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ResidualFOCuS

Incremental, causal technical analysis documentation

Summary

ResidualFOCuS applies the FOCuS mean changepoint detector to a residual or innovation series (model errors, hedge residuals, filter innovations, etc.). The engine is identical to FOCuS; the class name documents the intended input semantics.

Update API

import rtta

ind = rtta.ResidualFOCuS(threshold=10.0, mu0=0.0, sigma=1.0, max_candidates=200)
result = ind.update(residual)
# result.signal ∈ {-1, 0, +1}, result.statistic

advance(...) updates state without returning a result. Constructor parameters match FOCuS.

Theory Of Operation

Model-based monitoring often reduces to “is the residual still zero-mean noise?” Feeding pre-whitened or model residuals into FOCuS detects mean shifts that pure price-level CUSUM would confuse with trend. Typical pipelines:

  1. Fit a predictive model online (regression, Kalman, pairs residual).
  2. Stream \(r_t = y_t - \hat{y}_t\) (or a z-scored innovation) into ResidualFOCuS.
  3. Fire when the residual mean changes beyond the GLR threshold.

Under a correctly specified model, \(\mu_0 = 0\) and \(\sigma\) should match residual scale.

Recurrence

Identical to FOCuS with observation \(x_t\) replaced by residual \(r_t\):

\[y_t = r_t - \mu_0,\]

candidates \((S,n)\) updated as \((S+y_t, n+1)\) plus a new \((y_t,1)\), pruned by mean dominance, and

\[\Lambda_t = \max \frac{S^2}{2\sigma^2 n},\qquad \mathrm{signal}_t = \operatorname{sign}(S^\star)\ \text{if}\ \Lambda_t \ge h\ \text{else}\ 0.\]

See focus.md for the full FOCuS recurrence and pruning details.

Implementation Notes

The recurrence is implemented in src/rtta/indicator.cpp in class ResidualFOCuS as a member FOCuS focus_; update/advance/batch_array/last forward to that engine.

Reference