LogSaturation#

class pymc_marketing.mmm.components.saturation.LogSaturation(priors=None, prefix=None)[source]#

Logarithmic saturation for log-log models.

Applies \(\beta \, \log(1 + x)\) to the raw (unscaled) channel input, mapping spend through a concave logarithmic curve with diminishing returns.

When combined with link="log" in the MMM, the model becomes a log-log specification and \(\beta\) is an approximate elasticity – the percentage change in the response per one percent change in spend. For this interpretation to hold, the channel input must not be rescaled: an elasticity is dimensionless, so dividing spend by an arbitrary channel_scale would change \(\beta\) (because \(\log(1 + x)\) is not invariant under multiplicative rescaling of \(x\)). This class therefore sets requires_unscaled_input to True, which makes the MMM feed raw spend to the saturation and force channel_scale = 1 for the affected channels.

log(1 + x) (rather than log(x)) is used so that the transform is finite at x = 0 – common for paused or cold-start channels – while remaining numerically indistinguishable from log(x) once spend is large, where the elasticity interpretation is exact in the limit \(\partial \log y / \partial \log x \to \beta\).

(Source code, png, hires.png, pdf)

../../_images/pymc_marketing-mmm-components-saturation-LogSaturation-1.png

Methods

`LogSaturation.__init__`([priors, prefix])
`LogSaturation.apply`(x, *[, core_dim, idx])	Call within a model context.
`LogSaturation.from_dict`(data)	Reconstruct a transformation from a dict.
`LogSaturation.function`(x, beta, *[, dim])	Logarithmic saturation function: beta * log(1 + x).
`LogSaturation.plot_curve`(curve[, n_samples, ...])	Plot curve HDI and samples.
`LogSaturation.plot_curve_hdi`(curve[, ...])	Plot the HDI of the curve.
`LogSaturation.plot_curve_samples`(curve[, n, ...])	Plot samples from the curve.
`LogSaturation.sample_curve`([parameters, ...])	Sample the curve of the saturation transformation given parameters.
`LogSaturation.sample_prior`([coords])	Sample the priors for the transformation.
`LogSaturation.set_dims_for_all_priors`(dims)	Set the dims for all priors.
`LogSaturation.to_dict`([_orig])	Convert the transformation to a dictionary.
`LogSaturation.update_priors`(priors)	Update the priors for a function after initialization.
`LogSaturation.with_default_prior_dims`(dims)	Return a copy with default prior dims (dims=None) set to `dims` instead.
`LogSaturation.with_updated_priors`(priors)	Return a copy with updated priors.

Attributes

`LogSaturation.combined_dims`	Get the combined dims for all the parameters.
`LogSaturation.default_priors`
`LogSaturation.function_priors`	Get the priors for the function.
`LogSaturation.model_config`	Mapping from variable name to prior for the model.
`LogSaturation.prefix`
`LogSaturation.priors`	Get the priors for the function.
`LogSaturation.requires_unscaled_input`	Whether the saturation must receive raw (unscaled) channel inputs.
`LogSaturation.variable_mapping`	Mapping from parameter name to variable name in the model.