Scale-Wise Attribution Mechanism for Multi-Scale Time Series Analysis: A Methodological Framework

We present a comprehensive Scale-Wise Attribution Mechanism (SWAM) for quantifying the contribution of different temporal scales in time series analysis using discrete wavelet transforms. The methodology employs five complementary attribution methods—energy-based, variance-based, correlation-based, perturbation-based, and peak detection—to provide a holistic assessment of scale importance. Equal weighting across the five components is adopted as a principled symmetry baseline; a sensitivity analysis across alternative weighting schemes confirms that the principal findings are robust. We apply SWAM to two annual sub-periods of the Austrian EPEX day-ahead electricity market drawn from the Open Power System Data repository: 2015 (low-volatility regime) and 2017 (high-volatility regime. Three wavelet families (Haar, Daubechies-2, Daubechies-4) are evaluated with five-level decomposition. Across both regimes the approximation scale (A5, hours) dominates comprehensive attribution (56–62%), driven by the large mean price relative to variance—a finding substantiated by a pronounced energy–variance decoupling (A5 captures 92–94% of energy but only 50–60% of variance). The D3 detail scale (8–16 hours) is the second-ranked contributor in the 2015 regime (14.3%, Db4), while D4 (16–32 hours) rises to second rank in 2017 (15.8%, Db4), consistent with strengthened daily-cycle volatility in the higher-renewable year. Bootstrap uncertainty quantification yields coefficient-of-variation (CV) values of 2–14% across scales and a rank-order consistency exceeding 95% across bootstrap resamples, confirming stable and precise attribution profiles.