DSpace at VNU: On the performance evaluation of intuitionistic vector similarity measures for medical diagnosis

Intuitionistic fuzzy recommender system IFRS, which has been recently presented based on the theories of intuitionistic fuzzy sets and recommender systems, is an efficient tool for medic

IOS Press

On the performance evaluation

of intuitionistic vector similarity

Le Hoang Sona,∗and Pham Hong Phongb

aVNU University of Science, Vietnam National University, Hanoi, Vietnam

bNational University of Civil Engineering, Hanoi, Vietnam

Abstract Intuitionistic fuzzy recommender system (IFRS), which has been recently presented based on the theories of

intuitionistic fuzzy sets and recommender systems, is an efficient tool for medical diagnosis IFRS used the intuitionistic fuzzy similarity degree (IFSD) regarded as the generalization of the hard user-based, item-based and the rating-based similarity degrees in recommender systems to calculate the analogousness between patients in the system In this paper, we firstly extend IFRS by using a new term - the intuitionistic fuzzy vector (IFV) instead of the existing intuitionistic fuzzy matrix (IFM) in IFRS Then, the intuitionistic value similarity measure (IvSM) and the intuitionistic vector similarity measure (IVSM) are defined on the basis of the intuitionistic fuzzy vector Some mathematical properties of these new terms are examined, and several IVSM functions are proposed The performances of these IVSM functions for medical diagnosis are experimentally validated and compared with the existing similarity degrees of IFRS The suggestion and recommendation of this paper involve the most efficient IVSM function(s) that should be used for medical diagnosis

Keywords: Intuitionistic fuzzy recommender systems, intuitionistic fuzzy vector, intuitionistic vector similarity measure, medical diagnosis, performance evaluation

1 Introduction

Medical diagnosis is an important and

neces-sary process to issue appropriate medical figures for

patients in health care support systems It involves

the determination of the possible relations between

1 The authors are greatly indebted to the editor-in-chief: Prof.

Reza Langari and anonymous reviewers for their comments and

suggestions which improved the quality and clarity of the paper.

This research is funded by Vietnam National Foundation for

Sci-ence and Technology Development (NAFOSTED) under grant

number 102.05-2014.01.

∗Corresponding author Le Hoang Son, VNU University of

Science, Vietnam National University, Hanoi, Vietnam Tel.:

+84 904 171 284; E-mails: sonlh@vnu.edu.vn, chinhson2002@

gmail.com.

patients and diseases from those between patients and symptoms The answer of medical diagnosis for a cer-tain disease is often yes/no that eventually leads to the final specification of the most acquiring disease and appropriate treatments The medical diagnosis indeed must ensure the accuracy, which raises great interests of researchers to enhance as far as possible Recent advances of the health care support systems

have raised a great concentration to enhancing the

accuracy of medical diagnosis both in theory and

practice [2] An effort of this theme has presented

an efficient tool namely Intuitionistic Fuzzy Recom-mender System (IFRS), which was designed based

on the theories of intuitionistic fuzzy sets and

rec-ommender systems [9] In IFRS, the intuitionistic

1064-1246/16/$35.00 © 2016 – IOS Press and the authors All rights reserved

fuzzy similarity degree (IFSD) is utilized as a

gen-eralization of the hard user-based, item-based and

the rating-based similarity degrees to calculate the

analogousness between patients A hybrid similarity

degree between IFSD and the degree produced by a

picture fuzzy clustering method has been proposed

to enhance the accuracy of prediction These relevant

researches mostly investigated on improving the

sim-ilarity degree of IFRS to ensure the high accuracy of

the system

In this paper we propose intuitionistic fuzzy

vector (IFV) instead of the existing intuitionistic

fuzzy matrix (IFM) in IFRS Then, a

generaliza-tion of the existing multi-criteria IFRS so-called

the Modified multi-criteria IFRS (MMC-IFRS) that

takes into account the IFV is presented Two new

measures namely the intuitionistic value similarity

measure (IvSM) and the intuitionistic vector

similar-ity measure (IVSM) are defined Some mathematical

properties of these new terms are examined, and

sev-eral IVSM functions are proposed The performances

of these IVSM functions for medical diagnosis are

experimentally validated and compared with the

existing similarity degrees of IFRS

The rests of the paper are organized as follows

Sec-tion 2 recalls some previous works SecSec-tion 3 presents

the new contributions of this paper Section 4

vali-dates the proposed model by experiments Section 5

gives the conclusions and future works of the paper

2 Preliminaries

2.1 Related works

Assume that P, S and D being the sets of patients,

symptoms and diseases, respectively Each patient P i

(i = 1, n) (resp symptom S j , j = 1, m) is assumed

to have some features (resp characteristics) For

the simplicity, we consider the recommender system

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 D k (k = 1, p) also contains s

intuitionistic linguistic labels Thus, the definition

of Multi-criteria Intuitionistic Fuzzy Recommender

Systems (MC-IFRS) was given as follows.

