Media Transforms
Media enters Epsilon through a two-part transformation path: carryover followed by diminishing returns. The transformed media is then multiplied by the media coefficient and added to the model mean.
For channel j, the usual time-series path is:
\[f_j(x_{1:T,j})_t = S_j\left(A_j(x_{1:T,j})_t\right),\]
where A_j is the adstock transform and S_j is the saturation transform. Panel models apply the same idea for each flattened panel cell.
Adstock
Adstock converts current and previous media into a carryover signal:
\[A_j(x)_t = \sum_{\ell=0}^{L-1} w_{j,\ell} x_{t-\ell,j},\]
where L is l_max and the boundary behaviour is controlled by the configured convolution mode. With the usual trailing carryover mode, past spend can affect the current observation, but future spend cannot.
Supported adstock families in the maintained model path are:
| Type | Lag weights |
|---|---|
none | $A_j(x)_t = x_{t,j}$ |
geometric | $w_{\ell} = \alpha^\ell$ |
delayed | $w_{\ell} = \alpha^{(\ell - \theta)^2}$ |
binomial | $w_{\ell} = (1 - \ell / (L + 1))^{1 / \alpha - 1}$ |
weibull_pdf | Weibull-PDF-shaped finite-lag kernel |
weibull_cdf | Weibull-survival-style finite-lag kernel with a leading self-retention term |
When normalize=true, Epsilon normalises the constructed lag weights by their sum. Normalisation changes the scale of the carryover signal and therefore the interpretation of the media coefficient.
Saturation
Saturation maps the adstocked signal to a nonlinear response scale. It is where diminishing marginal returns enter the model.
Supported saturation families in the maintained model path are:
| Type | Functional form |
|---|---|
none | $S(x) = x$ |
logistic | $S(x; \lambda) = \tanh(\lambda x / 2)$ |
tanh | $S(x; b, c) = b \tanh(x / (bc))$ |
michaelis_menten | $S(x; a, \lambda) = ax / (\lambda + x)$ |
hill | $S(x; s, \kappa) = 1 - \kappa^s / (\kappa^s + x^s)$ |
Media spend must be nonnegative at public model boundaries. The low-level tanh primitive can accept signed values, but the MMM media path treats negative media as invalid input.
Contribution
After transformation, each channel contributes:
\[\mathrm{contribution}_{t,j} = \beta_j S_j(A_j(x_j)_t).\]
The total media contribution is:
\[m_t = \sum_{j=1}^{J} \mathrm{contribution}_{t,j}.\]
Panel models use the same expression with panel-cell-specific indexing:
\[\mathrm{contribution}_{t,j,p} = \beta_{j,p} S_{j,p}(A_{j,p}(x_{j,p})_t).\]
Why A Response Curve Can Look Linear
A response curve can look almost linear even when the configured saturation function is nonlinear. The visible curvature depends on the part of the curve covered by the plotted spend range and posterior draws.
For the centered-logistic form:
\[S(x; \lambda) = \tanh(\lambda x / 2).\]
Near zero, the first-order approximation is:
\[S(x; \lambda) \approx \lambda x / 2.\]
So if the fitted posterior places observed spend in a low-saturation region, the plotted response curve will be close to a straight line. Stronger curvature appears when the observed or counterfactual spend range reaches the shoulder of the saturation curve. This is a modelling result, not automatically a plotting bug.
The practical checks are:
- inspect the posterior for the saturation parameter,
- compare observed spend with the grid range used by the response plot,
- check whether the channel was scaled to a range where $\lambda x$ remains small,
- compare marginal response curves, not just total response curves.
Reading Adstock Curves
Adstock curves show carryover, not diminishing returns. A geometric or normalised finite-lag adstock curve can be close to linear over a plotted total spend range because adstock is a weighted sum. Curvature in MMM usually comes from the saturation layer, not the carryover layer by itself.
When interpreting Epsilon's plotted artifacts, treat adstock, saturation, response, contribution, and optimisation plots as different projections of the same fitted model. They answer related but distinct questions.