Journal of Software Engineering and Applications, 2011, 4, 411-416
doi:10.4236/jsea.2011.47047 Published Online July 2011 (http://www.SciRP.org/journal/jsea)
411
Propo sing a New Me t ric for Collaborativ e
Filtering
Arash Bahreh mand 1*, Reza Rafeh2
1Department of Computer Engineering, Islamic Azad University, Arak, Iran; 2Deputy of Computer Engineering, Arak University,
Arak, Iran.
Received May 21st, 2011; revised June 29th, 2011; accept ed June 30th, 2011.
ABSTRACT
The aim of a recommender system is filtering the enormous quantity of information to obtain useful information based
on the user’s interest. Collaborative filtering is a technique which improves the efficiency of recommendation systems
by considering the similarity between users. The similarity is based on the given rating to data by similar users. How-
ever, user’s interest may change over time. In this paper we propose an adaptive metric which considers the time in
measuring the similarity of users. The experimental results show that our approach is more accurate than the tradition-
al collaborative filterin g algorithm.
Keywords: Recommendation Systems, Collaborative Filtering, Similarity Metric
1. Introduction
In recent years, collaborative filtering has become a
widely adopted algorithm in recommendation systems.
“Collaborative” means that a group of people with the
same interests define their preferences in order to set up
the system. CF1 is used to create “recommendation sys-
tems” that can, for insta nce, enha nce your experience on
a Web site by suggesting music or movies that users
might like. This algorithm uses the similarity of other
users who have the same history as target user to rec-
ommend some items that may be related to his personal
interest. The history holds all feedbacks which can be
achieved from users by the system. In fact, feedbacks
simulate the behavior of users. By analyzing the behavior
of users, the similarity values between target user and
other members would be recognized and based on these
similarities, the items that he has not purchased yet but
his similar user s have, will be rec ommended to him.
The better you can measure the similarity of users, the
better result you a chi e ve i n p re d ict ing the user’s be ha vi or .
In the field of social psychology there are many candi-
date aspec ts for similarity. But out o f the infinite number
of qualities that each person has, some characteristics are
more important in determining the similarity of two per-
sons. In general, the similarity depends on the system
point of view on interactions of users. However, more
knowledge about personalities of an individual helps us
to make more confident decisions on how similar are
humans. In recommendation systems it is also subject to
the extent to whic h the syste m is able to capture. M ost of
the collaborative filtering algorithms have focused on
rating information of users. They believe, ratings deter-
mine t he leve l of agre ement or disagr eement o f users for
a specific item. But the user interest may change over the
time. By increa sing the numbe r of interactions and inte r-
connections of people, internet, the probability of transi-
tory of interests, personal and social behavior is not
far-fetched, therefore; maybe what people want now is
likely to variation (is not what they have explored in the
past).In general, people have various reactions to events
happen in the society. The reflections of these reactions
can be observed in e-commerce. Also, the interpretation
of each action is related to its time. For instance, suppose
a science fiction movie has added to the list of an online
movie show site. Users, who are fan of these kinds of
movies, would watch and rate it as soon as possible, but
those who are not very interested in mentioned type of
movie maybe watch it later or when there is no one left
in the list which is fascinating from their point of view.
Hence we can infer, this group of users is different from
pervious users and one of the most important factors
which help us to recognize this issue is measuring the
1Collaborative Filtering

