BIOMETRICS CSE190 Fall 2006 Assignment 2 Due: October 31, 2006 1. Duda, Hart, Stork 3.35. Let the sample mean μn and the sample covariance matrix Cn for a set of n samples x1…xn (each of which is d-dimensional) be defined by ∑∑−−−===tnininniinxxnCn))((1111μμμx We call these the “nonrecursive” formulae. (a) What is the computational complexity of calculating μn and Cn by these formulae? (b) Show that the alternative” recursive techniques based on successive addition of new samples xn+1 can be derived using the recursive relations tnnnnnnnnnnxxnCnnCn))((111)(111111μμμμμ−−++−=−++=++++x (c) What is the computational complexity of finding μn and Cn by these recursive methods? (d) Describe situations where you might prefer to use the recursive method for computing μn and Cn,, and ones where you might prefer the nonrecursive method? 2. Consider a normal p(x)=N(μ,σ2) and Parzen window function φ(x) = N(μ,1). Show that the Parzen window estimate ∑=⎟⎟⎠⎞⎜⎜⎝⎛−=nininnhxxnhxp11)(ϕ has the following property E[pn(x)] = N(μ,σ2+hn2)3. Consider the following set of two dimensional vectors from three categories: ω1 ω2 ω3 X1 X2 X1 X2 X1 X2 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 ω1 and ω2. Find the sample mean m1 and m2 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 ω1 and ω3. (c) Repeat part (a) for categorizing only ω2 and ω3. (d) Repeat part (a) for three-category classifier, classifying ω1, ω2 and
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