Generalized Pareto/NBD Probability Mass Function
Generalized probability mass function for the Pareto/NBD.
pnbd.pmf.General(params, t.start, t.end, x)
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.
start of time period for which probability is being calculated. It can also be a vector of values.
end of time period for which probability is being calculated. It can also be a vector of values.
number of repeat transactions by a random customer in the period defined by (t.start, t.end]. It can also be a vector of values.
P(X(t.start, t.end)=x | r, alpha, s, beta). Returns the probability that a customer makes x repeat transactions in the time interval (t.start, t.end].
It is impossible for a customer to make a negative number of
repeat transactions. This function will return an error if it is
given negative times or a negative number of repeat
transactions. This function will also return an error if
t.end is less than
x 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 a vector of probabilities.
Probability of x transaction occuring between t.start and t.end conditional on model parameters. If t.start, t.end, and/or x has a length greater than one, a vector of probabilities will be returned.
Fader, Peter S., and Bruce G.S. Hardie. “Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model.” Sept. 2006. Web. http://www.brucehardie.com/notes/012/
Fader, Peter S., Bruce G.S. Hardie, and Kinshuk Jerath. “Deriving an Expression for P (X(t, t + tau) = x) Under the Pareto/NBD Model.” Sept. 2006. Web. http://www.brucehardie.com/notes/013/
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params <- c(0.55, 10.56, 0.61, 11.64) # probability that a customer will make 10 repeat transactions in the # time interval (1,2] pnbd.pmf.General(params, t.start=1, t.end=2, x=10) # probability that a customer will make no repeat transactions in the # time interval (39,78] pnbd.pmf.General(params, t.start=39, t.end=78, x=0)