Definition 1 [9] (Multi-criteria Intuitionistic Fuzzy

Recommender Systems – MC-IFRS) The utility

func-tion R is a mapping specified on (X, Y ) as in

Equation (1)

R : X × Y → D1× · · · × D p ,

(μ 1X (x) , γ 1X (x))

· · ·

(μ sX (x) , γ sX (x))



×

(μ 1Y (y) , γ 1Y (y))

· · ·

(μ sY (y) , γ sY (y))



→

p



k=1

(μ 1D (D k ) , γ 1D (D k))

· · ·

(μ sD (D k ) , γ sD (D k))



In Equation (1), (μ iX (x) , γ iX (x)) is an intuitionistic fuzzy value (IFv) of the patient to the i-th linguistic label of feature X, (μ iY (y) , γ iY (y)) represents the IFv of the symptom to the i-th linguistic label of char-acter Y , and (μ iD (D k ) , γ iD (D k)) stands for the IFv

of the disease D k to the linguistic label i-th (i = 1, s,

k = 1, p).

MC-IFRS is the system that provides two basic functions below

a) Prediction: determine the values of

μ iD (D k ) , γ iD (D k ) , i = 1, s, k = 1, p b) Recommendation: choose i∗= 1, s which

max-imize(s) the expression

p



k=1

w k (μ iD (D k)+ μ iD (D k ) π iD (D k )),

where π iD (D k)= 1 − μ iD (D k)− γ iD (D k)

and w k ∈ [0, 1] is the weight of D k satisfying the constraint:p

k=1w k = 1

MC-IFRS could be compressed in a matrix form

as in Definition 2

Definition 2 [9] An intuitionistic fuzzy matrix (IFM)

Zin MC-IFRS is defined as,

Z=

⎛

⎜

⎜

⎜

⎝

a11a12· · · a 1s b21 b22 · · · b 2s

c31 c32 · · · c 3s

· · · ·

c t1 c t2 · · · c ts

⎞

⎟

⎟

⎟

⎠

In Equation (2), t = p + 2 where p ∈ N∗is the

num-ber of diseases in Definition 1 The value s∈ N∗is the

number of intuitionistic linguistic labels a 1i , b 2i , c hi,

h = 3, t, i = 1, s are the IFvs consisting of the

mem-bership and non-memmem-bership values as in Definition

1: a 1i = (μ iX (x) , γ iX (x)), b 2i = (μ iY (y) , γ iY (y))

and c hi = (μ iD (D h−2) , γ iD (D h−2)) , i = 1, s,

h = 3, t Each line from the third one to the last in

Equation (2) is related to a given disease

Based on IFM, the intuitionistic fuzzy similarity

matrix (IFSM) and the intuitionistic fuzzy similarity

degree (IFSD) were defined as in Definitions 3 and 4.

Definition 3 [9] Suppose that Z1 and Z2 are two

IFM in MC-IFRS The intuitionistic fuzzy

similar-ity matrix (IFSM) between Z1 and Z2is defined as

follows

˜S=

⎛

⎜

⎜

⎜

⎝

˜S11 ˜S12 · · · ˜S 1s

˜S21 ˜S22 · · · ˜S 2s

˜S31 ˜S32 · · · ˜S 3s

· · · ·

˜S t1 ˜S t2 · · · ˜S ts

⎞

⎟

⎟

⎟

⎠

,

where ˜S 1i = sim a(1)1i , a(2)1i



, ˜S 2i = sim b(1)2i , b(2)2i

 ,

and ˜S hi = sim c hi(1), c(2)hi



, i = 1, s, h = 3, t sim is

a measure specifying the similarity between two

intu-itionistic values u = (μ u , γ u ) and v = (μ v , γ v),

sim (u, v)

= 1 − 1− exp



−1

2√

μ u − √μ v+√

γ u − √γ v

(3)

Definition 4 [9] Suppose that Z1 and Z2 are two

IFM in MC-IFRS The intuitionistic fuzzy similarity

degree (IFSD) between Z1 and Z2is

SIM (Z1, Z2)= α

s



i=1

w 1i ˜S 1i + β

s



i=1

w 2i ˜S 2i

+χ t



h=3

s



i=1

w hi ˜S hi , (4)

where ˜S is the IFSM between Z1and Z2 W=w ji

(j = 1, t, i = 1, s) is the weight matrix of IFSM

between Z1and Z2satisfying

s



i=1

w ji = 1, j = 1, t,

α + β + χ = 1.

