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Knowledge-Based Systems 74 (2015) 133–150
Contents lists available at ScienceDirect
Knowledge-Based Systems
journal homepage: www.elsevier.com/locate/knosys
Intuitionistic fuzzy recommender systems: An effective tool for medical
diagnosis
Le Hoang Son ⇑, Nguyen Tho Thong
VNU University of Science, Vietnam National University, Viet Nam
a r t i c l e
i n f o
Article history:
Received 12 May 2014
Received in revised form 6 October 2014
Accepted 10 November 2014
Available online 20 November 2014
Keywords:
Accuracy
Fuzzy sets
Intuitionistic fuzzy collaborative filtering
Intuitionistic fuzzy recommender systems
Medical diagnosis
a b s t r a c t
Medical diagnosis has been being considered as one of the important processes in clinical medicine that
determines acquired diseases from some given symptoms. Enhancing the accuracy of diagnosis is the
centralized focuses of researchers involving the uses of computerized techniques such as intuitionistic
fuzzy sets (IFS) and recommender systems (RS). Based upon the observation that medical data are often
imprecise, incomplete and vague so that using the standalone IFS and RS methods may not improve the
accuracy of diagnosis, in this paper we consider the integration of IFS and RS into the proposed methodology and present a novel intuitionistic fuzzy recommender systems (IFRS) including: (i) new definitions
of single-criterion and multi-criteria IFRS; (ii) new definitions of intuitionistic fuzzy matrix (IFM) and
intuitionistic fuzzy composition matrix (IFCM); (iii) proposing intuitionistic fuzzy similarity matrix
(IFSM), intuitionistic fuzzy similarity degree (IFSD) and the formulas to predict values on the basis of
IFSD; (iv) a novel intuitionistic fuzzy collaborative filtering method so-called IFCF to predict the possible
diseases. Experimental results reveal that IFCF obtains better accuracy than the standalone methods of
IFS such as De et al., Szmidt and Kacprzyk, Samuel and Balamurugan and RS, e.g. Davis et al. and Hassan
and Syed.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
In this section, we formulate the medical diagnosis problem and
give some illustrated examples in Section 1.1. Section 1.2 describes
the relevant works using the intuitionistic fuzzy sets for the medical diagnosis problem. Section 1.3 summarizes the limitations of
those relevant works, and based on these facts the motivation
and ideas of the proposed approach are highlighted in Section 1.4.
Section 1.5 demonstrates our contributions in details, and their
novelty and significance are discussed in Section 1.6. Lastly,
Section 1.7 elaborates the organization of the paper.
1.1. The medical diagnosis problem
Medical diagnosis has been being considered as one of the most
important and necessary processes in clinical medicine that
determines acquired diseases of patients from given symptoms.
According to Kononenko [20], diagnosis commonly relates to the
probability or risk of an individual developing a particular state
of health over a specific time, based on his or her clinical and
⇑ Corresponding author at: 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam. Tel.:
+84 904171284; fax: +84 0438623938.
E-mail addresses: , (L.H. Son).
/>0950-7051/Ó 2014 Elsevier B.V. All rights reserved.
non-clinical profile. It is useful to minimize the risk of associated
health complications such as osteoporosis, small bowel cancer
and increased risk of other autoimmune diseases. Mathematically,
its definition is stated as follows.
Definition 1 (Medical diagnosis). Given three lists: P = {P1, . . ., Pn},
S = {S1, . . ., Sm} and D = {D1, . . ., Dk} where P is a list of patients, S a
list of symptoms and D a list of diseases, respectively. Three values
n, m, k 2 N+ are the numbers of patients, symptoms and
diseases, respectively. The relation between the patients and the
symptoms is characterized by the set- RPS ¼ fRPS ðP i ; Sj Þj8i ¼
1; . . . ; n; 8j ¼ 1; . . . ; mg where RPS(Pi, Sj) shows the level that patient
Pi acquires symptom Sj and is represented by either a numeric
value or a (intuitionistic) fuzzy value depending on the domain of
the problem. Analogously, the relation between the symptoms and
the diseases is expressed as RSD ¼ fRSD ðSi ; Dj Þj8i ¼ 1; . . . ; m; 8j ¼
1; . . . ; kg where RSD(Si, Dj) reflects the possibility that symptom Si
would lead to disease Dj. The medical diagnosis problem aims to
determine the relation between the patients and the diseases
described by the set- RPD ¼ fRPD ðP i ; Dj Þj8i ¼ 1; . . . ; n; 8j ¼ 1; . . . ; kg
where RPD(Pi, Dj) is either 0 or 1 showing that patient Pi acquires
disease Dj or not. The medical diagnosis problem can be shortly
represented by the implication fRPS ; RSD g ! RPD .
134
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Example 1. Consider the dataset in [31] having four patients
namely P = {Ram, Mari, Sugu, Somu}, five symptoms S = {Temperature, Headache, Stomach-pain, Cough, Chest-pain} and five diseases D = {Viral-Fever, Malaria, Typhoid, Stomach, Heart}. The
relations between the patients – the symptoms and the symptoms
– the diseases are illustrated in Tables 1 and 2, respectively.
The relation between the patients and the diseases determined
by the medical diagnosis is illustrated in Table 3. Since the domain
of the problem is the intuitionistic fuzzy values, this relation is also
expressed in this form. The most acquiring disease that the
patients suffer is expressed in Table 4, which is converted from
Table 3 by a trivial defuzzification method considering the maximal membership degree of disease among all.
Medical diagnosis is considered as an efficient support tool for
clinicians to make the right therapeutical decisions especially in
the cases that medicine extends its predictive capacities using
genetic data [5]. As being observed in Table 3, medical diagnosis
could assist the clinicians to enumerate the possible diseases of
patients accompanied with certain membership values. Thus, it is
convenient for clinicians, who are experts in this field, to quickly
diagnose and give proper medicated figures. This fact clearly shows
the importance of medical diagnosis in medicine sciences
nowadays.
P
Temperature
Headache
Stomach_pain
Cough
Chest_pain
Ram
Mari
Sugu
Somu
(0.8, 0.1)
(0, 0.8)
(0.8, 0.1)
(0.6, 0.1)
(0.6,
(0.4,
(0.8,
(0.5,
(0.2, 0.8)
(0.6, 0.1)
(0, 0.6)
(0.3, 0.4)
(0.6,
(0.1,
(0.2,
(0.7,
(0.1, 0.6)
(0.1, 0.8)
(0, 0.5)
(0.3, 0.4)
0.1)
0.7)
0.7)
0.2)
S
Viral_fever
Malaria
Typhoid
Stomach
Heart
Temperature
Headache
Stomach_pain
Cough
Chest_pain
(0.4,
(0.3,
(0.1,
(0.4,
(0.1,
(0.7, 0)
(0.2, 0.6)
(0, 0.9)
(0.7, 0)
(0.1, 0.8)
(0.3,
(0.6,
(0.2,
(0.2,
(0.1,
(0.1,
(0.2,
(0.8,
(0.2,
(0.2,
(0.1, 0.8)
(0, 0.8)
(0.2, 0.8)
(0.2, 0.8)
(0.8, 0.1)
0.3)
0.1)
0.7)
0.6)
0.9)
0.7)
0.4)
0)
0.7)
0.7)
Table 3
The relation between the patients and the diseases – RPD expressed by intuitionistic
fuzzy values.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
(0.4,
(0.3,
(0.4,
(0.4,
(0.7,
(0.2,
(0.7,
(0.7,
(0.6,
(0.4,
(0.6,
(0.5,
(0.2,
(0.6,
(0.2,
(0.3,
(0.2,
(0.1,
(0.2,
(0.3,
0.1)
0.5)
0.1)
0.1)
0.1)
0.6)
0.1)
0.1)
0.1)
0.4)
0.1)
0.3)
Definition 2. A Fuzzy Set (FS) [49] in a non-empty set X is a
function
ð1Þ
where l(x) is the membership degree of each element x 2 X. A fuzzy
set can be alternately defined as,
A ¼ fhx; lðxÞijx 2 Xg:
ð2Þ
An extension of FS that is widely applied to the medical prognosis problem is Intuitionistic Fuzzy Set (IFS), which is defined as
follows.
Definition 3. An Intuitionistic Fuzzy Set (IFS) [4] in a non-empty set
X is,
Table 2
The relation between the symptoms and the diseases – RSD.
0)
0.5)
0.7)
0.3)
0.7)
Computerized techniques for medical diagnosis such as fuzzy
set, genetic algorithms, neural networks, statistical tools and recommender systems aiming to enhance the accuracy of diagnosis
have been being introduced widely [20]. Nonetheless, an important issue in medical diagnosis is that the relations between the
patients – the symptoms (RPS) and the symptoms – the diseases
(RSD) are often vague, imprecise and uncertain. For instance, doctors could faced with patients who are likely to have personal
problems and/or mental disorders so that the crucial patients’
signs and symptoms are missing, incomplete and vague even
though the supports of patients’ medical histories and physical
examination are provided within the diagnosis. Even if information
of patients are clearly provided, how to give accurate evaluation to
given symptoms/diseases is another challenge requiring welltrained, copious-experienced physicians. These evidences raise
the need of using fuzzy set or its extension to model and assist
the techniques that improve the accuracy of diagnosis. The definition of fuzzy set is stated below.
l : X ! ½0; 1
x # lðxÞ;
Table 1
The relation between the patients and the symptoms – RPS.
0.1)
0.4)
0.1)
0.4)
1.2. The previous works
0.4)
0.1)
0.4)
0.4)
0.6)
0.7)
0.5)
0.4)
Table 4
The most acquiring diseases of patients.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
0
0
0
0
1
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
e¼
A
nD
E
o
x; le ðxÞ; ce ðxÞ jx 2 X ;
A
A
ð3Þ
where leðxÞ and ce ðxÞ are the membership and non-membership
A
A
degrees of each element x 2 X, respectively.
leA ðxÞ; ceA ðxÞ 2 ½0; 1; 8x 2 X;
0 6 leðxÞ þ ce ðxÞ 6 1; 8x 2 X:
A
A
ð4Þ
ð5Þ
The intuitionistic fuzzy index of an element showing the non-determinacy is denoted as,
peA ðxÞ ¼ 1 À leA ðxÞ þ ceA ðxÞ; 8x 2 X:
ð6Þ
When pe ðxÞ ¼ 0 for "x 2 X, IFS returns to the FS set of Zadeh.
A
Some extensions of fuzzy sets are not appropriate for modeling
uncertainty in the medical diagnosis such as the rough set [28],
rough soft sets [11,12,16], intuitionistic fuzzy rough sets [50] and
soft rough fuzzy sets & soft fuzzy rough sets [23]. The limitations
of these sets, as pointed out by Yao [48], Rodriguez et al. [30], Jafarian and Rezvani [17] and many other authors lie to their intrinsic
nature and how they are organized and operated such as (i) The
positive and the boundary rules are considered in rough sets and
their variants so that in cases of many concepts, the negative rules
would be redundant; (ii) The modeling of linguistic information is
limited due to the elicitation of single and simple terms that should
encompass and express the information provided by the experts
regarding the a linguistic variable; (iii) if exact membership
degrees cannot be determined due to insufficient information then
it is impossible to consider the uncertainty on the membership
135
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
function. Thus, these types of fuzzy sets could not be used for the
application of medical diagnosis.
