Calculates the probability of a customer being alive at the end of the calibration period, based on a customer's past transaction behavior and the Pareto/NBD model parameters.
pnbd_nocov_PAlive
P(alive) for the Pareto/NBD model without covariates
pnbd_staticcov_PAlive
P(alive) for the Pareto/NBD model with static covariates
pnbd_nocov_PAlive(r, alpha_0, s, beta_0, vX, vT_x, vT_cal)
pnbd_staticcov_PAlive(
r,
alpha_0,
s,
beta_0,
vX,
vT_x,
vT_cal,
vCovParams_trans,
vCovParams_life,
mCov_trans,
mCov_life
)
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.
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.
Vector of estimated parameters for the transaction covariates.
Vector of estimated parameters for the lifetime covariates.
Matrix containing the covariates data affecting the transaction process. One column for each covariate.
Matrix containing the covariates data affecting the lifetime process. One column for each covariate.
Returns a vector with the PAlive for each customer.
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