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Relationship Between Standard and General Changepoint Parameterizations
Rewrite standard,
\( E \left[ y \| x \right] = \beta_0 +\ \beta_1 x +\ \delta_1 \left( x - \tau_1 \right)^+ + \ldots +\ \delta_k \left( x - \tau_k \right)^+\)
\(= \beta_0 +\ \beta_1 x + \sum_{j=1}^k \delta_j \left( x - \tau_j \right) I \left( x > \tau_j \right)\)
\(= \beta_0 +\ \beta_1 x + \sum_{j=1}^k \delta_j x I \left( x > \tau_j \right) - \sum_{j=1}^k \delta_j \tau_j I \left( x > \tau_j \right)\)
Rewrite GCP,
\(E \left[ y \| x \right] = \sum_{j=1}^{k+1} \left( \beta_{0j} + \beta_{1j} x \right) \left\{ I \left( x > \tau_{j-1} \right) - I \left( x > \tau_j \right) \right\}\)
\( = \sum_{j=1}^{k+1} \left( \beta_{0j} + \beta_{1j} x \right) I \left( x > \tau_{j-1} \right) - \sum_{j=1}^{k+1} \left( \beta_{0j} + \beta_{1j} x \right) I \left( x > \tau_j \right)\)
\( = \sum_{j=0}^k \left( \beta_{0,j+1} + \beta_{1,j+1} x \right) I \left( x > \tau_j \right) - \sum_{j=1}^k \left( \beta_{0j} + \beta_{1j} x \right) I \left( x > \tau_j \right)\)
\( = \beta_{01} + \beta_{11} x + \sum_{j=1}^k I \left( x > \tau_j \right) \left\{ \left( \beta_{0,j+1} - \beta_{0j} \right) + \left( \beta_{1,j+1} - \beta_{1j} \right) x \right\}\)
Standard General Changepoint Parameterizations
β = β01
β = β11
δj = β1,j+1 - β1j
\(T_j = - \frac{\beta_{0,j+1} - \beta_{0,j}}{\beta_{1,j+1} - \beta_{1,j}}\)