Recommendation Method
for Extending Subscription Periods
Tomoharu Iwata,

Kazumi Saito,

Takeshi Yamada

NTT Communication Science Laboratories
2-4, Hikaridai, Seika-cho, Keihanna Science City
Kyoto 619-0237 Japan{iwata,saito,yamada}@cslab.kecl.ntt.co.jp

ABSTRACT

1.

Online stores providing subscription services need to extend
user subscription periods as long as possible to increase theirprofits. Conventional recommendation methods recommend
items that best coincide with user’s interests to maximize
the purchase probability, which does not necessarily contribute to extend subscription periods. We present a novel
recommendation method for subscription services that maximizes the probability of the subscription period being extended. Our method finds frequent purchase patterns in thelong subscription period users, and recommends items fora new user to simulate the found patterns. Using survival
analysis techniques, we efficiently extract information fromthe log data for finding the patterns. Furthermore, we inferuser’s interests from purchase histories based on maximum
entropy models, and use the interests to improve the recommendations. Since a longer subscription period is the resultof greater user satisfaction, our method benefits users as wellas online stores. We evaluate our method using the real log
data of an online cartoon distribution service for cell-phone
in Japan.

Recommender systems are widely used by online stores [13]because they can improve both user convenience and profits. The charges that online store users pay relate to thetype of service they use and these can be categorized as either measured or subscription services. With a measured
service, users pay for purchased items individually. With asubscription service, users pay for the periodic (e.g. monthly
or yearly) use of, for example, magazines, music, movies,
software, and cell-phone services. To increase profits, online
stores providing measured services need to encourage users
to purchase many items. On the other hand, online storesproviding subscription services need to encourage users toextend their subscription periods. Therefore, the desireduser behavior for online stores are different between measured and subscription services.

Categories and Subject Descriptors
H.2.8 [Database Management]: Database Applications—
Data Mining; I.2.6 [Artificial Intelligence]: Learning; I.5.1[Pattern Recognition]: Model—Statistical

General Terms
Algorithms

Keywords
Recommendation, User Purchase Pattern, Subscription Service, Maximum Entropy, Survival Analysis

INTRODUCTION

Recommendation is one way to influence user behavior for
online stores. Since the desired user behavior differs formeasured and subscription services, online stores should recommend items using different strategies. Conventional recommendation methods recommend items that best coincidewith user’s interests to maximize the purchase probability[5, 8, 11]. With a measured service, these methods can increase profits. With a subscription service, however, thosemethods do not necessarily extend subscription periods, andtherefore do not necessarily increase profits.
In this paper, we propose a novel recommendation methodthat maximizes the probability of the subscription period being extended. Our method finds frequent purchase patterns
in long subscription period users, and recommends items
for a new user to simulate the found patterns. For example,
we may find a pattern in long subscription period users, inwhich they purchase items from a wide variety of genres.
Since the difficulties involved in changing user purchase behavior differ depending on user’s interests, we take user’s
interests into consideration so that we can make effective
recommendations. To find the patterns, we use Cox proportional hazards models with survival analysis techniques.
To estimate user’s interests, we use maximum entropy models. Then, we combine the found patterns and the estimateduser’s interests in a probabilistically principled framework.
Since a longer subscription period is the result of higher
user satisfaction, our method benefits both users and online stores. Therefore, our method can be seen as a tool forcustomer relationship management (CRM) [2]. CRM is im-portant in terms of improving relationships between online
stores and the users.

2. RELATED WORK
A number of recommendation methods have been proposed,such as collaborative filtering [11], content filtering [8], andtheir hybrids [5]. Collaborative filtering is a method for predicting user’s interests using other user’s interests. Content
filtering predicts interests using item information. These
approaches recommend items that best coincide with user’s
interests to maximize the purchase probability. Our aim,
which is to extend subscription periods, is different fromtheirs.
The prediction of subscription periods and the identificationof users that unsubscribe within a period of time, which
are known as churn analysis, have been studied by various
researchers using survival analysis and data mining techniques [1, 9, 14]. Customer Lifetime Value (LTV) is related to the subscription period. LTV is usually defined as
a combination of two components: tenure (or subscription
period) and value over time [7]. In subscription services, thevalue does not change over time because of the fixed charge
per given period, and LTV corresponds to the subscription
period. Models of LTV have been proposed to facilitate
CRM [7]. But they are not used for recommendations.
Piatetsky-Shapiro and Masand [10] and Rosset et al. [12]proposed models for estimating the effects of marketing activity on subscription periods or LTV. A recommendation
can be considered a marketing activity. By focusing on therecommendation of items, our method becomes an automatic recommendation framework that can learn from thelog data.

