Individual and Global Assessments with Signed Distance Defuzziﬁcation,

and Characteristics of the Output Distributions based on an Empirical

Analysis

edina Berkachy and Laurent Donz

Applied Statistics and Modelling, Department of Informatics, Faculty of Economics and Social Sciences,

University of Fribourg, Boulevard de P

erolles 90, 1700 Fribourg, Switzerland

Keywords:

Individual Assessment, Finanzplatz Z

urich: Umfrage 2010, Fuzzy Logic, Fuzzy Statistics, Global As-

sessment, Linguistic Questionnaire, Normality, Output Distribution, Sampling Weights, Signed Distance

Defuzziﬁcation Method.

Abstract:

Considering a particular kind of questionnaires called linguistic questionnaires, we apply fuzzy logic to pro-

vide individual and global weighted evaluations in the case of the signed distance defuzziﬁcation method. We

test our method on real data coming from a survey of the ﬁnancial place of Zurich (Switzerland). Furthermore,

we have been enable to give a look at the output distributions, and put into evidence their statistical properties.

In particular, normality of distributions draws our attention. One of our main ﬁndings is that the individual

evaluations calculated with the signed distance defuzziﬁcation method tend to be normally distributed.

1 INTRODUCTION AND

MOTIVATION

A linguistic questionnaire is a set of items called

“questions”, coded as categorical variables. Indeed,

the modalities of the variables are deﬁned as linguis-

tics. This type of questionnaires has been used fre-

quently in many ﬁelds such as in satisfaction stud-

ies or in different economic studies, etc. Due to this

particular kind of questions, the answers are deeply

market by imprecision and vagueness, for the human

response often weaving between different answers.

Fuzzy logic was precisely introduced to deal with

such imprecision.

The treatment of the data provided by the survey

via the questionnaires can be done in different ways.

We support below the notion of assessment or evalu-

ation of questionnaires. This approach aims to bring

out more information about the overall questionnaires

compared to classical methods of treatment, often giv-

ing partial information. Fuzzy logic is a great tool to

produce quantitative outcomes — the evaluation —

from a given linguistic questionnaire. For instance,

(Lin and Lee, 2009) and (Lin and Lee, 2010) pre-

sented the global evaluation of a linguistic question-

naire, and in the same way, (Berkachy and Donz

2015) and (Berkachy and Donz

e, 2016a) displayed

the individual ones.

The third part of a fuzzy process is the defuzzi-

ﬁcation, i.e. the way of producing quantitative crisp

outcomes from the given fuzzy sets, (Yager, 1996).

Among several known defuzziﬁcation methods seen

in (Runkler, 1997), one has especially retained our

attention: the signed distance method defended ini-

tially by (Yao and Wu, 2000). We have already ex-

plored some of the nice features of the signed dis-

tance method in (Berkachy and Donz

e, 2015) and

(Berkachy and Donz

e, 2016a). In addition, (Berkachy

and Donz

e, 2016b) presented a comparison between

the signed distance and other defuzziﬁcation methods

in order to highlight some statistical characteristics of

distributions obtained by applying this method.

We intend in the following to present the measures

for the individual and global evaluations of question-

naires. The aim is to adapt the method described

in (Berkachy and Donz

e, 2015) and (Berkachy and

Donz

e, 2016a) by introducing the sampling weights

in the calculations. In addition, we emphasize the

relations between these two assessments’ methods.

The empirical part of the study, based on data col-

lected during a regional survey concerning the ﬁnan-

cial place of Zurich (Switzerland), shows how one can

implement these measures. Furthermore, the estima-

tion gives us the possibility to analyse the properties

of the distributions produced by the defuzziﬁcation

process, and in particular if the resulting distributions

Berkachy, R. and Donzé, L.

Individual and Global Assessments with Signed Distance Defuzziﬁcation, and Characteristics of the Output Distributions based on an Empirical Analysis.

DOI: 10.5220/0006036500750082

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 2: FCTA, pages 75-82

ISBN: 978-989-758-201-1

are normal.

