pnbd_LL {CLVTools}

Pareto/NBD: Log-Likelihood functions

Description

Calculates the Log-Likelihood values for the Pareto/NBD model with and without covariates.

The function pnbd_nocov_LL_ind calculates the individual log-likelihood values for each customer for the given parameters.

The function pnbd_nocov_LL_sum calculates the log-likelihood value summed across customers for the given parameters.

The function pnbd_staticcov_LL_ind calculates the individual log-likelihood values for each customer for the given parameters and covariates.

The function pnbd_staticcov_LL_sum calculates the individual log-likelihood values summed across customers.

Usage

pnbd_nocov_LL_ind(vLogparams, vX, vT_x, vT_cal)

pnbd_nocov_LL_sum(vLogparams, vX, vT_x, vT_cal)

pnbd_staticcov_LL_ind(vParams, vX, vT_x, vT_cal, mCov_life, mCov_trans)

pnbd_staticcov_LL_sum(vParams, vX, vT_x, vT_cal, mCov_life, mCov_trans)

Arguments

vLogparams

vector with the Pareto/NBD model parameters at log scale. See Details.

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.

vParams

vector with the parameters for the Pareto/NBD model at log scale and the static covariates at original scale. See Details.

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.

Details

vLogparams is a vector with model parameters r, alpha_0, s, beta_0 at log-scale, in this order.

vParams is vector with the Pareto/NBD model parameters at log scale, followed by the parameters for the lifetime covariates at original scale and then followed by the parameters for the transaction covariates at original scale

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 vParams 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 vParams at the respective position.

Value

Returns the respective Log-Likelihood value(s) for the Pareto/NBD model with or without covariates.

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