For a given media channel \(q\), the incremental revenue is defined as:

\[IncrementalSales_q = IncrementalSales \left(\Bigl\{ x_{g,t,m} \Bigr\}, \Bigl\{ x_{g,t,m}^{(0,q)} \Bigr\} \right)\]

Where:

- \(\left\{ x_{g,t,m} \right\}\) are the observed media values
- \(\left\{ x_{g,t,m}^{(0,q)} \right\}\) denotes the observed media values
for all channels except channel \(q\), which is set to zero everywhere.
More specifically:
- \(x_{g,t,q}^{(0,q)}=0\ \forall\ g,t\)
- \(x_{g,t,m}^{(0,q)}=x_{g,t,m}\ \forall\ g,t,m \neq q\)

The ROI of channel \(q\) is defined as:

\[ROI_q = \dfrac{IncrementalSales_q}{Cost_q}\]

Where \(Cost_q= \sum\limits _{g,t} \overset \sim x_{g,t,q}\)

Note that the ROI denominator represents media cost over a specified time period that aligns with the time period over which the incremental revenue is defined. As a result, the incremental revenue in the numerator includes the lagged effect of media executed prior to this time window, and similarly excludes the future effect of media executed during this time window. So, the incremental revenue in the numerator does not perfectly align with the cost in the denominator. However, this misalignment will be less material over a reasonably long time window.

Generalizing the incremental revenue definition, the response curve is defined for channel \(q\) as a function which returns the incremental revenue as a function of the spend on channel \(q\):

\[IncrementalSales_q (\omega \cdot Cost_q) = IncrementalSales \left(\left\{ x^{(\omega,q)}_{g,t,m} \right\}, \left\{ x^{(0,q)}_{g,t,m} \right\}\right)\]

Where \(\left\{ x^{(\omega,q)}_{g,t,m} \right\}\) denotes the observed media values for all channels except channel \(q\), which is multiplied by a factor of \(\omega\) everywhere. More specifically:

- \(x^{(\omega,q)}_{g,t,q}=\omega \cdot x_{g,t,q}\ \forall\ g,t\)
- \(x^{(\omega,q)}_{g,t,m}=x_{g,t,m} \forall\ g,t,m \neq q\)

The marginal ROI of channel \(q\) is defined as:

Where \(\delta\) is a small quantity, such as \(0.01\).

Note that the response curve and marginal ROI definitions implicitly assumes a constant cost per media unit that equals the historical average cost per media unit.