BTYD (version 2.4)

pnbd.DERT: Pareto/NBD Discounted Expected Residual Transactions

Description

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

Usage

pnbd.DERT(params, x, t.x, T.cal, d)

Arguments

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

the number of repeat transactions a customer made in the calibration period, or a vector of transaction frequencies.

t.x

recency: the time of the customer's last transaction in the calibration period, or a vector of recencies.

T.cal

the length of the calibration period, or a vector of calibration period lengths. Make sure that the lengths of time periods for all parameters match.

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).

Value

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

Details

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.

References

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).

Examples

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)

# 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)