Multivariate kernel density estimation: Difference between revisions

Revision as of 16:06, 24 September 2010

Revision as of 16:33, 24 September 2010

Line 15:

==Definition==

The previous figure is a graphical representation of kernel density estimate, which we now define in an exact manner. Let '''X'''<mathsub>1</sub>\bold{X}_1, \bold{'''X}_2'''₂, \dots…, \bold{'''X}_n'''<sub>''n''</mathsub> be a ''d''-variate random sample drawn from a common density function ''fƒ''. The kernel density estimate is defined to be

: <math>\hat{f}_\bold{H}(\bold{x})= n^{-1} \sum_{i=1}^n K_\bold{H} (\bold{x} - \bold{X}_i)</math>

Line 21:

where

<ul>

<li><math>\bold{{nowrap|'''x}''' {{=}} (x_1''x''₁, x_2''x''₂, \dots…, x_d''x_d'')^<sup>''T''</mathsup>}}, {{nowrap|'''X'''<mathsub>''i''</sub>\bold{X}_i {{=}} (X_{i1}''X''_''i''1, X_{i2}''X''_''i''2, \dots…, X_{''X_id}'')^^''T'', ''i'' {{=}} 1, 2, \dots…, ''n</math>''}} are ''d''-vectors

<li>''K'' is the kernel function which is a symmetric density function, with {{nowrap|''K''<mathsub>K_\bold{'''H}'''(\bold{</sub>'''x}''') {{=}} |\bold{H{!}|^}'''H'''{-1{!}}^−1/2}''K''(\bold{'''H}^{-1'''<sup>−1/2} \bold{x})</mathsup>'''x''')}}

<li><strong>H</strong> is the bandwidth (or smoothing) matrix which is a symmetric, [[positive definite matrix|positive definite]] ''d x d'' matrix.

</ul>

Line 60:

===Plug-in===

The plug-in (PI) estimate of the AMISE is formed by replacing '''Ψ'''<mathsub>\bold{\Psi}_44</mathsub> by its estimator <math>\hat{\bold{\Psi}}_4</math>

: <math>\operatorname{PI}(\bold{H}) = n^{-1} |\bold{H}|^{-1/2} R(K) + \tfrac{1}{4} m_2(K)^2

Line 66:

where <math>\hat{\bold{\Psi}}_4 (\bold{G}) = n^{-2} \sum_{i=1}^n

\sum_{j=1}^n [(\operatorname{vec} \, \operatorname{D}^2) (\operatorname{vec}^T \operatorname{D}^2)] K_\bold{G} (\bold{X}_i - \bold{X}_j)</math>. Thus <math>\hat{\bold{H}}_{\operatorname{PI}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{PI} (\bold{H})</math> is the plug-in selector.<ref>{{Cite journal| author1=Wand, M.P. | author2=Jones, M.C. | title=Multivariate plug-in bandwidth selection | journal=Computational Statistics | year=1994 | volume=9 | pages=97–177}}</ref><ref>{{Cite journal| doi=10.1080/10485250306039 | author1=Duong, T. | author2=Hazelton, M.L. | title=Plug-in bandwidth matrices for bivariate kernel density estimation | journal=Journal of Nonparametric Statistics | year=2003 | volume=15 | pages=17–30}}</ref> These references also contain algorithms on optimal estimation of the pilot bandwidth matrix <strong>G</strong> and establish that <math>\hat{\bold{H}}_{\operatorname{PI}}</math> [[convergence in probability|converges in probability]] to '''H'''<mathsub>\bold{H}_{\operatorname{AMISE}}</mathsub>.

===Smoothed cross validation===

Line 76:

Thus <math>\hat{\bold{H}}_{\operatorname{SCV}} = \operatorname{argmin}_{\bold{H} \in F} \, \operatorname{SCV} (\bold{H})</math> is the SCV selector.<ref>{{Cite journal| doi=10.1007/BF01205233 | author1=Hall, P. | author2=Marron, J. | author3=Park, B. | title=Smoothed cross-validation | journal=Probability Theory and Related Fields | year=1992 | volume=92 | pages=1–20}}</ref><ref>{{Cite journal| doi=10.1111/j.1467-9469.2005.00445.x | author1=Duong, T. | author2=Hazelton, M.L. | title=Cross validation bandwidth matrices for multivariate kernel density estimation | journal=Scandinavian Journal of Statistics | year=2005 | volume=32 | pages=485–506}}</ref>

These references also contain algorithms on optimal estimation of the pilot bandwidth matrix <strong>G</strong> and establish that <math>\hat{\bold{H}}_{\operatorname{SCV}}</math> converges in probability to '''H'''<mathsub>\bold{H}_{\operatorname{AMISE}}</mathsub>.

==Computer implementation==