The first approach for the medical diagnosis problem was
drawn from the Sanchez’s notion of medical knowledge [32]. Since
then several improvements of the Sanchez’s approach in association with IFS and other advanced fuzzy sets have been introduced.
De et al. [9] fuzzified the relations between the patients – the
symptoms and the symptoms – the diseases by intuitionistic fuzzy
memberships and derived the relation between the patients and
the diseases by means of intuitionistic fuzzy relations. The algorithm contains the following steps.
1. Calculate the relation between the patients and the diseases
by intuitionistic fuzzy relations with the membership and
non-membership functions being expressed in Eqs. (7) and
(8), respectively.
È
É
lPD ðPi ; Dj Þ ¼ max minflPS ðPi ; Sl Þ; lSD ðSl ; Dj Þg ;
Table 6
The SPD matrix where bold values imply the most possible disease.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
0.35
0.2
0.35
0.35
0.68
0.08
0.68
0.68
0.57
0.32
0.57
0.44
0.04
0.57
0.04
0.18
0.08
0.05
0.05
0.18
Table 7
The WPD matrix.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
(0.4,
(0.3,
(0.4,
(0.4,
(0.7,
(0.2,
(0.7,
(0.7,
(0.6,
(0.4,
(0.6,
(0.5,
(0.2,
(0.6,
(0.2,
(0.3,
(0.2,
(0.2,
(0.2,
(0.3,
0.9)
0.5)
0.9)
0.9)
0.9)
0.4)
0.9)
0.9)
0.9)
0.6)
0.9)
0.7)
0.6)
0.9)
0.6)
0.6)
0.4)
0.5)
0.5)
0.6)
l¼1;m
8i 2 f1; . . . ; ng;
8j 2 f1; . . . ; kg;
È
ð7Þ
É
Table 8
The reduction matrix where bold values imply the most possible disease.
cPD ðPi ; Dj Þ ¼ min maxfcPS ðPi ; Sl Þ; cSD ðSl ; Dj Þg ;
l¼1;m
8i 2 f1; . . . ; ng;
8j 2 f1; . . . ; kg:
ð8Þ
2. Perform the defuzzification through the SPD,
SPD ¼ lPD À cPD Â pPD :
ð9Þ
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
0.4
0.3
0.4
0.4
0.7
0.2
0.7
0.7
0.6
0.4
0.6
0.5
0.2
0.6
0.2
0.3
0.2
0.2
0.2
0.3
3. Determine the most acquiring diseases of patients based on
the maximal SPD and minimal pPD.
Example 2. Consider the dataset in Example 1. The relation
between the patients and the diseases calculated by Eqs. (7) and
(8) is expressed in Table 5. The SPD matrix is described in Table 6.
Based upon this table, Ram, Sugu and Somu suffer from the Malaria
and Mari acquires Stomach the most.
Samuel and Balamurugan [31] improved the method of De et al.
[9] by a new technique named intuitionistic fuzzy max–min composition. This method is analogous to that of De et al. [9] except
that Steps 2 & 3 are replaced by,
1. Compute W PD ¼ ðlPD ; 1 À cPD Þ.
2. For each Pi find maxj fminðlPD ðP i ; Dj Þ; 1 À cPD ðP i ; Dj ÞÞg and
conclude the most acquiring diseases.
1. Use the Hamming or Euclidean function to calculate the relation
between the patients and the diseases as in Eqs. (10) and (11),
respectively.
m
À
1 X
l ðP i ; Sl Þ À l ðSl ; Dj Þ þ jc ðPi ; Sl Þ
PS
PS
SD
2m l¼1
Á
ÀcSD ðSl ; Dj Þ þ jpPS ðPi ; Sl Þ À pSD ðSl ; Dj Þj ;
ð10Þ
RPD ðPi ; Dj Þ ¼
m
À
Á
1 X
lPS ðPi ; Sl Þ À lSD ðSl ; Dj Þ 2
2m l¼1
À
Á2
þ cPS ðP i ; Sl Þ À cSD ðSl ; Dj Þ
1=2
þðpPS ðPi ; Sl Þ À pSD ðSl ; Dj ÞÞ2
:
RPD ðPi ; Dj Þ ¼
ð11Þ
Example 3. Consider again the dataset in Example 1. The WPD
matrix is shown in Table 7. The reduction of WPD is presented in
Table 8. From this table, Ram, Sugu and Somu suffer from the
Malaria and Mari acquires Stomach the most.
2. Conclude the possible diseases of patients based on the minimal
distance criterion.
Another approach for the medical diagnosis is utilizing the
distance functions to calculate the relation between the patients
and the diseases from the relations between the patients – the
symptoms and the symptoms – the diseases as described in
[42–44,19,33]. The general activities of these algorithms are,
Example 4. Use this method for the dataset in Example 1, we have
the relations between the patients and the diseases by the Hamming (Table 9) or Euclidean function (Table 10). The most acquiring diseases of patients are highlighted in bold.
Table 5
The relation between the patients and the diseases – RPD in the method of De et al. [9]
expressed by intuitionistic fuzzy values.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
(0.4,
(0.3,
(0.4,
(0.4,
(0.7,
(0.2,
(0.7,
(0.7,
(0.6,
(0.4,
(0.6,
(0.5,
(0.2,
(0.6,
(0.2,
(0.3,
(0.2,
(0.2,
(0.2,
(0.3,
0.1)
0.5)
0.1)
0.1)
0.1)
0.6)
0.1)
0.1)
0.1)
0.4)
0.1)
0.3)
0.4)
0.1)
0.4)
0.4)
0.6)
0.5)
0.5)
0.4)
Besides these approaches, some authors have extended them
for special cases, e.g. multi-criteria medical diagnosis and the multiple time intervals modeling for the relation between the patients
and the symptoms. This requires the deployment on other
advanced fuzzy sets such as the type-2 fuzzy sets [26], the interval-valued intuitionistic fuzzy sets [2], fuzzy soft set [25,47] and
intuitionistic fuzzy soft set [1,21]. The combination of these fuzzy
sets with machine learning methods to handle the special cases
such as the fuzzy-neural automatic system [27,24] and the type2 fuzzy genetic algorithm [45,14] was also investigated.
136
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 9
The relation between the patients and the diseases by the Hamming function where
bold values imply the most possible disease.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
0.28
0.40
0.38
0.28
0.24
0.50
0.44
0.30
0.28
0.31
0.32
0.38
0.54
0.14
0.50
0.44
0.56
0.42
0.55
0.54
Table 10
The relation between the patients and the diseases by the Euclidean function where
bold values imply the most possible disease.
decision support systems, can give users information about predictive ‘‘rating’’ or ‘‘preference’’ that they would like to assess an item;
thus helping them to choose the appropriate item among numerous possibilities. This kind of expert systems is now commonly
popularized in numerous application fields such as books, documents, images, movie, music, shopping and TV programs personalized systems. The mathematical definition of RS is stated below.
Definition 4 (Recommender Systems – RS [29]). Suppose U is a set
of all users and I is the set of items in the system. The utility
function R is a mapping specified on U1 & U and I1 & I as follows.
R : U 1 Â I1 ! P
ð12Þ
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
ðu1 ; i1 Þ # Rðu1 ; i1 Þ;
Ram
Mari
Sugu
Somu
0.29
0.43
0.36
0.25
0.25
0.56
0.41
0.29
0.32
0.33
0.32
0.35
0.53
0.14
0.52
0.43
0.58
0.46
0.57
0.50
where R(u1, i1) is a non-negative integer or a real number within a
certain range. P is a set of available ratings in the system. Thus, RS is
the system that provides two basic functions below.
Ã
1.3. The limitations of the previous works
Considering the relevant works involving the usage of the IFS
set, we clearly recognize that IFS was used mainly for the applications of medical diagnosis among the advanced fuzzy sets. Nonetheless, these works have the following disadvantages.
(a) The previous works calculate the relation between the
patients and the diseases (RPD) solely from those between
the patients – the symptoms (RPS) and the symptoms –
the diseases (RSD). In some practical cases where the relation
between the patients – the symptoms or the symptoms – the
diseases is missing, those works could not be performed.
This fact is happened in reality since clinicians somehow
do not accurately express the values of membership and
non-membership degrees of symptoms to diseases or vive
versa;
(b) The information of previous diagnoses of patients could not
be utilized. That is to say, a patient has had some records in
the patients-diseases databases (RPD) beforehand. Nevertheless, the calculation of the next records of this patient is
made solely on the basis of both RPS and RSD. Historic diagnoses of patients are not taken into account so that the accuracy of diagnosis may not be high as a result;
(c) The determination of the most acquiring disease is dependent from the defuzzification method. For instance, De
et al. [9] used SPD for the defuzzification, Samuel and Balamurugan [31] relied on the reduction matrix from WPD and
Szmidt and Kacprzyk [42–44], Khatibi and Montazer [19]
and Shinoj and John [33] employed the distance functions.
Independent determination from the defuzzification method
should be investigated for the stable performance of the
algorithm.
(d) Mathematical properties of operations such as the fuzzy
implication in De et al. [9], Samuel and Balamurugan [31]
and the distance function in Szmidt and Kacprzyk [42–44],
Khatibi and Montazer [19] and Shinoj and John [33] were
not discussed in the equivalent articles. Readers could not
know the theoretical bases of these operations and why they
were selected for the medical diagnosis problem.
1.4. The motivation and ideas
From the disadvantages of the previous works, our idea in this
article is using the hybrid method between Recommender Systems
(RS) and the IFS set to handle them. RS, which are a subclass of
Ã
(a) Prediction: determine Rðuà ; i Þ for any ðuà ; i Þ 2 ðU; IÞ n ðU 1 ; I1 Þ.
(b) Recommendation: choose i⁄ 2 I satisfying i⁄ = arg maxi2IR(u, i)
for all u 2 U.
RS has been applied to the medical diagnosis problem. Davis et al.
[8] proposed CARE, a Collaborative Assessment and Recommendation Engine, which relies only on a patient’s medical history in order
to predict future diseases risks and combines collaborative filtering
methods with clustering to predict each patient’s greatest disease
risks based on their own medical history and that of similar patients.
An iterative version of CARE so-called ICARE that incorporates
ensemble concepts for improved performance was also introduced.
These systems required no specialized information and provided
predictions for medical conditions of all kinds in a single run. Hassan
and Syed [13] employed a collaborative filtering framework that
assessed patient risk both by matching new cases to historical
records and by matching patient demographics to adverse outcomes
so that it could achieve a higher predictive accuracy for both sudden
cardiac death and recurrent myocardial infraction than popular
classification approaches such as logistic regression and support
vector machines. More works on the applications of RS could be referenced in Duan et al. [10], Meisamshabanpoor and Mahdavi [22]
and our previous works in [7,38,40,39,41,34–37].
