Norton-Bass model

Description

Nortonbass fits a generational Bass model proposed by Norton and Bass (1987). Each subsequent generation influences the sales of the previous generation. The set of equation is estimated simulataneously.

Usage

Nortonbass(
  x,
  startval.met = c("2ST", "BB", "iBM"),
  estim.met = c("BOBYQA", "OLS", "SUR", "2SLS", "3SLS"),
  gstart = NULL,
  startval = NULL,
  flexpq = F
)

Arguments

Details

For starting values the Vector values need to be named in the case flexpq == T p_1,\dots,p_j,q_1,\dots,q_j,m_1,\dots,m_j. In the case of flexpq == F p_1, q_1, m_1,\dots, m_j.

If gstart is not provided, the generation starting points will be detected automatically selecting the first value that is non-zero.

Value

coef: coefficients for p, q and m

Author(s)

Oliver Schaer, info@oliverschaer.ch

References

Norton, J.A. and Bass, F.M., 1987. A Diffusion Theory Model of Adoption and Substitution for Successive Generations of High-Technology Products.

Islam, T. and Meade, N., 1997. The Diffusion of Successive Generations of a Technology: A More General Model. Technological Forecasting and Social Change, 56, 49-60.

Examples

 ## Not run: 
   fitNB1 <- Nortonbass(tsIbm, startval.met = "2ST", estim.met = "OLS",
                        startval = NULL, flexpq = F, gstart = NULL)
   fitNB2 <- Nortonbass(tsIbm, startval.met = "2ST", estim.met = "SUR",
                        startval = NULL, flexpq = F, gstart = NULL)
   # using BOBYQA algorithm
   fitNB3 <- Nortonbass(tsIbm, startval.met = "2ST", estim.met = "BOBYQA",
                        startval = NULL, flexpq = F, gstart = NULL)
   # Create some plots
   plot(tsibm[, 1],type = "l", ylim=c(0,35000))
   lines(tsibm[, 2],col ="blue")
   lines(tsibm[, 3],col ="green")
   lines(tsibm[, 4],col ="pink")
   lines(fitNB1$fit$fitted[[1]], col = "black", lty = 2)
   lines(fitNB1$fit$fitted[[2]], col = "blue", lty = 2)
   lines(fitNB1$fit$fitted[[3]], col = "green", lty = 2)
   lines(fitNB1$fit$fitted[[4]], col = "pink", lty = 2)
   lines(fitNB2$fit$fitted[[1]], col = "black", lty = 3)
   lines(fitNB2$fit$fitted[[2]], col = "blue", lty = 3)
   lines(fitNB2$fit$fitted[[3]], col = "green", lty = 3)
   lines(fitNB2$fit$fitted[[4]], col = "pink", lty = 3)
   lines(fitNB3$fit$fitted[[1]], col = "black", lty = 4)
   lines(fitNB3$fit$fitted[[2]], col = "blue", lty = 4)
   lines(fitNB3$fit$fitted[[3]], col = "green", lty = 4)
   lines(fitNB3$fit$fitted[[4]], col = "pink", lty = 4)
   # read out RMSE
   fitNB1$fit$RMSE[[1]]
   fitNB1$fit$RMSE[[2]]
   fitNB1$fit$RMSE[[3]]
   fitNB1$fit$RMSE[[4]]
   fitNB2$fit$RMSE[[1]]
   fitNB2$fit$RMSE[[2]]
   fitNB2$fit$RMSE[[3]]
   fitNB2$fit$RMSE[[4]]
   fitNB3$fit$RMSE[[1]]
   fitNB3$fit$RMSE[[2]]
   fitNB3$fit$RMSE[[3]]
   fitNB3$fit$RMSE[[4]]
 
## End(Not run)
 
  

Fits Norton Bass curve and estimated RMSE

Description

Fits Norton Bass curve and estimated RMSE

Usage

Nortonbass_error(x, param, gstart = NULL, flexpq = F)

Arguments

Value

yhat, the predicted values

actuals, the actual values

RMSE, the root mean squared error for each generation

Author(s)

Oliver Schaer, info@oliverschaer.ch

Fits Norton Bass curve and estimated RMSE

Description

Fits Norton Bass curve and estimated RMSE

Usage

Nortonbass_startvalgen(x, gstart, flexpq, startval.met)

Arguments

Value

starting values for all parameters

Author(s)

Oliver Schaer, info@oliverschaer.ch

Calculates the values for various diffusion curves, given some parameters.

