Fits Pareto/NBD models on transactional data with and without covariates.
# S4 method for class 'clv.data'
pnbd(
clv.data,
start.params.model = c(),
use.cor = FALSE,
start.param.cor = c(),
optimx.args = list(),
verbose = TRUE,
...
)
# S4 method for class 'clv.data.static.covariates'
pnbd(
clv.data,
start.params.model = c(),
use.cor = FALSE,
start.param.cor = c(),
optimx.args = list(),
verbose = TRUE,
names.cov.life = c(),
names.cov.trans = c(),
start.params.life = c(),
start.params.trans = c(),
names.cov.constr = c(),
start.params.constr = c(),
reg.lambdas = c(),
...
)
# S4 method for class 'clv.data.dynamic.covariates'
pnbd(
clv.data,
start.params.model = c(),
use.cor = FALSE,
start.param.cor = c(),
optimx.args = list(),
verbose = TRUE,
names.cov.life = c(),
names.cov.trans = c(),
start.params.life = c(),
start.params.trans = c(),
names.cov.constr = c(),
start.params.constr = c(),
reg.lambdas = c(),
...
)The data object on which the model is fitted.
Named start parameters containing the optimization start parameters for the model without covariates.
Whether the correlation between the transaction and lifetime process should be estimated.
Start parameter for the optimization of the correlation.
Additional arguments to control the optimization which are forwarded to optimx::optimx.
If multiple optimization methods are specified, only the result of the last method is further processed.
Show details about the running of the function.
Ignored
Which of the set Lifetime covariates should be used. Missing parameter indicates all covariates shall be used.
Which of the set Transaction covariates should be used. Missing parameter indicates all covariates shall be used.
Named start parameters containing the optimization start parameters for all lifetime covariates.
Named start parameters containing the optimization start parameters for all transaction covariates.
Which covariates should be forced to use the same parameters for the lifetime and transaction process. The covariates need to be present as both, lifetime and transaction covariates.
Named start parameters containing the optimization start parameters for the constraint covariates.
Named lambda parameters used for the L2 regularization of the lifetime and the transaction covariate parameters. Lambdas have to be >= 0.
Depending on the data object on which the model was fit, pnbd returns either an object of
class clv.pnbd, clv.pnbd.static.cov, or clv.pnbd.dynamic.cov.
The function summary can be used to obtain and print a summary of the results.
The generic accessor functions coefficients, vcov, fitted,
logLik, AIC, BIC, and nobs are available.
Model parameters for the Pareto/NBD model are r, alpha, s, and beta. s: shape parameter of the Gamma distribution for the lifetime process.
The smaller s, the stronger the heterogeneity of customer lifetimes. beta: rate parameter for the Gamma distribution for the lifetime process. r: shape parameter of the Gamma distribution of the purchase process.
The smaller r, the stronger the heterogeneity of the purchase process.alpha: rate parameter of the Gamma distribution of the purchase process.
Based on these parameters, the average purchase rate while customers are active is r/alpha and the average dropout rate is s/beta.
Ideally, the starting parameters for r and s represent your best guess concerning the heterogeneity of customers in their buy and die rate. If covariates are included into the model additionally parameters for the covariates affecting the attrition and the purchase process are part of the model.
If no start parameters are given, r=0.5, alpha=15, s=0.5, beta=10 is used for all model parameters and 0.1 for covariate parameters. The model start parameters are required to be > 0.
The Pareto/NBD is the first model addressing the issue of modeling customer purchases and
attrition simultaneously for non-contractual settings. The model uses a Pareto distribution,
a combination of an Exponential and a Gamma distribution, to explicitly model customers'
(unobserved) attrition behavior in addition to customers' purchase process.
In general, the Pareto/NBD model consist of two parts. A first process models the purchase
behavior of customers as long as the customers are active. A second process models customers'
attrition. Customers live (and buy) for a certain unknown time until they become inactive
and "die". Customer attrition is unobserved. Inactive customers may not be reactivated.
For technical details we refer to the original paper by Schmittlein, Morrison and Colombo
(1987) and the detailed technical note of Fader and Hardie (2005).