Example 5. Consider the training dataset in Table 11. Taking a
simple encoded method by multiplying the membership degree by
10 and adding the non-membership degree to it, we have a crisp
training in Table 12.
The method of Hassan and Syed [13] employed a collaborative
filtering including the traditional Pearson coefficient to calculate
the similarity between users and the k-nearest neighbor approximation function to predict the blank values in Table 12. The results
are shown in Table 13. If taking the maximal value among all for a
given patient in Table 13 then we can conclude that Ram, Sugu and
Somu are suffered from Malaria and Mari acquires Stomach.
Analogously, Table 14 shows the results of the method of Davis
Table 11
The training dataset with ⁄ being the values to be predicted.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
(0.4, 0.1)
(0.3, 0.5)
(0.4, 0.1)
⁄
(0.7, 0.1)
(0.2, 0.6)
(0.7, 0.1)
⁄
(0.6, 0.1)
(0.4, 0.4)
⁄
(0.5, 0.3)
(0.2, 0.4)
(0.6, 0.1)
⁄
(0.3, 0.4)
(0.2, 0.6)
(0.1, 0.7)
⁄
⁄
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 12
The crisp training dataset with ⁄ being the values to be predicted.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
4.1
3.5
4.1
⁄
7.1
2.6
7.1
⁄
6.1
4.4
⁄
5.3
2.4
6.1
⁄
3.4
2.6
1.7
⁄
⁄
et al. [8] where Ram is suffered from Malaria, Mari acquires Stomach and Sugu and Somu have Typhoid.
From Example 5, we clearly recognize the following facts:
(a) RS could be applied to the medical diagnosis. Yet in cases
that the relations are expressed by fuzzy memberships as
in Table 11, the accuracy of diagnosis in RS is dependent
on the encoded method. In the other words, RS is effective
with the crisp dataset such as Table 12 but not the fuzzy
one, e.g. Table 11;
(b) The problem of the previous researches about the dependence of the determination of the most acquiring disease
from the defuzzification method, e.g. the maximal function
in Example 5 still exists;
(c) RS works only if the training dataset is provided. That is to
say, we must have the historic diagnoses of patients for
the prediction.
From Sections 1.3 and 1.4 and illustrated examples, we clearly
recognize that the IFS and RS approaches have their own advantages
and disadvantages. Thus, a combination of these approaches in order
to combine the advantages and eliminate the disadvantages could
handle the mentioned issues. Scanning the literature, we realize that
some hybrid methods were also designed for the medical diagnosis
problem, to name but a few such as Davis et al. [8] combined collaborative filtering methods with clustering; Kala et al. [18] integrated
genetic algorithms with modular neural network; Hosseini et al.
[14] joined a type-2 fuzzy logic with genetic algorithm. These evidences show that the combination of groups of methods such as
between RS and IFS is a trendy approach for medical diagnosis.
1.5. The contributions of this work
Based upon the observations, our contribution in this paper is a
novel intuitionistic fuzzy recommender system (IFRS) for medical
diagnosis consisting of the following components:
Table 13
The full dataset derived by the method of Hassan & Syed [13] where bold values imply
the most possible disease.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
4.1
3.5
4.1
5.9
7.1
2.6
7.1
9.8
6.1
4.4
4.8
5.3
2.4
6.1
1.9
3.4
2.6
1.7
3.2
5.5
Table 14
The full dataset derived by the method of Davis et al. [8] where bold values imply the
most possible disease.
P
Viral_fever
Malaria
Typhoid
Stomach
Heart
Ram
Mari
Sugu
Somu
4.1
3.5
4.1
2.6
7.1
2.6
7.1
4.7
6.1
4.4
7.3
5.3
2.4
6.1
5.2
3.4
2.6
1.7
2.1
0.2
137
(a) The new definitions of single-criterion IFRS (SC-IFRS) and
multi-criteria IFRS (MC-IFRS) that extend the definition of
RS (Definition 4) taking into account a feature of a user
and a characteristic of an item expressed by intuitionistic
linguistic labels (See Section 2.1). These definitions are the
basis for the deployment of similarity degrees used for the
prediction of RPD(Pi, Dj) (Definition 1);
(b) The new definitions of intuitionistic fuzzy matrix (IFM),
which is a representation of SC-IFRS and MC-IFRS in the
matrix format and the intuitionistic fuzzy composition
matrix (IFCM) of two IFMs with the intersection/union operation. Some interesting theorems and properties of IFM and
IFCM are presented (See Section 2.2);
(c) Some new similarity degrees of IFMs such as the intuitionistic fuzzy similarity matrix (IFSM) and the intuitionistic fuzzy
similarity degree (IFSD). The formulas to predict RPD(Pi, Dj)
on the basis of IFSD accompanied with an interesting theorem is proposed (See Section 2.3);
(d) From the predicting formulas, a novel intuitionistic fuzzy
collaborative filtering method so-called IFCF is presented
for the medical diagnosis problem (See Section 2.4);
(e) The validation of the IFCF method in comparison with the
standalone methods of IFS such as De et al. [9], Szmidt and
Kacprzyk [44], Samuel and Balamurugan [31] and RS, e.g.
Davis et al. [8], Hassan and Syed [13] is made by both a
numerical illustration on the dataset in Example 1 and the
experiments on benchmark medical diagnosis datasets from
UCI Machine Learning Repository in terms of the accuracy of
diagnosis (See Section 3).
1.6. The novelty and significance of the proposed work
According to the contributions in Section 1.5 and the limitations
of IFS and RS in Sections 1.3 and 1.4, respectively, the novel and the
significance of the proposed work are stressed as follows.
(a) The proposed work is different from the previous ones especially the standalone IFS and RS methods. Specifically, it
employs the ideas of both the IFS set and RS in the definitions of SC-IFRS and MC-IFRS, which are the basis to develop
some new terms and similarity degrees for the IFCF algorithm. Furthermore, as being observed from Example 1 to
3, the determination of the relation between patients and
diseases in the standalone IFS methods is performed by
some operations such as the fuzzy implication in De et al.
[9], Samuel and Balamurugan [31] and the distance function
in Szmidt and Kacprzyk [42–44]. In the proposed work, this
can be done through the intuitionistic fuzzy similarity
degree (IFSD) in Section 2.3, which is developed based on
SC-IFRS and MC-IFRS. Comparing with the standalone RS
methods such as Davis et al. [8] and Hassan and Syed [13],
the similarity degree – IFSD in the proposed work is constructed from the light of the IFS set but not by the Pearson
coefficient from the hard values such as in Table 12. Additionally, the formulas to predict RPD(Pi, Dj) are also made
according to the membership and non-membership functions but not by the hard values above. These proofs demonstrate the novel of the proposed work;
(b) The proposed hybrid method could handle the issues of the
standalone IFS and RS methods. For instance, the limitations
of IFS relating to the missing relations and the historic diagnoses of patients stated in Section 1.3(a) and (b) and the limitations of RS relating to the crisp and training datasets stated
in Section 1.4(a) and (c) are solved by the integration of IFS
and RS. The deficiency of mathematical properties of operations in Section 1.3(d) is resolved by a number of interesting
138
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
theorems and properties in Section 2. Lastly, when predicting
RPD(Pi, Dj), users could find a suitable defuzzification method
for the determination of the most acquiring disease;
(c) The proposal of this work is significance in terms of both
theory and practice. In the theoretical aspect, the proposed
work motivates researching on advanced algorithms of IFS
and RS especially the hybrid method between them to
enhance the accuracy of the algorithm. Looking for details
in Section 1.5, we recognize that the proposed method is
constructed on a well-defined mathematical foundation,
which is not paid much attention in the previous researches.
Thus, this guarantees the further deployment of other
advanced methods of both IFS and RS on such the mathematical foundation. In the practical side, the proposed work
contributes greatly to the medical diagnosis problem and
some extensions and variants of this method could be
quickly deployed for other socio-economic problems. This
clearly affirms the significance of the proposed work.
where liX(x) 2 [0, 1] (resp. ciX(x) 2 [0, 1]), "i 2 {1, . . ., s} is the
membership (resp. non-membership) value of the patient to the
linguistic label ith of feature X. ljY(y) 2 [0, 1] (resp. cjY (y) 2 [0, 1]),
"j 2 {1, . . ., s} is the membership (resp. non-membership) value of
the symptom to the linguistic label jth of characteristicY. Finally,
llD(D) 2 [0, 1] (resp. clD(D) 2 [0, 1]), "l 2 {1, . . ., s} is the membership
(resp. non-membership) value of disease D to the linguistic label lth.
SC-IFRS provides two basic functions:
(a) Prediction: determine the values of ðllD ðDÞ; clD ðDÞÞ, "l 2 {1,
. . ., s};
Ã
(b) Recommendation: choose i⁄ 2 [1, s] satisfying i ¼ arg
maxi¼1;s fliD ðDÞ þ liD ðDÞð1 À liD ðDÞ À ciD ðDÞÞg.
Remark 1
(a) From Definition 5 and Eq. (13), the medical diagnosis is represented by the implication {Patient, Symtomp} ? Disease,
which is identical to that of Definition 1. Thus, we clearly
recognize that of SC-IFRS in Definition 5 is another representation and an extension of medical diagnosis in Definition 1
inspired by the ideas of RS in Definition 4;
(b) SC-IFRS in Definition 5 could be regarded as the extension of
the traditional RS in Definition 4 in cases that
$i: liX(x) = 1 ^ ciX(x) = 0; "j – i: ljX(x) = 0 ^ cjX(x) = 1,
$i: liY(y) = 1 ^ ciY(y) = 0; "j – i: ljY(y) = 0 ^ cjY(y) = 1,
$i: liD(D) = 1 ^ ciD(D) = 0; "j – i: ljD(D) = 0 ^ cjD(D) = 1,
1.7. The organization of the paper
The rest of the paper is organized as follows. Section 2 presents
the main contribution including the IFRS and its elements stated
in Section 1.5. Section 3 validates the proposed approach through
a set of experiments involving benchmark medical diagnosis data.
Section 4 draws the conclusions and delineates the future research
directions.
2. Intuitionistic fuzzy recommender systems
In this section, we present the new definitions of single-criterion IFRS (SC-IFRS) and multi-criteria IFRS (MC-IFRS) in Section 2.1;
the new definitions of intuitionistic fuzzy matrix (IFM) and the
intuitionistic fuzzy composition matrix (IFCM) of two IFMs with
the intersection/union operation in Section 2.2; the intuitionistic
fuzzy similarity matrix (IFSM), the intuitionistic fuzzy similarity
degree (IFSD) and the formulas to predict RPD(Pi, Dj) on the basis
of IFSD in Section 2.3; a novel intuitionistic fuzzy collaborative filtering method so-called IFCF in Section 2.4.
