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Weibull adstock weights

Source: R/adstock.R
adstock_weights_weibull.Rd

A two-parameter kernel that, unlike the geometric kernel, can place its peak after the period in which the money was spent. Television, cinema and out-of-home routinely behave this way: the response builds for a week or two before it turns over. A geometric kernel cannot represent that shape at any decay rate.

Usage

adstock_weights_weibull(
  max_lag,
  shape,
  scale,
  type = c("cdf", "pdf"),
  normalise = TRUE
)

Arguments

max_lag

Number of periods the kernel spans, including the current period. Unlike the geometric kernel, Weibull adstock has no infinite form: max_lag is required and must be finite.

shape

Weibull shape parameter, positive. In the "pdf" form, shape > 1 produces a delayed peak and shape <= 1 produces a monotonically decaying kernel. In the "cdf" form, larger values produce a flatter plateau followed by a sharper drop.

scale

Weibull scale parameter, positive, measured in periods. Larger values stretch the kernel over more periods.

type

Either "cdf" (default) or "pdf". See Details.

normalise

Should the weights sum to 1? When FALSE, weights are scaled so the largest is 1, matching the unnormalised geometric convention.

Value

A numeric vector of length max_lag, ordered from the current period to the most distant lag.

Details

The two forms answer different questions.

The "cdf" form builds the kernel as a cumulative product of Weibull survival values. Numbering lags from 1 so that \(w_1\) is the period of spend, \(w_i = \prod_{j<i} (1 - F(j))\), where \(F\) is the Weibull distribution function, so \(w_1 = 1\) and each later weight is the previous one times \(1 - F(j)\). Read \(1 - F(j)\) as a time-varying retention rate: it plays the role that the constant \(\theta\) plays in the geometric kernel, which is the sense in which this form generalises geometric decay. It is always monotonically decreasing, but the rate of decay can itself change over time. Note that the kernel is the running product of these values, not the Weibull survival function itself, so it falls faster than \(1 - F(i)\). The construction is the one Robyn calls "weibull_cdf"; Robyn, however, expresses scale as a quantile of the window length rather than in periods, so its fitted scale values are not directly comparable with this function's.

The "pdf" form uses the Weibull density directly, \(w_i \propto f(i)\). This is the form that permits a delayed peak, and it is the reason to reach for Weibull adstock at all. With shape <= 1 it collapses back to a monotone decay.

See also

adstock_weibull(), adstock_weights()

Examples

# Monotone decay, a flexible generalisation of geometric
round(adstock_weights_weibull(8, shape = 2, scale = 3, type = "cdf"), 4)
#> [1] 0.3680 0.3293 0.2111 0.0777 0.0131 0.0008 0.0000 0.0000

# Delayed peak: most weight lands one period after the spend, not in the
# period of spend itself. No geometric decay rate can do this.
w <- adstock_weights_weibull(8, shape = 2, scale = 3, type = "pdf")
round(w, 4)
#> [1] 0.2027 0.2905 0.2500 0.1531 0.0704 0.0249 0.0069 0.0015
which.max(w)
#> [1] 2