IFSD is used to calculate the analogousness

between patients in the system, and to make the

pre-diction of possible diseases for a patient It is obvious

that the better the IFSD is, the higher of accuracy the

health care support system may be achieved Thus, in

another recent paper [11], the authors have defined a hybrid similarity degree between IFSD and the degree produced by a picture fuzzy clustering method [8] to enhance the accuracy of prediction as in Definition 5

Definition 5 [11] Let us denote the IFSD in Equation

(4) as SIM history (a, b) The hybrid similarity degree is

then calculated as

SIM (a, b) = (1 − λ) SIM

history (a, b) + λSIM

group (a, b) , where λ ∈ [0, 1] is an adjustable coefficient, and SIM

group (a, b) is the similarity degree from the picture

fuzzy clustering [8] as in the equations below

SIM

group (a, b)

= 1

N C

N C



i=1



P (a, i) − P (a) P (b, i) − P (b),

P (j, k)= 1 − CS (j, k)

max

i {CS (i, k)} ,

CS (j, k)= 1 −



i

X i j V k i

X jV k,

where P (j, k) is the possibility of patient j belong-ing to the cluster k, CS (j, k) is the counter similarity between the patient j and the cluster k with X j and V k being the patient j and the center of cluster k respec-tively P (a) is the mean value of P (a, k), k = 1, N C

N Cis the number of groups used in the picture fuzzy clustering – DPFCM method [8]

2.2 Some remarks

The methods recalled in sub-section 2.1 achieved better accuracies than the relevant ones such as the standalone algorithms of intuitionistic fuzzy sets namely [4, 7, 10] and recommender systems, e.g [3, 5] These relevant researches mostly investigated

on improving the similarity degree of IFRS to ensure the high accuracy of the system Being noticed that the most important assumption in IFRS is the num-bers of intuitionistic linguistic labels in the features

of the patients, in the characteristics of the symp-toms and in the diseases being the same and denoted

as s (See some lines before Definition 1) In

practi-cal applications, this situation may not happen and brings out the difficulty to apply IFSD in Definition

4 and the hybrid similarity degree in Definition 5 to

them This motivates us to extend MC-IFRS in

Def-inition 1 and the equivalent similarity degrees to the

new context Thus, in this paper we firstly extend

MC-IFRS by using a new term – the intuitionistic

fuzzy vector (IFV) instead of the existing

intuitionis-tic fuzzy matrix (IFM) in IFRS Then, the intuitionisintuitionis-tic

value similarity measure (IvSM) and the

intuitionis-tic vector similarity measure (IVSM) are defined on

the basis of the IFV Some mathematical properties

of these new terms are examined, and several IVSM

functions are proposed The performances of these

IVSM functions for medical diagnosis are

experi-mentally validated and compared with the existing

similarity degrees of IFRS The suggestion and

rec-ommendation of this paper involve the most efficient

IVSM function(s) that should be used for

medi-cal diagnosis Hence, the contributions of this paper

occupy an important role to not only the theoretical

aspects of recommender systems but also the

appli-cable roles to the health care support system

3 The proposed method

In this section, we firstly propose a new MC-IFRS

so-called the Modified MC-IFRS (MMC-IFRS) to

handle the problem of different numbers of

intuition-istic linguintuition-istic labels in the features of patients, the

characteristics of symptoms and the diseases in

sub-section 3.1 An illustrated example of MMC-IFRS

and the conversion of MMC-IFRS to the

intuitionis-tic fuzzy value (IFV) are also given herein Secondly,

we define the intuitionistic value similarity measure

(IvSM) and the intuitionistic vector similarity

mea-sure (IVSM) accompanied with some mathematical

properties in sub-section 3.2 Several IVSM functions

for the validation in the experiments are also proposed

in this sub-section

3.1 Modified multi-criteria intuitionistic fuzzy

recommender system

Recall that P, S and D being the sets of patients,

symptoms and diseases having the cardinalities of n,

m and p, respectively Each patient P i (i = 1, n) is

assumed to have N features X1, , X N Each feature

X e consists of r e linguistic labels (e = 1, N) Each

symptom S j (j = 1, m) is assumed to have M

char-acteristics Y1, , Y M Each characteristic Y f consists

of s f linguistic labels (j = 1, m ) Each disease D g

contains t intuitionistic linguistic labels (g = 1, p).