The standard Pareto/NBD model captures heterogeneity was solely using Gamma distributions. However, often exogenous knowledge, such as for example customer demographics, is available. The supplementary knowledge may explain part of the heterogeneity among the customers and therefore increase the predictive accuracy of the model. In addition, we can rely on these parameter estimates for inference, i.e. identify and quantify effects of contextual factors on the two underlying purchase and attrition processes. For technical details we refer to the technical note by Fader and Hardie (2007).
In many real-world applications customer purchase and attrition behavior may be influenced by covariates that vary over time. In consequence, the timing of a purchase and the corresponding value of at covariate a that time becomes relevant. Time-varying covariates can affect customer on aggregated level as well as on an individual level: In the first case, all customers are affected simultaneously, in the latter case a covariate is only relevant for a particular customer. For technical details we refer to the paper by Bachmann, Meierer and Näf (2020).
The Pareto/NBD model with dynamic covariates can currently not be fit with data that has a temporal resolution
of less than one day (data that was built with time unit hours).
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
clvdata to create a clv data object, SetStaticCovariates
to add static covariates to an existing clv data object.
gg to fit customer's average spending per transaction with the Gamma-Gamma model
predict to predict expected transactions, probability of being alive, and customer lifetime value for every customer
plot to plot the unconditional expectation as predicted by the fitted model
pmf for the probability to make exactly x transactions in the estimation period, given by the probability mass function (PMF).
newcustomer to predict the expected number of transactions for an average new customer.
The generic functions vcov, summary, fitted.
SetDynamicCovariates to add dynamic covariates on which the pnbd model can be fit.
# \donttest{
data("apparelTrans")
clv.data.apparel <- clvdata(apparelTrans, date.format = "ymd",
time.unit = "w", estimation.split = 52)
# Fit standard pnbd model
pnbd(clv.data.apparel)
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD Standard Model
#>
#> Call:
#> pnbd(clv.data = clv.data.apparel)
#>
#> Coefficients:
#> r alpha s beta
#> 1.3080 46.3352 0.4794 70.2292
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
# Give initial guesses for the model parameters
pnbd(clv.data.apparel,
start.params.model = c(r=0.5, alpha=15, s=0.5, beta=10))
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD Standard Model
#>
#> Call:
#> pnbd(clv.data = clv.data.apparel, start.params.model = c(r = 0.5,
#> alpha = 15, s = 0.5, beta = 10))
#>
#> Coefficients:
#> r alpha s beta
#> 1.3080 46.3352 0.4794 70.2292
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
# pass additional parameters to the optimizer (optimx)
# Use Nelder-Mead as optimization method and print
# detailed information about the optimization process
apparel.pnbd <- pnbd(clv.data.apparel,
optimx.args = list(method="Nelder-Mead",
control=list(trace=6)))
#> Starting estimation...
#> fn is fn1
#> Looking for method = Nelder-Mead
#> Methods to be used:[1] "Nelder-Mead"
#> optcfg:$fname
#> [1] "fn1"
#>
#> $npar
#> [1] 4
#>
#> $ctrl
#> $ctrl$follow.on
#> [1] FALSE
#>
#> $ctrl$save.failures
#> [1] TRUE
#>
#> $ctrl$trace
#> [1] 6
#>
#> $ctrl$kkt
#> [1] TRUE
#>
#> $ctrl$all.methods
#> [1] FALSE
#>
#> $ctrl$starttests
#> [1] FALSE
#>
#> $ctrl$maximize
#> [1] FALSE
#>
#> $ctrl$dowarn
#> [1] TRUE
#>
#> $ctrl$usenumDeriv
#> [1] FALSE
#>
#> $ctrl$kkttol
#> [1] 0.001
#>
#> $ctrl$kkt2tol
#> [1] 1e-06
#>
#> $ctrl$badval
#> [1] 8.988466e+307
#>
#> $ctrl$scaletol
#> [1] 3
#>
#> $ctrl$have.bounds
#> [1] FALSE
#>
#>
#> $usenumDeriv
#> [1] FALSE
#>
#> $ufn
#> function (par)
#> fn(par, ...)
