model#

Bass diffusion model for product adoption.

Adapted from Wiki: https://en.wikipedia.org/wiki/Bass_diffusion_model

The Bass diffusion model, developed by Frank Bass in 1969, is a mathematical model that describes the process of how new products get adopted in a population over time. It is widely used in marketing, forecasting, and innovation studies to predict the adoption rates of new products and technologies.

Mathematical Formulation#

The model is based on a differential equation that describes the rate of adoption:

\[\frac{f(t)}{1-F(t)} = p + q F(t)\]

Where:

\(F(t)\) is the installed base fraction (cumulative proportion of adopters)
\(f(t)\) is the rate of change of the installed base fraction (\(f(t) = F'(t)\))
\(p\) is the coefficient of innovation or external influence
\(q\) is the coefficient of imitation or internal influence

The solution to this equation gives the adoption curve:

\[F(t) = \frac{1 - e^{-(p+q)t}}{1 + (\frac{q}{p})e^{-(p+q)t}}\]

The adoption rate at time t is given by:

\[f(t) = (p + q F(t))(1 - F(t))\]

Key Parameters#

The model has three main parameters:

\(m\): Market potential (total number of eventual adopters)
\(p\): Coefficient of innovation (external influence) - typically 0.01-0.03
\(q\): Coefficient of imitation (internal influence) - typically 0.3-0.5

Parameter Interpretation#

A higher \(p\) value indicates stronger external influence (advertising, marketing)
A higher \(q\) value indicates stronger internal influence (word-of-mouth, social interactions)
The ratio \(q/p\) indicates the relative strength of internal vs. external influences
The peak of adoption occurs at time \(t^* = \frac{\ln(q/p)}{p+q}\)

Applications#

The Bass model has been applied to forecast the adoption of various products and technologies:

Consumer durables (TVs, refrigerators)
Technology products (smartphones, software)
Pharmaceutical products
Entertainment products
Services and subscriptions

This implementation provides a Bayesian version of the Bass model using PyMC, allowing for: - Uncertainty quantification through prior distributions - Hierarchical modeling for multiple products/markets - Extension to incorporate additional factors

Examples#

Create a basic Bass model for multiple products:

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

../../_images/pymc_marketing-bass-model-1.png

Functions

`F`(p, q, t)	Installed base fraction (cumulative adoption proportion).
`create_bass_model`(t, observed, priors, coords)	Define a Bass diffusion model for product adoption forecasting.
`f`(p, q, t)	Installed base fraction rate of change (adoption rate).