Function | Values for h and r | Transfer function | Response i at times 0 through k post-intervention | Form of response | Interpretation |
---|---|---|---|---|---|
Step function \( {S}_t=\left\{\begin{array}{c}0, if\ t<T\\ {}1, if\ t\ge T\end{array}\right. \) | h = 0, r = 0 | ω0 | i0 = ω0 i1 = ω0 i2 = ω0 … ik = ω0 | The time series increases by ω0 immediately following the intervention, and remains at this new level for the duration of the study period. | |
h = 0, r = 1 | \( \frac{\omega_0}{\left(1-{\delta}_1B\right)} \) (|δ1| < 1) | i0 = ω0 i1 = ω0(1 + δ1) \( {i}_2={\omega}_0\left(1+{\delta}_1+{\delta}_1^2\right) \) … \( {i}_k={\omega}_0\left(1+{\delta}_1+{\delta}_1^2+\dots +{\delta}_1^k\right) \) | The time series increases by ω0 immediately following the intervention, and increases by \( {\omega}_0{\delta}_1^k \) each subsequent time point until it reaches a new level, calculated by \( \frac{\omega_0}{\left(1-{\delta}_1\right)} \). | ||
Pulse function \( {P}_t=\left\{\begin{array}{c}0, if\ t\ne T\\ {}1, if\ t=T\end{array}\right. \) | h = 0, r = 0 | ω0 | i0 = ω0 i1 = 0 i2 = 0 … ik = 0 | The time series increases by ω0 immediately following the intervention and returns to baseline immediately afterwards. | |
h = 0, r = 1 | \( \frac{\omega_0}{\left(1-{\delta}_1B\right)} \) (|δ1| < 1) | i0 = ω0 i1 = ω0δ1 \( {i}_2={\omega}_0{\delta}_1^2 \) … \( {i}_k={\omega}_0{\delta}_1^k \) | The time series increases by ω0 the time of the intervention, and decays by (1 − δ1) each subsequent time point. | ||
Ramp function \( {R}_t=\left\{\begin{array}{c}0, if\ t<T\\ {}t-T+1, if\ t\ge T\end{array}\right. \) | h = 0, r = 0 | ω0 | i0 = ω0 i1 = 2ω0 i2 = 3ω0 … ik = (k + 1)ω0 | The time series increases by ω0 at each time point. |