#> <bytecode: 0x11dce02d8>
#> <environment: 0x12952b610>
#>
#> $have.bounds
#> [1] FALSE
#>
#> $method
#> [1] "Nelder-Mead"
#>
#> Method: Nelder-Mead
#> Nelder-Mead direct search function minimizer
#> function value for initial parameters = 3577.490724
#> Scaled convergence tolerance is 5.33088e-05
#> Stepsize computed as 0.270805
#> BUILD 5 3618.102926 3557.499818
#> EXTENSION 7 3588.766454 3515.206187
#> LO-REDUCTION 9 3577.490724 3515.206187
#> LO-REDUCTION 11 3560.629019 3515.206187
#> LO-REDUCTION 13 3557.499818 3515.206187
#> EXTENSION 15 3538.949426 3502.124657
#> EXTENSION 17 3522.301702 3491.315153
#> LO-REDUCTION 19 3520.130144 3491.315153
#> LO-REDUCTION 21 3515.206187 3491.315153
#> EXTENSION 23 3502.124657 3487.029141
#> LO-REDUCTION 25 3494.722953 3487.029141
#> REFLECTION 27 3493.410647 3485.989378
#> LO-REDUCTION 29 3491.315153 3485.989378
#> HI-REDUCTION 31 3487.841273 3485.989378
#> HI-REDUCTION 33 3487.231847 3485.989378
#> HI-REDUCTION 35 3487.029141 3485.989378
#> EXTENSION 37 3486.635771 3484.509002
#> LO-REDUCTION 39 3486.256159 3484.509002
#> LO-REDUCTION 41 3486.135308 3484.509002
#> LO-REDUCTION 43 3485.989378 3484.509002
#> LO-REDUCTION 45 3485.602665 3484.509002
#> REFLECTION 47 3485.131743 3484.022296
#> LO-REDUCTION 49 3485.094658 3484.022296
#> REFLECTION 51 3484.734465 3483.944565
#> EXTENSION 53 3484.509002 3483.051738
#> HI-REDUCTION 55 3484.390375 3483.051738
#> EXTENSION 57 3484.022296 3482.672127
#> LO-REDUCTION 59 3483.944565 3482.672127
#> LO-REDUCTION 61 3483.885871 3482.672127
#> HI-REDUCTION 63 3483.183497 3482.672127
#> LO-REDUCTION 65 3483.051738 3482.605147
#> LO-REDUCTION 67 3482.869919 3482.605147
#> HI-REDUCTION 69 3482.678491 3482.605147
#> LO-REDUCTION 71 3482.677209 3482.586838
#> HI-REDUCTION 73 3482.672127 3482.586838
#> LO-REDUCTION 75 3482.626645 3482.586838
#> HI-REDUCTION 77 3482.606469 3482.584373
#> REFLECTION 79 3482.605147 3482.570883
#> HI-REDUCTION 81 3482.591298 3482.570883
#> HI-REDUCTION 83 3482.586838 3482.570883
#> REFLECTION 85 3482.584373 3482.562448
#> HI-REDUCTION 87 3482.578339 3482.562448
#> LO-REDUCTION 89 3482.574087 3482.562448
#> EXTENSION 91 3482.570883 3482.546872
#> HI-REDUCTION 93 3482.569656 3482.546872
#> LO-REDUCTION 95 3482.564846 3482.546872
#> REFLECTION 97 3482.562448 3482.545725
#> HI-REDUCTION 99 3482.559624 3482.545725
#> EXTENSION 101 3482.552132 3482.529225
#> LO-REDUCTION 103 3482.548141 3482.529225
#> LO-REDUCTION 105 3482.546872 3482.529225
#> EXTENSION 107 3482.545725 3482.524888
#> REFLECTION 109 3482.539489 3482.523911
#> EXTENSION 111 3482.532682 3482.508336
#> LO-REDUCTION 113 3482.529225 3482.508336
#> LO-REDUCTION 115 3482.524888 3482.508336
#> LO-REDUCTION 117 3482.523911 3482.508336
#> LO-REDUCTION 119 3482.518928 3482.508336
#> LO-REDUCTION 121 3482.518699 3482.508336
#> EXTENSION 123 3482.516119 3482.501974