412 Proposing a New Metric for Collaborative Filtering
level of time distances in interactions of users. An intel-
ligent recommender system must be able to adapt itself
with current requirements of users. On the other hand,
the order of interactions can be a source of semantics
which help us in simulating the mentally evolution
pro cess of user s.
In this research we have presented a new similarity
metric which combine different aspect of user behavior
in order to better understand who are more similar to
each other and who are different from other. Experiment
results demonstrate that our contribution improves the
accuracy of predictions and evaluate what approach is
more effective to satisfy users.
The rest of the paper is organized as follows. In Sec-
tion 2, we review previous research in the field of colla-
borative filtering and analyze most significant metrics
which are applied in recommendation system. In Section
3, we propose a novel method to measure the similarity
of users by conside ri ng different effective aspects such as
time. In Section 4 the experimental results are shown to
evaluate the proposed approach. And Section 5 con-
cludes the paper.
2. Related Work
For a long time recommender systems did not deal with
changing user’s interest in their p r e dictions. For example,
similarity of users was constant in different time periods
and training datasets were considered as a static resource.
In memory-based collaborative filtering, predictions are
based on preferences of neighbor users or items. A
widespread approach in memory-based collaborative
filtering is the k-nearest neighbor algorithm. In [1] two
imputed neighborhood based collaborative filtering algo-
rithms are proposed that improved the performance of
recommender system in very sparse rating data. Another
popular technique is the top-N recommendations which
recommends a set of N top-sorted items which probably
will be interesting items by a target user [2,3]. However
hybrid approaches are recognized as a way to alleviate
the scaling and sparsity problem without any optimized
solution when the content of environment changes dy-
namically over the time.
In [4] the history period is divided into intervals with
different length the similarity is computed in each inter-
val separately. But the way of determining initial time
interval and length of each time inter val, consideri ng the
conditions of each system is not described.
A critical step in K-nearest algorithm is the neighbor-
hood formation which means finding a special subset of
the co mmuni ty for target user by recognizing others with
analogous behavior to act as recommenders. In order to,
every pair of users profile to be computed, to measure the
degree of similarity,
,uv
Sim
, shared between all pairs a
and b [5]. The core objective of similarity metrics is
evaluating potential relationship between users with a
numeric value.
Jaccard similarity has focused on the amount of over-
lap between users based on the amount of shared items in
each pair. However, the value of the user’s rate is neg-
lected [6]. Equation (1) describes this metric in more
details in which dete rmine s the rat ing o f user u
for item i.
,,
,
,,
ui vi
uv
ui vi
RR
Sim RR
∩
=∪
(1)
One o f the most widespread methods in measuring the
similarity of t wo users is Pea rson Correlation Coe fficient
(PCC) which aims to analyzing the behavior of user,
considering ratings input by each user [7]. As you see in
Equation (2) ratings are normalized by subtracting the
user’s mean rating,
u
R
, because each user may have a
particular rating scale. I is the set of common items be-
twee n user u and v. De spite the pop ularity of PCC, it has
been the subject to a number of modifications that will be
discussed later in a more exact wa y .
( )()
() ( )
,,
,22
,,
ui uvi v
iI
uv
ui uvi v
iI iI
RRR R
Sim
RR RR
∈
∈∈
−−
=
−−
∑
∑∑
(2)
Another si milarity metric s is co sine similarit y as in (3)
that two users are compared as two vectors, while the
similarity between them depends on their cosine angle
[8]. The main application of cosine similarity is evaluat-
ing the similarity of two documents by tracing each
document as a vector of word frequencies and computing
the cosine of the angle formed by the frequency vectors
[9].In collaborative filtering, users or items which are
corresponding to documents and ratings are used instead
of word frequencies.
,,
,22
,,
ui vi
iI
uv
ui vi
iI iI
RR
Sim RR
∈
∈∈
=∑
∑∑
(3)
Shanle Ma [10] presented a hybrid recommender sys-
tem to cope with the interest d rifting proble m that used a
time-sensitive-f unction. One of the most related works to
our study is [5] that investigated on the performance of
collaborative filtering over time in which the similarity
of user s change s with ti me. But it was almo st an anal yti-
cal research but a clear way to cope relevant problem
was not presented.
In all mentioned approaches, there is no difference in
similarity of users considering items rated either in the
same timestamps or dissimilar timestamps. In other
iu
R
,