2.1. The single-criterion and multi-criteria intuitionistic fuzzy
recommender systems
Recall P, S and D from Definition 1 being the sets of patients,
symptoms and diseases, respectively. Each patient Pi ("i 2 {1, . . .,
n}) (resp. symptom Sj, "j 2 {1, . . ., m}) is assumed to have some
features (resp. characteristics). For the simplicity, we consider RS
including a feature of the patient and a characteristic of the
symptom denoted as X and Y, respectively. X and Y both consist of
s intuitionistic linguistic labels. Analogously, disease Di ("i 2 {1,
. . ., k}) also contains s intuitionistic linguistic labels. A new
definition of RS under the lights of those of medical diagnosis
expressed by IFS and the traditional RS in Definition 1 & 4 respectively is stated as follows.
Definition 5 (Single-criterion Intuitionistic Fuzzy Recommender
Systems – SC-IFRS). The utility function R is a mapping specified
on (X, Y) as follows.
R:XÂY !D
*
ðl1X ðxÞ; c1X ðxÞÞ;
ðl2X ðxÞ; c2X ðxÞÞ;u
...
ðlsX ðxÞ; csX ðxÞÞ
+ *
Â
ðl1Y ðyÞ; c1Y ðyÞÞ;
ðl2Y ðyÞ; c2Y ðyÞÞ;
...
ðlsY ðyÞ; csY ðyÞÞ
+
*
!
À
Á
l1D ðDÞ; c1D ðDÞ ; +
ðl2D ðDÞ; c2D ðDÞÞ;
...
;
then the mapping in (13) could be re-written as,
R:PÂS!D
ð14Þ
ððp; XÞ; ðs; YÞÞ # RPD :
Now we extend SC-IFRS to handle the cases of multiple diseases
D = {D1, . . ., Dk}.
Definition 6 (Multi-criteria Intuitionistic Fuzzy Recommender Systems – MC-IFRS). The utility function R is a mapping specified on
(X, Y) below.
R : X Â Y ! D1 Â Á Á Á Â Dk
*
ðl1X ðxÞ; c1X ðxÞÞ;
ðl2X ðxÞ; c2X ðxÞÞ; u
...
+
*
Â
ðlsX ðxÞ; csX ðxÞÞ
*
!
ðl1Y ðyÞ; c1Y ðyÞÞ;
ðl2Y ðyÞ; c2Y ðyÞÞ;
...
ðlsY ðyÞ; csY ðyÞÞ
ðl1D ðD1 Þ; c1D ðD1 ÞÞ;
ðl2D ðD1 Þ; c2D ðD1 ÞÞ;
...
ðlsD ðD1 Þ; csD ðD1 ÞÞ
+
+
*
 ÁÁÁ Â
ðl1D ðDk Þ; c1D ðDk ÞÞ;
ðl2D ðDk Þ; c2D ðDk ÞÞ;
...
ð15Þ
+
:
ðlsD ðDk Þ; csD ðDk ÞÞ
MC-IFRS is the system that provides two basic functions below.
(a) Prediction: determine the values of ðllD ðDi Þ; clD ðDi ÞÞ, "l 2 {1,
. . ., s}, "i 2 {1, . . ., k};
Ã
(b) Recommendation: choose i⁄ 2 [1, s] satisfying i ¼
nP
o
À
Á
k
arg maxi¼1;s
j¼1 wj liD ðDj Þ þ liD ðDj Þð1 À liD ðDj Þ À ciD ðDj ÞÞ
where wj 2 [0, 1] is the weight of Dj satisfying the constraint:
Pk
j¼1 wj ¼ 1.
ðlsD ðDÞ; csD ðDÞÞ
ð13Þ
139
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Example 6. In a medical diagnosis system, there are 4 patients
whose feature X is ‘‘Age’’ consisting of 3 linguistic labels {low, medium, high} (s = 3). The symptom‘s characteristic Y is ‘‘Temperature’’
including 3 linguistic labels {cold, medium, hot}. The diseases (D1,
D2) are {‘‘Flu’’, ‘‘Headache’’}, and both of them contain 3 linguistic
labels {Level 1, Level 2, Level 3}. We would like to verify which ages
of users and types of temperature are likely to cause the diseases of
flu and headache. In this case we have a MC-IFRS system. By using
the trapezoidal intuitionistic fuzzy number – TIFN ([3]) characterized
À
Á
by a1 ; a2 ; a3 ; a4 ; a01 ; a04 with a01 6 a1 6 a2 6 a3 6 a4 6 a04 , the membership (non-membership) functions of patients to the linguistic
label ith of feature X are:
8
1
x 6 20
>
<
llow ðxÞ ¼ ð35 À xÞ=15 20 < x 6 35 ;
>
:
0
x > 35
8
0
x 6 20
>
<
v low ðxÞ ¼ > ðx À 20Þ=15 20 < x 6 35 ;
:
1
x > 35
8
0
x 6 20; x > 60
>
>
>
>
< ðx À 20Þ=15 20 < x 6 35
lmedium ðxÞ ¼
;
>
1
35 < x 6 45
>
>
>
:
ð60 À xÞ=15 45 < x 6 60
8
1
x 6 20; x > 60
>
>
>
>
< ð35 À xÞ=15 20 < x 6 35
v medium ðxÞ ¼ >
;
0
35 < x 6 45
>
>
>
:
ðx À 45Þ=15 45 < x 6 60
8
0
x 6 45
>
<
lhigh ðxÞ ¼ ðx À 45Þ=15 45 < x 6 60 ;
>
:
1
x > 60
8
1
x 6 45
>
<
v high ðxÞ ¼ > ð60 À xÞ=15 45 < x 6 60 :
:
0
x > 60
ð16Þ
Joeð45tÞ : hhighð0; 1Þ; mediumð1; 0Þ; lowð0; 1Þi;
Tedð50tÞ : hhighð0:33; 0:67Þ; mediumð0:67; 0:33Þ; lowð0; 1Þi:
ð17Þ
ð18Þ
ð19Þ
ð29Þ
35 < x 6 40
ð30Þ
8
1
x 6 35
>
<
v hot ðxÞ ¼ > ð40 À xÞ=5 35 < x 6 40 :
:
0
x > 40
ð31Þ
ð4 CÞ : hcoldð1; 0Þ; mediumð0; 1Þ; hotð0; 1Þi;
ð32Þ
ð16 CÞ : hcoldð0:267; 0:733Þ; mediumð0:733; 0:267Þ; hotð0; 1Þi; ð33Þ
ð39 CÞ : hcoldð0; 1Þ; mediumð0:2; 0:8Þ; hotð0:8; 0:2Þi;
ð34Þ
ð35Þ
From Eqs. (22)–(25), (32)–(35) we have a MC-IFRS described by
Table 15.
In Table 15, the cells having question marks are needed to predict the intuitionistic fuzzy values ðllD ðDi Þ; clD ðDi ÞÞ; 8l 2 f1; 2; 3g;
8i 2 f1; 2g.
2.2. Intuitionistic fuzzy matrix and intuitionistic fuzzy composition
matrix
0
ð21Þ
B
B b21
B
B
B c31
Z¼B
B
B c41
B
B ...
@
ð22Þ
ð23Þ
ð24Þ
ð25Þ
8
x65
>
<0
v cold ðxÞ ¼ > ðx À 5Þ=15 5 < x 6 20 ;
:
1
x > 20
ð27Þ
a11
ct1
a12
b22
c32
c42
...
ct2
. . . a1s
1
C
. . . b2s C
C
C
. . . c3s C
C:
C
. . . c4s C
C
... ... C
A
...
ð36Þ
cts
In Eq. (36), t = k + 2 where k 2 N+ is the number of diseases in
Definition 6. The value s 2 N+ is the number of intuitionistic
linguistic labels. a1i, b2i, chi, " h 2 {3, . . ., t}, "i 2 {1, . . ., s} are the
intuitionistic fuzzy values (IFV) consisting of the membership and
non-membership values as in Definition 6. a1i ¼ ðliX ðxÞ; ciX ðxÞÞ,
"i 2 {1, . . ., s} represents for the IFV value of the patient to the
linguistic label ith of featureX. b2i ¼ ðliY ðyÞ; ciY ðyÞÞ, " i 2 {1, . . ., s}
stands for the IFV value of the symptom to the linguistic label ith
of characteristic Y. chi ¼ ðliD ðDhÀ2 Þ; ciD ðDhÀ2 ÞÞ, "i 2 {1, . . ., s},
"h 2 {3, . . ., t} is the IFV value of the disease to the linguistic label
ith. Each line from the third one to the last in Eq. (36) is related
to a given disease.
Example 7. The first line in Table 15 describing the information of
user Al (Age: 25) at the temperature 4°C can be expressed by the
IFM as follows.
1
ð0:0; 1:0Þ ð0:33; 0:67Þ ð0:67; 0:33Þ
C
B
ð0:0; 1:0Þ
ð0:0; 1:0Þ C
B ð1:0; 0:0Þ
C:
Z¼B
C
B
ð0:2; 0:6Þ
ð0:1; 0:9Þ A
@ ð0:8; 0:1Þ
0
x 6 5; x > 40
35 < x 6 40
;
8
0
x 6 35
>
<
lhot ðxÞ ¼ ðx À 35Þ=5 35 < x 6 40 ;
>
:
1
x > 40
ð20Þ
ð26Þ
20 < x 6 35
20 < x 6 35
Definition 7. An intuitionistic fuzzy matrix (IFM) Z in MC-IFRS is
defined as,
8
x65
>
<1
lcold ðxÞ ¼ ð20 À xÞ=15 5 < x 6 20 ;
>
:
0
x > 20
5 < x 6 20
0
>
>
>
:
ðx À 35Þ=5
ð25 CÞ : hcoldð0; 1Þ; mediumð1; 0Þ; hotð0; 1Þi:
Similarly, the membership (non-membership) functions of the
symptom to the linguistic label jth of characteristicY are:
8
0
>
>
>
< ðx À 5Þ=15
lmedium ðxÞ ¼
>1
>
>
:
ð40 À xÞ=5
v medium ðxÞ ¼ >
The information of symptom are shown as follows
Based on Eqs. (16)–(21), we calculate the information of
patients as follows.
Alð25tÞ : hhighð0; 1Þ; mediumð0:33; 0:67Þ; lowð0:67; 0:33Þi;
Bobð40tÞ : hhighð0; 1Þ; mediumð1; 0Þ; lowð0; 1Þi;
8
1
x 6 5; x > 40
>
>
>
>
< ð20 À xÞ=15 5 < x 6 20
;
ð28Þ
ð0:1; 0:8Þ
ð0:6; 0:35Þ
ð0:3; 0:55Þ
ð37Þ
140
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 15
A MC-IFRS for medical diagnosis with ⁄ being the values to be predicted.