#> EXTENSION 125 3482.511347 3482.491345
#> LO-REDUCTION 127 3482.509855 3482.491345
#> EXTENSION 129 3482.508336 3482.479883
#> LO-REDUCTION 131 3482.501974 3482.479883
#> EXTENSION 133 3482.491895 3482.451710
#> LO-REDUCTION 135 3482.491345 3482.451710
#> LO-REDUCTION 137 3482.485097 3482.451710
#> EXTENSION 139 3482.479883 3482.426861
#> LO-REDUCTION 141 3482.467825 3482.426861
#> EXTENSION 143 3482.464402 3482.392452
#> EXTENSION 145 3482.451710 3482.362389
#> EXTENSION 147 3482.434120 3482.326554
#> LO-REDUCTION 149 3482.426861 3482.326554
#> EXTENSION 151 3482.392452 3482.234085
#> LO-REDUCTION 153 3482.362389 3482.234085
#> LO-REDUCTION 155 3482.333002 3482.234085
#> LO-REDUCTION 157 3482.326554 3482.234085
#> LO-REDUCTION 159 3482.270427 3482.228485
#> EXTENSION 161 3482.249626 3482.187145
#> LO-REDUCTION 163 3482.243048 3482.187145
#> EXTENSION 165 3482.234085 3482.143757
#> LO-REDUCTION 167 3482.228485 3482.143757
#> EXTENSION 169 3482.187150 3482.076034
#> LO-REDUCTION 171 3482.187145 3482.076034
#> EXTENSION 173 3482.168268 3482.051621
#> LO-REDUCTION 175 3482.143757 3482.051621
#> REFLECTION 177 3482.100647 3482.024184
#> REFLECTION 179 3482.076034 3482.022426
#> HI-REDUCTION 181 3482.051696 3482.022426
#> REFLECTION 183 3482.051621 3481.999804
#> LO-REDUCTION 185 3482.030557 3481.999804
#> LO-REDUCTION 187 3482.024184 3481.999804
#> LO-REDUCTION 189 3482.022426 3481.999429
#> HI-REDUCTION 191 3482.014260 3481.999429
#> LO-REDUCTION 193 3482.005517 3481.998573
#> LO-REDUCTION 195 3482.005327 3481.998573
#> REFLECTION 197 3481.999804 3481.992755
#> HI-REDUCTION 199 3481.999429 3481.992755
#> LO-REDUCTION 201 3481.999171 3481.992755
#> REFLECTION 203 3481.998573 3481.992659
#> HI-REDUCTION 205 3481.996090 3481.992659
#> EXTENSION 207 3481.995202 3481.989353
#> EXTENSION 209 3481.994861 3481.987456
#> LO-REDUCTION 211 3481.992755 3481.987456
#> REFLECTION 213 3481.992659 3481.986721
#> EXTENSION 215 3481.989353 3481.980449
#> LO-REDUCTION 217 3481.987760 3481.980449
#> LO-REDUCTION 219 3481.987456 3481.980449
#> LO-REDUCTION 221 3481.986721 3481.980449
#> EXTENSION 223 3481.984187 3481.978655
#> LO-REDUCTION 225 3481.983425 3481.978655
#> LO-REDUCTION 227 3481.981552 3481.978655
#> REFLECTION 229 3481.980449 3481.978351
#> LO-REDUCTION 231 3481.979349 3481.978351
#> LO-REDUCTION 233 3481.978943 3481.978351
#> HI-REDUCTION 235 3481.978704 3481.978351
#> REFLECTION 237 3481.978655 3481.978253
#> LO-REDUCTION 239 3481.978629 3481.978253
#> LO-REDUCTION 241 3481.978532 3481.978253
#> HI-REDUCTION 243 3481.978351 3481.978253
#> HI-REDUCTION 245 3481.978345 3481.978253
#> LO-REDUCTION 247 3481.978321 3481.978252
#> LO-REDUCTION 249 3481.978295 3481.978236
#> Exiting from Nelder Mead minimizer
#> 251 function evaluations used
#> Post processing for method Nelder-Mead
#> Successful convergence!