Description

This function calculates the values of diffusion curves that can be of "bass", "gompertz", "gsgompertz" or "weibull" type, given some parameters.

Usage

difcurve(
  n,
  w = c(0.01, 0.1, 10),
  type = c("bass", "gompertz", "gsgompertz", "weibull"),
  curve = NULL
)

Arguments

Value

Returns a matrix of values with each row being a period.

Note

w needs to be provided for the Bass curve in the order of ("m", "p", "q"), 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 ("m", "a", "b"). 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 ("m", "a", "b", "c"). 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).

For the Weibull curve, vector w needs to be in the form of ("m", "a", "b"). Where "a" is the scale parameter, "b" determines the shape. Together, "a" and "b" determine the stepness of the curve. The "m" parameter sets the market potential (saturation point).

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

See Also

diffusion for fitting a diffusion curve.

Examples

  difcurve(w=c(0.01,0.1,10),20)
  

Fit various diffusion curves.

Description

This function fits diffusion curves that can be of "bass", "gompertz", "gsgompertz" (Gamma/Shifted Gompertz curve) or "Weibull" type.

Usage

diffusion(
  y,
  w = NULL,
  cleanlead = c(TRUE, FALSE),
  loss = 2,
  cumulative = c(TRUE, FALSE),
  verbose = c(FALSE, TRUE),
  type = c("bass", "gompertz", "gsgompertz", "weibull"),
  method = c("L-BFGS-B", "Nelder-Mead", "BFGS", "hjkb", "Rcgmin", "bobyqa"),
  maxiter = 500,
  opttol = 1e-06,
  multisol = c(FALSE, TRUE),
  initpar = c("linearize", "preset"),
  mscal = c(TRUE, FALSE),
  ...
)

Arguments

Value

Returns an object of class diffusion, which contains:

• type diffusion curve type used

• call calls function fitted

• w named vector of fitted parameters

• y 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

• init the initial values that have been used for the optimizer

Bass curve

The optimization of the Bass curve is initialized by the linear approximation suggested in Bass (1969).

Gompertz curve

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

Gamma/Shifted Gompertz

The curve is initialized 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.

Weibull

The initialization is obtained through by a linear approximation median-ranked OLS described in Sharif and Islam 1980.

Note

vector w needs to be provided for the Bass curve in the order of "m", "p", "q", 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 ("m", "a", "b"). Where "a" is the x-axis displacement coefficient, "b" determines the growth rate and "m" sets, similarly to the Bass curve, the market potential (saturation point).

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

For the Weibull curve, vector w needs to be in the form of ("m", "a", "b"). Where "a" is the scale parameter, "b" determines the shape. Together, "a" and "b" determine the steepness of the curve. The "m" parameter sets the market potential (saturation point).

Parameters are estimated by minimising the Mean Squared Error with a subplex algorithm from the optimx package. Optionally p-values of the coefficients can be determined via bootstraping. Furthermore, the bootstrapping allows removing insignificant parameters from the optimization process.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

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.

• Sharif, N.M. and Islam, M.N. 1980. The Weibull Distribution as a General Model for Forecasting Technological Change. Technological Forecasting and Social Change, 18, 247-256.

See Also

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

seqdiffusion for sequential diffusion model fitting across product generations.

Examples

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

Diffusion class checkers

Description

Functions to check if an object is of the specified class

Usage

is.diffusion(x)

is.bass(x)

Arguments

Details

The list of functions includes:

• is.diffusion() tests if the object was produced by a diffusion() function.

• is.bass() checks if the forecast was produced by the bass() function.

Value

TRUE if this is the specified class and FALSE otherwise.

Author(s)

Ivan Svetunkov, ivan@svetunkov.ru,

Oliver Schaer, info@oliverschaer.ch

Plot a fitted diffusion curve.

Description

Produces a plot of a fitted diffusion curve.

