Calculates the discounted expected residual transactions.

`pnbd_nocov_DERT`

Discounted expected residual transactions for the Pareto/NBD model without covariates`pnbd_staticcov_DERT`

Discounted 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 )

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

alpha_0 | scale parameter of the Gamma distribution 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 |

beta_0 | scale parameter for the Gamma distribution for the lifetime process. |

continuous_discount_factor | continuous discount factor to use |

vX | Frequency vector of length n counting the numbers of purchases. |

vT_x | Recency vector of length n. |

vT_cal | Vector of length n indicating the total number of periods of observation. |

mCov_life | Matrix containing the covariates data affecting the lifetime process. One column for each covariate. |

mCov_trans | Matrix containing the covariates data affecting the transaction process. One column for each covariate. |

vCovParams_life | Vector of estimated parameters for the lifetime covariates. |

vCovParams_trans | Vector of estimated parameters for the transaction covariates. |

Returns a vector with the DERT 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.

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.