The section 2 is devoted to a short presentation

of the signed distance measure, the formalisation of

a linguistic questionnaire in fuzzy sense, as well as

the global and individual evaluations. In Section 3,

we describe the survey “Finanzplatz Zurich: Umfrage

2010”, the database and the set up of the evaluations.

In addition, this section summarises some analysis on

an individual assessment. We ﬁnally give some ar-

guments about the use of evaluations through fuzzy

logic and classical methods of data analysis.

2 GLOBAL AND INDIVIDUAL

EVALUATIONS AND THE

SIGNED DISTANCE MEASURE

2.1 Deﬁnitions

Let us start by deﬁning the signed distance measure

for a fuzzy set as defended by (Yao and Wu, 2000)

and (Lin and Lee, 2010).

We consider a family F of fuzzy numbers on

R = (−∞,∞). Let

D ∈ F be a fuzzy set on R,

such as

D = {(x,µ

(x))|x ∈ R} where µ

(x) is the

membership function of x in

D which maps R to the

closed interval [0,1].

We denote by D

(α) and D

(α) the left and right α-

cuts of the fuzzy set

D, where 0 ≤ α ≤ 1. We assume

that D

(α) and D

(α) exist and are integrable for α ∈

[0,1]. Moreover, the fuzzy set

D can be represented

using its α-cuts as

D =

[

0≤α≤1

(α),D

(α);α].

We are now able to deﬁne the signed distance of the

fuzzy set

Deﬁnition 2.1. The signed distance of

D measured

from the fuzzy origin

0 is:

0) =

(α) + D

(α)]dα. (1)

In the case of a triangular fuzzy number

D =

(p, q,r), we can easily ﬁnd the corresponding signed

distance measure. For the left and right α-cuts

(α) and D

(α), where D

(α) = p + (q − p)α and

(α) = r − (r − q)α, the signed distance of

D mea-

sured from

0 is given by

0) =

(p + 2q + r). (2)

We will apply this particular signed distance later

in our calculations.

2.2 Notation and Conditions

Inspired by (Lin and Lee, 2010), we decompose a

linguistic questionnaire into r weighted main-items

, j = 1,... ,r, with weight b

, such as 0 ≤ b

≤ 1

and

∑

j=1

= 1, and the corresponding m

weighted

sub-items B

, k = 1,..., m

, with weight b

such as

0 ≤ b

≤ 1 and

∑

k=1

= 1. Each sub-item counts m

linguistic terms with the corresponding series of fuzzy

numbers

,. .. ,

, where q = 1, 2,...,m,

as seen in Table 1.

In addition, we required some other conditions:

• The fuzzy numbers are linearly ordered as well

as their signed distance measure i.e. d(

0) <

0) < . .. < d(

0),

• only one answer is possible by sub-item,

• missing values are not allowed,

• the answers are weighted,

• each unit sample i has a sampling weight α

, i =

1,. .. ,N.

We denote by δ

jkqi

an indicator of an answer at a

linguistic term L

jkqi











1 if the observation i has an answer

for the linguistic L

;

0 otherwise.

(3)

Let α

be the sampling weight. Then n

jkq•

is the total

number of weighted answers at the linguistic term L

of the sub-item B

where

jkq•

∑

i=1

jkqi

. (4)

Furthermore, under the condition that missing values

are not allowed, we have

jk••

∑

i=1

∑

q=1

jkqi

∑

i=1

, (5)

where j = 1,...,r and k = 1,..., m

2.3 Evaluation Measures

In their paper, (Lin and Lee, 2010) showed a so-called

global assessment in linguistic questionnaires, while

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

Table 1: Decomposition of a weighted main-item B

into m

sub-items and the signed distance measures corresponding to

their m fuzzy numbers.

Main-item Weight Sub-items Weights Linguistic terms Fuzzy numbers Signed Distance measures

(Berkachy and Donz

e, 2015) and (Berkachy and

Donz

e, 2016a) showed an individual one. Nev-

ertheless, we present in this part a revised global

evaluation measure taking into account the sampling

weight of each sample unit. We also display the

relation between the evaluations on both levels.

First, let us express the evaluation of a whole ques-

tionnaire in terms of global or aggregative evaluation.

