pnbd_PAlive {CLVTools} R Documentation

## Pareto/NBD: Probability of Being Alive

### Description

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

### Usage

```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
)
```

### Arguments

 `r` shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process `alpha_0` rate 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` rate parameter for the Gamma distribution for the lifetime process. `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. `vCovParams_trans` Vector of estimated parameters for the transaction covariates. `vCovParams_life` Vector of estimated parameters for the lifetime covariates. `mCov_trans` Matrix containing the covariates data affecting the transaction process. One column for each covariate. `mCov_life` Matrix containing the covariates data affecting the lifetime process. One column for each covariate.

### Details

`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.

### Value

Returns a vector with the PAlive for each customer.

### 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.

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

[Package CLVTools version 0.9.0 Index]