Forecasting S-shaped diffusion processes via response modelling methodology

Abstract

Diffusion processes abound in various areas of corporate activities, such as the time-dependent behaviour of cumulative demand of a new product, or the adoption rate of a technological innovation. In most cases, the proportion of the population that has adopted the new product by time t behaves like an S-shaped curve, which resembles the sigmoid curve typical to many known statistical distribution functions. This analogy has motivated the common use of the latter for forecasting purposes. Recently, a new methodology for empirical modelling has been developed, termed response modelling methodology (RMM). The error distribution of the RMM model has been shown to model well variously shaped distribution functions, and may therefore be adequate to forecast sigmoid-curve processes. In particular, RMM may be applied to forecast S-shaped diffusion processes. In this paper, forty-seven data sets, assembled from published sources by Meade and Islam, are used to compare the accuracy and the stability of RMM-generated forecasts, relative to current commonly applied models. Results show that in most comparisons RMM forecasts outperform those based on any individually selected distributional model.

Appendix

MI distinguish between symmetric models (inflexion point at P _t =0.5, like the normal and the logistic models), non-symmetric models (P _t <0.5 but constant, like Gompertz and linearized Gompertz), and flexible models (where the inflexion point can occur anywhere within the same model, including P _t =0.5; Typical examples are the log-normal, Weibull and the extended logistic). MI denote these classes by class II₁ (symmetric), class II₂ (asymmetric) and class II₃ (flexible). Based on a carefully selected subset of seven models that belong to the different classes (out of the original 29 models), MI introduce the following classification:

• Class II₁: Simple logistic, mansfield (AR),

• Class II₂: Gompertz, Floyd (AR),

• Class II_3A: Weibull, extended logistic (B),

• Class II_3B: Cumulative log-normal,

where the third class (II₃) is subdivided, based on results from stepwise discriminant-analysis, which showed that the log-normal ‘behaves differently from the other flexible curves’ (MI, p 1122).

These models are displayed below:

• 1) Simple Logistic:

  

• 2) Gompertz: P _t =exp(−c exp(−bt))+ɛ _t , b>0, c>0.

• 3) Cumulative log-normal:

  

• 4) Weibull: P _t =1−exp(−(t/α)^β)+ɛ _t , α>0, β>0.

• 5) Extended logistic (B):

  

• 6) Mansfield (AR): P _t −P _t−1=bP _t−1(1−P _t−1)+ɛ _t , b>0.

• 7) Floyd (AR): P _t −P _t−1=bP _t−1(1−P _t−1)²+ɛ _t , b>0