Proposing a New Metric for Collaborative Filtering
413
words, the behavior of users is not monitored w.r.t time.
There is still an open issue of measuring similarity in
which comparing prediction accuracy demonstrates that
one similarity measure can outperform another one based
on the particular dataset.
3. Proposed Metric
There is a public standard notation to describe CF prob-
lems. The input can be structured as an M × N interact ion
matrix, A, associated with M user
{ }
12 m
,, ,U uuu=
and N items
{ }
12 n
,, ,I iii=
. In this matrix,
,kj
a
can
take the value of either 0 or 1, which 1 represents an ob-
serving transaction between
k
u
and
j
i
(for example,
1
u
has purchased
1
i
.) and 0 represents the absence of any
transaction. By using this matrix the level of user’s in-
terest for different items can be shown. For analyzing the
behavior of users in a more exact way we need to know
the time that each interaction is occurred. As we men-
tioned earlier, one important factor on the performance of
similarity metrics is the ability of the recommender sys-
tem to record the different type of feedback from users.
There are several types of feedbacks which can be cap-
tured in these syste ms such as rating, listening habits for
music recommendation and comments in e-commerce
web site.
We analyze relation of two users in four phases and
then combine these factors to compute the final result as
the similarity amon g the m. Actua lly, the main p urpo se o f
this step is calculating the difference between user’s in-
terests. Four factors are important for the final weight:
a) The number of common ite ms .
b) The degree of user’s interest in common items.
c) Timestamp of rated ite ms.
d) Rating s equences.
However it is unclear which factor plays more impor-
tant role. Considering conditions and goals of our busi-
ness, each factor may have various effects on the final
weight between users. For example, if there is no appro-
priate rating definitio n to eval uate the user ’s interes t for a
specific item, the first factor plays more important ro le . If
the s yste m and use rs ha ve a d ynamic b ehav ior in vario us
time intervals, the third factor becomes more important.
In the following, we explain how each factor is con-
tributed in the proposed metric.
3.1. The Number of Common Items
The number of common items between users u and v,
,uv
NCI
, is first fac tor which is consid ere d in our calc ula-
tion. To normalize the value we use
,uv
MaxCI
. This
function returns the number of all possible common
items [11]. If one rates 10 items and the other one rates 5
items, they can have up to 5 common items. Therefore,
Proportion in (4) is a normal value (range from 0 to 1)
which repr e sents the first similarity factor.
,
,
Proportion
uv
uv
NCI
MaxCI
=
(4)
3.2. The Degree of User’s Interest in Common
Items
The Equation (5) is the sum of rate differences between
two users (u and v) and CI indicates the set of items
which are in common between them.
, ,,uvui vi
i CI
SORDR R
∈
= −
∑
(5)
To measure rating difference factor, Equation (6) is
applied as
,uv
MaxRateDiff
which determines the max-
imum difference that may occur between u and v w.r.t.
item’s rate. For example, if numerical feedback of the
system is 5-star, we have:
,,
4*
uv uv
MaxRateDiff NCI=
(6)
In order to measure the similarity of a pair by consi-
dering the first two factors, we use the following equa-
tion:
,
1/
,
,
Proportion*1
uv
EOR
uv
uv
BaseWeight
SORD
MaxRateDiff
=



−




(7)
EOR implies the effect of item’s rate factor and is
greater than one. By increasing the value of EOR, the
difference between users becomes more important.
3.3. Timestamps of Rated Items
The users’ similarity increases when they have more
common time distance. The Equation (8) implies sum of
time distance among u and v and is
,ui
T
the time of rat-
ing i by u s er u.
, ,,uvui vi
i CI
SOTDT T
∈
= −
∑
(8)
Consequently for the time factor we use (9) as an
adaptive decay function. In this equation α is exploited
for the purpose of adaptability which depends on the
system’s time s cale (α > 0).
( )
,
,
,
uv
uv
uv
DF SOTDSOTD
NCI
α
α
=
+


(9)
Thr ough a lear ning pr ocess we can tune α to make the
method more accurate. We will discuss this subject in
Section 4.
3.4. The Sequence of Rated Items
In this section the distance between users w.r.t. interac-

414 Proposing a New Metric for Collaborative Filtering
tions order is computed. If items rated by the user are
kept in a queue, each item has a priority number. There-
fore the distance can be calculated according to the fol-
lowing equation.
, ,,uvui vi
i CI
SOPDP P
∈
= −
∑
(10)
In above equation,
iu
P
,
is the priority of item i in u’s
item queue. For example in Figure 1,
1,uD
P
=4 In order
to determine the distance order (interaction sequence) of
u1from u2 and u3 in Figure 1, (10) is applied. As Figure
1 shows, 7 items are in common among all users. The
common items queue is illustrated as a link list. Accord-
ing to (10), we have
12
,uu
SOPD
=24 and
13
,uu
SOPD
=6,
therefore u3is more similar to u1 in comparison with u2
w.r.t. interactions order. A normal value for calculating
the impact of time in final result is needed. In (11),
,uv
NSOPD
is a no rmal value (rang from 0 a nd 1) that uses
()MSOPD n
functi on i n (1 2) which i mplie s the ma ximu m
dissimilarity of interactions sequence of a user to another
one when they have n common items.
,
,
()
uv
uv
SOPD
NSOPD MSOPD n
=
(11)
( )
2
2
2
2if is odd
2 ()
2 otherwise
n
nn n
MSOPD n
n

