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：我们考虑的是功能数据,其中潜在的光滑曲线不仅由错误组成,而且还由不规则的峰值组成。我们提出一种方法,将正常的滑动样条和预期-最大化算法结合起来,既能确定峰值,又能估计光滑部分。对错误分布作出一些假设,我们证明EM估计数的一致性。接下来,我们通过模拟来展示我们关于有限样本的建议的绩效及其强健性,通过模拟来证实假设的违规情况。最后,我们将我们的建议应用于美国年度热浪指数和爱尔兰每周电力消费的数据。在这两个数据集中,我们可以辨别潜在的光滑趋势并确定异常/极端行为。