BandSAD: Band-Decoupled Forecasting with ${\chi }^2$ Whitening for Label-Free Time-Series Anomaly Detection

Univariate time-series anomaly detection is dominated by deep autoencoding pipelines whose anomaly score has no interpretable scale, forcing practitioners to tune a decision threshold on labeled anomalies—a resource that streaming applications rarely have. We present BandSAD, a band-decoupled forecaster that (i) splits the forecasting target into $K$ frequency bands through a learnable partition-of-unity mask in the Fourier domain, (ii) whitens the resulting per-band errors with a robust MAD-based variance so that the window statistic is approximately ${\chi }^2$-distributed under the null, yielding a closed-form, label-free threshold ${\tau }_{\alpha }={\chi }_{KH,1-\alpha }^2/H$, and (iii) fuses several window scales by robust per-scale standardization and a max rule that encodes “anomalous at some scale”, with classical multiple-testing combinations (Sidak/Fisher) analyzed as calibrated alternatives. The entire pipeline consumes exactly one prior: an upper bound $\alpha$ on the false alarm rate. On three real NAB streams and two controlled synthetic benchmarks, BandSAD attains the best mean point-adjusted $F_1$ among six established baselines (Isolation Forest, PCA, Spectral Residual, AE, LSTM-AE, USAD) and the best AUC-PR on the strongly periodic stream, while remaining competitive on the others; more importantly, it is the only method whose decision threshold needs no labeled anomalies—the calibrated ${\tau }_{\alpha }$ recovers up to $99\%$ of the oracle-threshold $F_1$ on streams with adequate anomaly prevalence, and Q–Q analysis confirms the ${\chi }^2$ null in the body of the distribution. Ablations show that each component—band decoupling, out-of-sample MAD whitening, and multi-scale fusion—contributes measurably to the final performance.