Time evolving surfaces can be modeled as two-dimensional Functional time series, exploiting the tools of Functional data analysis. Leveraging this approach, a forecasting framework for such complex data is developed. The main focus revolves around Conformal Prediction, a versatile nonparametric paradigm used to quantify uncertainty in prediction problems. Building upon recent variations of Conformal Prediction for Functional time series, a probabilistic forecasting scheme for two-dimensional functional time series is presented, while providing an extension of Functional Autoregressive Processes of order one to this setting. Estimation techniques for the latter process are introduced and their performance are compared in terms of the resulting prediction regions. Finally, the proposed forecasting procedure and the uncertainty quantification technique are applied to a real dataset, collecting daily observations of Sea Level Anomalies of the Black Sea

翻译：随时间演变的曲面可建模为二维函数型时间序列，借助函数型数据分析工具加以处理。基于该方法，本文针对此类复杂数据开发了预测框架。核心关注点在于共形预测——一种用于量化预测问题不确定性的通用非参数范式。基于近期函数型时间序列共形预测的改进方法，本文提出了二维函数型时间序列的概率预测方案，并将一阶函数型自回归过程扩展至该场景。针对该过程引入了估计技术，并通过所得预测区域对其性能进行了比较。最后，将所提预测流程及不确定性量化技术应用于真实数据集，该数据集收集了黑海海面高度异常值的逐日观测数据。