It's my research as a graduate student and it's been submitted and accepted by the 12th FSKD (International Conference on Fuzzy Systems and Knowledge Discovery).
23. Assumptions
θ: the average dollar volume per re-purchase
Zi: the dollar value of record i, i=1,…, X, of a customer
• The set of Zi is normally distributed with mean θ and variance
• The average dollar spent per purchase across customers is
normally distributed with mean E[θ] and variance .
2
w
2
A
23
24. Assumptions
• Rates λ and μ don’t affect each other.
• The distribution of the average dollar amount is independent of the
distributions of the purchase rate and attrition rate.
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25. Key formulae
25
1 1 1 1
1
1 1 1 1
, , , ,t,T
1 , ; ;
, ; ;
r x s
s
P T r s X x
s T T
F a b c z t
r x s t t
T
F a b c z T
t
1a r x s 1 1b s 1 1c r x s 1z y
+y
Case 1:α > β
26. Key formulae
26
2 2 2 2
1
2 2 2 2
, , , ,t,T
1 , ; ;
, ; ;
r x s
r x
P T r s X x
s T T
F a b c z t
r x s t t
T
F a b c z T
t
Case 2:α < β
2a r x s 2b r x 2 1c r x s 2z y
+y
27. Key formulae
27
1
, , , ,t,T 1 1
r x s
s T
P T r s X x
r x s t
Case 3:α = β
28. Key formulae
28
* *
* * * * *
, ,s, , , , ,
, , , , , , , ,s , ,
E X r X t T T
P Active r s X t T E X r T
29. Key formulae
29
* *
2 2
* *
2 2 2 2
, , , , , , ,
, , , , , , ,A W
A W A W
E V r s X t T T
X
Z E E X r s X t T T
X X
30. Data processing
30
Identity of customer First transaction Last transaction
77 2013/4/5 2013/4/5
127 2013/4/11 2013/4/11
274 2013/4/28 2013/4/28
393 2013/5/8 2013/5/8
848 2013/6/30 2013/10/12
38. Case I Background
• Online retailing: A shoe manufacturer
• Time period: 2013/3/28-2014/11/30
Base: A year
Validation: Subsequent 8 months
• Records: 3,180
2,222 customers
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