Definition 6 (Modified Multi-criteria Intuitionistic

Fuzzy Recommender Systems – MMC-IFRS) The utility function R is a mapping:

N



e=1

X e



×

⎛

⎝M

f=1

Y f

⎞

⎠ →p

g=1

D g ,

N



e=1

μ 1X e (x e ) , γ 1X e (x e)

.



μ r e X e (x e ) , γ r e X e (x e)



×

M



f=1

 μ 1Y f

y f

, γ 1Y

f



y f

.

μ s

f Y f



y f

, γ s

f Y f



y f



→

p



g=1

 μ1D

D g



, γ1D

D g



.



μ t g D



D g



, γ t g D



D g





where

μ xX e (x e ) , γ xX e (x e)

is the IFv of the patient

to the x-th linguistic label of the feature X e (x=

1, r e , e = 1, N).μ yY f

y f

, γ yY f

y f

is the IFv

of the symptom to the y-th linguistic label of the characteristic Y f (y = 1, s f , f = 1, M) Finally,



μ zD



D g



, γ zD



D g



is the IFv of the disease D gto

the z-th linguistic label (z = 1, t g , g = 1, p)

MMC-IFRS provides two basic functions:

a) Prediction: determine the values of



μ zD



D g



, γ zD



D g



, z = 1, t g , g = 1, p b) Recommendation: choose z∗= 1, t g which maximize(s) the expression

p



g=1

w g

μ zD

D g

+ μ zD



D g

π zD

D g

,

where π zD



D g



= 1 − μ zD



D g



− γ zD



D g



and w g ∈ [0, 1] is the weight of D g satisfying the constraint:p

g=1w g= 1

It is obvious that MMC-IFRS in Definition 6 is a generalization of MC-IFRS in Definition 1 Consider the example below to illustrate the new definition to medical diagnosis

Example 1 In a medical diagnosis system, there

are 4 patients The feature X is “Age” consisting

of 5 linguistic labels: “VL=very low”, “L=low”,

“M=medium”, “H=high”, “VH=very high” By

using the trapezoidal intuitionistic fuzzy numbers –

TIFNs [1] characterized by 

a1, a2, a3, a4 ; a

1, a 4



with a

1≤ a1≤ a2≤ a3≤ a4≤ a4, the membership

(non-membership) functions of patients to the

lin-guistic labels of the feature X are:

μ VL (x)=

⎧

⎪

⎪

(20− x)/10 10 < x≤ 20

,

γ VL (x)=

⎧

⎪

⎪

(x − 10)/10 10 < x≤ 20

,

μ L (x)=

⎧

⎪

⎪

⎪

⎪

0 x ≤ 10, x > 40 (x − 10)/10 10 < x≤ 20

1 20 < x≤ 30 (40− x)/10 30 < x≤ 40

,

γ L (x)=

⎧

⎪

⎪

⎪

⎪

1 x ≤ 10, x > 40

(20− x)/10 10 < x≤ 20

0 20 < x≤ 30

(x − 30)/10 30 < x≤ 40

,

μ M (x)=

⎧

⎪

⎪

⎪

⎪

0 x ≤ 30, x > 60 (x − 30)/10 30 < x≤ 40

1 40 < x≤ 50 (60− x)/10 50 < x≤ 60

,

γ M (x)=

⎧

⎪

⎪

⎪

⎪

1 x ≤ 30, x > 60

(40− x)/10 30 < x≤ 40

0 40 < x≤ 50

(x − 50)/10 50 < x≤ 60

,

μ H (x)=

⎧

⎪

⎪

⎪

⎪

0 x ≤ 50, x > 80 (x − 50)/10 50 < x≤ 60

1 60 < x≤ 70 (80− x)/10 70 < x≤ 80

,

γ H (x)=

⎧

⎪

⎪

⎪

⎪

1 x ≤ 50, x > 80

(60− x)/10 50 < x≤ 60

0 60 < x≤ 70

(x − 70)/10 70 < x≤ 80

,

μ VH (x)=

⎧

⎪

⎪

(x − 70)/10 70 < x≤ 80

,

γ VH (x)=

⎧

⎪

⎪

(80− x)/10 70 < x≤ 80

.

