Add Optional Delayed Adstock Transformation for More Flexible Carryover Effects

[yujiosaka]

resolves #385

This PR implements an optional delayed adstock function to address the feature request for more flexible carryover effects, especially when dealing with channels that exhibit longer conversion periods. In my experience with travel, real estate, and other high-involvement purchases, the impact of marketing spend can peak days or even weeks after the initial touchpoint.

Previous workarounds, such as adjusting the max_lag parameter or shifting the input data manually, have proven too inflexible and ad hoc.

Key Changes:

1. ModelSpec Modification:

   • Added a new field adstock (defaulting to "geometric") in the ModelSpec class. When set to "delayed", the model will switch to using the delayed adstock transformation.

   • Users can now activate delayed adstock by configuring the model as follows:

     model_spec = meridian.model.spec.ModelSpec(adstock="delayed")

2. Delayed Adstock Implementation:

   • Introduced a new transformer class, DelayedAdstockTransformer, which computes adstock weights according to the following formula:

     $$w_d(l; alpha, theta) = alpha^((l - theta)^2)$$

     where theta is the delay parameter. This formulation is proposed by Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects.

3. Prior Distributions:

   • New Uniform priors have been added for the delay parameters (theta_m, theta_rf, etc.) in the PriorDistribution class.
   • By default, theta is assigned a Uniform prior over the range [0, max_lag - 1]. This choice is noninformative and allows the model to learn the appropriate delay from data. In our reading (e.g., Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects, section 2.1), a uniform prior for delay is suitable when no strong prior assumptions are available.

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