3. PROPOSED METHOD
3.1 Preliminaries
The log data usually possessed by online stores providing
subscription services are the subscription and the purchaselogs. The subscription log consists of the subscribed time,
the status (still subscribing or already unsubscribed) and,
where relevant, the unsubscribed time of each user. The
purchase log consists of the user, the item, and the time ofeach purchase.
{un }N
n=1

Consider a set of users U =and a set of items
S = {si }Vi=1 . Each user un has subscription period tn andstatus en (en = 1 if unsubscribed, en = 0 if subscribing).
The subscription period tn is obtained from the subscription
log as follows:(dend− dstart
if en = 1,
n
n
tn =
(1)
dend − dstart
if en = 0,nwhere dstartis the subscribed time of user un , dendis thennunsubscribed time of user un , and dend is the last time atwhich the log was modified.
Each user is represented by the sequence of purchased itemsnnun = (sn
1 , s2 , . . .), where sk ∈ S is the k-th item in thesequence of user un . The sequence is obtained from thepurchase log. We have two sets of features of purchase history x(un ) and y(un , sj ), where x(un ) is used for estimating

the user subscription period, and y(un , sj ) is used for estimating the user’s interests. Examples of features include:
whether the user has purchased item si , and whether theuser has purchased item si next to item sj . We rewrite aset of features x(un ) in the form of a column vector xn =(x1 (un ), x2 (un ), . . .)T for convenient notation, where xj (un )is the j-th feature of x(un ). We use the subscription periods
at which items are purchased for treating purchase histories
as time-dependent variables when we estimate the subscription periods.

3.2

Recommendation for Extending Subscription Periods

Our method recommends item sî that maximizes P (l|u, ri ),which is the probability of extending the subscription periodof user u when item si is recommended, as follows:î = arg max P (l|u, ri ),
i∈S

(2)

where l represents an event that the subscription period is
extended, and ri represents an event that item si is recommended. We can also recommend m highest P (l|u, ri ) items,
if we want to recommend m items. In real applications, candidates for recommendation would be a subset of item set
S, for example, only items not yet purchased by the user.
Let sj be the item purchased by user u when item si is recommended. If the recommendation does not influence theuser purchase behavior, it is natural to think that the recommendation does not influence the subscription period, either. Therefore, we assume that extending subscription period l and recommendation ri are independent conditioned
on purchased item sj and user u. By using this assumption, P (l|u, ri ) can be decomposed into two components:
Q(l|u, sj ), which is the probability of the subscription period being extended when user u purchases item sj , andR(sj |u, ri ), which is the probability of item sj being purchased when item si is recommended to user u, as follows:X
P (l|u, ri ) =P (l, sj |u, ri )j∈SX=
Q(l|u, sj )R(sj |u, ri ).
(3)j∈S
We estimate Q(l|u, sj ) and R(sj |u, ri ) using Cox proportional hazards models and maximum entropy models, respectively. We summarize the framework of our method in
Figure 1.

3.3

Cox Proportional Hazards Models

It is generally considered that the subscription period depends on the purchase history. Let x be the feature vectorof the purchase history of user u, and we refer to x as thepurchase history for the simplicity. We estimate the extension probability given the purchased item Q(l|u, sj ) from thehazard function h(t|x), which is the rate that users with purchase history x subscribing until t unsubscribe at t. We estimate h(t|x) using survival analysis techniques [4]. If users
are still subscribing, we cannot know their true subscription
period. These samples are called censored samples. Using survival analysis techniques, we can effectively estimateh(t|x) using information found in the data including censored samples.

Subscription Log

Purchase Log

Input Data

Cox Proportional Hazards Model

Extension Probability
given the Purchased Item

Maximum Entropy Model

Purchase Probability
given the Recommendation

Q( l | u, s )R( s | u, r )We explain the estimation of Q(l|u, sj ), which is the probability of extending the subscription period when user u purchases item sj , using hazard function h(t|x). Let x be thepurchase history of user u, and x+sj be the updated purchase history when item sj is purchased. For the simplicity,
we refer to the user when item sj is purchased as u+sj . We
assume that either one of u or u+sj unsubscribed at t while
the other being still subscribing. At t, the hazard functionof u and u+sj are h(t|x) and h(t|x+sj ), respectively. The
probability that u is the one who unsubscribed at t is equal
to the probability of extending the subscription period whenuser u purchases item sj as follows:

Extension Probability
given the Recommendation

Q(l|u, sj ) =P( l | u, r )=Recommendation

Figure 1: The framework of our method.

