pnbd.DERT: Pareto/NBD Discounted Expected Residual Transactions
In BTYD: Implementing BTYD Models with the Log Sum Exp Patch

Description Usage Arguments Details Value References Examples

Calculates the discounted expected residual transactions of a customer, given their behavior during the calibration period.

1	pnbd.DERT(params, x, t.x, T.cal, d, hardie = TRUE)

`params`	Pareto/NBD parameters - a vector with r, alpha, s, and beta, in that order. r and alpha are unobserved parameters for the NBD transaction process. s and beta are unobserved parameters for the Pareto (exponential gamma) dropout process.
`x`	number of repeat transactions in the calibration period T.cal, or a vector of transaction frequencies.
`t.x`	time of most recent repeat transaction, or a vector of recencies.
`T.cal`	length of calibration period, or a vector of calibration period lengths.
`d`	the discount rate to be used. Make sure that it matches up with your chosen time period (do not use an annual rate for monthly data, for example).
`hardie`	if TRUE, use `h2f1` instead of `hypergeo`.

DERT(d | r, alpha, s, beta, X = x, t.x, T.cal)

x, t.x, T.cal may be vectors. The standard rules for vector operations apply

if they are not of the same length, shorter vectors will be recycled (start over at the first element) until they are as long as the longest vector. It is advisable to keep vectors to the same length and to use single values for parameters that are to be the same for all calculations. If one of these parameters has a length greater than one, the output will be also be a vector.

The number of discounted expected residual transactions for a customer with a particular purchase pattern during the calibration period.

Fader, Peter S., Bruce G.S. Hardie, and Ka L. Lee. "RFM and CLV: Using Iso-Value Curves for Customer Base Analysis." Journal of Marketing Research Vol.42, pp.415-430. November. 2005. http://www.brucehardie.com/papers.html

See equation 2.

Note that this paper refers to what this package is calling discounted expected residual transactions (DERT) simply as discounted expected transactions (DET).

# elog <- dc.ReadLines(system.file("data/cdnowElog.csv", package="BTYD2"),2,3)
# elog[, 'date'] <- as.Date(elog[, 'date'], format = '%Y%m%d')
# cal.cbs <- dc.ElogToCbsCbt(elog)$cal$cbs
# params <- pnbd.EstimateParameters(cal.cbs, hardie = TRUE)
params <- c(0.5629966, 12.5590370, 0.4081095, 10.5148048)

# 15% compounded annually has been converted to 0.0027 compounded continuously,
# as we are dealing with weekly data and not annual data.
d <- 0.0027

# calculate the discounted expected residual transactions of a customer
# who made 7 transactions in a calibration period that was 77.86
# weeks long, with the last transaction occurring at the end of
# the 35th week.
pnbd.DERT(params, 
          x = 7, 
          t.x = 35, 
          T.cal = 77.86, 
          d, 
          hardie = TRUE)

# We can also use vectors to compute DERT for several customers:
pnbd.DERT(params, 
          x = 1:10, 
          t.x = 30, 
          T.cal = 77.86, 
          d, 
          hardie = TRUE)