Math 01 · The Mathematics of Adstock

When you spend on advertising in week \(t\), the effect does not vanish in week \(t\). A TV flight seen this week still nudges a purchase three weeks from now; a brand campaign builds for a while before it peaks. Adstock (carryover) is the function that turns a raw spend series \(x_t\) into an effective exposure series that carries the past forward. This page derives every formula the framework uses, charts every kernel live in your browser, and ends with the single most-misread fact in the codebase: the legacy model's adstock_<channel> parameter is a blend weight between two fixed kernels — not a decay rate (the default since June 2026 estimates the kernel itself in-graph). It mirrors nbs/math/math_01_adstock.ipynb, where every identity below is encoded as an executable assert against the real framework functions.

The framework offers three related representations of adstock, plus a fourth thing that is easy to misread — the legacy blend mechanism that was the default before June 2026 (the current default is the parametric path, (c)):

(a) Carryover as a convolution: the geometric recurrence

The classic form is a one-line recurrence — an infinite impulse response (IIR) filter, exactly what geometric_adstock(x, alpha) computes:

The decay rate \(\alpha\) is the share of last week's effective exposure that survives into this week. \(\alpha = 0\) means no carryover (the transform is the identity); \(\alpha \to 1\) means the effect barely fades.

Unrolling it: where the lag weights come from

Substitute the recurrence into itself, starting from \(a_0 = x_0\):

So the recurrence is secretly a causal convolution with the closed-form kernel \(w_k = \alpha^k\). A unit impulse at \(t=0\) — one unit of spend, then nothing — therefore produces the sequence \(1,\ \alpha,\ \alpha^2,\ \alpha^3,\dots\): the impulse response is the kernel, literally readable off the output.

Computed live in your browser — the same math as the notebook. Each line is geometric_adstock applied to a unit impulse at \(t=0\). The response at lag \(k\) is exactly \(\alpha^k\) — the notebook asserts this to \(10^{-9}\) against the real framework function.

The total carryover multiplier \(1/(1-\alpha)\)

Summing the kernel over all lags is a geometric series:

This is the framework docstring's claim that "the total effect of a unit spend is scaled by \(1/(1-\alpha)\)." At \(\alpha = 0.5\) a single dollar of spend contributes the equivalent of two dollars of effective exposure once you add up all its future echoes; the notebook's measured sums are \(\{\alpha{=}0.2 \to 1.25,\ \alpha{=}0.5 \to 2.0,\ \alpha{=}0.8 \to 5.0\}\). Crucially this kernel is un-normalized: its weights sum to \(1/(1-\alpha) > 1\), not to \(1\). Hold that thought for part (b).

Computed live — same math as the notebook. The multiplier explodes as \(\alpha \to 1\), which is why \(\alpha\) is constrained to \([0,1)\) and why an estimated \(\alpha\) near the boundary is a red flag (infinite memory). The marked points are the notebook's asserted values.

(b) Normalized FIR kernels: separating timing from size

The recurrence in (a) has infinite memory and can only ever peak at lag 0 — the current period is always the largest weight. The framework's adstock_weights(kind, l_max, ...) instead builds an explicit finite weight vector \(w = (w_0, w_1, \dots, w_{L-1})\) of length l_max, and apply_adstock(x, w) performs the causal convolution

with zero left-padding (weeks before the series start count as \(0\)). The wrapper parametric_adstock(x, kind, l_max=8, ...) does both steps at once.

normalize=True vs normalize=False — a magnitude convention, not a shape change

This is the conceptual hinge between (a) and (b). The shape of a geometric kernel is the same either way; what differs is where the total magnitude lives:

• normalize=False keeps the raw weights \(w_k = \alpha^k\). They sum to \(1/(1-\alpha) > 1\) — exactly the IIR multiplier from part (a). The carryover magnitude is inside the kernel.
• normalize=True rescales to \(\tilde w_k = w_k \big/ \sum_j w_j\), so the weights sum to exactly \(1\). The kernel becomes a weighted moving average that only redistributes spend across time; the total magnitude is folded into the channel coefficient \(\beta\) downstream — which keeps \(\beta\) interpretable as the channel's total effect size.

Computed live — same math as the notebook. Raw (\(\sum = 1/(1-\alpha)\)) vs normalized (\(\sum = 1\)) geometric kernels over a 20-lag window. Drag \(\alpha\): the bars rescale but the silhouettes stay proportional. At \(\alpha = 0.7\) the notebook measures raw sum 3.331 vs the infinite-series value 3.333 — the tiny gap is the truncation tail, the subject of the l_max section below.

Normalizing is the standard MMM choice because it cleanly separates timing (the kernel) from size (the coefficient) — but, as the identifiability section shows, that separation is exactly what creates the decay-vs-saturation tug-of-war.

(c) The kernel families: geometric, delayed, Weibull

Three kernel shapes are available, all returned by adstock_weights and all mirrored symbolically in transforms.adstock_pt (adstock_weights_pt, apply_adstock_pt, parametric_adstock_pt) so NUTS can differentiate through them and learn the parameters when use_parametric_adstock=True:

Geometric — \(w_k = \alpha^k\). Same shape as (a) but truncated to \(L\) lags. Peak always at lag \(0\); it cannot express a delayed build.

Delayed geometric (Jin et al., 2017) — \(w_k = \alpha^{(k-\theta)^2}\). The squared exponent makes the weight maximal where \((k-\theta)^2\) is smallest, so the peak moves to lag \(\theta\). At \(\theta = 0\) the peak returns to lag 0, but the weights are then \(\alpha^{k^2}\) — decaying faster than geometric, so it is not literally the geometric kernel, only the same peak location. This is the kernel that can model a genuinely delayed peak.

Weibull (PDF form, lags shifted by 1 so lag 0 is well-defined for every shape):

with shape \(s\) and scale \(\lambda\). The shape governs the silhouette: \(s < 1\) front-loads (most mass at lag 0, then a long thin tail — heavier than any geometric), \(s = 1\) is exponential, and \(s > 1\) pushes the peak out to a later lag as a smooth hump.

Computed live in your browser — the same math as the notebook's adstock_weights. Normalized weights inside the chosen window (faded bars: lags the window cuts off, shown out to lag 26). Watch the delayed kernel's mass walk right as \(\theta\) grows (peak lag \(= \theta\), asserted in the notebook) and the Weibull flip from front-load (\(s<1\)) through exponential (\(s=1\)) to a delayed hump (\(s>2\)). The "mass beyond window" chip turns red when l_max amputates more than 10% of the kernel's untruncated mass.

(d) The legacy blend path — read this carefully

Here is the single most common point of confusion. Since June 2026 the model's default is the parametric path of part (c) (use_parametric_adstock=True): it estimates the actual kernel parameters in-graph. But the framework still ships — and many older fits used — the legacy blend path (use_parametric_adstock=False, the default before the change), and on that path the model does not estimate a decay rate at all. Instead it:

1. Precomputes geometric adstock at two fixed decay rates — a low \(\alpha_\text{low}\) and a high \(\alpha_\text{high}\), taken from adstock_alphas (defaults \([0.0,\ 0.3,\ 0.5,\ 0.7,\ 0.9]\), so \(\alpha_\text{low}=0\) is no carryover and \(\alpha_\text{high}=0.9\) is strong carryover) — giving two exposure series \(x^\text{low}\) and \(x^\text{high}\) per channel.
2. Learns a blend weight per channel, \(m \sim \text{Beta}(2,2)\) — the model variable named adstock_<CHANNEL> — and mixes the two precomputed series:

Because convolution is linear, blending the series is the same as convolving with the blended kernel:

a convex combination of a spike at lag 0 and one fixed normalized geometric tail. So adstock_<CHANNEL> is a mixing weight between two fixed-\(\alpha\) geometric adstocks — it is not a decay rate. \(m \to 0\) pins the channel to the short-memory series, \(m \to 1\) to the long-memory series. The notebook asserts the endpoints exactly: \(m=0\) recovers \(x^\text{low}\), \(m=1\) recovers \(x^\text{high}\), and any \(m \in (0,1)\) stays pointwise between the two. The \(\text{Beta}(2,2)\) prior is symmetric and gently concentrated toward \(m = 0.5\). This design is cheap (no per-draw recurrence) and well-behaved for the sampler.

Computed live — the literal model formula \((1-m)\,x^{low} + m\,x^{high}\) on a seeded bursty spend plan (flights of activity separated by dark weeks, as in the notebook). At \(m=0\) the exposure is the raw flights; at \(m=1\) each flight keeps "glowing" for weeks after it ends. Every intermediate curve is a convex blend of these two endpoints — nothing else is reachable.

The reachable kernel set — and what lives outside it

Computed live. The shaded region is every kernel the legacy blend can express as \(m\) sweeps \([0,1]\) (spike-at-0 plus a scaled \(\alpha=0.9\) tail). The dashed lines are a delayed kernel (\(\theta=3\)) and a Weibull hump (\(s=2.5\)): their peaks sit far above the band at lags 2–5. No value of \(m\) can bend the blend into a delayed peak — the family is structurally monotone from lag 0.

The parametric path is the alternative: set use_parametric_adstock=True and the model estimates the actual kernel parameters in-graph — \(\alpha\) (adstock_alpha_<CHANNEL> \(\sim \text{Beta}(2,2)\)) for geometric/delayed, plus \(\theta\) (adstock_theta_<CHANNEL> \(\sim\) HalfNormal) for delayed, or \((s, \lambda)\) (adstock_shape/scale_<CHANNEL> \(\sim\) Gamma) for Weibull. Both are legitimate; they answer different questions. The blend asks "how much memory, on a 2-point scale?"; the parametric path asks "what is the precise decay shape?" The framework even reuses the learned-kernel closure at predict time, so perturbed spend is re-adstocked with the same kernel for marginal-ROAS analysis. Workshop 03 fits a first model with these settings end-to-end. This recommendation is stress-validated, not aesthetic: Stress 01 measures exactly what happens when the world's carryover is truly Weibull but the model assumes geometric.

IIR vs FIR truncation: what l_max cuts off

The recurrence (a) carries mass to infinity; an FIR kernel stops at lag \(L-1\). For the geometric family the amputated tail has a closed form: the un-normalized mass beyond the window is

That fraction is what the notebook's measured 3.331 vs 3.333 (\(\alpha=0.7\), \(L=20\)) is showing: \(0.7^{20} \approx 0.08\%\) of the mass — negligible. But at the framework default l_max=8 with \(\alpha = 0.9\), \(0.9^{8} \approx 43\%\) of the kernel's mass is cut off. Normalization then silently redistributes the surviving 57% to sum to 1, so the shape inside the window is distorted too, not just shortened. The explorer above reports this number live for all three families.

Computed live — exact formula \(\alpha^{L}\). Fraction of geometric kernel mass beyond an l_max window, for four decay rates. The vertical line marks the framework default l_max=8: harmless for fast decay, severe for \(\alpha \ge 0.9\). Rule of thumb: choose \(L\) so \(\alpha^{L}\) is small — at \(\alpha = 0.9\) you need \(L \approx 22\) just to get below 10%.

Identifiability: why decay and saturation fight each other

The framework's own adstock_weights docstring flags this, and it matters more than any single formula. In an additive MMM each channel contributes

and with a normalized kernel the carryover magnitude lives entirely in \(\beta\). That leaves three knobs — decay shape, saturation strength, and \(\beta\) — competing to explain the same wiggle in the data. A long-carryover + weak-saturation fit and a short-carryover + strong-saturation fit can be nearly indistinguishable in-sample. This is equifinality: inherent to additive MMM, not a bug.

Computed live (seeded) — same construction as the notebook. Regime A: longer carryover (\(\alpha=0.70\)) + weak saturation (\(\lambda=4\)); regime B: shorter carryover (\(\alpha=0.50\)) + strong saturation (\(\lambda=8\)), both peak-normalized (\(\beta\) would absorb the scale). The notebook's seeded run measures their shape correlation at 0.9509 — the likelihood can barely tell them apart from observational data alone.

The framework's recommended mitigations, in increasing order of power:

• Informative priors on \(\alpha\)/saturation — don't let them roam to the boundary.
• Anchoring saturation to data percentiles (e.g. SaturationConfig.compute_kappa_bounds_from_data for the Hill path), so the half-saturation point is tied to observed spend levels — see Math 02 for the curve this feeds.
• Experiment-calibrated coefficient priors (mmm_framework.calibration) — the most powerful fix. A geo-lift test pins \(\beta\) directly, which breaks the trade-off: once magnitude is nailed down by randomized evidence, the decay/saturation split is far better determined. Math 05 derives the machinery; Stress 05 measures it under fire.

Recap: the four representations, reconciled