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 | ω_{0} |
i_{0} = ω_{0} i_{1} = ω_{0} i_{2} = ω_{0} … i_{k} = ω_{0} | The time series increases by ω_{0} immediately following the intervention, and remains at this new level for the duration of the study period. | |
h = 0, r = 1 |
\( \frac{\omega_0}{\left(1-{\delta}_1B\right)} \) (|δ_{1}| < 1) |
i_{0} = ω_{0} i_{1} = ω_{0}(1 + δ_{1}) \( {i}_2={\omega}_0\left(1+{\delta}_1+{\delta}_1^2\right) \) … \( {i}_k={\omega}_0\left(1+{\delta}_1+{\delta}_1^2+\dots +{\delta}_1^k\right) \) | The time series increases by ω_{0} 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 | ω_{0} |
i_{0} = ω_{0} i_{1} = 0 i_{2} = 0 … i_{k} = 0 | The time series increases by ω_{0} immediately following the intervention and returns to baseline immediately afterwards. | |
h = 0, r = 1 |
\( \frac{\omega_0}{\left(1-{\delta}_1B\right)} \) (|δ_{1}| < 1) |
i_{0} = ω_{0} i_{1} = ω_{0}δ_{1} \( {i}_2={\omega}_0{\delta}_1^2 \) … \( {i}_k={\omega}_0{\delta}_1^k \) | The time series increases by ω_{0} the time of the intervention, and decays by (1 − δ_{1}) 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 | ω_{0} |
i_{0} = ω_{0} i_{1} = 2ω_{0} i_{2} = 3ω_{0} … i_{k} = (k + 1)ω_{0} | The time series increases by ω_{0} at each time point. |