A weighted fuzzy number is

jkq•

jk••

, and the signed

distance d

jkq

of this quantity can be written as

jkq

= d((n

jkq•

jk••

)

0) = (n

jkq•

jk••

)d(

0).

Using (4) and (5), we can derive the aggregative eval-

uation P of the linguistic questionnaire in the follow-

ing way:

P =

∑

j=1

∑

k=1

∑

q=1

jkq

∑

j=1

∑

k=1

∑

q=1

∑

i=1

jkqi

∑

i=1

0). (6)

On an individual level, we deﬁne the individual eval-

uation P

of the questionnaire for an individual i as:

∑

j=1

∑

k=1

∑

q=1

jkqi

0). (7)

Proposition 2.1. The weighted mean of the individ-

ual evaluations P

,i = 1, .. ., N, is equal to the global

evaluation P.

Proof 2.1. The weighted mean of the individual eval-

uations is equal to

∑

i=1

∑

i=1

Then, from (7), we have

∑

i=1

∑

i=1

∑

j=1

∑

k=1

∑

q=1

jkqi

0) =

∑

j=1

∑

k=1

∑

q=1

∑

i=1

jkqi

∑

i=1

0) = P.

As a corollary, if all sample units have the same

weight α

= . .. = α

= α, then the

arithmetic mean of the individual evaluations P

,i =

1,. .. ,N, is equal to the global evaluation P.

3 EMPIRICAL APPLICATION

3.1 Description of the Questionnaire

We apply the individual and global assessments pre-

sented previously to the survey ”Finanzplatz Z

urich:

Umfrage 2010” (BAK, 2010), which was conducted

by the Ofﬁce of Economy of the canton of Zurich

(Switzerland) in 2010. The survey was intended to

understand the situation on the ﬁnancial market of

the canton of Zurich. A sample of actors of the

place (mostly ﬁnancial enterprises, e.g. banks or

insurances) were asked through a written question-

naire about their expectations of demand, income and

Individual and Global Assessments with Signed Distance Defuzziﬁcation, and Characteristics of the Output Distributions based on an

Empirical Analysis

employment for a foreseeable future. The question-

naire relative to this survey consisted principally of

21 questions divided into 4 groups. We note that 19

of these questions are linguistic ones with ﬁve possi-

ble answers going from 1 (bad) to 5 (good) while only

2 are quantitative.

3.2 Database and Set up of the

Evaluations

Our database is composed of 234 observations, with

among others company size and branch variables,

and the sampling weight. Considering our aim to

transform linguistics into a more precise form, the

19 ﬁve-level Likert questions have been treated only.

Finally, we have excluded the missing values and

treated our database as a complete cases one. The

consequence is that we will refer to 9 questions ex-

clusively. The questionnaire was decomposed into 3

equally weighted main-items where 2 of them have 2

sub-items and the third 5 sub-items.

The individual and global evaluations in a linguis-

tic questionnaire with the signed distance are imple-

mented in R (R Core Team, 2015). Assuming tri-

angular isosceles membership functions, the signed

distance of the corresponding fuzzy number

q−1

q+1

) is given by

0) =

q−1

+ 2t

q+1

and it follows that the expression of the individual as-

sessment (7) is

( j)

∑

k=1

∑

q=1

jkqi

q−1

+ 2t

q+1

). (8)

We consider the ﬁve linguistic terms L

,. .. ,L

with the triangular fuzzy numbers

,. .. ,

where

= ((q −1)(

m+1

,(q + 1)(

m+1

)), q = 1,..., 5.

One can easily compute the individual evaluation of a

main-item. In this case, we ﬁx t

m+1

to 30, which gives

the following fuzzy numbers

= (0,5, 10)

= (5,10, 15)

= (10,15, 20)

= (15,20, 25)

= (20,25, 30).

Table 2 gives an example of the individual evalua-

tion of an observation i from the treated database. We

note that in our case, we can easily prove that chang-

ing the value of t

m+1

do not affect the interpretation of

the assessments. In fact, it is nothing but a translation

to the new fuzzy numbers system.