Age
*
Alð25Þ :
*
Alð25Þ :
highð0; 1Þ;
mediumð0:33; 0:67Þ;
lowð0:67; 0:33Þ
highð0; 1Þ;
mediumð0:33; 0:67Þ;
lowð0:67; 0:33Þ
*
highð0; 1Þ;
mediumð1; 0Þ;
lowð0; 1Þ
Bobð40Þ :
*
highð0; 1Þ;
mediumð1; 0Þ;
lowð0; 1Þ
Joeð45Þ :
*
Tedð50Þ :
Temperature
*
+
coldð1; 0Þ;
ð4 CÞ : mediumð0; 1Þ;
hotð0; 1Þ
+
+
*
ð16 CÞ :
*
+
ð39 CÞ :
*
+
ð16 CÞ :
highð0:33; 0:67Þ;
mediumð0:67; 0:33Þ;
lowð0; 1Þ
*
+
ð25 CÞ :
coldð0:267; 0:733Þ;
mediumð0:733; 0:267Þ;
hotð0; 1Þ
coldð0; 1Þ;
mediumð0:2; 0:8Þ;
hotð0:8; 0:2Þ
ð1Þ
a
B 11
B ð1Þ
Bb
B 21
B
B ð1Þ
B c31
B
B
B cð1Þ
B 41
B
B
B ...
@
ð1Þ
ct1
0
ð1Þ
a12
ð1Þ
b22
ð1Þ
c32
ð1Þ
c42
...
ð1Þ
ct2
ð12Þ
a11
B
B ð12Þ
Bb
B 21
B
B ð12Þ
B c31
¼B
B
B cð12Þ
B 41
B
B
B ...
@
ð12Þ
ct1
ð1Þ
. . . a1s
1 0
ð2Þ
a
C B 11
C B
ð1Þ C B ð2Þ
. . . b2s C B b21
C B
ð1Þ C B ð2Þ
. . . c3s C B c31
CB
C B
ð1Þ C B ð2Þ
. . . c4s C B c41
C B
C B
... ... C B ...
A @
ð1Þ
ð2Þ
. . . cts
ct1
1
ð12Þ
ð12Þ
a12
. . . a1s
C
ð12Þ
ð12Þ C
b22
. . . b2s C
C
C
ð12Þ
ð12Þ C
c32
. . . c3s C
C;
C
ð12Þ
ð12Þ
c42
. . . c4s C
C
C
C
... ... ... C
A
ð12Þ
ð12Þ
ct2
. . . cts
ð2Þ
a12
ð2Þ
ð2Þ
. . . a1s
b22
...
c32
ð2Þ
...
ð2Þ
c42
...
...
...
ð2Þ
...
ct2
1
C
C
ð2Þ
b2s C
C
C
ð2Þ C
c3s C
C
C
ð2Þ C
c4s C
C
C
... C
A
ð2Þ
cts
ð12Þ
ð12Þ
b2i
¼
ð1Þ
b2i
^
ð2Þ
b2i
ð12Þ
chi
ð1Þ
Level2(0.6, 0.35);
Level3(0.3,0.55)
Level1(0.4, 0.5);
Level1(0.0, 0.9);
Level2(0.6,0.2);
Level3(0.1,0.9);
Level2(0.2, 0.75);
Level3(0.7,0.2)
Level1(0.8, 0.2);
Level1(0.8, 0.1);
Level2(0.1,0.8);
Level3(0.0,0.95);
Level2(0.1,0.9);
Level3(0.0,0.9)
Level1(0.0, 1.0);
Level1(0.0, 0.9);
Level2(0.2, 0.7);
Level3(1.0,0.0);
Level2(0.7, 0.3);
Level3(0.1,0.85);
Level1(⁄, ⁄);
Level1(⁄, ⁄);
Level2(⁄, ⁄);
Level3(⁄, ⁄);
Level2(⁄, ⁄);
Level3(⁄, ⁄);
Example 8. Given 2 IFM below.
1
ð0:0; 1:0Þ ð0:33; 0:67Þ ð0:67; 0:33Þ
B ð1:0; 0:0Þ
ð0:0; 1:0Þ
ð0:0; 1:0Þ C
C
B
Z1 ¼ B
C;
@ ð0:8; 0:1Þ
ð0:2; 0:6Þ
ð0:1; 0:9Þ A
0
ð0:7; 0:3Þ
ð42Þ
ð43Þ
ð0:1; 0:85Þ
0
ð0:0; 1:0Þ
ð0:33; 0:67Þ
B ð0:267; 0:733Þ
ð0:0; 1:0Þ
B
Z ¼ Z1 Z2 ¼ B
@ ð0:0; 1:0Þ
ð0:2; 0:7Þ
ð0:0; 0:9Þ
ð0:0; 1:0Þ
1
ð0:0; 1:0Þ C
C
C:
ð0:1; 0:9Þ A
ð0:6; 0:35Þ
ð44Þ
ð0:1; 0:85Þ
ð38Þ
Definition 9. Suppose that Z1 and Z2 are two IFM in MC-IFRS. The
intuitionistic fuzzy composition matrix (IFCM) of Z1 and Z2 with the
union operation is defined as follows.
0
ð40Þ
ð2Þ
¼ chi ^ chi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ min liD ðDhÀ2 Þ; liD ðDhÀ2 Þ ; max ciD ðDhÀ2 Þ; ciD ðDhÀ2 Þ ;
8h 2 f3; . . . ; tg:
Level2(0.2, 0.6);
Level3(0.1, 0.9);
The IFCM of Z1 and Z2 with the intersection operation is:
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ min liY ðyÞ; liY ðyÞ ; max ciY ðyÞ; ciY ðyÞ ;
8i 2 f1; . . . ; sg;
Level1(0.1, 0.8)
ð0:0; 0:9Þ
ð39Þ
8i 2 f1; . . . ; sg;
Level1(0.8, 0.1);
ð0:1; 0:8Þ ð0:6; 0:35Þ
ð0:3; 0:55Þ
1
ð0:0; 1:0Þ
ð1:0; 0:0Þ
ð0:0; 1:0Þ
B ð0:267; 0:733Þ ð0:733; 0:267Þ ð0:0; 1:0Þ C
C
B
Z2 ¼ B
C:
@ ð0:0; 1:0Þ
ð0:2; 0:7Þ
ð1:0; 0:0Þ A
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ a1i ^ a1i ¼ min liX ðxÞ; liX ðxÞ ; max ciX ðxÞ; ciX ðxÞ ;
8i 2 f1; . . . ; sg;
Headache
0
where
a1i
+
+
Definition 8. Suppose that Z1 and Z2 are two IFM in MC-IFRS. The
intuitionistic fuzzy composition matrix (IFCM) of Z1 and Z2 with the
intersection operation is,
0
+
coldð0:267; 0:733Þ;
mediumð0:733; 0:267Þ;
hotð0; 1Þ
coldð0; 1Þ;
mediumð1; 0Þ;
hotð0; 1Þ
+
Flu
ð41Þ
ð1Þ
a11
B ð1Þ
Bb
B 21
B ð1Þ
Bc
B 31
B ð1Þ
B c41
B
B ...
@
ð1Þ
ct1
0
ð1Þ
a12
ð1Þ
b22
ð1Þ
c32
ð1Þ
c42
...
ð1Þ
ct2
ð12Þ
a
B 11
B bð12Þ
B 21
B ð12Þ
Bc
31
¼B
B ð12Þ
B c41
B
B ...
@
ð12Þ
ct1
ð1Þ
. . . a1s
1 0
ð2Þ
a11
C B ð2Þ
ð1Þ
B
. . . b2s C
C B b21
C
B ð2Þ
ð1Þ
B
. . . c3s C
C B c31
B ð2Þ
ð1Þ C
B
. . . c4s C
C B c41
C
... ... A B
@ ...
ð1Þ
ð2Þ
ct1
. . . cts
1
ð12Þ
ð12Þ
a12
. . . a1s
C
ð12Þ
ð12Þ
b22
. . . b2s C
C
C
ð12Þ
ð12Þ
c32
. . . c3s C
C;
ð12Þ
ð12Þ C
c42
. . . c4s C
C
... ... ... C
A
ð12Þ
ð12Þ
ct2
. . . cts
ð2Þ
a12
ð2Þ
b22
ð2Þ
c32
ð2Þ
c42
...
ð2Þ
ct2
ð2Þ
. . . a1s
1
C
ð2Þ
. . . b2s C
C
C
ð2Þ
. . . c3s C
C
ð2Þ C
. . . c4s C
C
... ... C
A
ð2Þ
. . . cts
ð45Þ
141
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
where
ð12Þ
a1i
It follows that,
ð1Þ
ð2Þ
¼ a1i _ a1i
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ max liX ðxÞ; liX ðxÞ ; min ciX ðxÞ; ciX ðxÞ ;
8i 2 f1; . . . ; sg;
ð12Þ
b2i
ð1Þ
ð2Þ
¼ b2i _ b2i
ð46Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ max liY ðyÞ; liY ðyÞ ; min ciY ðyÞ; ciY ðyÞ ;
8i 2 f1; . . . ; sg;
ð12Þ
chi
ð1Þ
ð47Þ
8h 2 f3; . . . ; tg:
ð48Þ
1
0
ð0:0; 1:0Þ
ð1:0; 0:0Þ
ð0:67; 0:33Þ
B ð1:0; 0:0Þ ð0:733; 0:267Þ
ð0:0; 1:0Þ C
C
B
Z ¼ Z1 Z2 ¼ B
C:
@ ð0:8; 0:1Þ
ð0:2; 0:6Þ
ð1:0; 0:0Þ A
ð0:1; 0:8Þ
ð0:7; 0:3Þ
ð49Þ
ð0:3; 0:55Þ
Theorem 1. The IFCM of Z1 and Z2 with the intersection (union) operation is an IFM.
Proof 1. We prove the theorem with the intersection operation
only. The theorem with the union operation is proven analogously.
From Definition 8, we know that
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ a1i ^ a1i ¼ min liX ðxÞ; liX ðxÞ ; max ciX ðxÞ; ciX ðxÞ ;
8i 2 f1; . . . ; sg;
ð12Þ
b2i
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ b2i ^ b2i ¼ min liY ðyÞ; liY ðyÞ ; max ciY ðyÞ; ciY ðyÞ ;
8i 2 f1; . . . ; sg;
ð12Þ
chi
B
B ð12Þ
B b21
B
B ð12Þ
B c31
Z1 Z2 ¼ B
B ð12Þ
Bc
B 41
B
B ...
@
ð12Þ
ct1
ð1Þ
ð2Þ
¼ chi ^ chi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ min liD ðDhÀ2 Þ; liD ðDhÀ2 Þ ; max ciD ðDhÀ2 Þ; ciD ðDhÀ2 Þ ;
8i 2 f1; . . . ; sg;
ð1Þ
a1i
8h 2 f3; . . . ; tg:
ð52Þ
ð2Þ
a1i
ð12Þ
a1i
Since
and
are two IFV, the function of them –
is also an
ð12Þ
ð12Þ
IFV. Similar conclusions are found for b2i and chi . Thus, the IFCM
of Z1 and Z2 with the intersection operation is an IFM. The proof is
complete. h
Property 1. Given Z1, Z2 and Z3 being IFM. The following properties
hold for these IFM.
ð12Þ
b22
ð12Þ
c32
ð12Þ
c42
...
