eq-e9d994889ed7d8mathematical expression
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The average uses a symmetric window of w observations around time t. For odd w, there are (w−1)/2 samples on each side and one center sample. Because future samples enter the window, the smoothed series is a retrospective descriptive summary rather than an immediately available online detector.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| g_t | Recorded gradient norm |
| w | Odd window length |
| g_bar | Smoothed gradient norm |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
End-of-record handling must specify whether incomplete windows are omitted or treated differently.
Original context
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4.1 Primary descriptive summaries
Let gt be the reported gradient norm. The first summary is its raw maximum over the acquired interval. Centered moving averages are also computed:
gtw=1wj=−w−1/2w−1/2gt+j,
for odd widths w∈{5,11,21,31}. Only complete windows are used. These centered summaries use observations after their nominal center and are not real-time detectors. They assess whether the visible episode depends on one unusually large point.
Adjacent 20-step windows summarize the approach to and departure from the published transition vicinity. The principal comparison is steps 161–180 against 181–200. Its percent change is:
1001−meang181:200meang161:180.Exact original notation
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}},Related builds
Sources & provenance
01
Declaration_and_State_Change_Pilot.docx
paragraph 62, math 1 · Sep 17, 2026
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}},
02
Declaration_and_State_Change_Pilot.docx
paragraph 62, math 1 · Sep 17, 2026
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}},