Based on the membership and non-membership func-tions, we calculate the information of patients as follows

Al(18) :

H (0, 1) , VH (0, 1) ,

Bob(39) :

H (0, 1) , VH (0, 1) ,

Joe(53) :

H (0.3, 0.7) , VH (0, 1) ,

Ted(74) :

H (0.4, 0.6) , VH (0, 1) The symptom’s characteristic Y is

“Tempera-ture” including three linguistic labels: “C=cold”,

“M=medium”, “H=hot” Similarly, the membership (non-membership) functions of the symptom to the linguistic labels of characteristic are defined using TIFNs as follows

μ C (x)=

⎧

⎪

⎪

(20− x) /15 5 < x≤ 20

,

γ C (x)=

⎧

⎪

⎪

(x − 5) /15 5 < x≤ 20

,

μ M (x)=

⎧

⎪

⎪

⎪

⎪

0 x ≤ 5, x > 40 (x − 5) /15 5 < x≤ 20

1 20 < x≤ 35 (40− x) /5 35 < x≤ 40

,

γ M (x)=

⎧

⎪

⎪

⎪

⎪

1 x ≤ 5, x > 40

(20− x) /15 5 < x≤ 20

0 20 < x≤ 35

(x − 35) /5 35 < x≤ 40

,

μ H (x)=

⎧

⎪

⎪

(x − 35) /5 35 < x≤ 40

,

γ H (x)=

⎧

⎪

⎪

(40− x) /5 35 < x ≤ 40

.

The information of symptom is shown as follows



4◦C

:



16◦C

:

M (0.733, 0.267) , H (0, 1),



39◦C

:



25◦C

:

The diseases (D1, D2) are “Flu” and “Headache”,

where D1contains four linguistic labels: “L1=Level

1”, “L2=Level 2”, “L3=Level 3” and “L4=Level

4”, D2 contains six linguistic labels: “L1=Level

1”, “L2=Level 2”, “L3=Level 3”, “L4=Level 4”,

“L5=Level 5” and “L6=Level 6” 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 MMC-IFRS system We have

a MMC-IFRS described in Table 1 In this table, the

cells having asterisk marks are needed to predict the

intuitionistic fuzzy values 

μ zD



D g



, γ zD



D g



(z = 1, t g , g = 1, 2]) A compression form of

MMC-IFRS is shown in Definition 7

Definition 7 An intuitionistic fuzzy vector (IFV) in

MMC-IFRS is defined as follows

V = (v1, v2, , v K ) ,

where K = K1+ K2+ K3, K1=N

e=1r e , K2=

M

f=1s f , K3=p

g=1t g The first K1elements of

V are

a11, , a 1r , a e1, , a e r e , a N1, , a Nr ,

Table 1

A MMC-IFRS for medical diagnosis with ∗ being the values to

be predicted

Al(18) :

VL (.2, 8)

L (.8, 2)

M (0, 1)

H (0, 1)

VH (0, 1)

4 ◦C:

C (1, 0)

M (0, 1)

H (0, 1)

L 1 (.8, 1)

L 2 (.6, 3)

L 3 (.2, 6)

L 4 (.1, 9)

L 1 (.1, 8)

L 2 (.2, 7)

L 3 (.5, 35)

L 4 (.6, 2)

L 5 (.4, 5)

L 6 (.3, 55)

Bob(39) :

VL (0, 1)

L (.1, 9)

M (.9, 1)

H (0, 1)

VH (0, 1)

39 ◦C:

C (0, 1)

M (.2, 8)

H (.8, 2)

L 1 (.4, 5)

L 2 (.6, 2)

L 3 (.3, 6)

L 4 (.1, 9)

L 1 (0, 9)

L 2 (.2, 75)

L 3 (.4, 55)

L 4 (.55, 35)

L 5 (.7, 2)

L 6 (.6, 3)

Joe(53) :

VL (0, 1)

L (0, 1)

M (.7, 3)

H (.3, 7)

VH (0, 1)

16 ◦C:

C (.267, 733)