ð12Þ
ct2
ð12Þ
. . . a1s
ð1Þ
ð2Þ
the intersection operation. Since a1i and a1i are two IFV, we obtain
ð12Þ
¼ a1i ^ a1i ¼ a1i ^ a1i ¼ a1i ;
ð53Þ
ð12Þ
b2i
ð12Þ
chi
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð21Þ
b2i ^ b2i ¼ b2i ^ b2i ¼ b2i ;
ð1Þ
ð2Þ
ð2Þ
ð1Þ
ð21Þ
chi ^ chi ¼ chi ^ chi ¼ chi :
ð54Þ
¼
¼
ð2Þ
ð2Þ
ð1Þ
0
ð21Þ
a11
C B
B ð21Þ
ð12Þ C
. . . b2s C B b21
C B
B ð21Þ
ð12Þ C
B
. . . c3s C
C ¼ B c31
C B ð21Þ
ð12Þ C
. . . c4s C B
B c41
C B
C
... ... A B
@ ...
ð12Þ
ð21Þ
. . . cts
ct1
ð21Þ
a12
ð21Þ
b22
ð21Þ
c32
ð21Þ
c42
...
ð21Þ
ct2
ð21Þ
. . . a1s
1
C
ð21Þ C
. . . b2s C
C
ð21Þ C
. . . c3s C
C
C
ð21Þ
. . . c4s C
C
C
... ... C
A
ð21Þ
. . . cts
¼ Z2 Z1 :
ð56Þ
The proof is analogously performed with the IFCM of Z1 and Z2
equipped with the union operation.(b) Suppose that the IFCM of
Z1 and Z2 is equipped with the intersection operation. We have,
0
ð123Þ
a11
ð123Þ
a12
ð123Þ
. . . a1s
1
C
B
B ð123Þ
ð123Þ
ð123Þ C
B b21
b22
. . . b2s C
C
B
B ð123Þ
ð123Þ
ð123Þ C
C
B c31
c
.
.
.
c
32
3s
C:
ðZ 1 Z 2 Þ Z 3 ¼ B
C
B ð123Þ
ð123Þ
ð123Þ
C
Bc
c
.
.
.
c
C
B 41
42
4s
C
B
C
B ...
.
.
.
.
.
.
.
.
.
A
@
ð123Þ
ð123Þ
ð123Þ
ct2
. . . cts
ct1
ð123Þ
ð1Þ
ð2Þ
ð3Þ
a1i ¼ a1i ^ a1i ^ a1i ; 8i 2 f1; . . . ; sg;
ð123Þ
ð1Þ
ð2Þ
ð3Þ
b2i ¼ b2i ^ b2i ^ b2i ; 8i 2 f1; . . . ; sg;
ð123Þ
ð1Þ
ð2Þ
ð3Þ
chi ¼ chi ^ chi ^ chi ; 8i 2 f1; . . . ; sg; 8h 2 f3; . . . ; tg:
ð57Þ
ð58Þ
ð59Þ
ð60Þ
Because
ð1Þ
ð2Þ
ð3Þ
ð1Þ
ð2Þ
ð3Þ
a1i ^ a1i ^ a1i ¼ a1i ^ a1i ^ a1i ;
ð1Þ
ð2Þ
ð3Þ
ð1Þ
ð2Þ
ð3Þ
b2i ^ b2i ^ b2i ¼ b2i ^ b2i ^ b2i ;
ð1Þ
ð2Þ
ð3Þ
ð1Þ
ð2Þ
ð3Þ
chi ^ chi ^ chi ¼ chi ^ chi ^ chi :
ð61Þ
ð62Þ
ð63Þ
It follows that
ðZ 1 Z 2 Þ Z 3 ¼ Z 1 ðZ 2 Z 3 Þ:
ð64Þ
The proof is analogously performed with the IFCM of Z1 and Z2
equipped with the union operation. h
2.3. The intuitionistic fuzzy similarity matrix and intuitionistic fuzzy
similarity degree
Motivated by the ideas of Hung and Yang [15], we present the
definition of intuitionistic fuzzy similarity matrix as follows.
0
Proof 2. (a) Suppose that the IFCM of Z1 and Z2 is equipped with
ð1Þ
1
Definition 10. Suppose that Z1 and Z2 are two IFM in MC-IFRS. The
intuitionistic fuzzy similarity matrix (IFSM) between Z1 and Z2 is
defined as follows.
(a) Z1Z2 = Z2Z1,
(b) (Z1Z2)Z3 = Z1(Z2Z3).
a1i
ð12Þ
a12
ð2Þ
Example 9. Given 2 IFM in Example 8. The IFCM of Z1 and Z2 with
the union operation is:
ð12Þ
ð12Þ
a11
¼ chi _ chi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ max liD ðDhÀ2 Þ; liD ðDhÀ2 Þ ; min ciD ðDhÀ2 Þ; ciD ðDhÀ2 Þ ;
8i 2 f1; . . . ; sg;
a1i
0
ð21Þ
ð55Þ
e
S 11
B
Be
B S 21
Be
B S 31
e
S¼B
Be
B S 41
B
B ...
@
e
S t1
e
S 12
e
S 22
...
e
S 32
e
S 42
...
...
e
S t2
...
...
...
...
1
e
S 1s
C
e
S 2s C
C
C
e
S 3s C
C;
C
e
S 4s C
C
... C
A
e
S ts
ð65Þ
142
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
where,
Since
e
S 1i ¼ 1 À
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
1 À exp À1=2 liX ðxÞ À liX ðxÞ þ ciX ðxÞ À ciX ðxÞ
1 À expðÀ1Þ
8i 2 f1;.. .;sg;
;
ð66Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
1 À exp À1=2 liY ðyÞ À liY ðyÞ þ ciY ðyÞ À ciY ðyÞ
e
S 2i ¼ 1 À
À
SIMðPu ; P2 Þ Â
À
Á
lPiDv ðDj Þ þ cPiDv ðDj Þ 6 SIMðPu ; P1 Þ;
l
Pv
iD ðDj Þ
Pv
iD ðDj Þ
þc
Á
ð78Þ
6 SIMðPu ; P2 Þ;
ð79Þ
lPiDv ðDj Þ þ cPiDv ðDj Þ 6 SIMðPu ; Pn Þ:
ð80Þ
...
SIMðPu ; Pn Þ Â
À
Á
1 À expðÀ1Þ
8i 2 f1; . . . ;sg;
e
S hi ¼ 1 À
SIMðPu ; P1 Þ Â
ð67Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ
ð2Þ
ð1Þ
ð2Þ
1 À exp À1=2 liD ðDhÀ2 Þ À liD ðDhÀ2 Þ þ ciD ðDhÀ2 Þ À ciD ðDhÀ2 Þ
1 À expðÀ1Þ
8i 2 f1; . . . ; sg;
;
8h 2 f3; . . . ; tg:
ð68Þ
Definition 11. Suppose that Z1 and Z2 are two IFM in MC-IFRS. The
intuitionistic fuzzy similarity degree (IFSD) between Z1 and Z2 is
Pu
iD ðDj Þ
l
s
s
t X
s
X
X
X
S 1i þ b w2i e
S 2i þ v
S hi ;
SIMðZ 1 ; Z 2 Þ ¼ a w1i e
whi e
i¼1
i¼1
It follows that
Pu
iD ðDj Þ
þc
ð69Þ
h¼3 i¼1
where e
S is the IFSM between Z1 and Z2. W = (wij) ("i 2 {1, . . ., t},
"j 2 {1, . . ., s}) is the weight matrix of IFSM between Z1 and Z2
satisfying,
s
X
w1i ¼ 1;
s
X
w2i ¼ 1;
s
X
whi ¼ 1;
i¼1
i¼1
i¼1
8h 2 f3; . . . ; tg;
ð70Þ
a þ b þ v ¼ 1:
ð71Þ
Pn
v ¼1 SIMðP u ; P v Þ Â
À
Pn
v ¼1 SIMðP u ; P v Þ
Pn
SIMðPu ; Pv Þ
6 Pvn ¼1
¼ 1:
v ¼1 SIMðP u ; P v Þ
¼
The proof is complete.
Á
lPiDv ðDj Þ þ cPiDv ðDj Þ
ð81Þ
h
2.4. The intuitionistic fuzzy collaborative filtering method
Fig. 1)
3. Evaluation
Remark 2. The formula of IFSD in Eq. (68) can be recognized as the
generalization of the hard user-based, item-based and the ratingbased similarity degrees in recommender systems [29] when
b = v = 0, a = v = 0 and a = b = 0, respectively.
In this section, we describe the experimental environment in
Section 3.1. The database for experiments is given in Section 3.2.
Definition 12. The formulas to predict the values of linguistic
labels of patient Pu ("u 2 {1, . . ., n}) to symptom Sj (" j 2 {1, . . .,
m}) according to diseases (D1, D2, . . ., Dk) in MC-IFRS are:
lPiDu ðDj Þ ¼
Pn
Pv
SIMðPu ; Pv Þ Â liD ðDj Þ
v ¼1P
; 8i 2 f1; . . . ; sg;
n
v ¼1 SIMðP u ; P v Þ
8j 2 f1; . . . ; kg;
cPiDu ðDj Þ ¼
Pn
8u 2 f1; . . . ; ng;
ð72Þ
Pv
SIMðP u ; P v Þ Â ciD ðDj Þ
v ¼1P
;
n
v ¼1 SIMðP u ; P v Þ
8j 2 f1; . . . ; kg;
8i 2 f1; . . . ; sg;
8u 2 f1; . . . ; ng:
ð73Þ
Theorem 2. The predictive IFM results in Definition 12 are an IFV.
Proof 3. We have the following fact.
lPiDu ðDj Þ þ cPiDu ðDj Þ ¼
Pn
v ¼1 SIMðP u ; P v Þ Â
Pn
À
lPiDv ðDj Þ þ cPiDv ðDj Þ
v ¼1 SIMðP u ; P v Þ
Á
;
ð74Þ
0 6 lPiDv ðDj Þ þ cPiDv ðDj Þ 6 1;
ð75Þ
0 6 SIMðPu ; Pv Þ 6 1:
ð76Þ
It is obvious that
lPiDu ðDj Þ þ cPiDu ðDj Þ P 0:
ð77Þ
Fig. 1. The IFCF algorithm.
143
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 16
The descriptions of experimental datasets.
Dataset
No.
elements
No.
attributes
No.
classes
Elements in each
classes
HEART
RHC
270
5735
13
8
2
2
(150, 120)
(3804, 1931)
Section 3.3 illustrates the activities of IFCF on the intuitionistic
medical diagnosis dataset in [31]. Lastly, Section 3.4 presents the
experimental results on the benchmark medical diagnosis datasets
namely HEART and RHC.
