Orthogonal Self-Similarity Decomposition (OSSD): A Delay-Based Framework for Multiscale Time Series Analysis with Applications in Hydrological Forecasting

Decomposition of nonlinear, nonstationary multicomponent signals remains challenging for existing decomposition strategies, including frequency-based, data-driven, and subspace methods, which can suffer from mode mixing, leakage across components, and unreliable isolation of transients. Motivated by this gap, this study proposes Orthogonal Self-Similarity Decomposition (OSSD), which exploits a self-similarity structure in delay-embedded orbit geometry so that temporal organization, rather than spectrum alone, guides component construction. OSSD-Basic introduces three algorithmic novelties within a single pipeline: (1) an adaptive proxy-correlation band merging on the delay axis, (2) a dominant-component cascade that prevents energy-dominant carriers from masking weaker components, and (3) a double MGS + LS reprojection that collapses the inter-mode orthogonality index to numerical zero, regardless of merging and pruning operations. Synthetic experiments with known ground truth show that OSSD-Basic provides a parsimonious four-mode representation with exact inter-mode orthogonality (OI = 9.4 × 10⁻¹⁸), the highest reconstruction SNR among the evaluated baselines (27.14 dB), and the highest ground-truth diagonal correlation sum (3.038) among the tested methods, while using two fewer modes than EMD, VMD, and SSA. Daily streamflow forecasting on a U.S. Geological Survey discharge record further shows that augmenting OSSD-derived inputs with fractal descriptors and fractional-order differencing features yields progressive accuracy gains over the AR-ANN baseline, with R² improving from 0.855 to 0.915 at one-step-ahead and from 0.388 to 0.699 at four-step-ahead forecasting in the single-input setting, within a single-station case study on USGS 01554000. Overall, OSSD-Basic offers an interpretable multiscale decomposition with guaranteed inter-mode orthogonality and a structured feature pathway for oscillatory–transient mixtures.