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)

Arguments

r

shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process

s

shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes

alpha_0

rate parameter of the Gamma distribution of the purchase process

beta_0

rate parameter for the Gamma distribution for the lifetime process.

vT_i

Number of periods since the customer came alive

vAlpha_i

Vector of individual parameters alpha

vBeta_i

Vector of individual parameters beta

Value

Returns the expected transaction values according to the chosen model.

References

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.

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 BG (2007). “Incorporating time-invariant covariates into the Pareto/NBD and BG/NBD models.” URL http://www.brucehardie.com/notes/019/time_invariant_covariates.pdf.