#> Compute Hessian approximation at finish of Nelder-Mead
#> Compute gradient approximation at finish of Nelder-Mead
#> Save results from method Nelder-Mead
#> $par
#> log.r log.alpha log.s log.beta
#> 0.2678451 3.8353761 -0.7332306 4.2554487
#>
#> $value
#> [1] 3481.978
#>
#> $message
#> NULL
#>
#> $convcode
#> [1] 0
#>
#> $fevals
#> function
#> 251
#>
#> $gevals
#> gradient
#> NA
#>
#> $nitns
#> [1] NA
#>
#> $kkt1
#> [1] TRUE
#>
#> $kkt2
#> [1] TRUE
#>
#> $xtimes
#> user.self
#> 0.047
#>
#> Assemble the answers
#> Estimation finished!
# estimated coefs
coef(apparel.pnbd)
#> r alpha s beta
#> 1.3071447 46.3108420 0.4803547 70.4884348
# summary of the fitted model
summary(apparel.pnbd)
#> Pareto/NBD Standard Model
#>
#> Call:
#> pnbd(clv.data = clv.data.apparel, optimx.args = list(method = "Nelder-Mead",
#> control = list(trace = 6)))
#>
#> Fitting period:
#> Estimation start 2005-01-02
#> Estimation end 2006-01-01
#> Estimation length 52.0000 Weeks
#>
#> Coefficients:
#> Estimate Std. Error z-val Pr(>|z|)
#> r 1.3071 0.2123 6.157 7.42e-10 ***
#> alpha 46.3108 7.5368 6.145 8.01e-10 ***
#> s 0.4804 1.0819 0.444 0.657
#> beta 70.4884 203.3985 0.347 0.729
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Optimization info:
#> LL -3481.9782
#> AIC 6971.9565
#> BIC 6989.5442
#> KKT 1 TRUE
#> KKT 2 TRUE
#> fevals 251.0000
#> Method Nelder-Mead
#>
#> Used Options:
#> Correlation FALSE
# predict CLV etc for holdout period
predict(apparel.pnbd)
#> Predicting from 2006-01-02 until (incl.) 2010-12-20 (259.14 Weeks).
#> Estimating gg model to predict spending...
#> Starting estimation...
#> Estimation finished!
#> Key: <Id>
#> Id period.first period.last period.length actual.x
#> <char> <Date> <Date> <num> <int>
#> 1: 1 2006-01-02 2010-12-20 259.1429 5
#> 2: 10 2006-01-02 2010-12-20 259.1429 2
#> 3: 100 2006-01-02 2010-12-20 259.1429 9
#> 4: 101 2006-01-02 2010-12-20 259.1429 0
#> 5: 102 2006-01-02 2010-12-20 259.1429 0
#> ---
#> 596: 95 2006-01-02 2010-12-20 259.1429 5
#> 597: 96 2006-01-02 2010-12-20 259.1429 3
#> 598: 97 2006-01-02 2010-12-20 259.1429 0
#> 599: 98 2006-01-02 2010-12-20 259.1429 0
#> 600: 99 2006-01-02 2010-12-20 259.1429 1
#> actual.total.spending PAlive CET DERT
#> <num> <num> <num> <num>
#> 1: 524.10 0.9163171 4.080260 0.21731960
#> 2: 63.31 0.8411806 3.745685 0.19949974
#> 3: 333.35 0.6578365 1.659621 0.08839341
#> 4: 0.00 0.8946531 3.983793 0.21218163
#> 5: 0.00 0.6578365 1.659621 0.08839341
#> ---
#> 596: 113.80 0.9400910 4.186123 0.22295799
#> 597: 253.61 0.6578365 1.659621 0.08839341
#> 598: 0.00 0.6578365 1.659621 0.08839341
#> 599: 0.00 0.9655447 4.299465 0.22899475
#> 600: 13.99 0.9384998 5.990383 0.31905509
#> predicted.mean.spending predicted.total.spending predicted.CLV
#> <num> <num> <num>
#> 1: 63.77959 260.23732 13.860556
#> 2: 38.10316 142.72242 7.601570
#> 3: 37.12434 61.61231 3.281547
#> 4: 33.08404 131.79995 7.019826
#> 5: 37.12434 61.61231 3.281547
#> ---
#> 596: 28.29614 118.45113 6.308851
#> 597: 37.12434 61.61231 3.281547
#> 598: 37.12434 61.61231 3.281547
#> 599: 34.87739 149.95411 7.986738
#> 600: 16.11315 96.52392 5.140981
# predict CLV etc for the next 15 periods
predict(apparel.pnbd, prediction.end = 15)
#> Predicting from 2006-01-02 until (incl.) 2006-04-16 (15 Weeks).