We assume Cox proportional hazards models for h(t|x) as
follows:h(t|x) = λ0 (t) exp(β T x),(4)
T

where β is the unknown parameter vector, β is the transpose of β, and λ0 (t) is the baseline hazard function, which
represents the hazard function for users with x = 0. We
employ Cox proportional hazards models since the global
optimum of the estimation is guaranteed and Q(l|u, sj ) canbe written in a closed form as described below.
The log partial likelihood with Breslow approximation [4] isdefined as follows:QYh(t|xn (t))
P n∈D(t)P L(β) = log|D(t)|(
h(t|xm (t)))m∈R(t)t
X X T=
β xn (t)t−Xtn∈D(t)|D(t)| logXexp(β T xm (t)),(5)

m∈R(t)

where D(t) is the set of users unsubscribed at t and |D(t)| is
its size, R(t) is the set of users subscribing at t, and xn (t) is
the feature vector of user u at t. We can estimate unknown
parameters β by maximizing the log partial likelihood using
optimization methods such as quasi-Newton methods [6].
In Cox proportional hazards models, features that have highexp(−βb ) (> 1) are characteristic purchase patterns in thelong subscription users, and features that have low exp(−βb )(< 1) are characteristic patterns in the short subscription
users. These patterns are informative for the online store,
for example, to understand the relationship between purchase histories and subscription periods, or to determine
new items to be distributed for extending subscription periods.

3.4 Extension Probability given the Purchased
Itemh(t|x)
h(t|x) + h(t|x+sj )1,T
1 + exp(−β (x − x+sj ))(6)which is a sigmoid function. We can recommend an itemthat maximizes Q(l|u, sj ), however, users may not purchaserecommended items. In this case, the recommendation isuseless as regards extending subscription periods. Therefore, we need to consider whether the recommended item ispurchased by the user taking user’s interests into consideration.

3.5

Purchase Probability given the Recommendation

We explain the estimation of R(sj |u, ri ), the probability thatuser u purchases item sj when item si is recommended touser u . Let R(sj |u) be the probability that user u purchases
P
item sj without recommendations, where Vj=1 R(sj |u) = 1.
The recommendation of item si increases the probability ofitem si being purchased. We assume that the probability
increase γ times as follows:(γ
R(si |u) j = i,i)
R(sj |u, ri ) = Z(u,r(7)
1
R(sj |u) j 6= i,Z(u,ri )where Z(u, ri ) = 1 + (γ − 1)R(si |u) is the normalization
term, and γ ≥ 1. γ represents the recommendation effect onuser purchase behavior, and depends on the way the recommendation is presented in the online store, such as the sizeor position.
If an item matches the user’s interests, the probability of theuser purchasing the item is high, and if it does not match,
the probability is low. Therefore, R(sj |u) represents thedegree of coincidence between the interests of user u anditem sj . Since conventional recommendation methods suggest items that coincide with user’s interests, we can use
conventional methods to obtain R(sj |u). We employ maximum entropy models [5], which estimate a probabilistic distribution that maximizes entropy under the constraints ofthe given data. Maximum entropy models can integrate
arbitrary features such as purchase histories and user attributes. In maximum entropy models, the probability thatuser u purchases item sj is:X1exp(
αc yc (u, sj )),(8)
R(sj |u) =
Z(u)
c
PV
P
where Z(u) =j=1 exp(c αc yc (u, sj )) is the normalization term, yc is a feature of the purchase history, αc is anTable 1: The number of subscribers, unsubscribers and features.
2005/06/30
2005/07/31
2005/08/31
learning
test
learning
test
learning
test
number of subscribers
13,284
7,221
14,6699,608
28,40917,028
number of unsubscribers
4,988
6,063
8,802
5,061
9,765
11,381
number of features
3,711
4,455
5,250

Table 2: The average log partial likelihoods of the model that does not use purchase histories, h(t), and theCox proportional hazards models, h(t|x).
2005/06/30
2005/07/31
2005/08/31
learning
test
learning
test
learning
test
model that does not use purchase histories h(t)-8.865-9.845-9.165-9.465-9.513-9.904Cox proportional hazards model h(t|x)-8.604-9.129-9.048-9.351-9.325-9.798

whether user u has purchased item sj next to item si :8>
<1 if user u has purchased
xsi →sj (u) =(10)item sj next to item si ,>:0 otherwise,

unknown parameter, and c is an index of each feature.
The log likelihood of the maximum entropy model is:XXnlog R(snL(α) =
k |uk )
n

=

n

−

k

XXX
k

XX
n

k

c

log

where we omitted features that occurred fewer than ten
times in the purchase histories.

nαc yc (unk , sk )

X
j

exp(

X

αc yc (unk , sj )),(9)cwhere snk is the k-th item in the purchase sequence of user
un , and un
k is the purchase history of user un before the kth item is purchased. We can estimate unknown parameters
α that maximize log likelihood L(α) by using optimization
techniques such as quasi-Newton methods. In maximum entropy models, we can obtain the global optimum solution.
By using a Gaussian prior with a zero mean on unknown parameters α, overfitting can be reduced [3]. We use a Gaussian prior in our experiments.

