The paper motivates high dimensional smoothing with penalized splines and its numerical calculation in an efficient way. If smoothing is carried out over three or more covariates the classical tensor product spline bases explode in their dimension bringing the estimation to its numerical limits. A recent approach by Siebenborn and Wagner(2019) circumvents storage expensive implementations by proposing matrix-free calculations which allows to smooth over several covariates. We extend their approach here by linking penalized smoothing and its Bayesian formulation as mixed model which provides a matrix-free calculation of the smoothing parameter to avoid the use of high-computational cross validation. Further, we show how to extend the ideas towards generalized regression models. The extended approach is applied to remote sensing satellite data in combination with spatial smoothing.

翻译：本文以有效的方式鼓励高维平滑,使用受罚的样条和数字计算。如果平滑使用三个或三个以上,则古典高压产品样条基在三个或三个以上的共变中爆炸,使估计达到其数量限度。Siebenborn和Wagner(2019年)最近采取的办法绕过存储费用昂贵的操作,提出无基数计算,使若干共变平滑。我们通过将受罚的平滑和巴耶斯式配方作为混合模型,提供平滑参数的无基数计算,以避免使用高分子截取交叉验证。此外,我们展示了如何将想法推广到普遍回归模型。扩大的办法适用于遥感卫星数据,同时进行空间平滑。