#> Estimating gg model to predict spending...
#> Starting estimation...
#> Estimation finished!
#> Key: <Id>
#> Id period.first period.last period.length actual.x
#> <char> <Date> <Date> <int> <int>
#> 1: 1 2006-01-02 2006-04-16 15 2
#> 2: 10 2006-01-02 2006-04-16 15 0
#> 3: 100 2006-01-02 2006-04-16 15 0
#> 4: 101 2006-01-02 2006-04-16 15 0
#> 5: 102 2006-01-02 2006-04-16 15 0
#> ---
#> 596: 95 2006-01-02 2006-04-16 15 0
#> 597: 96 2006-01-02 2006-04-16 15 0
#> 598: 97 2006-01-02 2006-04-16 15 0
#> 599: 98 2006-01-02 2006-04-16 15 0
#> 600: 99 2006-01-02 2006-04-16 15 0
#> actual.total.spending PAlive CET DERT
#> <num> <num> <num> <num>
#> 1: 217.5 0.9163171 0.3136059 0.21731960
#> 2: 0.0 0.8411806 0.2878907 0.19949974
#> 3: 0.0 0.6578365 0.1275573 0.08839341
#> 4: 0.0 0.8946531 0.3061915 0.21218163
#> 5: 0.0 0.6578365 0.1275573 0.08839341
#> ---
#> 596: 0.0 0.9400910 0.3217425 0.22295799
#> 597: 0.0 0.6578365 0.1275573 0.08839341
#> 598: 0.0 0.6578365 0.1275573 0.08839341
#> 599: 0.0 0.9655447 0.3304539 0.22899475
#> 600: 0.0 0.9384998 0.4604166 0.31905509
#> predicted.mean.spending predicted.total.spending predicted.CLV
#> <num> <num> <num>
#> 1: 63.77959 20.001658 13.860556
#> 2: 38.10316 10.969545 7.601570
#> 3: 37.12434 4.735479 3.281547
#> 4: 33.08404 10.130052 7.019826
#> 5: 37.12434 4.735479 3.281547
#> ---
#> 596: 28.29614 9.104071 6.308851
#> 597: 37.12434 4.735479 3.281547
#> 598: 37.12434 4.735479 3.281547
#> 599: 34.87739 11.525368 7.986738
#> 600: 16.11315 7.418761 5.140981
# }
# \donttest{
# Estimate correlation as well
pnbd(clv.data.apparel, use.cor = TRUE)
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD Standard Model
#>
#> Call:
#> pnbd(clv.data = clv.data.apparel, use.cor = TRUE)
#>
#> Coefficients:
#> r alpha s beta
#> 1.308e+00 4.635e+01 4.812e-01 7.070e+01
#> Cor(life,trans)
#> 1.337e-04
#> KKT1: TRUE
#> KKT2: FALSE
#>
#> Used Options:
#> Correlation: TRUE
# }
# \donttest{
# To estimate the pnbd model with static covariates,
# add static covariates to the data
data("apparelStaticCov")
clv.data.static.cov <-
SetStaticCovariates(clv.data.apparel,
data.cov.life = apparelStaticCov,
names.cov.life = c("Gender", "Channel"),
data.cov.trans = apparelStaticCov,
names.cov.trans = c("Gender", "Channel"))
# Fit pnbd with static covariates
pnbd(clv.data.static.cov)
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD with Static Covariates Model
#>
#> Call:
#> pnbd(clv.data = clv.data.static.cov)
#>
#> Coefficients:
#> r alpha s beta life.Gender
#> 1.64611 103.81807 0.37820 308.80294 -0.01539
#> life.Channel trans.Gender trans.Channel
#> 2.75180 0.47034 0.73633
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
#> Constraints: FALSE
#> Regularization: FALSE
# Give initial guesses for both covariate parameters
pnbd(clv.data.static.cov, start.params.trans = c(Gender=0.75, Channel=0.7),
