Fits Pareto/NBD models on transactional data with and without covariates.
# S4 method for clv.data
pnbd(
clv.data,
start.params.model = c(),
use.cor = FALSE,
start.param.cor = c(),
optimx.args = list(),
verbose = TRUE,
...
)
# S4 method for 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 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 alpha, r, beta, and s. 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, 1.0 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).
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 = 40)
# 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
#> 0.7866 5.3349 0.3570 11.6152
#> 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
#> 0.7858 5.3312 0.3606 11.8221
#> 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: 0x7fb5826c0950>
#> <environment: 0x7fb58724c140>
#>
#> $have.bounds
#> [1] FALSE
#>
#> $method
#> [1] "Nelder-Mead"
#>
#> Method: Nelder-Mead
#> Nelder-Mead direct search function minimizer
#> function value for initial parameters = 3324.183952
#> Scaled convergence tolerance is 4.95342e-05
#> Stepsize computed as 0.100000
#> BUILD 5 3370.858623 3309.338871
#> EXTENSION 7 3349.894066 3246.484880
#> LO-REDUCTION 9 3324.183952 3246.484880
#> EXTENSION 11 3311.326140 3192.670595
#> EXTENSION 13 3309.338871 3136.995127
#> EXTENSION 15 3246.890840 3070.661973
#> LO-REDUCTION 17 3246.484880 3070.661973
#> EXTENSION 19 3192.670595 2962.711423
#> LO-REDUCTION 21 3136.995127 2962.711423
#> LO-REDUCTION 23 3092.779244 2962.711423
#> EXTENSION 25 3070.661973 2904.899192
#> EXTENSION 27 2979.692293 2901.790592
#> REFLECTION 29 2971.137892 2896.215344
#> LO-REDUCTION 31 2962.711423 2890.642122
#> LO-REDUCTION 33 2906.533400 2890.642122
#> LO-REDUCTION 35 2904.899192 2890.642122
#> HI-REDUCTION 37 2901.790592 2890.642122
#> LO-REDUCTION 39 2896.854291 2890.642122
#> LO-REDUCTION 41 2896.733398 2890.642122
#> LO-REDUCTION 43 2896.215344 2890.642122
#> EXTENSION 45 2895.169640 2884.464824
#> LO-REDUCTION 47 2891.545912 2884.464824
#> LO-REDUCTION 49 2890.755169 2884.464824
#> EXTENSION 51 2890.642122 2882.953045
#> REFLECTION 53 2888.700681 2882.446051
#> LO-REDUCTION 55 2884.570203 2882.446051
#> HI-REDUCTION 57 2884.464824 2882.446051
#> LO-REDUCTION 59 2883.270678 2882.446051
#> HI-REDUCTION 61 2882.953045 2882.396690
#> HI-REDUCTION 63 2882.895319 2882.314190
#> HI-REDUCTION 65 2882.482512 2882.248928
#> HI-REDUCTION 67 2882.446051 2882.248928
#> HI-REDUCTION 69 2882.396690 2882.240024
#> LO-REDUCTION 71 2882.314190 2882.191572
#> LO-REDUCTION 73 2882.251157 2882.163573
#> LO-REDUCTION 75 2882.248928 2882.163573
#> HI-REDUCTION 77 2882.240024 2882.163573
#> HI-REDUCTION 79 2882.191572 2882.162231
#> HI-REDUCTION 81 2882.167468 2882.140209
#> LO-REDUCTION 83 2882.166824 2882.140209
#> HI-REDUCTION 85 2882.163573 2882.140209
#> LO-REDUCTION 87 2882.162231 2882.138172
#> REFLECTION 89 2882.144473 2882.137304
#> LO-REDUCTION 91 2882.141601 2882.135061
#> HI-REDUCTION 93 2882.140209 2882.132257
#> REFLECTION 95 2882.138172 2882.129096
#> EXTENSION 97 2882.137304 2882.117858
#> LO-REDUCTION 99 2882.135061 2882.117858
#> LO-REDUCTION 101 2882.132257 2882.117858
#> EXTENSION 103 2882.129096 2882.106725
#> EXTENSION 105 2882.126478 2882.091916
#> LO-REDUCTION 107 2882.119912 2882.091916