3.1. Experimental design
In this part, we describe the experimental environments such
as,
Experimental tools: We have implemented the proposed hybrid
algorithm – IFCF in addition to the typical standalone methods
of IFS such as De et al. [9], Szmidt and Kacprzyk [44], Samuel
and Balamurugan [31] and RS such as Davis et al. [8] and Hassan
and Syed [13] in PHP programming language and executed
them on a PC Intel(R) core(TM) 2 Duo CPU T6400 @ 2.00 GHz
2 GB RAM. The results are taken as the average value of 50 runs.
Evaluation indices: Mean Absolute Error (MAE) and the computational time.
Objective:
To illustrate the activities of IFCF on an illustrated dataset;
To evaluate the IFCF in comparison with the relevant algorithms in terms of accuracy through evaluation indices.
3.2. Database
In the evaluation, we use three kinds of datasets for
experiments.
The intuitionistic medical diagnosis dataset in [31], which was
used from Example 1 to 5 to illustrate the activities of the relevant algorithms in Section 1;
The benchmark medical diagnosis dataset namely HEART from
UCI Machine Learning Repository [46] (Table 16 and Fig. 2);
A large benchmark medical diagnosis dataset namely RHC
(Right Heart Catheterization) including 5735 critically ill adult
patients receiving care in ICU [6] (Table 16 and Fig. 3).
The cross-validation method for the experiments is the k-fold
validation with k from 2 to 10. The aim for various folds is to
observe the changes of MAE and computational time of algorithms
so that this could help us better analysis of experimental results.
Besides testing with the k-fold validation, the random experiments
with the cardinalities of the testing being from 10 to 100 random
elements are also performed. In order to validate the results with
accurate classes, the intuitionistic defuzzification method [3] is
used for experimental algorithms.
3.3. An illustration of IFCF
In this section, we illustrate the activities of IFCF on the intuitionistic medical diagnosis dataset in [31] described from Tables
1–3. Similar to Example 5, the training dataset is demonstrated
in Table 17 where ⁄ values in this table are needed to be predicted.
From Tables 1 and 17 we have extracted the SC-IFRS dataset in
Table 18 similar to that in Table 15 where ⁄ values in this table are
needed to be predicted.
From Table 18, we realize that a semi-SC-IFRS with the hard
patients being used instead of their features. Thus, the parameters
of the IFCF algorithm are automatically updated as a = 0, b = c = 1/2
and w1i = w2i = w3i = 0.2. From Definition 11, we calculate the IFSD
between Sugu (Somu) and Ram and Mari as follows.
IFSDðSugu; RamÞ ¼ 0:87;
IFSDðSugu; MariÞ ¼ 0:57;
ð82Þ
ð83Þ
IFSDðSomu; RamÞ ¼ 0:83;
ð84Þ
IFSDðSomu; MariÞ ¼ 0:58:
ð85Þ
From Eqs. (82)–(85), we used Definition 12 to calculate the predictive IFM results of Sugu and Somu.
Viral feverð0:49; 0:38Þ;
* Malariað0:52; 0:22Þ
DiseaseðSuguÞ ¼
Fig. 2. The 2D distribution of HEART.
Typhoidð0:36; 0:52Þ;
Stomach problemð0:40; 0:34Þ
Chest problemð0:10; 0:68Þ
+
;
ð86Þ
144
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 17
The training dataset for IFCF with ⁄ being the values to be predicted.
Table 18
The extracted SC-IFRS dataset with ⁄ being the values to be predicted.
P
Viral_fever
Malaria
Typhoid
Stomach
Chest
Ram
Mari
Sugu
Somu
(0.4, 0.1)
(0.3, 0.5)
⁄
⁄
(0.7, 0.1)
(0.2, 0.6)
⁄
⁄
(0.6, 0.1)
(0.4, 0.4)
⁄
⁄
(0.2, 0.4)
(0.6, 0.1)
⁄
⁄
(0.2, 0.6)
(0.1, 0.7)
⁄
⁄
P
S
D
Ram
Temperatureð0:8; 0:1Þ;
*
+
Headacheð0:6; 0:1Þ
Stomach painð0:2; 0:8Þ;
Coughð0:6; 0:1Þ
Chest painð0:1; 0:6Þ
Temperatureð0:0; 0:8Þ;
*
+
Headacheð0:4; 0:4Þ
Stomach painð0:6; 0:1Þ;
Coughð0:1; 0:7Þ
Chest painð0:1; 0:8Þ
Temperatureð0:8; 0:1Þ;
*
+
Headacheð0:8; 0:1Þ
Viral feverð0:4; 0:1Þ;
+
Malariað0:7; 0:1Þ
Typhoidð0:6; 0:1Þ;
Stomach problemð0:2; 0:4Þ
Chest problemð0:2; 0:6Þ
Viral feverð0:3; 0:5Þ;
*
+
Malariað0:2; 0:6Þ
Typhoidð0:4; 0:4Þ;
Stomach problemð0:6; 0:1Þ
Chest problemð0:1; 0:7Þ
⁄
Mari
Viral feverð0:47; 0:39Þ;
* Malariað0:52; 0:22Þ
DiseaseðSomuÞ ¼
Typhoidð0:36; 0:51Þ;
+
:
ð87Þ
Sugu
Stomach problemð0:39; 0:47Þ
Chest problemð0:10; 0:68Þ
Based on the recommendation function of Definition 5 and Eqs. (86)
and (87), we recommend the disease those patients suffer the most
as in Table 19.
From Table 19, we conclude that Sugu and Somu both suffer
from the Malaria.
Somu
Stomach painð0:0; 0:6Þ;
Coughð0:2; 0:7Þ
Chest painð0:0; 0:5Þ
Temperatureð0:6; 0:1Þ;
*
+
Headacheð0:5; 0:4Þ
Stomach painð0:3; 0:4Þ;
Coughð0:7; 0:2Þ
Chest painð0:3; 0:4Þ
*
⁄
Remark 3
(a) The result of IFCF expressed in Table 19 is identical to those
of De et al. [9] in Example 2, Samuel and Balamurugan [31]
in Example 3 and Hassan and Syed [13] in Example 5.
Besides these methods, if we perform other relevant algorithms such as Own [26] and Shinoj and John [33] then the
same results would be given. This proves the correctness
of the proposed method;
(b) IFCF is capable to perform the prediction and recommendation with more types of datasets including the hard and
(intuitionistic) fuzzy data, not only the semi-SC-IFRS dataset
like what being experimented in this section. This affirms
the flexibility and usefulness of IFCF in comparison with
other relevant methods;
(c) The limitation of the defuzzification method stated in
Sections 1.3(c) and 1.4(b) of the relevant works such as De
et al. [9], Szmidt and Kacprzyk [44], Samuel and
Table 19
The recommended diseases where the most suffered one is highlighted in bold.
P
Viral_fever
Malaria
Typhoid
Stomach
Chest
Sugu
Somu
0.5537
0.5358
0.6552
0.6552
0.4032
0.4068
0.504
0.4446
0.122
0.122
Balamurugan [31], Davis et al. [8] and Hassan and Syed
[13] is handled by the recommendation function in SC-IFRS
(Definition 5) and MC-IFRS (Definition 6);
(d) IFCF used both the data about the relations between the
patients – the symptoms and the patients – the diseases
for the prediction, not solely on the data of the patients –
the diseases as in the relevant methods relating to RS such
as Davis et al. [8] and Hassan and Syed [13]. This enhances
the accuracy of diagnosis since more information is count
for the calculation of the algorithm.
Fig. 3. The 2D distribution of RHC.
145
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 20
The results of random experiments on the HEART dataset.
Data sets
10
20
30
40
50
60
70
80
90
100
MAE
IFCF
DAVIS
HASSAN
DE
SAMUEL
SZMIDT
0.49641
0.492051
0.490383
0.488195
0.492451
0.490892
0.491429
0.493946
0.494265
0.493219
0.500286
0.495541
0.492238
0.49185
0.494166
0.494094
0.494952
0.497251
0.49741
0.495757
0.494036
0.488739
0.486775
0.483692
0.487049
0.486138
0.487727
0.490348
0.490197
0.488414
0.489147
0.496361
0.493604
0.47396
0.490438
0.476327
0.480793
0.491512
0.484721
0.483999
0.508045
0.504834
0.509998
0.526556
0.510221
0.523458
0.519985
0.507438
0.517029
0.516403
0.491955
0.495166
0.490002
0.473444
0.489779
0.476542
0.480015
0.492562
0.482971
0.483597
0.646215
0.703126
0.739453
1.192436
1.234968
2.084581
2.360242
3.014339
3.081387
3.305678
0.089682
0.152622
0.211139
0.255561
0.290003
0.304465
0.306179
0.323535
0.324501
0.328335
0.011868
0.028907
0.032683
0.043212
0.043985
0.048735
0.058691
0.078036
0.081523
0.127595
0.009564
0.0347
0.036562
0.037371
0.043722
0.052548
0.053144
0.060445
0.11539
0.11964
0.014055
0.023194
0.029526
0.046678
0.053152
0.058788
0.072443
0.072655
0.082507
0.102687
Computational time (s)
10
20
30
40
50
60
70
80
90
100
0.327558
0.796293
1.166132
1.221915
1.232003
1.260366
1.278752
1.469952
1.613406
2.210751
Table 21
The results of k-fold cross validation on the HEART dataset.
Fold
2
3
4
5
6
7
8
9
10
MAE
IFCF
DAVIS
HASSAN
DE
SAMUEL
SZMIDT
0.495733
0.494624
0.490572
0.488722
0.489378
0.492201
0.492132
0.488487
0.487778
0.498977
0.497455
0.49394
0.492826
0.493186
0.496148
0.496581
0.492575
0.491602
0.491243
0.489978
0.486429
0.485395
0.484268
0.488301
0.488893
0.485452
0.484024
0.479124
0.474681
0.491674
0.474125
0.474017
0.468833
0.492033
0.486527
0.490158
0.521357
0.525323
0.510025
0.525475
0.526214
0.527847
0.509502
0.515524
0.514568
2.060489
0.474677
0.489975
0.474525
0.474768
0.472153
0.490498
0.484476
0.485432
4.059911
4.039289
3.383637
3.208319
2.952161
2.342186
2.186523
2.150605
0.492826
1.572734
1.431444
1.335837
1.137519
1.108358
1.101027
1.097188
0.956852
0.863702
0.213536
0.191975
0.09894
0.092019
0.088419
0.076773
0.057825
0.047176
0.045803
0.21604
0.103076
0.100901
0.083503
0.04564
0.043674
0.034734
0.029806
0.019865
0.207523
0.144031
0.128371
0.086687
0.075412
0.07241
0.06922
0.0414
0.037433
Data sets
Computational time (s)
2
3
4
5
6
7
8
9
10
3.225795
3.158217
2.821404
2.70524
2.310253
2.166859
2.088267
2.060489
1.889909
3.4. Assessment
In this section, we perform the experiments on the benchmark
medical diagnosis datasets namely HEART and RHC. The experimental results are described from Tables 20–23. The MAE results
of random experiments (resp. k-fold cross validation) on the HEART
dataset is illustrated in Fig. 4 (resp. Fig. 5). Analogously, the MAE
results of random experiments (resp. k-fold cross validation) on
the RHC dataset is illustrated in Fig. 6 (resp. Fig. 7). The discussion
of the experimental results is demonstrated in Remark 4.
