Media saturation and lagging

The effects of media execution on KPI are governed by two mechanisms: a lagged effect and a saturation effect. Lagged effects refer to the way in which the effect of a media channel on KPI has a lagged effect that tapers off slowly over time. Saturation effects refer to diminishing marginal returns with increased media execution.

Adstock function

Meridian's model architecture is designed to capture lagged effects through an Adstock function.

In the Adstock function, the cumulative media effect at time \(t\) is a weighted average of media execution at times \(t, t-1, ..., t-L\) with weights determined by a weight function \(w(s; \alpha)\). Here, \(L\) is the maximum duration of the lagged effect.

Meridian offers the Adstock function with two weight functions \(w(s; \alpha)\): geometric and binomial. For more details on the functions, see Set the adstock_decay_spec parameter. For more information on the Adstock function, see A Hierarchical Bayesian Approach to Improve Media Mix Models Using Category Data and Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects.

The Adstock function is defined as follows:

$$ \text{Adstock}(x_t, x_{t-1}, \cdots, x_{t-L};\ \alpha)\ = \dfrac{\sum\limits_{s=0}^L\ w(s; \alpha)x_{t-s}} {\sum\limits _{s=0}^L\ w(s; \alpha)} $$

where:

• \(w(s; \alpha) \) is the decay function

• \(x_s \geq 0\) is media execution at time \(s\)

• \(\alpha\ \in\ [0, 1]\) is the decay parameter

• \(L\) is the maximum lag duration.

Hill function

Meridian's model architecture is designed to capture saturation effects through a Hill function.

It is intuitive that as spending on a given media channel within any given time period increases, you eventually see diminishing marginal returns, for example, saturation. Meridian models this saturation effect through a two-parameter function known as the Hill function.

The Hill function is defined as follows:

$$ \text{Hill}(x; ec, \text{slope}) = \frac{1}{1+\left( \frac{x}{ec} \right)^ {- \text{slope}}} $$

where:

• \(x \geq 0\)

• \(ec > 0\) is the half saturation point, meaning that \(\text{Hill}(x=ec; ec, \text{slope}) = 0.5\)

• \(\text{slope} > 0\) is a parameter that controls the function shape:

  • \(\text{slope} \leq 1\) corresponds to a concave shape
  • \(\text{slope} > 1\) corresponds to an S shaped function that is convex for \( x < ec \) and concave for \( x > ec \).

Important: The model's estimation of the Hill function parameters is based on the observed range of media data. The fitted response curve can be extrapolated outside this range, but results based on extrapolation should be interpreted with an appropriate level of caution.

The Hill function can be applied either before or after the Adstock transformation, depending on the boolean hill_before_adstock argument of the ModelSpec. The default setting is hill_before_adstock = False, which makes the media effect of channel \(m\) within geo \(g\) and time period \(t\) equal to \(\beta_{g,m} \text{Hill}(\text{Adstock}(x_t,x_{t-1},\cdots,x_{t-L};\ \alpha_m) ;ec_m, \text{slope}_m)\).