CLVTools (version 0.11.2)

ggomnbd: Gamma-Gompertz/NBD model

Description

Fits Gamma-Gompertz/NBD models on transactional data with static and without covariates.

Usage

# S4 method for clv.data
ggomnbd(
  clv.data,
  start.params.model = c(),
  optimx.args = list(),
  verbose = TRUE,
  ...
)
# S4 method for clv.data.static.covariates
ggomnbd(
  clv.data,
  start.params.model = c(),
  optimx.args = list(),
  verbose = TRUE,
  names.cov.life = c(),
  names.cov.trans = c(),
  start.params.life = c(),
  start.params.trans = c(),
  names.cov.constr = c(),
  start.params.constr = c(),
  reg.lambdas = c(),
  ...
)

Value

Depending on the data object on which the model was fit, ggomnbd returns either an object of class clv.ggomnbd or clv.ggomnbd.static.cov.

The function summary can be used to obtain and print a summary of the results. The generic accessor functions coefficients, vcov, fitted, logLik, AIC, BIC, and nobs are available.

Arguments

clv.data

The data object on which the model is fitted.

start.params.model

Named start parameters containing the optimization start parameters for the model without covariates.

optimx.args

Additional arguments to control the optimization which are forwarded to optimx::optimx. If multiple optimization methods are specified, only the result of the last method is further processed.

verbose

Show details about the running of the function.

...

Ignored

names.cov.life

Which of the set Lifetime covariates should be used. Missing parameter indicates all covariates shall be used.

names.cov.trans

Which of the set Transaction covariates should be used. Missing parameter indicates all covariates shall be used.

start.params.life

Named start parameters containing the optimization start parameters for all lifetime covariates.

start.params.trans

Named start parameters containing the optimization start parameters for all transaction covariates.

names.cov.constr

Which covariates should be forced to use the same parameters for the lifetime and transaction process. The covariates need to be present as both, lifetime and transaction covariates.

start.params.constr

Named start parameters containing the optimization start parameters for the constraint covariates.

reg.lambdas

Named lambda parameters used for the L2 regularization of the lifetime and the transaction covariate parameters. Lambdas have to be >= 0.

Details

Model parameters for the GGompertz/NBD model are r, alpha, beta, b and s.
r: shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process.
alpha: scale parameter of the Gamma distribution of the purchase process.
beta: scale parameter for the Gamma distribution for the lifetime process.
b: scale parameter of the Gompertz distribution (constant across customers).
s: shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes.

If no start parameters are given, r=0.5, alpha=2, b=0.1, s=1, beta=0.1 is used. All model start parameters are required to be > 0. If no start values are given for the covariate parameters, 0.1 is used.

Note that the DERT expression has not been derived (yet) and it consequently is not possible to calculated values for DERT and CLV.

The Gamma-Gompertz/NBD model

There are two key differences of the gamma/Gompertz/NBD (GGompertz/NBD) model compared to the relative to the well-known Pareto/NBD model: (i) its probability density function can exhibit a mode at zero or an interior mode, and (ii) it can be skewed to the right or to the left. Therefore, the GGompertz/NBD model is more flexible than the Pareto/NBD model. According to Bemmaor and Glady (2012) can indicate substantial differences in expected residual lifetimes compared to the Pareto/NBD. The GGompertz/NBD tends to be appropriate when firms are reputed and their offerings are differentiated.

References

Bemmaor AC, Glady N (2012). “Modeling Purchasing Behavior with Sudden “Death”: A Flexible Customer Lifetime Model” Management Science, 58(5), 1012-1021.

Adler J (2022). “Comment on “Modeling Purchasing Behavior with Sudden “Death”: A Flexible Customer Lifetime Model” Management Science 69(3):1929-1930.

The expression for the PMF was derived by Adler J (2024). (unpublished)

See Also

clvdata to create a clv data object, SetStaticCovariates to add static covariates to an existing clv data object.

gg to fit customer's average spending per transaction with the Gamma-Gamma model

predict to predict expected transactions, probability of being alive, and customer lifetime value for every customer

plot to plot the unconditional expectation as predicted by the fitted model

pmf for the probability to make exactly x transactions in the estimation period, given by the probability mass function (PMF).

newcustomer to predict the expected number of transactions for an average new customer.

The generic functions vcov, summary, fitted.

Examples

# \donttest{
data("apparelTrans")
clv.data.apparel <- clvdata(apparelTrans, date.format = "ymd",
                            time.unit = "w", estimation.split = 52)

# Fit standard ggomnbd model
ggomnbd(clv.data.apparel)

# Give initial guesses for the model parameters
ggomnbd(clv.data.apparel,
     start.params.model = c(r=0.5, alpha=15, b=5, beta=10, s=0.5))


# pass additional parameters to the optimizer (optimx)
#    Use Nelder-Mead as optimization method and print
#    detailed information about the optimization process
apparel.ggomnbd <- ggomnbd(clv.data.apparel,
                     optimx.args = list(method="Nelder-Mead",
                                        control=list(trace=6)))

# estimated coefs
coef(apparel.ggomnbd)

# summary of the fitted model
summary(apparel.ggomnbd)

# predict CLV etc for holdout period
predict(apparel.ggomnbd)

# predict CLV etc for the next 15 periods
predict(apparel.ggomnbd, prediction.end = 15)
# }
# \donttest{
# To estimate the ggomnbd model with static covariates,
#   add static covariates to the data
data("apparelStaticCov")
clv.data.static.cov <-
 SetStaticCovariates(clv.data.apparel,
                     data.cov.life = apparelStaticCov,
                     names.cov.life = c("Gender", "Channel"),
                     data.cov.trans = apparelStaticCov,
                     names.cov.trans = c("Gender", "Channel"))

# Fit ggomnbd with static covariates
ggomnbd(clv.data.static.cov)

# Give initial guesses for both covariate parameters
ggomnbd(clv.data.static.cov, start.params.trans = c(Gender=0.75, Channel=0.7),
                   start.params.life  = c(Gender=0.5, Channel=0.5))

# Use regularization
ggomnbd(clv.data.static.cov, reg.lambdas = c(trans = 5, life=5))

# Force the same coefficient to be used for both covariates
ggomnbd(clv.data.static.cov, names.cov.constr = "Gender",
                   start.params.constr = c(Gender=0.5))

# Fit model only with the Channel covariate for life but
# keep all trans covariates as is
ggomnbd(clv.data.static.cov, names.cov.life = c("Channel"))
# }