#> LO-REDUCTION 109 2882.117858 2882.091916
#> EXTENSION 111 2882.106725 2882.071824
#> LO-REDUCTION 113 2882.104616 2882.071824
#> EXTENSION 115 2882.099252 2882.044612
#> EXTENSION 117 2882.091916 2882.012574
#> LO-REDUCTION 119 2882.079287 2882.012574
#> EXTENSION 121 2882.071824 2881.969055
#> EXTENSION 123 2882.044612 2881.898069
#> LO-REDUCTION 125 2882.032117 2881.898069
#> EXTENSION 127 2882.012574 2881.780423
#> LO-REDUCTION 129 2881.969055 2881.780423
#> EXTENSION 131 2881.918692 2881.640737
#> LO-REDUCTION 133 2881.898069 2881.640737
#> EXTENSION 135 2881.809312 2881.425073
#> LO-REDUCTION 137 2881.780423 2881.425073
#> EXTENSION 139 2881.647928 2881.303066
#> EXTENSION 141 2881.640737 2881.229749
#> REFLECTION 143 2881.430767 2881.166043
#> LO-REDUCTION 145 2881.425073 2881.166043
#> HI-REDUCTION 147 2881.303066 2881.166043
#> EXTENSION 149 2881.279120 2881.119888
#> EXTENSION 151 2881.277302 2881.042370
#> LO-REDUCTION 153 2881.229749 2881.042370
#> EXTENSION 155 2881.166043 2880.967832
#> EXTENSION 157 2881.119888 2880.879994
#> LO-REDUCTION 159 2881.087359 2880.879994
#> REFLECTION 161 2881.042370 2880.879853
#> LO-REDUCTION 163 2880.967832 2880.879853
#> LO-REDUCTION 165 2880.901343 2880.879853
#> HI-REDUCTION 167 2880.882262 2880.869537
#> REFLECTION 169 2880.881546 2880.868296
#> REFLECTION 171 2880.879994 2880.855871
#> HI-REDUCTION 173 2880.879853 2880.854640
#> HI-REDUCTION 175 2880.869537 2880.854640
#> EXTENSION 177 2880.868296 2880.818013
#> LO-REDUCTION 179 2880.857420 2880.818013
#> LO-REDUCTION 181 2880.855871 2880.818013
#> LO-REDUCTION 183 2880.854640 2880.818013
#> EXTENSION 185 2880.842474 2880.781988
#> LO-REDUCTION 187 2880.842315 2880.781988
#> LO-REDUCTION 189 2880.832743 2880.781988
#> EXTENSION 191 2880.818013 2880.720303
#> EXTENSION 193 2880.795624 2880.670354
#> LO-REDUCTION 195 2880.787343 2880.670354
#> EXTENSION 197 2880.781988 2880.617628
#> EXTENSION 199 2880.720303 2880.439926
#> LO-REDUCTION 201 2880.680023 2880.439926
#> EXTENSION 203 2880.670354 2880.358120
#> EXTENSION 205 2880.617628 2880.277481
#> EXTENSION 207 2880.473779 2879.924264
#> LO-REDUCTION 209 2880.439926 2879.924264
#> LO-REDUCTION 211 2880.358120 2879.924264
#> LO-REDUCTION 213 2880.277481 2879.924264
#> EXTENSION 215 2880.105129 2879.758291
#> EXTENSION 217 2880.090475 2879.608276
#> LO-REDUCTION 219 2879.960968 2879.608276
#> LO-REDUCTION 221 2879.924264 2879.608276
#> HI-REDUCTION 223 2879.758291 2879.608276
#> REFLECTION 225 2879.713263 2879.550139
#> LO-REDUCTION 227 2879.630410 2879.550139
#> EXTENSION 229 2879.610548 2879.511643
#> LO-REDUCTION 231 2879.608276 2879.511643
#> REFLECTION 233 2879.568497 2879.480952
#> LO-REDUCTION 235 2879.550139 2879.480952
#> LO-REDUCTION 237 2879.526502 2879.480952
#> LO-REDUCTION 239 2879.511643 2879.480932
#> REFLECTION 241 2879.492292 2879.478755
#> LO-REDUCTION 243 2879.484230 2879.476099
#> LO-REDUCTION 245 2879.480952 2879.475526
#> REFLECTION 247 2879.480932 2879.471443
#> HI-REDUCTION 249 2879.478755 2879.471443
#> HI-REDUCTION 251 2879.476099 2879.471443
#> LO-REDUCTION 253 2879.475526 2879.471441
#> LO-REDUCTION 255 2879.473131 2879.470985
#> LO-REDUCTION 257 2879.472498 2879.470985
#> HI-REDUCTION 259 2879.471443 2879.470914
#> HI-REDUCTION 261 2879.471441 2879.470638
