Model specification

The standard Meridian model is a geo-level hierarchical model with non-linear parametric transformations on the media variables. Random coefficients account for heterogeneity across geographical regions.

We recommend using geo-level data as it provides more reliable estimation. A national model, essentially a single-geo model, is available for cases where geo-level data is unavailable. See National-level modeling for more details.

The Meridian model is an extension of Bayesian models previously proposed ( Jin et al. 2017 and Sun et al. 2017 ). Extensions include the use of reach and frequency data ( Zhang et al. 2023 ), the inclusion of time-varying intercept terms to model trend and seasonality (related to trend and seasonality modeling by Ng et al. 2021 ) and the addition of non-media treatments and organic media variables.

The model is represented as follows:

Basic details

The basic details are:

• Index variables, as defined in Input data :

  • \(g=1,\ldots,G\) indexes the geographical units
  • \(t=1,\ldots,T\) indexes the time units
  • \(i=1,\ldots,N_C\) indexes the control variables
  • \(i=1,\ldots,N_N\) indexes the non-media treatments
  • \(i=1,\ldots,N_M\) indexes the paid media channels without reach and frequency data
  • \(i=1,\ldots, N_{OM}\) indexes the organic media channels without reach and frequency data
  • \(i=1,\ldots,N_{RF}\) indexes the paid media channels with reach and frequency data
  • \(i=1,\ldots, N_{ORF}\) indexes the organic media channels with reach and frequency data

• \(\tau_b = 0\) for identifiability for some baseline geo \(b\). Any geo can be set as the baseline using the baseline_group argument.

• \(\{q_{t-s}\}^L_{s=0}\) denotes the vector \((q_t, q_{t-1}, \ldots, q_{t-L})\). This notation is used to denote Adstock function input values.

• The integer value \(L\) is the media maximum lag duration, such as the maximum media effect duration. This parameter can be set using the max_lag argument.

• Note the following about the \(\text{Hill}()\) and \(\text{Adstock}()\) functions. For more information, see Media saturation and lagging and Set the adstock decay spec parameter .

  $$ \text{Adstock} \left( \left\{ q_{t-s} \right\}^L_{s=0},\ \alpha \right) = \dfrac{\sum\limits^{L}_{s=0}\ w(s; \alpha) q_{t-s} }{\sum\limits^L_{s=0}\ w(s; \alpha)} $$

  Where:

  • \(q>0\)
  • \(w(s; \alpha) \) is the decay parameterization
  • \(\alpha\ \in\ [0, 1]\) is the decay parameter.
  $$ \text{Hill} \left( q, ec, \text{slope} \right) = \left( 1 + \left( \dfrac{q}{ec} \right)^{- \text{slope} } \right)^{-1} $$

  Where:

  • \(q>0,\ ec>0,\ \text{slope} > 0\)
  • \(ec, \text{slope}\) are Hill function shape and slope parameters.

• The \(\text{HillAdstock}()\) function depends on the hill_before_adstock argument.

  • If the default hill_before_adstock = False , then \(\text{HillAdstock}(q;\ \alpha, ec, \text{slope}) = \text{Hill}(\text{Adstock}(q;\ \alpha);\ ec, \text{slope})\)
  • If hill_before_adstock = True , then \(\text{HillAdstock}(q;\ \alpha, ec, \text{slope}) = \text{Adstock}(\text{Hill}(q;\ ec, \text{slope}); \alpha)\)

\( \mu_t \) parameters

The \(\mu_t\) parameters are time-varying intercepts that contribute a trend and seasonality component to the model:

• \(\mu_t\) are determined by a series of knot parameters \(b_1,b_2,\dots,b_K\) located at corresponding time points \(s_1,s_2,\dots,s_K\).

• The time points $s_1,\dots,s_K$ are located between \(1\) and \(T\) and are specified by the knots argument.

  • You can specify a list of knot locations, or just the number of knots.

  • If the number of knots is specified, then knots are evenly spaced with two of the knots at the endpoints ($s_1=1$ and $s_K=T$).

  • When there are multiple geos (\(G>1\)), the default places a knot at each time point ($s_1=1,s_2=2,\dots,s_K=T$, where $K=T$).

  • When \(G=1\) (such as a national-level model), the default is a single knot, which is effectively a common intercept across all time periods.

• The \(\mu_t\) values are a weighted average of the two closest neighboring knots, with more weight given to the closer knot. (For example, say there are knots at time 9 and time 18. The estimate for $\mu_{16}$ will be influenced by both the knot at time 9 and the knot at time 18, with more weight given to the knot at time 18.) The precise weight calculation is as follows. For any \(t\), define the following:

  • $\ell(t)$ and $u(t)$ represent the knot indices of the nearest neighboring knots. Usually $\ell(t) < u(t)$, but note that $\ell(t) = u(t)$ if $t$ is precisely one of the knot locations, or if $t$ is before the first knot or after the last knot (as can be the case when custom knot locations are set, or if only one knot is used).