start.params.life = c(Gender=0.5, Channel=0.5))
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD with Static Covariates Model
#>
#> Call:
#> pnbd(clv.data = clv.data.static.cov, start.params.life = c(Gender = 0.5,
#> Channel = 0.5), start.params.trans = c(Gender = 0.75, Channel = 0.7))
#>
#> Coefficients:
#> r alpha s beta life.Gender
#> 1.64557 103.94063 0.37849 324.01338 -0.01451
#> life.Channel trans.Gender trans.Channel
#> 2.79911 0.47025 0.73813
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
#> Constraints: FALSE
#> Regularization: FALSE
# Use regularization
pnbd(clv.data.static.cov, reg.lambdas = c(trans = 5, life=5))
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD with Static Covariates Model
#>
#> Call:
#> pnbd(clv.data = clv.data.static.cov, reg.lambdas = c(trans = 5,
#> life = 5))
#>
#> Coefficients:
#> r alpha s beta life.Gender
#> 1.317e+00 4.693e+01 4.769e-01 6.951e+01 -7.733e-04
#> life.Channel trans.Gender trans.Channel
#> -6.011e-05 6.173e-03 6.701e-03
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
#> Constraints: FALSE
#> Regularization: TRUE
# Force the same coefficient to be used for both covariates
pnbd(clv.data.static.cov, names.cov.constr = "Gender",
start.params.constr = c(Gender=0.5))
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD with Static Covariates Model
#>
#> Call:
#> pnbd(clv.data = clv.data.static.cov, names.cov.constr = "Gender",
#> start.params.constr = c(Gender = 0.5))
#>
#> Coefficients:
#> r alpha s beta life.Channel
#> 1.6171 104.1847 0.3552 563.3726 3.1050
#> trans.Channel constr.Gender
#> 0.7324 0.4923
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
#> Constraints: TRUE
#> Regularization: FALSE
# Fit model only with the Channel covariate for life but
# keep all trans covariates as is
pnbd(clv.data.static.cov, names.cov.life = c("Channel"))
#> Starting estimation...
#> Estimation finished!
#> Pareto/NBD with Static Covariates Model
#>
#> Call:
#> pnbd(clv.data = clv.data.static.cov, names.cov.life = c("Channel"))
#>
#> Coefficients:
#> r alpha s beta life.Channel
#> 1.6470 103.9005 0.3775 309.2000 2.7485
#> trans.Gender trans.Channel
#> 0.4710 0.7364
#> KKT1: TRUE
#> KKT2: TRUE
#>
#> Used Options:
#> Correlation: FALSE
#> Constraints: FALSE
#> Regularization: FALSE
# }
# Add dynamic covariates data to the data object
# add dynamic covariates to the data
# \donttest{
if (FALSE) { # \dontrun{
data("apparelDynCov")
clv.data.dyn.cov <-
SetDynamicCovariates(clv.data = clv.data.apparel,
data.cov.life = apparelDynCov,
data.cov.trans = apparelDynCov,
names.cov.life = c("High.Season", "Gender", "Channel"),
names.cov.trans = c("High.Season", "Gender", "Channel"),
name.date = "Cov.Date")
# Fit PNBD with dynamic covariates
pnbd(clv.data.dyn.cov)
# The same fitting options as for the
# static covariate are available
pnbd(clv.data.dyn.cov, reg.lambdas = c(trans=10, life=2))
} # }
# }