Rolling correlation & tail dependence

Rolling windows in time

Rolling statistics look at a fixed-width window (e.g., 90 days) that slides through time. At each step we compute a summary from only the values inside the window, producing a new cube aligned to the window center. This approach keeps temporal context while remaining responsive to recent dynamics.

Correlation vs a reference pixel

When studying spatial synchrony we often compare every pixel to a single reference location. In cubedynamics the default reference is the center pixel in the requested climate cube. The function cubedynamics.stats.correlation.rolling_corr_vs_center computes the Pearson correlation between each pixel and the reference pixel within every rolling window. The resulting cube reveals how tightly each pixel's climate fluctuations track the anchor over time.

Tail dependence

Mean correlation can miss asymmetric extremes. Tail dependence focuses on the degree to which two pixels vary together during unusually low or high conditions. The helper cubedynamics.stats.tails.rolling_tail_dep_vs_center implements a rolling Spearman correlation after splitting each pixel and the center reference at their respective window quantiles. With b=0.5, the bottom set contains dates when both values are at or below their medians, while the top set contains dates when both values are above their medians. The helper returns bottom, top, and bottom-minus-top synchrony cubes.

Conceptual snippets

from cubedynamics.stats.correlation import rolling_corr_vs_center
from cubedynamics.stats.tails import rolling_tail_dep_vs_center

corr_cube = rolling_corr_vs_center(tmin, window_days=90, min_t=10)

bottom_tail, top_tail, diff_tail = rolling_tail_dep_vs_center(
    tmin, window_days=90, min_t=10, b=0.5
)

For temperature analyses, compare below-median synchrony from tmin with above-median synchrony from tmax to distinguish coordinated cold conditions from coordinated hot conditions.

The lower-level helper returns Spearman correlations in [-1, 1] and their signed lower-minus-upper difference in [-2, 2]. Negative values are valid.

The pipe verb v.rolling_tail_dep_vs_center(...) is a retained historical name for a different signed statistic: upper-tail variance minus full-window variance. It is not a probability or a [0, 1] dependence coefficient. Negative values mean variability inside the selected upper tail is smaller than variability across the full window. Its units are squared source units and its valid range is the unbounded real line.

v.rolling_median_split_synchrony(...) returns bottom_synchrony, top_synchrony, and bottom_minus_top. Result attrs record the per-series split rule, center-pixel reference, inclusive trailing window, evaluated end labels, stride, and incomplete-edge behavior.