#> REFLECTION 263 2879.471127 2879.470444
#> HI-REDUCTION 265 2879.470985 2879.470444
#> HI-REDUCTION 267 2879.470914 2879.470444
#> HI-REDUCTION 269 2879.470638 2879.470444
#> REFLECTION 271 2879.470543 2879.470242
#> LO-REDUCTION 273 2879.470526 2879.470242
#> EXTENSION 275 2879.470450 2879.470087
#> LO-REDUCTION 277 2879.470444 2879.470087
#> LO-REDUCTION 279 2879.470311 2879.470087
#> LO-REDUCTION 281 2879.470242 2879.470087
#> REFLECTION 283 2879.470158 2879.470001
#> HI-REDUCTION 285 2879.470109 2879.470001
#> LO-REDUCTION 287 2879.470101 2879.470001
#> LO-REDUCTION 289 2879.470087 2879.470001
#> Exiting from Nelder Mead minimizer
#> 291 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
#> -0.238902 1.675864 -1.030490 2.452082
#>
#> $value
#> [1] 2879.47
#>
#> $message
#> NULL
#>
#> $convcode
#> [1] 0
#>
#> $fevals
#> function
#> 291
#>
#> $gevals
#> gradient
#> NA
#>
#> $nitns
#> [1] NA
#>
#> $kkt1
#> [1] TRUE
#>
#> $kkt2
#> [1] TRUE
#>
#> $xtimes
#> user.self
#> 0.122
#>
#> Assemble the answers
#> Estimation finished!
# estimated coefs
coef(apparel.pnbd)
#> r alpha s beta
#> 0.787492 5.343412 0.356832 11.612502
# 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 2005-01-03
#> Estimation end 2005-10-10
#> Estimation length 40.0000 Weeks
#>
#> Coefficients:
#> Estimate Std. Error z-val Pr(>|z|)
#> r 0.7875 0.1326 5.941 2.83e-09 ***
#> alpha 5.3434 0.9039 5.912 3.38e-09 ***
#> s 0.3568 0.1833 1.947 0.0515 .
#> beta 11.6125 10.6334 1.092 0.2748
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Optimization info:
#> LL -2879.4700
#> AIC 5766.9400
#> BIC 5781.0258
#> KKT 1 TRUE
#> KKT 2 TRUE
#> fevals 291.0000
#> Method Nelder-Mead
#>
#> Used Options:
#> Correlation FALSE
# predict CLV etc for holdout period
predict(apparel.pnbd)
#> Predicting from 2005-10-11 until (incl.) 2006-07-16 (39.86 Weeks).
#> Estimating gg model to predict spending...
#> Starting estimation...
#> Estimation finished!
#> Id period.first period.last period.length actual.x actual.total.spending
#> 1: 1 2005-10-11 2006-07-16 39.85714 0 0.00
#> 2: 10 2005-10-11 2006-07-16 39.85714 0 0.00
#> 3: 100 2005-10-11 2006-07-16 39.85714 23 737.53
#> 4: 1000 2005-10-11 2006-07-16 39.85714 23 1069.91
#> 5: 1001 2005-10-11 2006-07-16 39.85714 11 364.00
#> ---
#> 246: 1219 2005-10-11 2006-07-16 39.85714 14 413.76
#> 247: 122 2005-10-11 2006-07-16 39.85714 0 0.00
#> 248: 1220 2005-10-11 2006-07-16 39.85714 0 0.00
#> 249: 1221 2005-10-11 2006-07-16 39.85714 9 302.65
#> 250: 1222 2005-10-11 2006-07-16 39.85714 0 0.00
#> PAlive CET DERT predicted.mean.spending predicted.CLV
#> 1: 0.3571652 0.2214654 0.05854369 39.95483 2.339103
#> 2: 0.4226928 0.9277460 0.24524673 55.23031 13.545053
#> 3: 0.9156048 13.5446396 3.58048291 43.57390 156.015602
#> 4: 0.9967799 13.1757590 3.48297048 41.60921 144.923640
#> 5: 0.5100298 3.5289953 0.93287881 45.58153 42.522040
#> ---
#> 246: 0.9579466 3.6111037 0.95458391 33.58728 32.061875
#> 247: 0.3571652 0.2214654 0.05854369 39.95483 2.339103
#> 248: 0.3571652 0.2214654 0.05854369 39.95483 2.339103
#> 249: 0.9434610 4.2993712 1.13652527 34.28958 38.970974
#> 250: 0.4137006 0.5822653 0.15391999 47.35500 7.288881
# predict CLV etc for the next 15 periods
predict(apparel.pnbd, prediction.end = 15)
#> Predicting from 2005-10-11 until (incl.) 2006-01-23 (15 Weeks).
