Multivariate kernel density estimation: Difference between revisions

Revision as of 16:51, 24 September 2010

Revision as of 17:15, 24 September 2010

Line 28:

==Optimal bandwidth matrix selection==

The most commonly used optimality criterion for selecting a bandwidth matrix is the MISE or [[mean integrated squared error | Mean Integrated Squared Error]]

: <math>\operatorname{MISE} (\bold{H}) = \operatorname{E}\!\left[\, \int [(\hat{f}_\bold{H} (\bold{x}) - f(\bold{x})])^2 \, d\bold{x} \;\right].</math>

This in general does not possess a closed form expression, so it is usual to use its asymptotic approximation (AMISE) as a proxy

Line 41:

<li><math>R(K) = \int K(\bold{x})^2 \, d\bold{x}</math>, with {{nowrap|''R''(''K'') {{=}} (4''π'')^''−d''/2}} when ''K'' is a normal kernel

<li><math>\int \bold{x} \bold{x}^T K(\bold{x})^2 \, d\bold{x} = m_2(K) \bold{I}_d</math>,

with <strong>I</strong>_d being the ''d x× d'' [[identity matrix]], with ''m''₂ = 1 for the normal kernel

<li>D²''ƒ'' is the ''d x× d'' Hessian matrix of second order partial derivatives of ''ƒ''

<li><math>\bold{\Psi}_4 = \int (\operatorname{vec} \, \operatorname{D}^2 f(\bold{x})) (\operatorname{vec}^T \operatorname{D}^2 f(\bold{x})) \, d\bold{x}</math> is a ''d''² x× ''d''² matrix of integrated fourth order

partial derivatives of ''fƒ''

<li>vec is the vector operator which stacks the columns of a matrix into a single vector e.g. <math>\operatorname{vec}\begin{bmatrix}a & c \\ b & d\end{bmatrix} = \begin{bmatrix}a & b & c & d\end{bmatrix}^T.</math>

</ul>

Line 56:

where ''F'' is the space of all symmetric, positive definite matrices.

Since this ideal selector contains the unknown density function ''fƒ'', it cannot be used directly. The many different varieties of data-based bandwidth selectors arise from the different estimators of the AMISE. We concentrate on two classes of selectors which have been shown to be the most widely applicable in practise: smoothed cross validation and plug-in selectors.

===Plug-in===