def probability_of_n_purchases_up_to_time(self, t, n):
r"""
Compute the probability of n purchases up to time t.
.. math:: P( N(t) = n | \text{model} )
where N(t) is the number of repeat purchases a customer makes in t
units of time.
Parameters
----------
t: float
number units of time
n: int
number of purchases
Returns
-------
float:
Probability to have n purchases up to t units of time
"""
r, alpha, a, b = self._unload_params("r", "alpha", "a", "b")
_j = np.arange(0, n)
first_term = (beta(a, b + n + 1) / beta(a, b) * gamma(r + n) /
gamma(r) / gamma(n + 1) * (alpha / (alpha + t))**r *
(t / (alpha + t))**n)
finite_sum = (gamma(r + _j) / gamma(r) / gamma(_j + 1) *
(t / (alpha + t))**_j).sum()
second_term = beta(a + 1, b + n) / beta(
a, b) * (1 - (alpha / (alpha + t))**r * finite_sum)
return first_term + second_term
def probability_of_n_purchases_up_to_time(
self,
t,
n
):
r"""
Compute the probability of n purchases.
.. math:: P( N(t) = n | \text{model} )
where N(t) is the number of repeat purchases a customer makes in t
units of time.
Comes from equation (8) of [2]_.
Parameters
----------
t: float
number units of time
n: int
number of purchases
Returns
-------
float:
Probability to have n purchases up to t units of time
References
----------
.. [2] Fader, Peter S., Bruce G.S. Hardie, and Ka Lok Lee (2005a),
"Counting Your Customers the Easy Way: An Alternative to the
Pareto/NBD Model," Marketing Science, 24 (2), 275-84.
"""
r, alpha, a, b = self._unload_params("r", "alpha", "a", "b")
first_term = (
beta(a, b + n)
/ beta(a, b)
* gamma(r + n)
/ gamma(r)
/ gamma(n + 1)
* (alpha / (alpha + t)) ** r
* (t / (alpha + t)) ** n
)
if n > 0:
j = np.arange(0, n)
finite_sum = (gamma(r + j) / gamma(r) / gamma(j + 1) * (t / (alpha + t)) ** j).sum()
second_term = beta(a + 1, b + n - 1) / beta(a, b) * (1 - (alpha / (alpha + t)) ** r * finite_sum)
else:
second_term = 0
return first_term + second_term
defvjp(
gammaincln,
central_difference_of_(gammaincln),
lambda ans, a, x: unbroadcast_f(
x,
lambda g: g * np.exp(-x + np.log(x) *
(a - 1) - gammaln(a) - gammaincln(a, x)),
),
)
defvjp(
betainc,
central_difference_of_(betainc, argnum=0),
central_difference_of_(betainc, argnum=1),
lambda ans, a, b, x: unbroadcast_f(
x, lambda g: g * np.power(x, a - 1) * np.power(1 - x, b - 1) / beta(
a, b)),
)
defvjp(
betaincln,
central_difference_of_(betaincln, argnum=0),
central_difference_of_(betaincln, argnum=1),
lambda ans, a, b, x: unbroadcast_f(
x,
lambda g: g * np.power(x, a - 1) * np.power(1 - x, b - 1) / beta(a, b)
/ betainc(a, b, x),
),
)
defvjp(
gammaincln,
central_difference_of_log(gammainc),
lambda ans, a, x: unbroadcast_f(
x,
lambda g: g * np.exp(-x + np.log(x) *
(a - 1) - gammaln(a) - gammaincln(a, x)),
),
)
defvjp(
betainc,
central_difference_of_(betainc, argnum=0),
central_difference_of_(betainc, argnum=1),
lambda ans, a, b, x: unbroadcast_f(
x, lambda g: g * np.power(x, a - 1) * np.power(1 - x, b - 1) / beta(
a, b)),
)
defvjp(
betaincinv,
central_difference_of_(betaincinv, argnum=0),
central_difference_of_(betaincinv, argnum=1),
lambda ans, a, b, y: unbroadcast_f(
y, lambda g: g * 1 / (np.power(betaincinv(a, b, y), a - 1) * np.power(
1 - betaincinv(a, b, y), b - 1) / beta(a, b))),
)
defvjp(
betaincln,
central_difference_of_log(betainc, argnum=0),
central_difference_of_log(betainc, argnum=1),