Computes the expected number of repeat transactions in the interval (0, vT_i] for a randomly selected customer, where 0 is defined as the point when the customer came alive.
pnbd_nocov_expectation(r, s, alpha_0, beta_0, vT_i)
pnbd_staticcov_expectation(r, s, vAlpha_i, vBeta_i, vT_i)shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity 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 of the Gamma distribution of the purchase process
rate parameter for the Gamma distribution for the lifetime process.
Number of periods since the customer came alive
Vector of individual parameters alpha
Vector of individual parameters beta
Returns the expected transaction values according to the chosen model.
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