Beta Gamma Decay

Overview

Any execution of media spend is assumed to have instantaneous impact on the sales. This media spend is assumed to have persistant impact in the following time periods as well, though decreasing at decayed rate compared to instant impact($i_m$) till it reaches to zero. These decays are computed using two parameters Beta($\beta$) and Gamma($\gamma$).

Beta

For each media vehicle, $\beta_m$ is used to model the immediate decay for the following period of media spend in order to learn a sharp initial decay.

The default range of beta is $(0,1)$ but it changes depending on characteristics of a media spend. Typical media spends have lingering impact after the campaign as mentioned earlier. However, a few media spends result in a huge spike in sales due to stock-up of the product by the customers at the expense of our future sales. To incorporate such behavior, $\beta$ is assumed to be negative for such spends to model a sharp decline in sales below the baseline. Default range of $\beta$ in such scenario changes to $(-1,0)$.

Gamma

For each media vehicle, $\gamma_m$ is used to model an exponential decay parameter for the successive time periods after the applying $\beta_m$.

Decay & Total Impact

Decayed impact at time period() for a media spend in vehicle with instant impact of at time is given by, $Decay_{t,m} = i_m*\beta_m*\gamma_m^{t-1} \quad t>0,\quad \left\lvert \beta_m \right\rvert,\gamma\ \epsilon\ (0,1)$ Assuming the decays continue for an infinite time periods, theoritical total impact is calculated using geometric series given by, $Total\ Impact_m = i_m * \left( 1 + \frac{\beta_m}{1-\gamma_m} \right)$

Note: Currently, decays below a pre-defined threshold(a very low value) are ignored for computational efficiency. Overall margin of error due to this round-off is insignificant.

Additional restriction when Beta is negative:

Net total impact by a spend should never be negative. This is ensured by setting $Total\ Impact_m >0$ in the earlier equation which results in $\beta_m > \gamma_m-1$. The new range of $\beta_m$ is dynamically adjusted.