Compact Phase Histograms for Guided Exploration of Periodicity

TL;DR

The paper addresses discovering periodic patterns in long time-series data where the period is unknown. It introduces a phase-histogram widget that aggregates phase distributions across candidate period lengths $\tau$, visualized as a heatmap with guided exploration through precomputed data and on-demand calculations. Two quality measures, Shannon entropy $H$ and von Mises vector strength $r$, guide users to promising periods, and phase information can be mapped to color or glyphs in other views to reveal multi-attribute periodicity. Case studies on NOAA tidal data demonstrate the method's ability to uncover known cycles and show practical visual mappings, arguing for improved interactivity and scalability over traditional automated analyses like STL or DMD.

Abstract

Periodically occurring accumulations of events or measured values are present in many time-dependent datasets and can be of interest for analyses. The frequency of such periodic behavior is often not known in advance, making it difficult to detect and tedious to explore. Automated analysis methods exist, but can be too costly for smooth, interactive analysis. We propose a compact visual representation that reveals periodicity by showing a phase histogram for a given period length that can be used standalone or in combination with other linked visualizations. Our approach supports guided, interactive analyses by suggesting other period lengths to explore, which are ranked based on two quality measures. We further describe how the phase can be mapped to visual representations in other views to reveal periodicity there.

Figures (2)

• Figure 1: A schematic explanation (a) of our approach: Event data gets binned over the temporal domain. For the Cartesian representations Kosara_2011vanWijk_1999Lammarsch_2009Silva_2021 the bins then get placed so that each row represents one period. Our approach shows an aggregated view on this, and varies the period durations slightly from row to row. The current period length is framed in red in the figure. Example patterns for periodic behavior in our approach for a sharp (b) and a less sharp (c) periodic pattern, and for uniform noise (d). As a comparison, periodic behavior (e) in the Cartesian binned representation is shown for the actual signal period length, and for multiples (f), integer fractions (g), and non-integer fractions (h, here $6/5$) thereof. Here, the periodicity manifests as vertical lines. Patterns appear as straight lines emanating from the center in the Archimedean binned representation (i)Carlis_1998Weber_2001, but are also visible if the period length is nearly right (k).
• Figure 2: Example mappings of phase to color (a, c) or shape of visual marks (b) in a scatter plot. Periodic behavior related to the spatial aspect of the data is revealed by uniform areas. The mapping is shown in our widget (\ref{['fig:teaser:widget']}) as a legend (d, e) that can be interactively adjusted to change the mapping of glyph or color to phase.