Detecting Stable Cross-Impact Patterns in Bivariate Time Series

Source: G. Andrienko, N. Andrienko, M. Akila, B. Kathirgamanathan and M. Ponce-de-Leon, "Detecting Stable Cross-Impact Patterns in Bivariate Time Series," in IEEE Transactions on Visualization and Computer Graphics, vol. 32, no. 7, pp. 5996-6013, 2026, doi: 10.1109/TVCG.2026.3676810

Workflow Summary

This workflow supports the detection, extraction, and analysis of stable cross-impact patterns between the two variables of a bivariate time series — population mobility and COVID-19 incidence — recorded for each spatial unit (province). For each province's bivariate time series, impact values are computed over a sliding time window across a range of time lags using several measures, including a novel tolerance-based variant of Kendall's tau that treats minor fluctuations as ties, yielding a time × lag matrix of impact values. An interactive "Ikat plot" visualises this matrix, coordinated with the underlying time series and a scatter plot, and supports dynamic queries on value level, trend, and impact strength. Analysts use a few representative provinces to iteratively explore how detected patterns depend on parameter settings (tolerance thresholds, window length, maximum lag, trend and significance thresholds), assisted by an automated parameter-sensitivity analysis that plots event counts against varied parameter values and flags bend points as candidate critical settings, until the settings reveal patterns aligned with the analysis goals. The calibrated settings are then applied to the full dataset to automatically extract "impact events" (stable cross-impact intervals linking a base-series trend to a lagged response-series trend, classified according to domain-expected relationship types) and "trend events" (sustained increases or decreases in each individual series). These events are visualised as timeline charts, with time on one axis and, in the case study, provinces ordered by geographic similarity (via Sammon mapping) on the other, allowing inspection of co-occurring trend and impact events across space and time. A quantitative event-context extraction step determines, for every trend event, whether it was "impacted" by a preceding cross-impact and/or "had impact" on a subsequent one, summarised in impact transition graphs. Finally, interpretable logistic regression models are built, per pandemic wave and overall, to identify which event features (e.g., average level, trend type, spatial location, amplitude) predict whether a trend event produces a measurable downstream impact, and the resulting patterns and model insights are synthesised into explicit findings about the strength, asymmetry, and temporal evolution of the cross-impacts. A case study on Spanish province-level mobility and COVID-19 data demonstrates the complete workflow, revealing strong early-pandemic mobility-to-case and case-to-mobility impacts that weaken over time, seasonal/holiday-driven mobility drops largely unrelated to case trends, and marked regional heterogeneity.

Graphical view

ATWL Representation