−−

=



(12)
Using linear combination which is shown in (13), the
distance between two users can be computed in w.r.t.
rated items timestamp and order of common items.
( )
( )
( )
,, ,
1
1
uv uvuv
TFDF SOTDNSOPD αβ
αβ
= +−
+=
(13)
Finally, the similarity between two users by consider-
ing all four factors can be measured through (14) in
whic h EOT deter mines ho w the val ue of
,uv
TF
can effect
on the final res ult.
Figure 1. u1, u2 and u3 have purchased same items (A

G)
in differ ent orders.
( ) ( )
( )
,, ,
* 1*
uv uvuv
SimBaseWeightEOTEOT TF= +−
(14)
In above equation, 0 < EOT < 1 and
,
01
uv
Sim<<
4. Evaluation
To evaluate the performance of our proposed metric, we
need to apply it in creating recommendation process and
compare the result with classic collaborative filtering
algorithm2. Therefore the Weighted Sum of Other Rat-
ings [7] is used to generate recommendations. The value
of
vu
Sim
,
is calculated based on our metrics in the pre-
vious step. In this formula U is the collection of k-most
similar user to u.
( )
,,
,,
.
v
vi uv
vU
u
ui uv
RR Sim
RR Sim
∈
−
= +
∑
(15)
Statistical Accuracy Metrics evaluate the accuracy of a
system by comparing the numerical recommendation rate
on training dataset against the real user provided value
for the user-item pairs in the test dataset [12,13]. In this
study Mean Absolute Error (MAE) is used as one of the
widely adopted metric for evaluation. Where n is the
total number of ratings over all users,
,ij
p
is the pre-
dicted rating for user i on item j, and
ji
r
,
is the actual
rating. A lower MAE means a b e tte r prediction [14].
,,
{, }ij ij ij
pr
MAE n
−
=∑
(16)
We have chosen MovieLens dataset as one of the most
prevalent and reliable datasets for evaluation and com-
parison of recommender systems. MovieLens was col-
lected through the MovieLens project, and was distri-
buted by GroupLens Research Group at the University of
Minnesota. However, each vote is accompanied by a
time stamp, but for our own purpose, it is not an appro-
priate source for evaluating our research. For example,
most of the time stamps belong to the time that the sys-
tem as ke d the o p inion o f t he u se r, no t to the first ti me the
user has seen the movie. Therefore, a user may rate 100
items in one da y. As it can be seen in Figure 2 t he resul t
of analyzing Movielens dataset shows that 55% of users
inter acted with the system in onl y one d ay.
As we mentioned before our approach can adopt itself
to different circumstances and limitations of the target
system. Because of this adaptively, the algorithm should
be customized to our qualifications and concepts of our
business. For achieving this purpose some parameters
such as EOR, EOT and α should be set up. Figure 3
shows the effect of each parameter with 200 neighbors
2
In classic approach, PCC is applied as the similarity metric and
Wei g hted Sum of O ther’s Ra ting s a s the re co m mendatio n m ethod .

Proposing a New Metric for Collaborative Filtering
415
Figure 2. In MovieLens dat aset 55 % of people have bought
items only in one day.
(a)
(b)
(c)
Figure 3. Evaluat ion of the pr opose d app roach, k = 200. (a)
α = 10, EOT = 0.7 and 0.25 ≤ EOR ≤ 6. (b) α = 10 , EO R = 2,
0.1 ≤ EOT ≤ 0.7. (c) EOT = 0.5, E O R = 2 and 1 ≤ α ≤ 15.
Figure 4. Compar ing the pr op osed a ppr oac h wi th the class ic
appr o ach.
(k). This process helps us finding the best values of each
parameter for our data set.
After achieving the best performance of our approach
for MovieLens dataset (α = 10, EOT = 0 .7 and EOR =
10), the accuracy of the proposed algorithm is compared
with classic colla borative filte ring algor ithm. In Figure 4,
the graph shows the progression of the MAE using the
PCC and Our metrics when the number of neighbors, k,
ranges from 30 to 400. The evaluation demonstrates the
best accuracy is achieved with k=200.
5. Conclusions and Future Works
In this article, we proposed a new metric using the time
factor as an important parameter for measuring the simi-
larity of users. Our experiments show that our approach
is more accurate than the classic one especially when
users’ interests change over the time.
We are currently working on tuning the required pa-
rameters such as EOR and EOT for a given dataset. In t h e
future, we intend to include implicit factors as user
comments in our approach.
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