The practical approximate bandwidth from Cross Validation j(h)=∫(fn)2dc-号=1f无-(c) 一般当h未知的时候，可以用更实用的方式选择窗宽， R(h)=(p-p(x))2dz =t-2mt+∫pre =J(h)+p2(x)dz. 注意到后面一项与无关，第一项可以用交叉验证方法估计： 其中，-)(c)是去掉第个观测值后对直方图的估计，J()称为交叉验证得分
The practical approximate bandwidth from Cross Validation
Parzen Windows(固定V) Parzen-window approach to estimate densities assume that the region n is a d- dimensional hypercube V=h(h length of the edge of) Let p(u)be the following window function: w=/asj=1,d 0 otherwise ((x-x)/h)is equal to unity if x;falls within the hypercube of volume V,centered at x and equal to zero otherwise
Parzen Windows（固定V） – Parzen-window approach to estimate densities assume that the region Rn is a ddimensional hypercube ((x-xi )/hn ) is equal to unity if xi falls within the hypercube of volume Vn centered at x and equal to zero otherwise. = = = 0 otherwise j 1,... ,d 2 1 1 u (u) Let (u) be the following window function : V h (h :length of the edge of ) j n n d n n
The number of samples in this hypercube is: -2 By substituting kn in equation(7),we obtain the following estimate: p.w-） P(x)estimates p(x)as an average of functions of x and the samples (x)(i= 1,...,n).These functions o can be general!
– The number of samples in this hypercube is: By substituting kn in equation (7), we obtain the following estimate: Pn (x) estimates p(x) as an average of functions of x and the samples (xi ) (i = 1,… ,n). These functions can be general! = = − = i n i 1 n i n h x x k − = = = n i i n i 1 n n h x x V 1 n 1 p (x)