M (.733, 267)

H (0, 1)

L 1 (0, 1)

L 2 (.2, 7)

L 3 (.4, 5)

L 4 (1, 0)

L 1 (0, 0.9)

L 2 (.4, 6)

L 3 (.4, 45)

L 4 (.7, 2)

L 5 (.3, 6)

L 6 (.1, 85)

Ted(74) :

VL (0, 1)

L (0, 1)

M (.6, 4)

H (.4, 6)

VH (0, 1)

25 ◦C:

C (0, 1)

M (1, 0)

H (0, 1)

L1 (∗, ∗)

L2 (∗, ∗)

L3 (∗, ∗)

L4 (∗, ∗)

L1 (∗, ∗)

L2 (∗, ∗)

L3 (∗, ∗)

L4 (∗, ∗)

L5 (∗, ∗)

L6 (∗, ∗)

with a exrepresents for an IFv of the patient to the

lin-guistic label x-th of feature X e (x = 1, r e , e = 1, N) The next K2elements of V are

b11, ,b1s1, , b f , , b fs f , , b M1, , bMs M ,

where b fy means an IFv of the symptom to the

linguistic label y-th of characteristic Y f (y = 1, s f,

f = 1, M) And the last K3elements of V are

c11, , c 1t1, , c g , , c gt g , , c p1, , c pt p ,

where c gz is an IFv of the disease D gto the linguistic

label z-th (z = 1, t g , g = 1, p).

3.2 Intuitionistic value similarity measure and intuitionistic vector similarity measure

In the following definition, θ denotes the set of all

intuitionistic fuzzy values (IFVs)

Definition 8. (Intuitionistic value similarity measure–IvSM) LetR be the set of all real number,

sim : θ × θ → R is called an intuitionistic value similarity measure (IvSM) if it satisfies the following

conditions:

(A1) sim (u, v) = sim (v, u), for all u, v ∈ θ;

(A2) 0≤ sim (u, v) ≤ 1, for all u, v ∈ θ;

(A3) sim (u, v) = 1 ⇔ u = v, for all u, v ∈ θ;

(A4) If u ≤ v ≤ w, then sim (u, v) ≥ sim (u, w)

and sim (v, w) ≥ sim (u, w), for all u, v, w ∈

θ (u ≤ v means μ u ≤ μ v and γ u ≥ γ v)

Theorem 1 For all u, v ∈ θ, we define:

sim1 (u, v)= 1 − 1

2(|μu − μ v | + |γ u − γ v|) ; (6)

sim2(u, v)= min{μ u , μ v } + min {γ u , γ v}

max{μ u , μ v } + max {γ u , γ v}; (7)

sim3 (u, v)

=exp −

1

2(|μu − μ v | + |γ u − γ v|)− exp (−1)

(8)

sim4(u, v)

= exp



− 1√μ

u − √μ v+√γ

u − √γ v− exp (−1)

(9)

Then, sim1, sim3, sim3 and sim4 are IvSMs Notice

that to avoid the denominator being zero, set 00 = 1

in the definition of sim2.

Proof We consider sim1, the remainders are also

proved by analogous calculation

(A1) and (A3) are straightforward

(A2) We have 0≤ |μ u − μ v | + |γ u − γ v| ≤ 2 It

follows that 0≤ sim1(u, v)≤ 1

(A4) We prove sim1(u, v) ≥ sim1(u, w) with

con-dition of u ≤ v ≤ w By the definition of the relation

≤ of IFvs, we get μ u ≤ μ v ≤ μ w and γ u ≥ γ v ≥ γ w

which implies

sim1 (u, v)= 1 −1

2((μ v − μ u)+ (γ u − γ v))

≥ 1 −1

2((μ w − μ u)+ (γ u − γ w))

= 1 −1

2(|μw − μ u | + |γ w − γ u|)

= sim1(u, w)

By similar argument, we get sim1(v, w)≥

In the following definition, denotes the set of all

intuitionistic fuzzy vectors (IFVs) having the lengths

of K in MMC-IFRS.

