hw2 - ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = n i n i n...

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BIOMETRICS CSE190 Fall 2006 Assignment 2 Due: October 31, 2006 1. Duda, Hart, Stork 3.35. Let the sample mean μ n and the sample covariance matrix C n for a set of n samples x 1 …x n (each of which is d-dimensional) be defined by = = = t n i n i n n i i n x x n C n ) )( ( 1 1 1 1 x We call these the “nonrecursive” formulae. (a) What is the computational complexity of calculating n and C n by these formulae? (b) Show that the alternative” recursive techniques based on successive addition of new samples x n+1 can be derived using the recursive relations t n n n n n n n n n n x x n C n n C n ) )( ( 1 1 1 ) ( 1 1 1 1 1 1 + + = + + = + + + + x (c) What is the computational complexity of finding n and C n by these recursive methods? (d) Describe situations where you might prefer to use the recursive method for computing n and C n, , and ones where you might prefer the nonrecursive method? 2. Consider a normal p(x)=N( μ , σ 2 ) and Parzen window function φ (x) = N( μ ,1). Show that the Parzen window estimate
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Unformatted text preview: ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = n i n i n n h x x nh x p 1 1 ) ( ϕ has the following property E[p n (x)] = N( μ , σ 2 +h n 2 ) 3. Consider the following set of two dimensional vectors from three categories: ω 1 ω 2 ω 3 X 1 X 2 X 1 X 2 X 1 X 2 10 0 5 10 2 8 0 -10 0 5 -5 2 5 -2 5 5 10 -4 (a) Plot the decision boundary resulting from the nearest neighbor rule just for categorizing ω 1 and ω 2. Find the sample mean m 1 and m 2 and on the same figure sketch the decision boundary corresponding to classifying x by assigning it to the category of the nearest sample mean. (b) Repeat part (a) for categorizing only ω 1 and ω 3. (c) Repeat part (a) for categorizing only ω 2 and ω 3. (d) Repeat part (a) for three-category classifier, classifying ω 1, ω 2 and ω 3....
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hw2 - ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = n i n i n...

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