ch02 — Gen1 Statistical Methods (1930s–2000s)

Companion notebook: notebooks/ch02_gen1_statistical.ipynb — PCA-T²/SPE on SMD

Control charts

The first "anomaly detection", born in industrial quality control.

• CUSUM (Page 1954): cumulative sums of standardized deviations, \(S^+_t = \max(0, S^+_{t-1} + z_t - k)\). Sensitive to small sustained mean shifts; the slack \(k\) is half the shift you want to detect.
• EWMA (Roberts 1959): \(z_t = \lambda x_t + (1-\lambda) z_{t-1}\). Weights recent observations — reacts faster than CUSUM to abrupt changes.

Implementation: tsad_forge/models/gen1_statistical/control_charts.py

Hotelling T² and PCA-T²/SPE

For multivariate normal data \(x \sim \mathcal{N}(\mu, \Sigma)\), the Mahalanobis distance \(T^2 = (x-\mu)^\top \Sigma^{-1} (x-\mu)\) follows a \(\chi^2\) distribution with D degrees of freedom.

When D is large, \(\Sigma^{-1}\) becomes unstable, so PCA splits the space:

• T² (principal subspace): normalized squared scores of the top-k components — "excessive movement within the normal directions of variation"
• SPE/Q (residual subspace): \(\|x - \hat{x}\|^2\) (reconstruction error) — "escaping outside the normal directions of variation"

The two statistics catch different kinds of anomalies. In semiconductor FDC, T² often maps to process drift and SPE to sensor faults and novel failure modes (ch09).

Sub-PCA — a baseline you underestimate at your peril

Slicing the series into sliding windows and scoring PCA reconstruction error is simple — and sits near the top of the TSB-AD leaderboard. It captures temporal structure through the window while keeping a linear model's stability. Check our leaderboard and see for yourself.

STL residuals

Remove trend and seasonality with STL (Season-Trend decomposition using Loess) and score |z| of the residual. The period is estimated automatically from the dominant ACF peak (estimate_period); aperiodic channels fall back to moving-average detrending.