Many real-world applications involve analyzing time-dependent phenomena, which are intrinsically functional---consisting of curves varying over a continuum, which is time in this case. When analyzing continuous data, functional data analysis (FDA) provides substantial benefits, such as the ability to study the derivatives and to restrict the ordering of data. However, continuous data inherently has infinite dimensions, and FDA methods often suffer from high computational costs. This is even more critical when we have new incoming data and want to update the FDA results in real-time. In this paper, we present a visual analytics approach to consecutively monitor and review the changing time-series data with a focus on identifying outliers by using FDA. To perform such an analysis while addressing the computational problem, we introduce new incremental and progressive algorithms that promptly generate the magnitude-shape (MS) plot, which reveals both the functional magnitude and shape outlyingness of time-series data. In addition, by using an MS plot in conjunction with an FDA version of principal component analysis, we enhance the analyst's ability to investigate the visually-identified outliers. We illustrate the effectiveness of our approach with three case studies using real-world and synthetic datasets.

中文翻译：

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使用增量和渐进功能数据分析来监视时间序列数据的可视化分析方法

许多现实世界的应用都涉及到分析与时间有关的现象，这些现象本质上是功能性的-由连续的曲线（在这种情况下为时间）组成。在分析连续数据时，功能数据分析（FDA）提供了很多好处，例如研究衍生工具和限制数据排序的能力。但是，连续数据固有地具有无限的维度，并且FDA方法经常遭受高计算成本的困扰。当我们有新的传入数据并想实时更新FDA结果时，这一点就显得尤为重要。在本文中，我们提出了一种视觉分析方法，可以连续监视和审查变化的时间序列数据，重点是通过使用FDA识别异常值。为了在解决计算问题的同时执行此类分析，我们引入了新的增量和渐进算法，这些算法可以迅速生成量级形状（MS）图，从而揭示了时间序列数据的功能量级和形状异常。此外，通过将MS图与FDA版本的主成分分析结合使用，我们可以提高分析员调查视觉识别异常值的能力。我们通过使用真实和综合数据集的三个案例研究来说明我们的方法的有效性。