Fits the Gamma-Gamma model on a given object of class clv.data to predict customers' mean
spending per transaction.
# S4 method for clv.data gg( clv.data, start.params.model = c(), optimx.args = list(), remove.first.transaction = TRUE, verbose = TRUE, ... )
Arguments
| clv.data | The data object on which the model is fitted. |
|---|---|
| start.params.model | Named start parameters containing the optimization start parameters for the model without covariates. |
| optimx.args | Additional arguments to control the optimization which are forwarded to |
| remove.first.transaction | Whether customer's first transaction are removed. If |
| verbose | Show details about the running of the function. |
| ... | Ignored |
Value
An object of class clv.gg is returned.
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.
Details
Model parameters for the G/G model are p, q, and gamma.
p: shape parameter of the Gamma distribution of the spending process.
q: shape parameter of the Gamma distribution to account for customer heterogeneity.
gamma: scale parameter of the Gamma distribution to account for customer heterogeneity.
If no start parameters are given, 1.0 is used for all model parameters. All parameters are required
to be > 0.
The Gamma-Gamma model cannot be estimated for data that contains negative prices. Customers with a mean spending of zero or a transaction count of zero are ignored during model fitting.
The G/G model
The G/G model allows to predict a value for future customer transactions. Usually, the G/G model is used in combination with a probabilistic model predicting customer transaction such as the Pareto/NBD or the BG/NBD model.
References
Colombo R, Jiang W (1999). “A stochastic RFM model.” Journal of Interactive Marketing, 13(3), 2–12.
Fader PS, Hardie BG, Lee K (2005). “RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.” Journal of Marketing Research, 42(4), 415–430.
Fader PS, Hardie BG (2013). “The Gamma-Gamma Model of Monetary Value.” URL http://www.brucehardie.com/notes/025/gamma_gamma.pdf.
See also
clvdata to create a clv data object.
predict to predict expected mean spending for every customer.
plot to plot the density of customer's mean transaction value compared to the model's prediction.
Examples
# \donttest{ data("apparelTrans") clv.data.apparel <- clvdata(apparelTrans, date.format = "ymd", time.unit = "w", estimation.split = 40) # Fit the gg model gg(clv.data.apparel)#>#>#> Gamma-Gamma Model #> #> Call: #> gg(clv.data = clv.data.apparel) #> #> Coefficients: #> p q gamma #> 2.305 17.148 279.974 #> KKT1: TRUE #> KKT2: TRUE# Give initial guesses for the model parameters gg(clv.data.apparel, start.params.model = c(p=0.5, q=15, gamma=2))#>#>#> Gamma-Gamma Model #> #> Call: #> gg(clv.data = clv.data.apparel, start.params.model = c(p = 0.5, #> q = 15, gamma = 2)) #> #> Coefficients: #> p q gamma #> 2.305 17.148 279.976 #> KKT1: TRUE #> KKT2: TRUE# pass additional parameters to the optimizer (optimx) # Use Nelder-Mead as optimization method and print # detailed information about the optimization process apparel.gg <- gg(clv.data.apparel, optimx.args = list(method="Nelder-Mead", control=list(trace=6)))#>#> fn is fn #> Looking for method = Nelder-Mead #> Methods to be used:[1] "Nelder-Mead" #> optcfg:$fname #> [1] "fn" #> #> $npar #> [1] 3 #> #> $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 (LL.params, LL.function.sum, use.interlayer.constr, #> use.interlayer.reg, reg.lambda.trans, reg.lambda.life, use.cor, #> ...) #> { #> all.other.args <- list(...) #> interlayers.to.use <- list(callLL = interlayer_callLL) #> if (use.cor == T & !anyNA(use.cor)) #> interlayers.to.use <- c(correlation = interlayer_correlation, #> interlayers.to.use) #> if (use.interlayer.reg == T & !is.null(reg.lambda.trans) & #> !anyNA(reg.lambda.trans) & !is.null(reg.lambda.life) & #> !anyNA(reg.lambda.life)) { #> interlayers.to.use <- c(regularization = interlayer_regularization, #> interlayers.to.use) #> } #> if (use.interlayer.constr == TRUE) { #> interlayers.to.use <- c(constraints = interlayer_constraints, #> interlayers.to.use) #> } #> interlayer.call.args <- list(next.interlayers = interlayers.to.use, #> LL.function.sum = LL.function.sum, LL.params = LL.params, #> reg.lambda.life = reg.lambda.life, reg.lambda.trans = reg.lambda.trans) #> interlayer.call.args <- modifyList(interlayer.call.args, #> all.other.args) #> return(do.call(what = interlayer_callnextinterlayer, args = interlayer.call.args)) #> } #> <bytecode: 0x7ffd4731d698> #> <environment: namespace:CLVTools> #> #> $have.bounds #> [1] FALSE #> #> $method #> [1] "Nelder-Mead" #> #> Method: Nelder-Mead #> Nelder-Mead direct search function minimizer #> function value for initial parameters = 1240.281219 #> Scaled convergence tolerance is 1.84816e-05 #> Stepsize computed as 0.100000 #> BUILD 4 1292.921244 1223.563989 #> EXTENSION 6 1240.281219 1128.661157 #> LO-REDUCTION 8 1223.804770 1128.661157 #> EXTENSION 10 1223.563989 1068.349439 #> EXTENSION 12 1142.444973 988.511082 #> LO-REDUCTION 14 1128.661157 988.511082 #> EXTENSION 16 1068.349439 956.324694 #> LO-REDUCTION 18 996.040505 956.324694 #> LO-REDUCTION 20 988.511082 954.177109 #> HI-REDUCTION 22 960.169361 954.177109 #> HI-REDUCTION 24 956.804304 954.177109 #> EXTENSION 26 956.324694 952.236519 #> EXTENSION 28 955.245822 949.900709 #> REFLECTION 30 954.177109 949.356210 #> EXTENSION 32 952.236519 937.295634 #> LO-REDUCTION 34 949.900709 937.295634 #> LO-REDUCTION 36 949.356210 937.295634 #> EXTENSION 38 938.805659 914.134919 #> LO-REDUCTION 40 938.686281 914.134919 #> EXTENSION 42 937.295634 907.757954 #> EXTENSION 44 924.489444 862.957919 #> LO-REDUCTION 46 914.134919 862.957919 #> LO-REDUCTION 48 907.757954 862.957919 #> EXTENSION 50 898.139478 819.596526 #> EXTENSION 52 874.530915 759.983315 #> LO-REDUCTION 54 862.957919 759.983315 #> LO-REDUCTION 56 819.596526 752.383313 #> HI-REDUCTION 58 774.707685 752.383313 #> HI-REDUCTION 60 768.293825 752.383313 #> REFLECTION 62 760.808365 751.244381 #> LO-REDUCTION 64 759.983315 749.622829 #> REFLECTION 66 752.383313 748.162755 #> LO-REDUCTION 68 751.244381 747.967701 #> HI-REDUCTION 70 749.622829 747.734357 #> HI-REDUCTION 72 748.162755 747.089316 #> HI-REDUCTION 74 747.967701 