Usage

## S3 method for class 'diffusion'
plot(x, cumulative = c(FALSE, TRUE), ...)

Arguments

Value

None. Function produces a plot.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

See Also

diffusion.

Examples

 fit <- diffusion(tsChicken[, 2])
 plot(fit)

Plot sequentially fitted diffusion curves.

Description

Produces a plot of sequentially fitted diffusion curves.

Usage

## S3 method for class 'seqdiffusion'
plot(x, cumulative = c(FALSE, TRUE), ...)

Arguments

Value

None. Function produces a plot.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

See Also

seqdiffusion.

Examples

 fit <- seqdiffusion(tsIbm)
 plot(fit)

Predict future periods of a fitted diffusion curve.

Description

Calculates the values for h future periods of a fitted diffusion curve.

Usage

## S3 method for class 'diffusion'
predict(object, h = 10, ...)

Arguments

Value

Returns an object of class diffusion, which contains:

• type diffusion curve type used

• call calls function fitted

• w named vector of fitted parameters

• y actuals

• fit fitted values of model

• frc forecasts for future periods.

• mse insample Mean Squared Error

• prew the w of the previous generation

• pval p-values for w

Note

This function populates the matrix frc of the diffusion object used as input.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikoloas Kourentzes, nikoloas@kourentzes.com

See Also

diffusion.

Examples

 fit <- diffusion(tsChicken[, 2])
 fit <- predict(fit, 20)
 plot(fit)

Print a fitted diffusion curve.

Description

Outputs the result of a fitted diffusion curve.

Usage

## S3 method for class 'diffusion'
print(x, ...)

Arguments

Value

None. Console output only.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

See Also

diffusion.

Examples

 fit <- diffusion(tsChicken[, 2])
 print(fit)

Print sequentially fitted diffusion curves.

Description

Outputs the result of sequentially fitted diffusion curves.

Usage

## S3 method for class 'seqdiffusion'
print(x, ...)

Arguments

Value

None. Console output only.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

See Also

seqdiffusion.

Examples

 fit <- seqdiffusion(tsIbm)
 print(fit)

Enables fitting various sequential diffusion curves.

Description

This function fits diffusion curves of the type "bass", "gompertz", gsgompertz or weibull across generations. Parameters are estimated for each generation individually by minimizing the Mean Squared Error with subplex algorithms from the optimx package. Optionally p-values of the coefficients can be determined via bootstraping. Furthermore, the bootstrapping allows removing insignificant parameters from the optimisation process.

Usage

seqdiffusion(
  y,
  w = NULL,
  cleanlead = c(TRUE, FALSE),
  loss = 2,
  cumulative = c(TRUE, FALSE),
  pvalreps = 0,
  eliminate = c(FALSE, TRUE),
  sig = 0.05,
  verbose = c(FALSE, TRUE),
  type = c("bass", "gompertz", "gsgompertz", "weibull"),
  method = c("L-BFGS-B", "Nelder-Mead", "BFGS", "hjkb", "Rcgmin", "bobyqa"),
  maxiter = 500,
  opttol = 1e-06,
  multisol = c(FALSE, TRUE),
  initpar = c("linearize", "preset"),
  mscal = c(TRUE, FALSE),
  bootloss = c("smthempir", "empir", "se"),
  ...
)

Arguments

Value

Returns an object of class seqdiffusion, which contains:

• type diffusion model type used

• diffusion returns model specification for each generation (see diffusion for details)

• call calls function fitted

• w named matrix of fitted parameters for each generation

• y matrix of actuals

• mse insample Mean Squared Error for each generation

• pval all p-values for w at each generation

Bass curve

The optimization of the Bass curve is initialized by the linear approximation suggested in Bass (1969).

Gompertz curve

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

Gamma/Shifted Gompertz

The curve is initialized 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.

Weibull

The initialization is obtained through by a linear approximation median-ranked OLS described in Sharif and Islam 1980.

Author(s)

Oliver Schaer, info@oliverschaer.ch,

Nikolaos Kourentzes, nikolaos@kourentzes.com

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.

• Sharif, N.M. and Islam, M.N. 1980. The Weibull Distribution as a General Model for Forecasting Technological Change. Technological Forecasting and Social Change, 18, 247-256.

