# project_1 - (d) (Optional) Dene M h,G ( x ) = n i =1 x i g...

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Project I (Due: 9/17/2010) Math Modeling II, Fall 2010 (Dr. Xin Li) Given a set of points { x i } n i =1 in the d dimensional space R d , the kernel density estimate (with kernel K and radius h ) is given by ˆ f K ( x ) = 1 nh d n X i =1 K ± x - x i h . Some special cases of the kernel K are: (i) the multivariate Epanechnikov kernel K E ( x ) = ( 1 2 c - 1 d ( d + 2)(1 - k x k 2 ) if k x k < 1 0 otherwise where c d is the volume of the unit d -dimensional sphere; (ii) the multivariate normal K N ( x ) = (2 π ) - d/ 2 exp ± - 1 2 k x k 2 . In general, K can be deﬁned in term of a proﬁle , k : a real valued function k ( t ) deﬁned for t 0 is called a proﬁle of K if K ( x ) = k ( k x k 2 ). (a) Find the proﬁles for K E and K N , respectively. (b) Assume that k is diﬀerentiable and let g ( t ) = - k 0 ( t ) . Show that ˆ f K ( x ) = 2 nh d +2 n X i =1 ( x i - x ) g ˆ ² ² ² ² x - x i h ² ² ² ² 2 ! . (c) Design an algorithm to ﬁnd maximum of ˆ f K ( x ) by using a steepest ascent method. Numerical experiments are welcomed but not required.

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Unformatted text preview: (d) (Optional) Dene M h,G ( x ) = n i =1 x i g x-x i h 2 n i =1 g x-x i h 2 -x 1 and (for some positive constant C ) f G ( x ) = C nh d n X i =1 g x-x i h 2 ! . Verify that f K ( x ) = f G ( x ) 2 /C h 2 M h,G ( x ) . (e) (Optional) Let y i , i = 1 , 2 , 3 ,... be related by y j +1 = y j + M h,G ( y j ) and let M j = f K ( y j ) , j = 1 , 2 , 3 ,... Show that M j +1 M j , j = 1 , 2 , 3 ,... (f) (Optional, continued from (e)) Prove or disprove that lim j y j exists. 2...
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## This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

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project_1 - (d) (Optional) Dene M h,G ( x ) = n i =1 x i g...

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