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Implementation of the shifted beta geometric (sBG) model from "How to Project Customer Retention" (Fader and Hardie 2006) http://www.brucehardie.com/papers/021/sbg_2006-05-30.pdf Apache 2 License
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""" | |
Implementation of the shifted beta geometric (sBG) model from "How to Project Customer Retention" (Fader and Hardie 2006) | |
http://www.brucehardie.com/papers/021/sbg_2006-05-30.pdf | |
Apache 2 License | |
""" | |
from math import log | |
import numpy as np | |
from scipy.optimize import minimize | |
__author__ = 'JD Maturen' | |
def generate_probabilities(alpha, beta, x): | |
"""Generate probabilities in one pass for all t in x""" | |
p = [alpha / (alpha + beta)] | |
for t in xrange(1, x): | |
pt = (beta + t - 1) / (alpha + beta + t) * p[t-1] | |
p.append(pt) | |
return p | |
def probability(alpha, beta, t): | |
"""Probability function P""" | |
if t == 0: | |
return alpha / (alpha + beta) | |
return (beta + t - 1) / (alpha + beta + t) * probability(alpha, beta, t-1) | |
def survivor(probabilities, t): | |
"""Survivor function S""" | |
s = 1 - probabilities[0] | |
for x in xrange(1, t + 1): | |
s = s - probabilities[x] | |
return s | |
def log_likelihood(alpha, beta, data, survivors=None): | |
"""Function to maximize to obtain ideal alpha and beta parameters""" | |
if alpha <= 0 or beta <= 0: | |
return -1000 | |
if survivors is None: | |
survivors = survivor_rates(data) | |
probabilities = generate_probabilities(alpha, beta, len(data)) | |
final_survivor_likelihood = survivor(probabilities, len(data) - 1) | |
return sum([s * log(probabilities[t]) for t, s in enumerate(survivors)]) + data[-1] * log(final_survivor_likelihood) | |
def survivor_rates(data): | |
s = [] | |
for i, x in enumerate(data): | |
if i == 0: | |
s.append(1 - data[0]) | |
else: | |
s.append(data[i-1] - data[i]) | |
return s | |
def maximize(data): | |
survivors = survivor_rates(data) | |
func = lambda x: -log_likelihood(x[0], x[1], data, survivors) | |
x0 = np.array([100., 100.]) | |
res = minimize(func, x0, method='nelder-mead', options={'xtol': 1e-8}) | |
return res | |
def predicted_retention(alpha, beta, t): | |
"""Predicted retention probability at t. Function 8 in the paper""" | |
return (beta + t) / (alpha + beta + t) | |
def predicted_survival(alpha, beta, x): | |
"""Predicted survival probability, i.e. percentage of customers retained, for all t in x. | |
Function 1 in the paper""" | |
s = [predicted_retention(alpha, beta, 0)] | |
for t in xrange(1, x): | |
s.append(predicted_retention(alpha, beta, t) * s[t-1]) | |
return s | |
def test(): | |
"""Test against the High End subscription retention data from the paper""" | |
example_data = [.869, .743, .653, .593, .551, .517, .491] | |
ll11 = log_likelihood(1., 1., example_data) | |
print np.allclose(ll11, -2.115, 1e-3) | |
res = maximize(example_data) | |
alpha, beta = res.x | |
print res.status == 0 and np.allclose(alpha, 0.668, 1e-3) and np.allclose(beta, 3.806, 1e-3) | |
print "real\t", map(lambda x: "{0:.1f}%".format(x*100), example_data) | |
print "pred\t", map(lambda x: "{0:.1f}%".format(x*100), predicted_survival(alpha, beta, 12)) | |
if __name__ == '__main__': | |
test() |
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