Read the equation
For each candidate split k, a separate straight line is fitted to the observations on each side, and their squared residuals are added. The preferred split minimizes this combined residual cost. Because the two lines are independently fitted, they need not meet at k; the procedure selects a descriptive fit boundary rather than proving a physical discontinuity.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| SSE | Sum of squared residuals |
| k | Candidate split index |
| g_t | Gradient norm |
| alpha | First line intercept |
| beta | First line slope |
| gamma | Second line intercept |
| zeta | Second line slope |
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
The source fits the first 300 observations; changing the search interval can change the selected split.
Original context
Read the surrounding source passage
Centered smoothing uses future observations relative to its labeled center. It is a descriptive robustness check, not a real time declaration detector. A gradient norm is also not the size of the actual optimizer update, accumulated information, or electrical energy. To preserve the earlier pilot comparison, I fit two independent lines to the first 300 observations. Each segment must contain at least 20 observations. For 20≤k≤280, the objective is SSE(k)=minα,β,γ,ζ[t=1k(gt−α−βt)2+t=k+1300(gt−γ−ζt)2]. (4.9) The minimizing k is the last step of the first segment. The fitted lines need not meet at the boundary. This summarizes a rise and decline within the original pilot domain; it does not establish a discontinuity or fit the entire record of 792 observations. For steps 161 through 180 and 181 through 200, I calculate the percentage change in mean gradient norm. These windows were selected retrospectively around the region discussed in the source work. For evaluation loss, I retain every recorded change between adjacent observations, including any increase.
Exact original notation
SSE(k)={min}_{\alpha ,\beta ,\gamma ,ζ}[\sum _{t=1}^{k}{{({g}_{t}-\alpha -\beta t)}^{2}}+\sum _{t=k+1}^{300}{{({g}_{t}-\gamma -ζt)}^{2}}].Related builds
Sources & provenance
SSE(k)={min}_{\alpha ,\beta ,\gamma ,ζ}[\sum _{t=1}^{k}{{({g}_{t}-\alpha -\beta t)}^{2}}+\sum _{t=k+1}^{300}{{({g}_{t}-\gamma -ζt)}^{2}}].
SSE(k)={min}_{\alpha ,\beta ,\gamma ,ζ}[\sum _{t=1}^{k}{{({g}_{t}-\alpha -\beta t)}^{2}}+\sum _{t=k+1}^{300}{{({g}_{t}-\gamma -ζt)}^{2}}].
SSE(k)={min}_{\alpha ,\beta ,\gamma ,ζ}[\sum _{t=1}^{k}{{({g}_{t}-\alpha -\beta t)}^{2}}+\sum _{t=k+1}^{300}{{({g}_{t}-\gamma -ζt)}^{2}}].
SSE(k)={min}_{\alpha ,\beta ,\gamma ,ζ}[\sum _{t=1}^{k}{{({g}_{t}-\alpha -\beta t)}^{2}}+\sum _{t=k+1}^{300}{{({g}_{t}-\gamma -ζt)}^{2}}].