3.3 Some Analysis on an Individual

Level

3.3.1 Company Size

Our ﬁrst analysis is made by company sizes: Small,

Medium and Big companies. Some statistical mea-

sures appear in Table 3. Examining global assess-

ments, we see that the evaluation given by small

companies (18.09547) is higher than respectively

the evaluations given by medium (17.58005) and

big (16.89257) companies. This result indicates

that small companies expect better business than the

medium and the big ones. The boxplot Figure 1(a)

displays this result as well.

3.3.2 Company Branch

A second analysis can be made by company branches.

Four branches were considered: Banks, Specialised

ﬁnancial services companies, Investment companies,

Insurance companies. The global assessment of the

Specialized ﬁnance services companies is the high-

est (18.27666) compared to the banks (17.43678),

Investment companies (17.38439) and Insurances

(16.98436), as seen in Table 4. Figure 1(b) shows

boxplots of the 4 types of companies where the Spe-

cialized ﬁnance services companies seem to have a

more positive vision to the future than other compa-

nies.

3.4 Normality of the Crisp Output

Distribution

(Berkachy and Donz

e, 2016b) emphasized different

statistical characteristics such as location, dispersion

and asymmetry measures of distributions obtained us-

ing the signed distance method compared to different

defuzziﬁcation methods. In the same way, since many

statistical tests rely on the normality of a sample, it

is advantageous to know whether the underlying dis-

tribution is symmetric or a fortiori tends to be nor-

mal. This can be done graphically via histograms and

QQ-plots or by calculating the skewness and excess

of kurtosis, and last but not least by using different

statistical normality tests.

The histogram and QQ-plot corresponding to the

output distribution (considering the whole dataset),

seen in Figure 2, give us a hint about the normality

hypothesis. We calculate as well the excess of kurto-

sis of this distribution and we get 0.14731 and a skew-

ness of -0.09489 signalizing a symmetric distribution.

At last, the normality tests used (Chi-square, Shapiro-

Wilk, Anderson-Darling, Kolmogorov-Smirnov and

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

●

●●

●

Big Small Medium

10 15 20 25

Boxplot of the output distributions obtained for different company sizes

Crisp output

(a) Classiﬁcation by company size

●

Banks Invest. comp. Specialised financial services comp. Insurances

10 15 20 25

Boxplot of the output distributions obtained for different company branchs

Crisp output

(b) Classiﬁcation by company branch

Figure 1: The output distributions classiﬁed by: company size (a) and company branch (b).

Histogram of output crisp distribution

Crisp output

Density

10 15 20 25

0.00 0.05 0.10 0.15

(a) Histogram of crisp distribution

●

● ●

● ● ● ●

●●●●●

●

●●

●●●

●●

●

●●

●

●●●●●●

●●●

●

●●●●●

●

●●●●●●●●●

●

●●●●●●●●●

●

●●●●●

●●●●●●●●●●●●●●

●●●●●●●

●●●●●

●●●●●●●

●●●●●●

●●●●●

●●●

●●

●●●●●

●●●●●●●●●

●●●●●

●●●

●●●●●●●

●●●●

●●

●●●●

●●●●●●

●

●●●

●●●●●●●●●●

●

●●●●●

●●●

●●●●●

●●

●

●●●

●●

● ●

● ● ●

●

−3 −2 −1 0 1 2 3

10 15 20 25

Normal Q−Q Plot

Theoretical Quantiles

Sample Quantiles

(b) QQ-plot of crisp distribution

Figure 2: The histogram (a) and qqplot (b) of the output distributions.

Lilliefors) couldn’t reject the hypothesis H

of the

normality of the distribution.

3.5 Analyses on Original Data vs.

Evaluations through Fuzzy Logic

Statistical inference is described as the most substan-

tial way of thinking regarding data analysis. Despite

the fact that evaluations noted previously are quan-

titative and therefore should present less imprecision

than the linguistic terms, some analyses can be done

on the original data. Nevertheless, a fuzzy approach

could be advantageous in several situation.

First of all, it has to be mentioned that evaluations

through fuzzy logic offer the possibility to charac-

terise units by their individual assessments and thus

give another way for interpretations. We have done

this kind of analyses for instance in section 3.3. Fur-

thermore, we note that the resulting distribution of the

outputs, as seen in section 3.4, tends to be normal,

which is a nice property for statistical inferences.

