Earlier post-declaration onset search

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This earlier definition searches only after t_D for the first window of persistent discrepancy. It therefore cannot discover a qualifying change that began before declaration and, by construction, makes every found lag positive. The final manuscript replaces that restriction with an independent search across the eligible full record.

Terms & definitions

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Original context

tD=inf{t:Dt=1}.

For a selected discrepancy and tolerance ε, a finite-horizon candidate onset is

tHW=inf{t>tD:δt+k>ε for k=0,…,W−1}.

This is a retrospective event time: at t one does not yet know whether the next W−1 observations will satisfy the condition. An online detector must wait for the required window and report that detection delay.

The associated declaration count can be defined without calling it a validated cognition measure:

NDT=t=tHWT1Dt=1 1δt>ε.

{t}_{H}^{\left(W\right)}=inf{t>{t}_{D}:{\delta }_{t+k}>\epsilon  for k=0,…,W-1}.