4. EXPERIMENTAL RESULTS
We evaluated our method with the log data of an online
cartoon distribution service for cell-phones in Japan. With
this service, users pay monthly to read cartoons on theircell-phones. Our method is applicable since the service employs a subscription log and a purchase log, and extensionof the subscription period increases the profit of the online
store. Some cartoons have several volumes, and some users
purchased an item more than once. We regarded a cartoonthat had several volumes as one item, and the unit time wasa day. This service began on 16 August 2004, and the lastmodification date of the log was 28 October 2005.

4.1 Evaluation of Cox Proportional Hazards
ModelsWith our method, we assume that we can estimate subscription periods more precisely using purchase histories. To
evaluate this assumption, we compared the Cox proportional
hazards models h(t|x) = λ0 (t) exp(β T x) that use purchasehistories described in Section 3.3 and models that do not
use purchase histories h(t) = λ00 (t). We used as features

We used three sets of learning and test samples. The learning samples were the log data up to 30 June 2005, 31 July
2005 and 31 August 2005. The test samples were the log
data of subscribers on the end date of the learning samples,
and the end date of the test samples was 28 October 2005.
The number of subscribers, unsubscribers and features wereas shown in Table 1.
For the evaluation measurements, we used the average log
partial likelihood, which is the log partial likelihood divided
by the number of unsubscribers. The higher average log
partial likelihood for the test samples indicates the higher
predictive performance of the model. Table 2 shows theresults. The average log partial likelihoods for the test samples of the Cox proportional hazards models were higher
than the model that does not use purchase histories. This
result shows that Cox proportional hazards models can predict subscription periods more precisely by using purchasehistories.

4.2 Evaluation of Maximum Entropy
We evaluated the maximum entropy models described in
Section 3.5, which estimate the probability that user u purchases item sj , R(sj |u). We used first-order Markov transitions as features, since we considered the last purchased
item revealed the user’s interests:
8><1 if item sa is the last purchased
ysa ,sb (u, sj ) =(11)
item of user u and sb = sj ,>:0 otherwise.
We used three sets of learning and test samples. The learning samples were the log data up to 30 June 2005, 31 July
2005 and 31 August 2005, from which we omitted transitions to the same item. The test samples were the log datafrom the end date of the learning samples to 28 OctoberTable 3: The number of transitions and items.
2005/06/30
2005/07/31
2005/08/31
learning
test
learning
test
learning
test
number of transitions 300,486 122,904 382,778 171,749 459,456 197,476
number of items
75
81
86

Table 4: The average log likelihoods of the uniform distribution, the multinomial distribution and the maximum entropy model.
2005/06/30
2005/07/31
2005/08/31
learning
test
learning
test
learning
test
uniform distribution-4.317-4.317-4.394-4.394-4.454-4.454
multinomial distribution-3.875-4.263-3.938-4.673-3.975-4.454maximum entropy model-3.554-3.551-3.581-3.732-3.605-3.762

2005, from which we omitted transitions to the same item
and transitions that contained items that had not been distributed during the learning sample period. The number oftransitions and items were as shown in Table 3. We compared maximum entropy models with uniform distributions
and multinomial distributions. Uniform distributions do not
use the information in the log data at all. Multinomial distributions use the information of the purchased number of
each item by all users, but do not consider the individual interests. The unknown parameters of the multinomial distribution were estimated by the maximum likelihood method.
We used the average log likelihood for the evaluation measurements. Table 4 shows the results. The average log
likelihoods of the maximum entropy models for the test
samples were higher than uniform and multinomial distributions. This means that maximum entropy models canpredict user purchase behavior and interest more precisely
by using purchase histories.

