BTYD (version 2.4)

bgbb.DERT: BG/BB Discounted Expected Residual Transactions

Description

Computes the number of discounted expected residual transactions by a customer, conditional on their behavior in the calibration period.

Usage

bgbb.DERT(params, x, t.x, n.cal, d)
bgbb.rf.matrix.DERT(params, rf.matrix, d)

Arguments

params

BG/BB parameters - a vector with alpha, beta, gamma, and delta, in that order. Alpha and beta are unobserved parameters for the beta-Bernoulli transaction process. Gamma and delta are unobserved parameters for the beta-geometric dropout process.

x

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

t.x

recency - the last transaction opportunity in which a customer made a transaction, or a vector of recencies.

n.cal

number of transaction opportunities in the calibration period, or a vector of calibration period transaction opportunities.

d

discount rate.

rf.matrix

recency-frequency matrix. It must contain columns for frequency ("x"), recency ("t.x"), and the number of transaction opportunities in the calibration period ("n.cal"). Note that recency must be the time between the start of the calibration period and the customer's last transaction, not the time between the customer's last transaction and the end of the calibration period.

Value

The present value of the expected future transaction stream for a particular customer.

Details

DERT(d | alpha, beta, gamma, delta, x, t.x, n). This is the present value of the expected future transaction stream for a customer with x transactions and a recency of t.x in n.cal transaction opportunities, discounted by a rate d.

x, t.x, and n.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 Jen Shang. “Customer-Base Analysis in a Discrete-Time Noncontractual Setting.” Marketing Science 29(6), pp. 1086-1108. 2010. INFORMS. http://www.brucehardie.com/papers/020/

See equation 14.

Examples

params <- c(1.20, 0.75, 0.66, 2.78)
# Compute DERT for a customer who made 3 transactions
# in the calibration period(consisting of 6 transaction
# opportunities), with the last transaction occurring
# during the 4th transaction opportunity, discounted at
# 10%.
bgbb.DERT(params, x=3, t.x=4, n.cal=6, d=0.1)

# We can also compare DERT for several customers:
bgbb.DERT(params, x=1:6, t.x=6, n.cal=6, d=0.1)

data(donationsSummary)

rf.matrix <- donationsSummary$rf.matrix
# donationsSummary$rf.matrix already has appropriate column names

# starting-point parameters
startingparams <- c(1, 1, 0.5, 3)
# estimated parameters
est.params <- bgbb.EstimateParameters(rf.matrix, startingparams)

# compute DERT for a customer from every row in rf.matrix,
# discounted at 10%.
bgbb.rf.matrix.DERT(est.params, rf.matrix, d=0.1)