Calculates the discounted expected residual transactions.
pnbd_nocov_DERTDiscounted expected residual transactions for the Pareto/NBD model without covariates
pnbd_staticcov_DERTDiscounted expected residual transactions for the Pareto/NBD model with static covariates
pnbd_nocov_DERT(
r,
alpha_0,
s,
beta_0,
continuous_discount_factor,
vX,
vT_x,
vT_cal
)pnbd_staticcov_DERT(
r,
alpha_0,
s,
beta_0,
continuous_discount_factor,
vX,
vT_x,
vT_cal,
mCov_life,
mCov_trans,
vCovParams_life,
vCovParams_trans
)
Returns a vector with the DERT for each customer.
shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process
rate parameter of the Gamma distribution of the purchase process
shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes
rate parameter for the Gamma distribution for the lifetime process.
continuous discount factor to use
Frequency vector of length n counting the numbers of purchases.
Recency vector of length n.
Vector of length n indicating the total number of periods of observation.
Matrix containing the covariates data affecting the lifetime process. One column for each covariate.
Matrix containing the covariates data affecting the transaction process. One column for each covariate.
Vector of estimated parameters for the lifetime covariates.
Vector of estimated parameters for the transaction covariates.
mCov_trans is a matrix containing the covariates data of
the time-invariant covariates that affect the transaction process.
Each column represents a different covariate. For every column a gamma parameter
needs to added to vCovParams_trans at the respective position.
mCov_life is a matrix containing the covariates data of
the time-invariant covariates that affect the lifetime process.
Each column represents a different covariate. For every column a gamma parameter
needs to added to vCovParams_life at the respective position.
Schmittlein DC, Morrison DG, Colombo R (1987). “Counting Your Customers: Who-Are They and What Will They Do Next?” Management Science, 33(1), 1-24.
Bachmann P, Meierer M, Naef, J (2021). “The Role of Time-Varying Contextual Factors in Latent Attrition Models for Customer Base Analysis” Marketing Science 40(4). 783-809.
Fader PS, Hardie BGS (2005). “A Note on Deriving the Pareto/NBD Model and Related Expressions.” URL http://www.brucehardie.com/notes/009/pareto_nbd_derivations_2005-11-05.pdf.
Fader PS, Hardie BGS (2007). “Incorporating time-invariant covariates into the Pareto/NBD and BG/NBD models.” URL http://www.brucehardie.com/notes/019/time_invariant_covariates.pdf.
Fader PS, Hardie BGS (2020). “Deriving an Expression for P(X(t)=x) Under the Pareto/NBD Model.” URL https://www.brucehardie.com/notes/012/pareto_NBD_pmf_derivation_rev.pdf