稀疏采样随机曲面的推理与计算
Inference and Computation for Sparsely Sampled Random Surfaces
摘要：
与一维情况相比，二维域上的函数数据的非参数推断需要额外的计算和统计挑战。协方差的可分性通常被假设为在密集观测的情况下解决这些问题。相反，我们考虑稀疏制度，其中的潜在表面观察到只有在几个不规则的位置与添加剂测量误差，并提出了一个估计的协方差的基础上局部线性平滑。因此，可分性的假设减少了固有的四维平滑问题到几个二维平滑，并允许建议的估计，以保持经典的最小最大最优收敛速度的二维平滑。即使可分性不成立，作为正则化的一种形式，强加它仍然是有利的。仿真研究表明，我们的方法实现了有利的偏差方差权衡和大规模的加速。最后，所提出的方法用于定性分析隐含波动率表面对应的看涨期权，并预测的潜在表面的基础上，从整个数据集的信息，允许不确定性量化。我们的交叉验证样本外的定量结果表明，所提出的方法优于通常的方法，预先平滑每个隐含波动率表面分别。
英文摘要：
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.
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