The definitions described in the previous sections can be extended for channels
with reach and frequency data. The potential outcomes can be written more
generally as \\( \\overset \\sim Y_{g,t}\^{ \\left( \\left\\{ x_{g,t,i}\^{(\\ast)}
\\right\\}, \\left\\{ r_{g,t,i}\^{(\\ast)} \\right\\}, \\left\\{ f_{g,t,i}\^{(\\ast)}
\\right\\} \\right) } \\)

The incremental outcome of the \\(q\^{th}\\) channel with reach and frequency data
is defined as:
$$ \\text{IncrementalOutcome}\^{\[RF\]}_q = E \\Biggl(\\sum\\limits_{g,t} \\biggl( \\overset \\sim Y_{g,t}\^{ \\left( \\left\\{ x_{g,t,i} \\right\\}, \\left\\{ r_{g,t,i} \\right\\}, \\left\\{ f_{g,t,i} \\right\\} \\right) } - \\overset \\sim Y_{g,t}\^{ \\left( \\left\\{ x_{g,t,i} \\right\\}, \\left\\{ r_{g,t,i}\^{(0,q)} \\right\\}, \\left\\{ f_{g,t,i} \\right\\} \\right) } \\biggr) \\bigg\| \\left\\{ z_{g,t,i} \\right\\} \\Biggr) $$

Where \\(r\^{(0)}_{g,t,i}\\) denotes the observed reach values for all channels
except channel \\(q\\), which is set to zero everywhere. More specifically:

- \\(r\^{(0,q)}_{g,t,q} = 0\\ \\forall\\ g,t\\)
- \\(r\^{(0,q)}_{g,t,i} = r_{g,t,i}\\ \\forall\\ g,t,i \\neq q\\)

Note that the frequency counterfactual values don't matter when the reach is
zero; the incremental outcome should be zero regardless. These are arbitrarily
set to historical values in this definition.

The ROI of the \\(q\^{th}\\) channel with reach and frequency data is defined as:

\\\[\\text{ROI}\^{\[RF\]}_q = \\dfrac{\\text{IncrementalOutcome}\^{\[RF\]}_q}{\\text{Cost}\^
{\[RF\]}_q}\\\]

Where \\(\\text{Cost}\^{\[RF\]}_q=\\sum\\limits_{g,t} \\overset \\sim r_{g,t,q}\\).

To define response curves, note that there are many ways to scale spend for
channels with reach and frequency data. For any given spend level, there are any
number of reach and frequency combinations that can result in that spend level.
Meridian focuses primarily on two types of response curve:

- A *reach* response curve where reach is scaled, holding frequency constant
  at the historical values for each geo and time period.

- A *frequency* response curve where frequency is scaled, holding reach
  constant at the historical values for each geo and time period.

The reach response curve is defined as the following function:
$$ \\text{IncrementalOutcome}_q\^{\[reach\]} \\left( \\omega \\cdot \\text{Cost}_q\^{\[RF\]} \\right) = E \\Biggl( \\sum\\limits_{g,t} \\biggl( \\overset \\sim Y_{g,t}\^{ (\\{x_{g,t,i}\\},\\{r_{g,t,i}\^{\\omega,q}\\},\\{f_{g,t,i}\\}) } - \\overset \\sim Y_{g,t}\^{ (\\{x_{g,t,i}\\},\\{r_{g,t,i}\^{(0,q)}\\},\\{f_{g,t,i}\\}) } \\biggr) \\bigg\| \\{z_{g,t,i}\\} \\Biggr) $$

Where \\(r_{g,t,i}\^{(\\omega,q)}\\) denotes the observed reach values for all
channels except channel \\(q\\), which is scaled by \\(\\omega\\) everywhere. More
specifically:

- \\(r_{g,t,q}\^{(\\omega,q)}=\\omega \\cdot r_{g,t,q}\\ \\forall\\ g,t\\)
- \\(r_{g,t,i}\^{(\\omega,q)}=r_{g,t,i}\\ \\forall\\ g,t,i \\neq q\\)

The frequency response curve is defined as the following function:
$$ \\text{IncrementalOutcome}_q\^{\[freq\]} \\left( \\omega \\cdot \\text{Cost}_q\^{\[RF\]} \\right) = E \\Biggl( \\sum\\limits_{g,t} \\biggl( \\overset \\sim Y_{g,t}\^{ (\\{x_{g,t,i}\\},\\{r_{g,t,i}\\},\\{f_{g,t,i}\^{(\\omega,q)}\\}) } - \\overset \\sim Y_{g,t}\^{ (\\{x_{g,t,i}\\},\\{r_{g,t,i}\^{(0,q)}\\},\\{f_{g,t,i}\\}) } \\biggr) \\bigg\| \\{z_{g,t,i}\\} \\Biggr) $$

Where \\(f_{g,t,i}\^{(\\omega,q)}\\) denotes the observed frequency values for all
channels except channel \\(q\\), which is scaled by \\(\\omega\\) everywhere. More
specifically:

- \\(f_{g,t,q}\^{(\\omega,q)}=\\omega \\cdot f_{g,t,q}\\ \\forall\\ g,t\\)
- \\(f_{g,t,i}\^{(\\omega,q)}=f_{g,t,i}\\ \\forall\\ g,t,i \\neq q\\)

Note that for \\(\\omega \< \\dfrac{1}{min_{g,t} f_{g,t}}\\), the counterfactual
frequency \\(\\omega \\cdot f_{g,t,q}\\) will be less than one for some combinations
of \\(g,t\\). Although it is not possible to have average frequency values below
one, Meridian's model specification allows incremental outcome to be
estimated for such implausible values. Be careful when interpreting response
curves for such small values of \\(\\omega\\).