Secondly, on a global level, we show that the mean

of the means works as well as our aggregative evalua-

tion. It is a useful property, especially when it comes

to illustrate broadly information about the question-

naire. Last but not least, and it is a recurrent draw-

back when missing values occur, the mean of means

applied on linguistic data cannot be “representative”,

especially if missingness is not ramdom and the esti-

mates are biased. We present our measures under the

assumption of no missing values. Indeed, it is not dif-

ﬁcult to show that the global evaluation measure can

be adapted in the case of missing values appearing in

a dataset.

4 CONCLUSION

Due to the particularities of the variables — qualita-

tive answers, the traditional treatment methods of lin-

guistic questionnaires could encounter many difﬁcul-

ties. For instance the non-normality of the distribution

could gravely impair the statistical inferences. Or, the

results appear difﬁcult to interpret. An approach with

the fuzzy logic, and individual and global assessments

could be an alternative way.

This study intends to present the individual and

global assessments in linguistic questionnaires, an ef-

ﬁcient method to manage uncertainty in this kind of

questionnaires. Based on the signed distance defuzzi-

Individual and Global Assessments with Signed Distance Defuzziﬁcation, and Characteristics of the Output Distributions based on an

Empirical Analysis

ﬁcation method, we estimated easily these evaluations

in the fuzzy sense, considering furthermore the sam-

pling weights in real databases. The signed distance

calculations are made with triangular isosceles fuzzy

numbers.

Moreover, in order to help using our method in

empirical research, we applied these evaluation meth-

ods on a real dataset coming from a survey of the

ﬁnancial place of Zurich (Switzerland). This empir-

ical part should give conﬁdence that the individual

and global assessments are easily implementable, and

could provide interesting results.

From another side, the non-normality of distribu-

tions is one of the most appealing topics in statisti-

cal inference. Yet, we examined the normality of the

output distributions and we found that the individual

evaluations with the signed distance method tend to

be normally distributed. This result can be proﬁt-

making in terms of adequacy of data in further sta-

tistical analysis. Applying individual evaluation will

enable us to assess easily topics in linguistic question-

naires per observation through fuzzy logic.

Finally, future work certainly encompasses the

understanding of the effect of normality and sym-

metry of such distributions in different statistical

modellings, investigating in particular databases with

missing values.

ACKNOWLEDGEMENT

The authors would like to thank the Ofﬁce of Econ-

omy of the canton of Zurich (Switzerland) which pro-

vided the permission to use the database.

REFERENCES

BAK (November 2010). Finanzplatz z

urich 2010, monitor-

ing, impact-analyse und ausblick. Technical report,

Amt fur Wirtschaft und Arbeit (AWA) des Kantons

urich.

Berkachy, R. and Donz

e, L. (2015). Linguistic question-

naire evaluation: global and individual assessment

with the signed distance defuzziﬁcation method. In

Advances in Computational Intelligence, Proceedings

of the 16th International Conference on Fuzzy Systems

FS’15, Rome, Italy, volume 34, pages 13–20.

Berkachy, R. and Donz

e, L. (2016a). Linguistic question-

naire evaluation: an application of the signed distance

defuzziﬁcation method on different fuzzy numbers.

the impact on the skewness of the output distributions.

International Journal of Fuzzy Systems and Advanced

Applications, 3:12 – 19.

Berkachy, R. and Donz

e, L. (2016b). Statistical characteris-

tics of distributions obtained using the signed distance

defuzziﬁcation method compared to other methods.

In Proceedings of the International Conference on

Fuzzy Management Methods ICFMSquare, Fribourg,

Switzerland, pages 48–58.

Lin, L. and Lee, H. (2009). Fuzzy assessment method on

sampling survey analysis. Expert Systems with Appli-

cations, 36(3):59555961.

Lin, L. and Lee, H. (2010). Fuzzy assessment for sampling

survey defuzziﬁcation by signed distance method. Ex-

pert Systems with Applications, 37(12):78527857.

R Core Team (2015). R: A Language and Environment for

Statistical Computing. R Foundation for Statistical

Computing, Vienna, Austria.