#> Estimating gg model to predict spending...
#> Starting estimation...
#> Estimation finished!
#> Id period.first period.last period.length actual.x actual.total.spending
#> 1: 1 2005-10-11 2006-01-23 15 0 0.00
#> 2: 10 2005-10-11 2006-01-23 15 0 0.00
#> 3: 100 2005-10-11 2006-01-23 15 10 224.71
#> 4: 1000 2005-10-11 2006-01-23 15 10 343.11
#> 5: 1001 2005-10-11 2006-01-23 15 6 206.65
#> ---
#> 246: 1219 2005-10-11 2006-01-23 15 11 325.05
#> 247: 122 2005-10-11 2006-01-23 15 0 0.00
#> 248: 1220 2005-10-11 2006-01-23 15 0 0.00
#> 249: 1221 2005-10-11 2006-01-23 15 7 158.46
#> 250: 1222 2005-10-11 2006-01-23 15 0 0.00
#> PAlive CET DERT predicted.mean.spending predicted.CLV
#> 1: 0.3571652 0.08876335 0.05854369 39.95483 2.339103
#> 2: 0.4226928 0.37184063 0.24524673 55.23031 13.545053
#> 3: 0.9156048 5.42869219 3.58048291 43.57390 156.015602
#> 4: 0.9967799 5.28084482 3.48297048 41.60921 144.923640
#> 5: 0.5100298 1.41442148 0.93287881 45.58153 42.522040
#> ---
#> 246: 0.9579466 1.44733052 0.95458391 33.58728 32.061875
#> 247: 0.3571652 0.08876335 0.05854369 39.95483 2.339103
#> 248: 0.3571652 0.08876335 0.05854369 39.95483 2.339103
#> 249: 0.9434610 1.72318818 1.13652527 34.28958 38.970974
#> 250: 0.4137006 0.23337194 0.15391999 47.35500 7.288881
# }
# \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
#> 0.790394 5.355977 0.278047 7.116957
#> Cor(life,trans)
#> -0.009087
#> 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
#> 1.4197 35.5729 0.2727 8.4407 1.5067
#> life.Channel trans.Gender trans.Channel
#> -1.7005 1.4210 0.4022
#> 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
#> 1.4181 35.5968 0.2730 8.6001 1.5230
#> life.Channel trans.Gender trans.Channel
#> -1.7011 1.4227 0.4023
#> 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
#> 0.7927218 5.4471984 0.3583809 11.6113953 -0.0005224
#> life.Channel trans.Gender trans.Channel
#> -0.0024106 0.0105961 0.0087095
#> 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
#> 1.424 35.470 0.274 7.867 -1.690
#> trans.Channel constr.Gender
#> 0.402 1.416
#> 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
#> 1.4917 32.1539 0.2496 1.7971 -1.6949
#> trans.Gender trans.Channel
#> 1.2723 0.3937
#> 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) {
data("apparelDynCov")
clv.data.dyn.cov <-
SetDynamicCovariates(clv.data = clv.data.apparel,
data.cov.life = apparelDynCov,
data.cov.trans = apparelDynCov,
names.cov.life = c("Marketing", "Gender", "Channel"),
names.cov.trans = c("Marketing", "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))
}
# }