Definition 9. (Intuitionistic vector similarity measure–IVSM) Let SIM : × → R SIM is called an intuitionistic vector similarity measure

(IVSM) if it satisfies the following conditions:

(B1) SIM (U, V ) = SIM (V, U), for all U, V ∈ ;

(B2) 0≤ SIM (U, V ) ≤ 1, for all U, V ∈ ; (B3) SIM (U, V ) = 1 ⇔ U = V , for all U,

V ∈ ;

(B4) If U ≤ V ≤ T , then SIM (U, V )≥

SIM (U, T ) and SIM (V, T ) ≥ SIM (U, T ), for all U, V , T ∈ (let U = (u1, , uK),

V = (v1, , vK ) U ≤ V means u ≤ v

= 1, K).

Definition 10 Let U, V ∈ , sim is an IvSM, and

W = (w1, , w K ) is weight vector satisfying w ≥

= 1, K) andK

=1w = 1 We define:

1) The quadric intuitionistic fuzzy similarity degree between U and V :

SIM Q (U, V )=

 K



=1

w (sim (u , v))2

1

.

(10)

2) The arithmetic intuitionistic fuzzy similarity degree between U and V :

SIM A (U, V )=

K



=1

w sim (u , v ). (11)

3) The geometric intuitionistic fuzzy similarity degree between U and V :

SIM G (U, V )=

K



=1

(sim (u , v ))w (12)

4) The harmonic intuitionistic fuzzy similarity degree between U and V :

SIM H (U, V )=

 K



=1

w sim (u , v )

−1

.

(13)

Theorem 2 Let U, V ∈ We have SIM Q (U, V )≥

SIM A (U, V ) ≥ SIM G (U, V ) ≥ SIM H (U, V ).

Proof The proof is done by using classical

inequalities: the Cauchy-Schwarz and the weighted AM-GM inequalities For example, we consider

SIM Q (U, V ) ≥ SIM A (U, V ) Using the

Cauchy-Schwarz inequality,

 K



=1

x2

 



= 1K y2



≥





= 1K x y

2

,

for all (x1, , xK ), (y1, , yK)∈ RK, we have

K



=1

w (sim (u , v ))2

=

 K



=1

w 1/2

2  K



=1

w 1/2 sim (u , v )

2

≥

 K



=1

w 1/2 w 1/2 sim (u , v)

2

=

 K



=1

w sim (u , v )

2

.

That means 

SIM Q (U, V )2

≥ (SIM A (U, V ))2, or

SIM Q (U, V ) ≥ SIM A (U, V ). 

Theorem 3 Assume that w > = 1, K.

SIM Q , SIM A , SIM G and SIM H are IVSM.

Proof Obviously, SIM Q , SIM A , SIM G and SIM H

satisfy (B1)

(B2) For all U, V ∈ By Theorem 2, it is

sufficient to prove that SIM Q (U, V )≤ 1 From

sim (u , v )

SIM Q (U, V )

=

 K



=1

w (sim (u , v ))2

1

≤

 K



=1

w

1

= 1.

(B3) For all U, V ∈ , it is easily to show that

SIM H (U, V ) = 1 ⇔ U = V.

By Theorem 2, if one in the values SIM Q (U, V ),

SIM A (U, V ) and SIM G (U, V ) equals to 1, then

SIM H (U, V ) equals to 1 Then, SIM Q (U, V ),

SIM A (U, V ) and SIM G (U, V ) satisfy (B3).

(B4) The condition U ≤ V ≤ T yields that

sim (u , v)≥ sim (u , t ) ,

Thus, (sim (u , v ))w ≥ (sim (u , t ))w =

1, K Hence

K



=1

w (sim (u , v ))2

1

≥

K



=1

w (sim (u , t))2

1

,

or SIM Q (U, V ) ≥ SIM Q (U, T ) The remainders of

Definition 11 Let SIM is an IVSM The formulas

to predict the values of linguistic labels of the patient

P∗to the diseases D

g (g = 1, p) in MMC-IFRS are:

μ P zD∗

D g

=

n



v=1SIM (P

∗, P v)× μ P v

zD



D g



n



v=1SIM (P

∗, P v)

,

γ zD P∗

D g

=

n



v=1SIM (P

∗, P v)× γ P v

zD



D g



n



v=1SIM (P

∗, P v)

,

for all∀z ∈ 1, t g , g = 1, p.

Theorem 4 For all z ∈ 1, t g , g = 1, p and patient

P∗, we have μ P∗

zD



D g

, γ zD P∗

D g

is an IFv.

Proof It is easily seen that μ P zD∗

D g



≥ 0, and

γ zD P∗

D g

≥ 0 Moreover,

μ P zD∗

D g

+ γ P∗

zD



D g

=

n



v=1

SIM (P∗, P

v)× μ P v

zD



D g

+ γ P v zD



D g

n



v=1SIM (P

∗, P v)

.