Runkler, T. A. (1997). Selection of appropriate defuzziﬁ-

cation methods using application speciﬁc properties.

IEEE Transactions on Fuzzy Systems.

Wagner, C., Miller, S., and Garibaldi, J. M. (2011). A fuzzy

toolbox for the r programming language. In IEEE In-

ternational Conference on Fuzzy Systems, Taipei, Tai-

wan, pages 1185–1192. IEEE.

Yager, R. R. (1996). Knowledge-based defuzziﬁcation.

Fuzzy Sets and Systems, 80(2):177–185.

Yao, J. and Wu, K. (2000). Ranking fuzzy numbers

based on decomposition principle and signed distance.

Fuzzy sets and Systems, 116(2):275–288.

Zadeh, L. (1996). Fuzzy Sets, Fuzzy Logic, and Fuzzy Sys-

tems: Selected Papers of Lotﬁ A. Zadeh, G. J. Klir and

B. Yuan, Eds., Advances in Fuzzy Systems – Applica-

tions and Theory, volume 6. World Scientiﬁc.

Zimmermann, H. (2010). Fuzzy set theory. Wiley Interdisci-

plinary Reviews: Computational Statistics, 2(3):317–

332.

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

APPENDIX

Table 2: Application of the individual evaluation on Finanzplatz Zurich Umfrage 2010: Answers of an observation i for the 3

main-items equally weighted using the signed distance of triangular fuzzy numbers.

Main-items Weights Sub-items Weights Linguistics Answers Signed Individual

jkqi

Distance Evaluation

The present state of 0.5 bad 0 20 14.333

business is L

fair 0

excellent 0

The expected state of 0.5 worst 0 15

business in 12 months is L

same 1

better 0

The demand for our services / products 0.5 smaller 0 10

compared to the last 12 months is L

same 0

bigger 0

The expected demand for our services / 0.5 worst 0 15

products in the next 12 months is L

same 1

better 0

The gross proﬁt compared 0.2 smaller 0 10

to the last 12 months is L

same 0

bigger 0

The employment compared 0.2 smaller 0 15

to the last 12 months is L

same 1

bigger 0

The expected gross proﬁt 0.2 worst 0 10

in the next 12 months is L

same 0

better 0

The expected employment in 0.2 worst 0 15

the next 12 months is L

same 1

better 0

The expected use of technical and 0.2 smaller 0 15

personal capacities in the next L

12 months is same 1

bigger 0

Individual and Global Assessments with Signed Distance Defuzziﬁcation, and Characteristics of the Output Distributions based on an

Empirical Analysis

Table 3: Statistical measures of the distribution obtained from individual evaluations: Classiﬁcation by company size.

Company size

Statistical measure Small Medium Big

Mean 18.09547 17.58005 16.89257

Median 17.83333 17.66667 16.66667

Variance 9.65144 7.21249 5.55479

Standard deviation 3.10667 2.68561 2.35686

Skewness -0.21248 -0.05683 0.32202

Excess of Kurtosis 0.14485 0.45263 -0.72025

Quantiles:

P = 0 10.33333 10 12.66667

P = 0.25 16.5 16 15

P = 0.5 17.83333 17.66667 16.66667

P = 0.75 20 19.33333 18.66667

P = 1 24.33333 25 21.66667

Table 4: Statistical measures of the distribution obtained from individual evaluations: Classiﬁcation by company branch.

Company branch

Statistical Banks Specialised ﬁnancial Investment Insurance

measure services companies companies companies

Mean 17.43678 18.27666 17.38439 16.98436

Median 17.66667 17.83333 16.83333 16.83333

Variance 9.17247 8.20156 10.11543 7.02378

Standard deviation 3.02861 2.86384 3.18048 2.65024

Skewness -0.43497 -0.03864 0.13030 0.31873

Excess of Kurtosis -0.21930 0.38951 -0.33553 1.65608

Quantiles:

P = 0 10.33333 10.66667 10.83333 10

P = 0.25 15.83333 16.66667 15.83333 15.16667

P = 0.5 17.66667 17.83333 16.83333 16.83333

P = 0.75 19.5 20 19 19

P = 1 24.33333 24.33333 23.83333 25

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications