Delayed adstock

The delayed-peak geometric kernel of Jin et al. (2017), \(w_l = \theta^{(l - \delta)^2}\) for lags \(l = 0, \ldots, L-1\), where \(\theta\) is the retention rate and \(\delta\) the lag at which the effect peaks. With peak = 0 the weights fall off as \(\theta^{l^2}\), faster than geometric; with peak > 0 the response builds for peak periods before it decays, the shape television and out-of-home often show. It is the delayed form used in Google's Bayesian MMM work, and a one-parameter-per-idea alternative to the Weibull kernel: peak is read directly in periods.

Usage

adstock_weights_delayed(max_lag, decay, peak = 0, normalise = TRUE)

adstock_delayed(
  x,
  decay,
  peak = 0,
  max_lag,
  normalise = TRUE,
  state = 0,
  by = NULL,
  na_action = c("error", "zero", "keep")
)

Arguments

max_lag

Number of periods the kernel spans, including the current period. Required and finite: the kernel has no recursive form. Jin et al. use 13 weeks for weekly data.

decay

Retention rate \(\theta\) in (0, 1).

peak

Lag of the peak effect, in periods, in [0, max_lag - 1]. Need not be a whole number.

normalise

Should the weights sum to 1? When FALSE the weights are left as \(\theta^{(l - \delta)^2}\), whose largest value is 1 when the peak falls on a whole period.

x

Numeric vector of media spend (or impressions, or GRPs) in time order. x must already be sorted by time and evenly spaced; the function has no index argument and cannot check this for you. Use step_adstock() if you want the time index validated.

state

The preceding max_lag - 1 values of x, oldest first, or 0 for a cold start; a named list of them when by is supplied.

by

Optional grouping vector, or data frame of grouping vectors, the same length as x. Adstock is applied independently within each group, which is what geo-level and panel models need. Never rely on dplyr::group_by() for this: grouping metadata does not reliably survive into every context where this function is called.

na_action

What to do about missing values in x. "error" (the default) refuses to guess. "zero" treats missing media as no media, which is usually right for spend but is a substantive assumption. "keep" lets NA propagate through the filter, which for the recursive form poisons every subsequent value.

Value

adstock_weights_delayed() returns the kernel, ordered from the current period outward. adstock_delayed() returns a numeric vector the same length as x, in the same order.

References

Jin, Y., Wang, Y., Sun, Y., Chan, D. and Koehler, J. (2017). Bayesian methods for media mix modeling with carryover and shape effects. Google Inc. https://research.google/pubs/pub46001/

See also

adstock_geometric(), adstock_weibull(), media_transform()

Examples

# Peak two weeks after the spend
round(adstock_weights_delayed(8, decay = 0.6, peak = 2), 3)
#> [1] 0.052 0.243 0.405 0.243 0.052 0.004 0.000 0.000

spend <- c(100, 0, 0, 0, 0, 0, 0, 0)
round(adstock_delayed(spend, decay = 0.6, peak = 2, max_lag = 8), 2)
#> [1]  5.25 24.30 40.49 24.30  5.25  0.41  0.01  0.00

# Through media_transform()
media_transform(spend, adstock = list(kernel = "delayed", decay = 0.6,
                                      peak = 2, max_lag = 8))
#> [1] 5.247893e+00 2.429580e+01 4.049300e+01 2.429580e+01 5.247893e+00
#> [6] 4.080761e-01 1.142352e-02 1.151228e-04