An ad's effect does not vanish when the flight ends - people remember. Geometric adstock encodes that memory as a running total that decays:

Each week the channel's effective pressure is this week's impressions plus a fraction α of everything carried from before. A burst of activity leaves a tail that fades geometrically: after one week a fraction α remains, after two weeks α², and so on. The natural way to read α is as a half-life - the number of periods for a burst's effect to halve:

Higher α means slower decay and a longer memory. The KT heuristic follows directly: TV adstock is the highest, most traditional media decays slower than digital, and intent-driven channels (search, e-com) barely carry at all.

Doubling spend rarely doubles sales - audiences run out, frequency stops adding persuasion. The program applies a Hill transformation to the adstocked series to encode this:

Two parameters, two distinct jobs:

• γ (the half-saturation point) is the media level at which the response reaches exactly half its maximum - set x = γ and the formula returns 0.5. Moving γ slides the curve along the spend axis: a higher γ means the channel saturates later, so there is more headroom before returns flatten.
• β (the shape) controls steepness. β ≤ 1 gives a concave curve - diminishing returns from the very first impression. β > 1 gives an S-shape - a slow start, a steep middle, then a plateau, the signature of channels that need threshold weight before they work.

One symbol caution: the methodology workbook's transformation images write this same function with different letters (its "alpha" is the Hill steepness and "theta" the half-saturation). The production config's vocabulary - which this course uses throughout - reserves alpha for adstock decay and beta for the Hill shape, matching alpha_range / beta_range in the YAML. Order of operations, per the pipeline: adstock first, Hill on the adstocked series. Control and base variables get neither - they receive plain min-max scaling (except variables that are already normalised, like percentages and indexes, listed in exclude_Scaling).

Open a real production config and find platform_cat_parameter_ranges: one entry per platform category, each holding an alpha_range and a beta_range. The structural fact that surprises every new analyst: despite the name, these are not [low, high] ranges - they are discrete candidate grids. The search only ever tries the listed values; a value between two grid points is never fitted.

platform_cat_parameter_ranges:
  tv:
    alpha_range: [ ...many candidates, highest values, finest grid... ]
    beta_range:  [ ...common grid... ]
  paid_social:
    alpha_range: [ ...three coarse low values... ]
    beta_range:  [ ...same common grid... ]

The alpha grids cluster into bands that mirror the funnel:

The beta grid is identical for every platform in this config family - roughly 14 candidates spanning from below 1 to well above 1, so both concave and S-shaped saturation are searchable everywhere. All per-platform differentiation is carried by alpha; beta is left free on a common grid. Editing one platform's beta "to match TV" is a conceptual no-op and a red flag in review.

Why grids per platform category rather than one free parameter? Three reasons. First, business knowledge is real: TV genuinely carries longer than search, and letting search borrow a TV-like alpha invites the model to launder baseline into media. Second, the search is combinatorial - every alpha × beta combination per channel is transformed, fed to the model builder, and scored (the pipeline's stages: build on all possible combinations of transformed variables, then iterate parameter combinations until statistical and business KPIs are satisfied). Discrete grids keep that explosion affordable. Third, comparability: a header rule in the config states the alpha/beta chosen at L1 are retained for L2 and L3 - the grid search happens once per channel, not per level.