μ P v

zD



D g

, γ P v

zD



D g

is an IFv, then μ P v

zD



D g +

γ P v

zD



D g

≤ 1 Thus, μ P∗

zD



D g

+ γ P∗

zD



D g

≤ 1 

4 Evaluation

4.1 Experimental design

In this part, we describe the experimental environ-ments such as,

Experimental tools: We have implemented 16

vari-ants of the prediction algorithm for medical diagnosis

by matching each IVSM function in Equations (6–9) with each IvSM function given in Equations (10– 13) in PHP programming language Notice that the variant combining Equations (9, 11) is exactly the IFSD function of MC-IFRS defined in Equations (3–4) of Definitions 3 & 4, respectively Thus, we clearly recognize that IFSD is a special case of

Table 2

The MAE values of the variants by k-fold cross validation with the best values being marked as bold

the proposed IVSM functions in this work The

variants are denoted from A1 to A16 with A1

being matched between Equations (6, 10), A2 being

matched between Equations (6, 11), and A16 being

matched between Equations (9, 13) A14 is replaced

with the IFSD function [9] as explained above Notice

that the hybrid similarity degree [9] described in

Definition 5 is just a derivative of IFSD with the

supplement of information from a picture fuzzy

clus-tering method so that for the accurate comparison

between the original similarity degrees, it should not

be mentioned herein Further hybridization between

the IVSM functions and the degree from a picture

fuzzy clustering method is considered in another

work These algorithms are executed on a PC Intel(R)

core(TM) 2 Duo CPU T6400 @ 2.00GHz 2GB RAM

The results are taken as the average value of 50 runs

Evaluation indices: Mean Absolute Error (MAE) and

the computational time

Datasets: The benchmark medical diagnosis da-taset

namely HEART from UCI Machine Learning

Repos-itory [12] consisting of 270 patients characterized by

13 attributes This dataset was also used for

experi-ments in [9, 11]

Cross validation: The cross-validation method for the

experiments is the k-fold validation with k from 2 to

10 Besides testing with the k-fold validation, the

ran-dom experiments with the cardinalities of the testing

being from 10 to 100 random elements are also

per-formed In order to validate the results with accurate

classes, the intuitionistic defuzzification method of [1] as in Example 1 is used for experimental algo-rithms

Parameter setting: the weights of the degrees are set

up as in [9, 11]

Objective: To validate the performance of IVSM

functions in terms of accuracy through evaluation indices

4.2 Assessment

In Tables 2 and 3, we illustrate the MAE values

and the computational time of the variants by k-fold

cross validation respectively From Table 2, we cal-culate the average MAE values of variants by the numbers of folds This Tab shows the MAE values

of the A7 variant is the best among all Besides A7, other variants such as A3, A11 and A15 should be used for the best MAE values of the algorithm It is clear that a large number of folds do not correspond

to the better MAE value of algorithm For the sake

of both the computational time and MAE values, the number of folds should be selected within the range

[8, 10] especially when it is equal to 9, the average and the best MAE values of all variants are 0.484 and 0.462 respectively, which hold the best trials among

all In Table 3, the average computational time of all variants by various numbers of folds are illustrated Apparently, the processing time of these algorithms

is from 0.68 to 1.44 seconds (sec) Furthermore,

Table 3

The computational time of the variants by k-fold cross validation with the best values being marked as bold (sec)

Table 4 The MAE values of the variants by random experiments with the best values being marked as bold

A2 is the best variant in term of the computational

time

In order to validate the efficiencies of variants, we

have made experiments on another cross validation

method Tables 4 and 5 demonstrate the MAE values

and the computational time of the variants by

ran-dom experiments respectively The remarks about the

superior of A7 and other variants such as A3, A11 and

A15 are kept intact The results have clearly shown

that the ideal cardinality of the testing set should be

selected as 40 or in the range [20, 60].

5 Conclusions

In this paper, we concentrated on improving the accuracy of medical diagnosis in the health care support system We have shown that Intuitionis-tic Fuzzy Recommender System (IFRS) and the hybridization between IFRS and a picture fuzzy clus-tering method are the efficient tools to achieve the desired goal Nonetheless, both these methods were relied on an important assumption in IFRS con-firming that the numbers of intuitionistic linguistic