diffusion: Fit various diffusion curves.

In diffusion: Forecast the Diffusion of New Products

Description

This function fits diffusion curves that can be of "bass", "gompertz" or "gsgompertz" type.

Usage

diffusion(x, w = NULL, cleanlead = c(TRUE, FALSE), prew = NULL, l = 2,
  cumulative = c(TRUE, FALSE), pvalreps = 0, eliminate = c(FALSE, TRUE),
  sig = 0.05, verbose = c(FALSE, TRUE), type = c("bass", "gompertz",
  "gsgompertz"), optim = c("nm", "hj"), maxiter = Inf, opttol = 1e-06)

Arguments

x	vector with adoption per period
w	vector of curve parameters (see note). If provided no estimation is done.
cleanlead	removes leading zeros for fitting purposes (default == TRUE)
prew	Experimental. Ignore!
l	the l-norm (1 is absolute errors, 2 is squared errors).
cumulative	If TRUE optimisation is done on cumulative adoption.
pvalreps	Experimental. Ignore!
eliminate	Experimental. Ignore!
sig	Experimental. Ignore!
verbose	if TRUE console output is provided during estimation (default == FALSE)
type	diffusion curve to use. This can be "bass", "gompertz" and "gsgompertz"
optim	optimization method to use. This can be "nm" for Nelder-Meade or "hj" for Hooke-Jeeves.
maxiter	number of iterations the optimser takes (default == 10000 for "nm" and Inf for "hj")
opttol	Tolerance for convergence (default == 1.e-06)

Value

Returns an object of class diffusion, which contains:

• type diffusion curve type used

• call calls function fitted

• w named vector of fitted parameters

• x actuals

• fit fitted values of model

• frc forecasts for future periods. This is NULL until predict.diffusion is called.

• mse insample Mean Squared Error

• prew the w of the previous generation

• pval p-values for w

Bass curve

The optimisation of the Bass curve is initialisated by the linear aproximation suggested in Bass (1969).

Gompertz curve

The initialisation of the Gompertz curve uses the approach suggested by Jukic et al. (2004), but is adapted to allow for the non-exponential version of Gompertz curve. This makes the market potential parameter equivalent to the Bass curves's and the market potential from Bass curve is used for initialisation.

Gamma/Shifted Gompertz

The curve is initialised by assuming the shift operator to be 1 and becomes equivalent to the Bass curve, as shown in Bemmaor (1994). A Bass curve is therefore used as an estimator for the remaining initial parameters.

Note

vector w needs to be provided for the Bass curve in the order of "p", "q", "m", where "p" is the coefficient of innovation, "q" is the coefficient of imitation and "m" is the market size coefficient.

For the Gompertz curve, vector w needs to be in the form of ("a", "b", "m"). Where "a" is the x-axis displacement coefficient, "b" determines the growth rate and "m" sets, similarly to Bass model, the market potential (saturation point).

For the Shifted-Gompertz curve, vector w needs to be in the form of ("a", "b", "c", "m"). Where "a" is the x-axis displacement coefficient, "b" determines the growth rate, "c" is the shifting parameter and "m" sets, similarly to Bass model, the market potential (saturation point).

Parameters are estimated by minimising the Mean Squared Error with a Subplex algorithm from the nloptr package.

Author(s)

Oliver Schaer, [e-mail],

Nikoloas Kourentzes, [e-mail]

References

• For an introduction to diffusion curves see: Ord K., Fildes R., Kourentzes N. (2017) Principles of Business Forecasting 2e. Wessex Press Publishing Co., Chapter 12.

• Bass, F.M., 1969. A new product growth for model consumer durables. Management Science 15(5), 215-227.

• Bemmaor, A. 1994. Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect versus Consumer Heterogeneity. In G. Laurent, G.L. Lilien and B. Pras (Eds.). Research Traditions in Marketing. Boston: Kluwer, pp. 201-223.

• Jukic, D., Kralik, G. and Scitovski, R., 2004. Least-squares fitting Gompertz curve. Journal of Computational and Applied Mathematics, 169, 359-375.

See Also

predict.diffusion, plot.diffusion and print.diffusion.

Examples

 fitbass <- diffusion(tsChicken[, 2], type = "bass")
 fitgomp <- diffusion(tsChicken[, 2], type = "gompertz")
 fitgsg <- diffusion(tsChicken[, 2], type = "gsgompertz")
 
 # Produce some plots
 plot(fitbass)
 plot(fitgomp)
 plot(fitgsg)

diffusion documentation built on May 2, 2019, 9:38 a.m.