statsmodels.nonparametric.kernel_density.KDEMultivariateConditional.imse¶
-
KDEMultivariateConditional.imse(bw)[source]¶ The integrated mean square error for the conditional KDE.
- Parameters
- bwarray_like
The bandwidth parameter(s).
- Returns
- CV
float The cross-validation objective function.
- CV
Notes
For more details see pp. 156-166 in [1]. For details on how to handle the mixed variable types see [2].
The formula for the cross-validation objective function for mixed variable types is:
\[CV(h,\lambda)=\frac{1}{n}\sum_{l=1}^{n} \frac{G_{-l}(X_{l})}{\left[\mu_{-l}(X_{l})\right]^{2}}- \frac{2}{n}\sum_{l=1}^{n}\frac{f_{-l}(X_{l},Y_{l})}{\mu_{-l}(X_{l})}\]where
\[G_{-l}(X_{l}) = n^{-2}\sum_{i\neq l}\sum_{j\neq l} K_{X_{i},X_{l}} K_{X_{j},X_{l}}K_{Y_{i},Y_{j}}^{(2)}\]where \(K_{X_{i},X_{l}}\) is the multivariate product kernel and \(\mu_{-l}(X_{l})\) is the leave-one-out estimator of the pdf.
\(K_{Y_{i},Y_{j}}^{(2)}\) is the convolution kernel.
The value of the function is minimized by the
_cv_lsmethod of the GenericKDE class to return the bw estimates that minimize the distance between the estimated and “true” probability density.References