Non-parametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. Even when separability fails to hold, imposing it can be still advantageous as a form of regularization. A simulation study reveals a favorable bias-variance trade-off and massive speed-ups achieved by our approach. Finally, the proposed methodology is used for qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our cross-validated out-of-sample quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.
翻译:与一维情况相比,对二维领域功能数据的非参数推论需要额外的计算和统计挑战。常假定共差是用来在密集观察的系统中解决这些问题的。相反,我们认为稀疏的制度,即只在少数非正常地点观察到潜伏表面,有添加度测量错误,并提议根据当地线性平滑剂估算共差值。因此,假定分离将内在的四维平滑问题降为几个二维平滑器,使拟议的估计者能够保留传统的二维平滑器小型和最佳汇合率。即使分离性无法维持,将它作为正规化的一种形式仍具有优势。模拟研究揭示出一种有利的偏差偏差交易和我们的方法所实现的大规模加速。最后,拟议的方法用于对隐含的波动表进行定性分析,并用于根据整个数据集的信息预测潜伏表面,从而可以进行不确定性的量化。我们共同的地表波动前的假设方法分别展示了各种拟议浮化法。