Algorithm 1 Simulate a user in a subscription service
1: Set t ← 0, u ← φ
2: loop3:
Sample r1 ∼ Bernoulli(h(t|x))4:
if r1 is success then
5:
break
6:
end if7:
Sample r2 ∼ Bernoulli(g)8:if r2 is success then9:
if u = φ then10:
Sample sj ∼ M ultinomial(R(sj ))
11:
else
12:
î ← arg maxi P (l|u, ri )13:
Sample sj ∼ M ultinomial(R(sj |u, rî ))14:
end if15:
Set u ← u+sj
16:
end if17:
Set t ← t + 1
18: end loop
19: Output t

4.3 Simulation
In Section 4.1, we showed that Cox proportional hazards
models can predict subscription periods, and in Section 4.2,
we revealed that maximum entropy models can predict userpurchase behavior. Here, we examined the effectiveness of
our method by simulation. We simulated user behavior using the Cox proportional hazards model and the maximum
entropy model that were estimated using the log data from
16 August 2004 to 28 October 2005. The log data comprised
107 items.
The function of Algorithm 1 is to generate a subscription
period t, where u is the sequence of purchased items, u+sj
is the updated sequence when item sj is purchased, φ isan empty sequence, Bernoulli(θ) is the Bernoulli distribution with success probability θ, and M ultinomial(ψ) is themultinomial distribution of one event with j’s success probability ψj . First, from line 3 to 4 in Algorithm 1, we decide whether the user unsubscribes or not in unit time using
the unsubscription probability in unit time of a subscriberh(t|x). Second, from line 7 to 8, we decide whether the userpurchases or not in unit time using the purchase probability
in unit time, g. We assumed that g is constant over subscription period t. The first item that the user purchases

is determined according to R(sj ), where R(sj ) is the probability of purchasing item sj first (line 10). If the user has
purchased some items, we make a recommendation using
our method (line 12), and the item that the user purchases
is determined according to R(sj |u, rî ) (line 13). We estimated unknown parameters λ0 (t), g and R(sj ) using thelog data by the maximum likelihood method.
We compared our method with following recommendation
methods:• Q Recommend recommends an item that is most
likely to extend the subscription period when the userpurchases it. Line 12 in Algorithm 1 is changed asfollows:î ← arg max Q(l|u, si ).
i(12)This recommendation does not take the user’s interests
into consideration.
• R Recommend recommends an item that best coincides with the user’s interests. Line 12 is changed as

Figure 2: The average subscription periods in simulations with parameters estimated from the log data.
200

Our Method

190

180

Q Recommend

170
R Recommend

160

No Recommend
150

1

2

3

4
5
6
7
Effect of recommendations (γ)

8

9

10

follows:î ← arg max R(si |u).
i

(13)This recommendation is the same as that of conventional methods.
• No Recommend does not recommend any items.
The item that the user purchases is determined solely
according to the user’s interests. Line 12 is omitted
and line 13 is changed as follows:
Sample sj ∼ M ultinomial(R(sj |u)).

(14)

This recommendation can be also achieved by using
γ = 1 with Algorithm 1, which means that the recommendation has no effect on user purchase behavior.
We generated 100, 000 user subscription periods with recommendations by each method in 1 ≤ γ ≤ 10. We set the
maximum subscription period as 365 days. Figure 2 shows
the average subscription periods. Our method was more
successful than the others in extending subscription periods.
The subscription periods became longer with increases in γ.
This result indicates that if recommendations can influenceon user behavior, or γ > 1, our method can extend subscription periods, and subscription periods can be extended evenfurther by improving the influence of the recommendations.
Q Recommend extended subscription periods, although theeffect was smaller than with our method. This is because Q
Recommend may recommend items that have no probability of being purchased by the user. On the other hand, ourmethod suggests items taking user’s interests into account toimprove the recommendations. R Recommend only slightly
extended subscription periods because of little correlation
between the subscription periods and the user’s interests.

5. CONCLUSIONSIn this paper, we proposed a novel recommendation methodthat encourages users to extend their subscription periods
for online stores providing subscription service. We usedbasic features in the experiments to make the novelty of
our framework easy to understand. Our method can use
other features such as long distance dependencies or user
attributes. Since our method is divided into two modules:
the estimation of subscription periods and the estimation ofuser’s interests, it can be further enhanced using survival
analysis or collaborative filtering techniques. For example,
we can combine our approach with content filtering to estimate user’s interests. Online stores providing measured
services are eager to prevent their users from switching toother stores. Therefore, our method can also be used forrecommendations in measured services by considering thesubscription period as a period of continuous use of the online store.
Although we have obtained encouraging results to date, somedirections remain in which we must extend our approach before it can become a useful tool for recommendation. First,
it is important to improve the Cox proportional hazards
model and the feature selection for finding informative purchase patterns in long subscription period users. Second, weneed to estimate the influence of recommendations on userpurchase behavior from the log data automatically. Thiscan be estimated by using the log data of purchase histories with and without recommendations. Finally, we want
to apply our method to an online store and show how it can
extend the subscription periods of real users.

6.

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