See Also

plot.seqdiffusion and print.seqdiffusion.

Examples

  fit <- seqdiffusion(tsIbm)
  plot(fit)

Time series: Assassins Creeds

Description

A dataset containing the weekly sales of Assassins Creeds game.

Format

A matrix with 380 observations and 8 variables

ac1

Assassins Creed 1

ac2

Assassins Creed 2

ac3

Assassins Creed 3

ac4

Assassins Creed 4

ac5

Assassins Creed 5

ac6

Assassins Creed 6

ac7

Assassins Creed 7

ac8

Assassins Creed 8

References

VGChartz

Time series: Stock of cars

Description

A dataset containing the yearly stock of cars in the Netherlands (1965-1989).

Format

A data frame with 25 observations and 3 variables

year

Year

raw

Raw stock numbers

smoothed

Smoothed stock numbers as described by Franses (1994)

References

Franses, P.H. 1994. Fitting a Gompertz curve. Journal of Operational Research Society, 45, 109-113.

Time series: Chicken weight

Description

A dataset containing the average weekly female chicken weight.

Format

A data frame with 13 observations and 2 variables

time

Weeks since birth

weight

Weight of the female chicken in Kg

References

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

Time series: COVID-19 confirmed cases US

Description

A dataset containing the number of confirmed COVID-19 cases in the US.

Format

A ts object with 107 days of observations

tsCovid

Daily confirmed COVDID-19 cases

Source

https://github.com/CSSEGISandData/COVID-19

References

COVID-19 Data Repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University

Time series: Sales of IBM Computers

Description

A dataset containing the first four generations of yearly IBM general-purpose computers installations in the USA.

Format

A data frame with 24 observations and 4 variables

SIU1

1st generation

SIU2

2nd generation (starts 6 years after first generation)

SIU3

3rd generation (starts 11 years after first generation)

SIU4

4th generation (starts 16 years after first generation)

Source

https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=8bbf197bc39a27ccf44cfd5ed22b5db3da0c7bb2

References

Bass, P.I. and Bass, F.M. 2004. IT Waves: Two Completed Generational Diffusion Models. Working Paper Basseconomics, 1-33.

Time series: U.S. Merchant Marine conversion to metal

Description

A dataset with conversion of U.S. Merchant Marine from wood to metal.

Format

A data frame with 17 observations and 2 variables

year

Year

substitution

Conversion to metal

References

Martino, J.P. 1993. Technological Forecasting for Decision Making. 3rd edition. New York: McGraw-Hill.

Time series: Safari Browser market share

Description

A dataset containing the monthly market share of Safari browser generations from Safari 4.0 to Safari 10.

Format

A data frame with 98 observations and 13 variables

Date

Log file date

Safari10.0

Usage of Windows 10

Safari9.1

Market share of Safari browser v 10.0

Safari9.0

Market share of Safari browser v 9.1

Safari8.0

Market share of Safari browser v 9.0

Safari7.1

Market share of Safari browser v 8.0

Safari7.0

Market share of Safari browser v 7.1

Safari6.1

Market share of Safari browser v 6.1

Safari6.0

Market share of Safari browser v 6.0

Safari5.1

Market share of Safari browser v 5.1

Safari5.0

Market share of Safari browser v 5.0

Safari4.1

Market share of Safari browser v 4.1

Safari4.0

Market share of Safari browser v 4.0

Source

https://gs.statcounter.com/browser-version-market-share

Time series: Windows OS Platform Statistics

Description

A dataset containing the 3WSchools monthly log files of Windows operating system usage from March 2003 until February 2017.

Format

A data frame with 168 observations and 9 variables

Date

Log file date

Win10

Usage of Windows 10

Win8

Usage of Windows 8

Win7

Usage of Windows 7

Vista

Usage of Windows Vista

WinXP

Usage of Windows XP

Win2000

Usage of Windows 2000

Win98

Usage of Windows 98

Win95

Usage of Windows 95

Note

From March 2003 until January 2008 log file is only available bi-monthly. To retain monthly consistency, values have been linearly interpolated

Source

https://www.w3schools.com/browsers/browsers_os.asp