746.544987 #> LO-REDUCTION 76 747.734357 746.544987 #> HI-REDUCTION 78 747.089316 746.544987 #> REFLECTION 80 746.857257 746.510007 #> LO-REDUCTION 82 746.802741 746.497484 #> HI-REDUCTION 84 746.544987 746.465219 #> REFLECTION 86 746.510007 746.355301 #> HI-REDUCTION 88 746.497484 746.355301 #> HI-REDUCTION 90 746.465219 746.355301 #> LO-REDUCTION 92 746.412387 746.355301 #> LO-REDUCTION 94 746.403207 746.355301 #> EXTENSION 96 746.367792 746.294665 #> HI-REDUCTION 98 746.362460 746.294665 #> LO-REDUCTION 100 746.355301 746.294665 #> EXTENSION 102 746.337978 746.219161 #> LO-REDUCTION 104 746.318441 746.219161 #> EXTENSION 106 746.294665 746.126237 #> EXTENSION 108 746.254455 745.982691 #> EXTENSION 110 746.219161 745.879542 #> EXTENSION 112 746.126237 745.588361 #> EXTENSION 114 745.982691 745.147114 #> LO-REDUCTION 116 745.879542 745.147114 #> EXTENSION 118 745.588361 744.502694 #> EXTENSION 120 745.182087 743.995863 #> EXTENSION 122 745.147114 743.677547 #> LO-REDUCTION 124 744.502694 743.677547 #> LO-REDUCTION 126 744.046453 743.677547 #> EXTENSION 128 743.995863 743.472399 #> LO-REDUCTION 130 743.863539 743.413764 #> REFLECTION 132 743.677547 743.213063 #> LO-REDUCTION 134 743.472399 743.182565 #> REFLECTION 136 743.413764 743.124013 #> LO-REDUCTION 138 743.213063 743.097778 #> EXTENSION 140 743.182565 742.867019 #> LO-REDUCTION 142 743.124013 742.867019 #> REFLECTION 144 743.097778 742.840287 #> LO-REDUCTION 146 742.929399 742.796612 #> LO-REDUCTION 148 742.867019 742.791727 #> REFLECTION 150 742.840287 742.736707 #> LO-REDUCTION 152 742.796612 742.736707 #> HI-REDUCTION 154 742.791727 742.736707 #> EXTENSION 156 742.745177 742.609240 #> LO-REDUCTION 158 742.738479 742.609240 #> LO-REDUCTION 160 742.736707 742.609240 #> EXTENSION 162 742.625110 742.507629 #> LO-REDUCTION 164 742.611615 742.507629 #> HI-REDUCTION 166 742.609240 742.507629 #> EXTENSION 168 742.558215 742.423200 #> LO-REDUCTION 170 742.555718 742.423200 #> LO-REDUCTION 172 742.507629 742.423200 #> LO-REDUCTION 174 742.499652 742.423200 #> EXTENSION 176 742.449455 742.333625 #> EXTENSION 178 742.435242 742.248018 #> EXTENSION 180 742.423200 742.119100 #> EXTENSION 182 742.333625 742.019486 #> EXTENSION 184 742.248018 741.605817 #> EXTENSION 186 742.119100 741.340371 #> LO-REDUCTION 188 742.019486 741.340371 #> EXTENSION 190 741.605817 740.345312 #> LO-REDUCTION 192 741.350110 740.345312 #> LO-REDUCTION 194 741.340371 740.345312 #> EXTENSION 196 740.747133 739.814178 #> REFLECTION 198 740.404009 739.677055 #> EXTENSION 200 740.345312 738.953603 #> HI-REDUCTION 202 739.814178 738.953603 #> LO-REDUCTION 204 739.677055 738.953603 #> EXTENSION 206 739.616030 737.942192 #> LO-REDUCTION 208 739.172099 737.942192 #> LO-REDUCTION 210 738.953603 737.942192 #> EXTENSION 212 738.031178 735.715065 #> LO-REDUCTION 214 737.944606 735.715065 #> LO-REDUCTION 216 737.942192 735.715065 #> EXTENSION 218 736.579820 732.512082 #> LO-REDUCTION 220 735.938126 732.512082 #> EXTENSION 222 735.715065 729.412594 #> EXTENSION 224 732.742526 728.118056 #> EXTENSION 226 732.512082 725.822141 #> LO-REDUCTION 228 729.412594 725.564258 #> HI-REDUCTION 230 728.118056 725.564258 #> REFLECTION 232 726.465121 724.197025 #> LO-REDUCTION 234 725.822141 724.197025 #> EXTENSION 236 725.564258 722.973395 #> LO-REDUCTION 238 725.075052 722.973395 #> LO-REDUCTION 240 724.197025 722.973395 #> REFLECTION 242 723.485003 722.577461 #> REFLECTION 244 723.268962 722.544540 #> LO-REDUCTION 246 722.973395 722.501957 #> REFLECTION 248 722.577461 722.332380 #> HI-REDUCTION 250 722.544540 722.332380 #> HI-REDUCTION 252 722.501957 722.332380 #> LO-REDUCTION 254 722.380874 722.332380 #> HI-REDUCTION 256 722.376559 722.315590 #> LO-REDUCTION 258 722.335225 722.308712 #> REFLECTION 260 722.332380 722.304639 #> LO-REDUCTION 262 722.315590 722.296588 #> HI-REDUCTION 264 722.308712 722.295781 #> HI-REDUCTION 266 722.304639 722.294565 #> HI-REDUCTION 268 722.296588 722.294556 #> HI-REDUCTION 270 722.295781 722.293049 #> HI-REDUCTION 272 722.294565 722.292873 #> HI-REDUCTION 274 722.294556 722.292483 #> HI-REDUCTION 276 722.293049 722.292483 #> HI-REDUCTION 278 722.292873 722.292421 #> HI-REDUCTION 280 722.292815 722.292418 #> REFLECTION 282 722.292483 722.292351 #> HI-REDUCTION 284 722.292421 722.292176 #> HI-REDUCTION 286 722.292418 722.292176 #> REFLECTION 288 722.292351 722.292138 #> LO-REDUCTION 290 722.292254 722.292100 #> HI-REDUCTION 292 722.292176 722.292100 #> REFLECTION 294 722.292138 722.292066 #> HI-REDUCTION 296 722.292132 722.292066 #> REFLECTION 298 722.292100 722.292014 #> HI-REDUCTION 300 722.292075 722.292014 #> EXTENSION 302 722.292066 722.291962 #> LO-REDUCTION 304 722.292045 722.291962 #> EXTENSION 306 722.292014 722.291879 #> EXTENSION 308 722.291975 722.291841 #> LO-REDUCTION 310 722.291962 722.291841 #> REFLECTION 312 722.291879 722.291778 #> LO-REDUCTION 314 722.291848 722.291778 #> LO-REDUCTION 316 722.291841 722.291778 #> LO-REDUCTION 318 722.291815 722.291778 #> LO-REDUCTION 320 722.291802 722.291775 #> Exiting from Nelder Mead minimizer #> 322 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.p log.q log.gamma #> 0.8346738 2.8420365 5.6350423 #> #> $value #> [1] 722.2918 #> #> $message #> NULL #> #> $convcode #> [1] 0 #> #> $fevals #> function #> 322 #> #> $gevals #> gradient #> NA #> #> $nitns #> [1] NA #> #> $kkt1 #> [1] TRUE #> #> $kkt2 #> [1] TRUE #> #> $xtimes #> user.self #> 0.058 #> #> Assemble the answers#>#> p q gamma #> 2.304062 17.150658 280.070758#> Gamma-Gamma Model #> #> Call: #> gg(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|) #> p 2.3041 0.4793 4.807 1.53e-06 *** #> q 17.1507 5.9500 2.882 0.00395 ** #> gamma 280.0708 149.3574 1.875 0.06077 . #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> Optimization info: #> LL -722.2918 #> AIC 1450.5835 #> BIC 1461.1479 #> KKT 1 TRUE #> KKT 2 TRUE #> fevals 322.0000 #> Method Nelder-Mead#> Id actual.mean.spending predicted.mean.spending #> 1: 1 0.00000 39.95506 #> 2: 10 0.00000 55.22646 #> 3: 100 32.06652 43.57362 #> 4: 1000 46.51783 41.60911 #> 5: 1001 33.09091 45.58074 #> --- #> 246: 1219 29.55429 33.58880 #> 247: 122 0.00000 39.95506 #> 248: 1220 0.00000 39.95506 #> 249: 1221 33.62778 34.29084 #> 250: 1222 0.00000 47.35301# }