    • \(\ell(t) = \max \{\{1\} \cup \{k: s_k \leq t\}\}\)

      • Note: $\ell(t) = 1$ if $t < s_1$.

    • \(u(t) = \min \{\{K\} \cup \{k: s_k > t\}\}\)

      • Note: $u(t) = K$ if $t > s_K$.

  • \(w(t) = 1\) if \(\ell(t) = u(t)\) ; otherwise \(w(t) = \dfrac{s_{u(t)}-t}{s_{u(t)}-s_{\ell(t)}}\).

  • \(\mu_t = w(t)b_{\ell(t)} + (1-w(t))b_{u(t)}\)

• The knot values \(b_1,b_2,\dots,b_K\) have a user-specified prior distribution.

This methodology is based on Bayesian Time Varying Coefficient Model with Applications to Marketing Mix Modeling , with key distinctions, including a different weight function, distinct priors, the absence of autoregression, and other differences.

For more information about setting knots, see How the knots argument works .

Other parameter distributions

Other parameter distributions are as follows:

• Note that here Meridian parameterizes the Normal distribution in terms of its mean and standard deviation.

  • \(\gamma_{g,i}^{[C]} \sim \text{Normal}(\gamma_i^{[C]},\xi_i^{[C]})\)
  • \(\gamma_{g,i}^{[N]} \sim \text{Normal}(\gamma_i^{[N]},\xi_i^{[N]})\)

• The \(\beta\) distributions depends on the media_effects_dist argument, as follows:

  • \(log(\beta_{g,i}^{[M]})\sim \text{Normal}( \beta_i^{[M]},\eta_i^{[M]})\)
  • \(log(\beta_{g,i}^{[OM]})\sim \text{Normal}( \beta_{i}^{[OM]},\eta_{i}^{[OM]})\)
  • \(log(\beta_{g,i}^{[RF]})\sim \text{Normal}( \beta_{i}^{[RF]},\eta_{i}^{[RF]})\)

  • \(log(\beta_{g,i}^{[ORF]})\sim \text{Normal}( \beta_{i}^{[ORF]},\eta_{i}^{[ORF]})\)

    if media_effects_dist = LOG_NORMAL

  • \(\beta_{g,i}^{[M]} \sim \text{Normal}( \beta_i^{[M]},\eta_i^{[M]})\)

  • \(\beta_{g,i}^{[OM]} \sim \text{Normal}( \beta_{i}^{[OM]},\eta_{i}^{[OM]})\)

  • \(\beta_{g,i}^{[RF]} \sim \text{Normal}( \beta_{i}^{[RF]},\eta_{i}^{[RF]})\)

  • \(\beta_{g,i}^{[ORF]} \sim \text{Normal}( \beta_{i}^{[ORF]},\eta_{i}^{[ORF]})\)

    if media_effects_dist = NORMAL

• \(\epsilon_{g,t}\sim \text{Normal}(0,\sigma_g)\):

  • The residuals are independent of each other, all media and control variables, and all model parameters.

  • If unique_sigma_for_each_geo = False (which is the default), then \(\sigma_1=\sigma_2=\cdots=\sigma_G=\sigma\).

• The remaining parameters all have user-specified prior distributions:

  \( \{\gamma_i^{[C]}\}, \{\gamma_i^{[N]}\}, \{\xi_i^{[C]}\}, \{\xi_i^{[N]}\}, \)

  \( \{\beta_i^{[M]}\}, \{ \beta_{i}^{[OM]} \}, \{ \beta_{i}^{[RF]} \}, \{ \beta_{i}^{[ORF]} \}, \)

  \( \{\eta_i^{[M]}\}, \{ \eta_{i}^{[OM]} \}, \{\eta_{i}^{[RF]}\}, \{ \eta_{i}^{[ORF]} \}, \)

  \( \{\alpha_i^{[M]}\}, \{ \alpha_{i}^{[OM]} \}, \{\alpha_{i}^{[RF]}\}, \{ \alpha_{i}^{[ORF]} \}, \)

  \( \{ec_i^{[M]}\}, \{ ec_{i}^{[OM]} \}, \{ec_{i}^{[RF]}\}, \{ ec_{i}^{[ORF]} \}, \)

  \( \{slope_i^{[M]}\}, \{ slope_{i}^{[OM]} \}, \{slope_{i}^{[RF]}\}, \{ slope_{i}^{[ORF]} \}, \)

  \( \{\tau_g\}, \{\sigma_g\}. \)

You can change some aspects of the model specification, based on modeling options covered in the following sections: Media saturation and lagging , Reach and frequency , Paid search , ROI priors for calibration , and Set the adstock decay spec . You can also customize the default prior distributions .