Remark 4
(a) Tables 20 and 21 have revealed that the MAE values of IFCF
are approximate to those of other algorithms. They are
better than those of Davis et al. [8] and Samuel and
Balamurugan [31] only for this dataset. Specifically, the
average MAE values of IFCF, Davis et al. [8], Hassan and Syed
[13], De et al. [9], Samuel and Balamurugan [31] and Szmidt
and Kacprzyk [44] in Table 20 are 0.4923241, 0.4953545,
0.4883115, 0.4860862, 0.5143967 and 0.4856033, respectively. These numbers in Table 21 are 0.491069667,
0.49481, 0.487109222, 0.481241333, 0.519537222 and
0.656332556, respectively. Figs. 4 and 5 illustrate this fact.
This shows that the proposed algorithm IFCF is not really
effective in case that the dataset contains hard values and
is small-sized & concentrative in a small range such as the
HEART dataset;
(b) Tables 22 and 23 show other results of MAE in cases of a large
dataset such as RHC. The average MAE values of IFCF, Davis
et al. [8], Hassan and Syed [13], De et al. [9], Samuel and Balamurugan [31] and Szmidt and Kacprzyk [44]
in Table 22 are 0.43716495, 0.4374099, 0.4504119,
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L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Table 22
The results of random experiments on the RHC dataset.
Data sets
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
MAE
IFCF
DAVIS
HASSAN
DE
SAMUEL
SZMIDT
0.442175
0.439844
0.438635
0.438282
0.436268
0.436115
0.437641
0.436992
0.440996
0.435963
0.436537
0.435991
0.435262
0.438705
0.437769
0.435194
0.434399
0.435067
0.436633
0.434831
0.44247
0.440324
0.438546
0.438322
0.436218
0.436708
0.438068
0.436864
0.441618
0.436172
0.436788
0.436306
0.435451
0.439116
0.437992
0.435462
0.434582
0.435319
0.436888
0.434984
0.441502
0.451435
0.450023
0.450814
0.444011
0.438171
0.453067
0.449501
0.450962
0.441205
0.453276
0.458081
0.438105
0.45581
0.463094
0.444957
0.451129
0.463573
0.448817
0.460705
0.479901
0.482438
0.481689
0.476943
0.474844
0.480948
0.480032
0.47773
0.481943
0.475933
0.480885
0.478958
0.478893
0.481182
0.481186
0.476458
0.479923
0.476773
0.483203
0.478549
0.52626
0.524069
0.524843
0.529039
0.531077
0.525275
0.526245
0.528056
0.523331
0.529638
0.525715
0.526851
0.527255
0.524615
0.524855
0.529806
0.526656
0.529274
0.523412
0.527785
0.47374
0.475931
0.475157
0.470961
0.468923
0.474725
0.473755
0.471944
0.476669
0.470362
0.474285
0.473149
0.472745
0.475385
0.475145
0.470194
0.473344
0.470726
0.476588
0.472215
33.57283
65.9519
87.0333
113.8669
135.3158
152.9428
168.4303
189.6871
219.4764
249.4249
262.0359
275.2695
291.7030
315.0818
333.7332
351.0495
385.3729
402.2348
421.2111
442.4772
12.06656
30.21522
51.16982
67.81981
83.36726
96.91099
113.4414
129.7520
142.0267
156.7367
169.7921
180.1441
191.0308
202.0023
204.5337
213.1942
217.8114
226.5664
262.3495
287.3276
0.010758
0.015019
0.021002
0.038148
0.049271
0.056363
0.061749
0.067974
0.081742
0.096007
0.108004
0.110438
0.128739
0.138693
0.153086
0.163775
0.169128
0.185551
0.205338
0.216944
0.010931
0.019594
0.028967
0.038453
0.047737
0.056865
0.066663
0.076921
0.087006
0.094882
0.112752
0.134178
0.143444
0.151820
0.166440
0.170800
0.187068
0.199256
0.208938
0.237930
0.014407
0.028265
0.041422
0.055456
0.068599
0.081633
0.095873
0.113722
0.123928
0.135023
0.154736
0.162411
0.188254
0.210567
0.214335
0.242015
0.252885
0.268898
0.273174
0.306299
Computational time (s)
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
40.06698
77.14874
113.0401
147.2283
180.2224
211.1297
242.7504
272.7885
319.4657
325.5207
344.3712
365.4371
385.1481
404.1568
421.5027
436.5416
453.7668
465.9425
478.8142
597.5560
Table 23
The results of k-fold cross validation on the RHC dataset.
Fold
MAE
IFCF
DAVIS
HASSAN
DE
SAMUEL
SZMIDT
2
3
4
5
6
7
8
9
10
0.440949
0.437283
0.434009
0.436070
0.435874
0.437098
0.434386
0.437215
0.441839
0.441186
0.437374
0.434141
0.436107
0.436164
0.437535
0.434495
0.437222
0.442047
0.469381
0.463187
0.458663
0.464382
0.436214
0.461758
0.44394
0.435230
0.457397
0.481136
0.480916
0.477842
0.479325
0.482992
0.479249
0.481571
0.479689
0.477239
0.524992
0.525445
0.528936
0.527072
0.523708
0.526433
0.525246
0.52695
0.527824
0.475008
0.474555
0.471064
0.472928
0.476292
0.473567
0.474754
0.47305
0.472176
Data sets
Computational time (s)
2
3
4
5
6
7
8
9
10
736.6445
701.3804
692.8024
629.4906
566.3853
508.6334
456.6042
436.1555
409.1319
477.9642
476.6182
450.9788
424.9808
396.0038
370.6705
349.1144
329.7963
319.6725
262.6263
262.9038
251.8981
238.3611
226.2907
215.6599
206.5204
198.37997
194.55609
0.791994
0.503385
0.366384
0.291197
0.229366
0.113842
0.051412
0.037128
0.013360
0.942148
0.712606
0.617685
0.615317
0.494319
0.346833
0.287202
0.277723
0.250653
1.651110
1.358187
1.215837
0.993458
0.780865
0.356283
0.321834
0.288569
0.275802
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
147
Fig. 4. The MAE results of random experiments on the HEART dataset.
Fig. 5. The MAE results of k-fold cross validation on the HEART dataset.
0.47942055, 0.52670285 and 0.47329715, respectively.
These numbers in Table 23 are 0.437191444, 0.437363444,
0.454461333, 0.479995444, 0.526289556 and 0.473710444,
respectively. In this case, the MAE value of IFCF is the smallest
among all. Looking for the comparisons of algorithms in
terms of MAE in Figs. 6 and 7 also illustrates this fact. This
show that IFCF is effective in cases of large medical diagnosis
datasets having wide ranges of values;
148
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
Fig. 6. The MAE results of random experiments on the RHC dataset.
Fig. 7. The MAE results of k-fold cross validation on the RHC dataset.
(c) The computational time is a drawback of the IFCF algorithm.
In cases of small-sized datasets such as HEART, the computational time of IFCF is ranged from 1.2 to 2.5 s on average
and is not quite larger than those of other algorithms that
take approximately from 0.3 to 1.8 s to process. Yet in cases
of large-size datasets such as RHC, the difference is getting
larger and obvious. Thus a better trade-off between the computational time and the accuracy of diagnosis should be paid
much attention for the assurance of the performance of the
algorithm.
L.H. Son, N.T. Thong / Knowledge-Based Systems 74 (2015) 133–150
4. Conclusions
In this paper, we concentrated on the problem of enhancing
the accuracy of medical diagnosis and presented a novel intuitionistic fuzzy recommender system (IFRS) consisting of the following components: (i) the new definitions of single-criterion
IFRS (SC-IFRS) and multi-criteria IFRS (MC-IFRS) that extend the
definition of traditional recommender systems (RS) taking into
account a feature of a user and a characteristic of an item
expressed by intuitionistic linguistic labels; (ii) the new definitions of intuitionistic fuzzy matrix (IFM), which is a representation of SC-IFRS and MC-IFRS in the matrix format and the
intuitionistic fuzzy composition matrix (IFCM) of two IFMs with
the intersection/union operation; (iii) some new similarity
degrees of IFMs such as the intuitionistic fuzzy similarity matrix
(IFSM) and the intuitionistic fuzzy similarity degree (IFSD) and
the formulas to predict diseases on the basis of IFSD accompanied
with an interesting theorem; (iv) a novel intuitionistic fuzzy collaborative filtering method relying on the basis of the predicting
formulas so-called IFCF. Some interesting theorems and properties of the proposed components were also investigated. The
proposed IFCF algorithm was used mainly for the medical diagnosis problem. Some numerical examples have been introduced
throughout the paper to illustrate the problem and the activities
of the algorithms.
In the Evaluation section, a numerical example on a small intuionistic fuzzy medical diagnosis data was presented. The next
experiments were conducted on both the small and large real hard
benchmark medical diagnosis data from UCI Machine Learning
Repository. The findings from the experiments are summarized
as follows: (i) the proposed IFCF algorithm is capable to perform
the prediction and recommendation with more types of datasets
including the hard and (intuitionistic) fuzzy data than other algorithms such as the standalone methods of intuitionistic fuzzy sets
(IFS) such as De et al. [9], Szmidt and Kacprzyk [44], Samuel and
Balamurugan [31] and RS such as Davis et al. [8] and Hassan and
Syed [13]; (ii) IFCF has better accuracy of prediction than those
algorithms in cases of the (intuitionistic) fuzzy data and the large
medical diagnosis datasets having wide ranges of values; (iii) IFCF
could handle the limitations of those works as pointed out in Section 1.6(b) of this article; (iv) Lastly, a better trade-off between the
computational time and the accuracy of diagnosis in IFCF should be
paid much attention for the assurance of the performance of the
algorithm. These findings clearly show the effectiveness of the proposed algorithm.
As being mentioned in the Section 1.6 stressing the importance
and significance of the proposed work, some solid further works
are necessary to impulse the development of the algorithm in this
field. Firstly, a variation of IFCF algorithm that tackles with the
deficiency of processing small-sized real datasets should be studied. Secondly, a trade-off between the computational time and
the accuracy of diagnosis in IFCF is examined. Thirdly, a hybrid
algorithm between IFCF and a fuzzy clustering method to enhance
the accuracy is considered. Fourthly, the theoretical analyses of the
IFRS especially the formulation of IFCM with other operations such
as t-norm and t-conorm are examined. Lastly, applications of IFRS
for other problems, e.g. the time series forecast and the nowcasting
could be performed. These future works will enrich the knowledge
of deploying advanced fuzzy recommender systems for practical
problems.
Acknowledgement
The authors are greatly indebted to the editors-in-chief, Prof. H.
Fujita, Prof. J. Lu and anonymous reviewers for their comments and
149
their valuable suggestions that improved the quality and clarity of
paper. This work is sponsored by the NAFOSTED under Contract
No. 102.05-2014.01.
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