ‘separates’ the two classes. Classification: A new point x is +ve if
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Lect15 Microarray analysis - L15:Microarray analysis...

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