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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
4.3 A deliberately limited change-point fit An exploratory two-line fit minimizes: SSEk=mina1,b1t≤kgt−a1−b1t2+mina2,b2t>kgt−a2−b2t2. The two lines are independently fitted; continuity at k is not imposed. The selected k is the final step assigned to the first segment. This parameter summarizes the rise-and-decline shape. It is not automatically a discontinuity in the underlying dynamics, and it is not a declaration detector. For the full series, each segment must contain at least 20 observations. Sensitivity checks change that minimum to 15, 30, 50, and 60, and repeat the analysis on shorter windows. Polynomial fits of degrees 1 through 5 provide a descriptive smooth-curve comparison. Their in-sample residuals are reported without treating lower residual error as proof of a phase transition; the selected split adds flexibility beyond its four line coefficients. 4.4 Statistical interpretation
Exact original notation
SSE\left(k\right)={min}_{{a}_{1},{b}_{1}}\sum _{t\le k}^{}{{\left({g}_{t}-{a}_{1}-{b}_{1}t\right)}^{2}}+{min}_{{a}_{2},{b}_{2}}\sum _{t>k}^{}{{\left({g}_{t}-{a}_{2}-{b}_{2}t\right)}^{2}}.Related builds
Sources & provenance
SSE\left(k\right)={min}_{{a}_{1},{b}_{1}}\sum _{t\le k}^{}{{\left({g}_{t}-{a}_{1}-{b}_{1}t\right)}^{2}}+{min}_{{a}_{2},{b}_{2}}\sum _{t>k}^{}{{\left({g}_{t}-{a}_{2}-{b}_{2}t\right)}^{2}}.
SSE\left(k\right)={min}_{{a}_{1},{b}_{1}}\sum _{t\le k}^{}{{\left({g}_{t}-{a}_{1}-{b}_{1}t\right)}^{2}}+{min}_{{a}_{2},{b}_{2}}\sum _{t>k}^{}{{\left({g}_{t}-{a}_{2}-{b}_{2}t\right)}^{2}}.