This document discusses modeling customer transaction data using the Pareto-Negative Binomial Distribution (PNBD) model. It loads transaction log data, splits it into calibration and validation sets, builds the calibration CBS object, estimates PNBD parameters from the calibration CBS, and evaluates model fit. Key outputs include the estimated PNBD parameters, log-likelihood, expected transactions, conditional expected transactions for a given customer, and plots of transaction rate heterogeneity and dropout rates. The document also references foundational papers on customer base analysis and the PNBD model.

1. 1

5. ü
ü

6. LTV
LTV
LTV

8. 1.
2. 1
3.
4.

9. A
B
C
0.9
0.7
0.4
2.0
2

10. P |

12. 1. λ
2. µ
3. λ
4. µ 3
5. Λ µ
5

13. 0 Tt
X
×
τ

14. 0 Tt
X
×
τ

15. 0 Tt
X
×
τ

16. 0 Tt
X
×
τ

17. 0 Tt
X
×
τ

18. 0 Tt
X
×
τ

19. 0 Tt
X
×
τ
λ, µ

20. 0 Tt
X
×
τ
r, s, α, β

21. 0 Tt
X
×
τ
T

24. ID
1 19970101 2 29.33
1 19970118 2 29.73
1 19970802 1 14.96
1 19971212 2 26.48
2 19970101 3 63.34
2 19970113 1 11.77
3 19970101 1 6.79
4 19970101 1 13.97

25. > library(dplyr)
> library(BTYD)
> cdnowElog<-system.file("data/cdnowElog.csv",package="BTYD")
> elog<-dc.ReadLines(cdnowElog,cust.idx=2,date.idx=3,sales.idx=5)
> elog$date<-as.Date(elog$date,"%Y%m%d")
> head(elog)
cust date sales
1 1 1997-01-01 29.33
2 1 1997-01-18 29.73
3 1 1997-08-02 14.96
4 1 1997-12-12 26.48
5 2 1997-01-01 63.34
6 2 1997-01-13 11.77

26. > nrow(elog)
[1] 6919
> elog<-dc.MergeTransactionsOnSameDate(elog);
Started merging same-date transactions...
... Finished merging same-date transactions.
> nrow(elog)
[1] 6696

27. > elog.cal<-elog[which(elog$date<= as.Date("1997-09-30"), ]
> split.data<-dc.SplitUpElogForRepeatTrans(elog.cal);
> freq.cbt<-dc.CreateFreqCBT(split.data$repeat.trans.elog);
> tot.cbt<-dc.CreateFreqCBT(elog)
> cal.cbt<-dc.MergeCustomers(tot.cbt, freq.cbt)
> birth.periods<-split.data$cust.data$birth.per
> last.dates<-split.data$cust.data$last.date
> cal.cbs.dates<-data.frame(birth.periods, last.dates,end.of.cal.period)
> cal.cbs<-dc.BuildCBSFromCBTAndDates(cal.cbt,cal.cbs.dates,per="week")
> head(cal.cbs)
x t.x T.cal
1 2 30.428571 38.85714
2 1 1.714286 38.85714
3 0 0.000000 38.85714
4 0 0.000000 38.85714
5 0 0.000000 38.85714
6 7 29.428571 38.85714
X, t, T

28. > params<-pnbd.EstimateParameters(cal.cbs);
> params
[1] 0.5533971 10.5801985 0.6060625 11.6562237
> LL<-pnbd.cbs.LL(params, cal.cbs);
> p.matrix<-c(params, LL);
> for(i in 1:2){
+ params<-pnbd.EstimateParameters(cal.cbs, params);
+ LL<-pnbd.cbs.LL(params, cal.cbs);
+ p.matrix.row<-c(params, LL);p.matrix<-rbind(p.matrix, p.matrix.row);
+ }
> colnames(p.matrix)<-c("r","alpha","s","beta","LL");
> rownames(p.matrix)<-1:3;p.matrix;
r alpha s beta LL
1 0.5533971 10.58020 0.6060625 11.65622 -9594.976
2 0.5534354 10.57952 0.6060276 11.65666 -9594.976
3 0.5533710 10.57952 0.6059074 11.65812 -9594.976

29. > pnbd.PlotTransactionRateHeterogeneity(params)
> pnbd.PlotDropoutRateHeterogeneity(params)

30. > pnbd.Expectation(params,t=52);
[1] 1.473434
> cal.cbs["1516",]
x t.x T.cal
26.00000 30.85714 31.00000
> x<-cal.cbs["1516","x"]
> t.x<-cal.cbs["1516","t.x"]
> T.cal<-cal.cbs["1516","T.cal"]
> pnbd.ConditionalExpectedTransactions(params,T.star=52,x, t.x, T.cal)
[1] 25.45647
> pnbd.PAlive(params, x, t.x, T.cal)
[1] 0.997874
← 52
←
← 52
←

31. > pnbd.PlotFrequencyInCalibration(params, cal.cbs, 7)

32. > comp<-bgnbd.PlotFreqVsConditionalExpectedFrequency(params,
T.star,cal.cbs, x.star, censor)

33. • Counting Your Customers: Who Are They and What Will They Do Next?
David C. Schmittlein, Donald G. Morrison and Richard Colombo, 1987)
• Buy ’Til You Die - A Walkthrough Daniel McCarthy, Edward Wadsworth,
2014

34. Thank you!