We consider functional data where an underlying smooth curve is composed not just with errors, but also with irregular spikes. We propose an approach that, combining regularized spline smoothing and an Expectation-Maximization algorithm, allows one to both identify spikes and estimate the smooth component. Imposing some assumptions on the error distribution, we prove consistency of EM estimates. Next, we demonstrate the performance of our proposal on finite samples and its robustness to assumptions violations through simulations. Finally, we apply our proposal to data on the annual heatwaves index in the US and on weekly electricity consumption in Ireland. In both datasets, we are able to characterize underlying smooth trends and to pinpoint irregular/extreme behaviors.

翻译：我们考虑一类函数型数据，其中潜在平滑曲线不仅包含误差，还包含不规则尖峰。本文提出一种结合正则化样条平滑与期望最大化算法的方法，既能识别尖峰又能估计平滑分量。在误差分布假设下，我们证明了EM估计量的一致性。通过模拟实验，我们展示了该方法在有限样本下的表现及其对假设违背的鲁棒性。最后，我们将该方法应用于美国年度热浪指数和爱尔兰周度电力消费数据。在这两个数据集中，我们成功刻画了潜在的平滑趋势，并精准定位了不规则/极端行为。