Categorizing Quantities using an Interactive

Fuzzy Membership Function

Liqun Liu and Romain Vuillemot

Ecole Centrale de Lyon, Univ. Lyon, LIRIS CNRS, France

Keywords:

Membership Function, Fuzzy Logic, Categorization.

Abstract:

In this paper, we investigate how an interactive version of the membership function from the Fuzzy Logic Theory

can be used to categorize quantitative data. This function is simple and similar to a line chart and provides

an explicit mapping of the categorization process. We ﬁrst review the requirements for such quantitative

values partitioning process and provide the fuzzy logic mathematical foundations related to the membership

function. We then report on the implementation of the interactive function for several quantitative datasets case

studies (e. g., age, temperature, speed). We expect this interactive function to provide more control over the

categorization process, as well as a way to make the categorization more explicit.

1 INTRODUCTION

When reasoning on quantitative values—like people’s

age—analysts tend to use categories like

YOUNG

and

OLD

. Such a categorization process aims at mimicking

the logic of human thoughts and reasoning (Clifford

et al., 1975). In the situation where such categories

are not in the dataset, it is likely they are subjective,

from the analyst’s head, based on familiarity with the

domain or prior knowledge. Such a process raises the

following issues:

•

An explicit mapping for those categories is miss-

ing: the mapping function between quantities and

categories should be clearly deﬁned.

•

The mapping properties may vary across analy-

sis sessions and analysts: the mapping should be

consistent and may not change.

•

The transfer between analysts and a user should

be possible: by some means of communication

like a legend or a visual representation.

Besides, there are also some alternative automatic

categorization methods and clustering. Those meth-

ods categorize quantities based on the distribution of

quantitative values, e. g., KNN (Cover and Hart, 1967),

K-Means (MacQueen et al., 1967), DBSCAN (Ester

et al., ). However, those approaches do not provide

enough control over the mapping process to let humans

customize and capture uncertainty.

In this paper, our goal is to address those issues

with an interactive visualization that captures such

knowledge. It relies upon Fuzzy Logic Theory, created

back in the 1960s to model domains with imprecise

information (Zadeh, 1965; Pedrycz and Gomide, 2007)

which we argue provides the theoretical framework to

address the above issues. In particular, we rely upon

a visual representation from this theory called mem-

bership function which is a line chart of the mapping

function between a quantitative scale and categories,

which makes this mapping explicit, which captures

various points of view using a conﬁdence factor and

ﬁnally which visual can easily be transferred.

Our main contribution is that we designed an in-

teractive version of fuzzy membership function as an

interactive chart (Fig. 1), which is able to help users

categorize quantitative values with conﬁdence belong-

ing to categories. We provide an implementation that

demonstrates how it supports the categorization of

multiple datasets. Our implementation exposes many

parameters of the function that can be adjusted to gen-

erate new shapes of membership function for ﬁne-

grained categorization. The generated categories from

this membership function can be assigned names that

are then used as category names in the updated dataset.

We conclude with an experimental design for further

evaluation of this tool to validate its usability. We ex-

pect this technique to be not only used as a categories-

generation tool, but also as an interactive legend (Riche

et al., 2010) to communicate the generated categories

properties and let the user adjust them dynamically

when analyzing a visual representation.

Liu, L. and Vuillemot, R.

Categorizing Quantities using an Interactive Fuzzy Membership Function.

DOI: 10.5220/0010270801950202

In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 3: IVAPP, pages

195-202

ISBN: 978-989-758-488-6

195

2 RELATED WORK

This paper focuses on how to categorize quantities

with fuzzy logic theory and quantify those generated

categories with membership degree. In this section,

we review papers in probabilistic classiﬁcation, fuzzy

visualization, and uncertainty visualization.

2.1 Visualization of Probabilistic

Classiﬁcation

(Alsallakh et al., 2014) proposed a visual interactive

analysis tool to evaluate the effectiveness of classiﬁers

to help machine learning experts ﬁnd out the possible

reasons for incorrect classiﬁcation. This tool is able to

emphasize the classiﬁcation probabilities of items and

make relations with a false negative and false positive.

Similar to Alsallakh’s study, Cao and Lin proposed

UnTangle Map (Cao et al., 2015; Lin et al., 2014)

by a web of the connected triangles, which is able

to make the efﬁcient relation between data items and

their probabilistic labels. Based on RadViz, Seifert pro-

posed a visual system to help understand the process

of classiﬁcation and results, which handles multiple

classes, nominal and numeral data format (Seifert and

Lex, 2009). Zhao conducted an experiment to eval-

uate the effectiveness and efﬁciency of fuzzy cluster

analysis, objective questionnaires are designed to com-

pare the accuracy and subjective questionnaires are

designed to collect the experience of a user based on

using four-dimensional visualization technique (Zhao

et al., 2018). (Rheingans and Desjardins, 2000) is de-

signed to visualize high-dimensional predictive results

with richer representation, e.g., classiﬁcation accuracy

and confusion matrices to help a user understand high-

dimensional data space.

2.2 Fuzzy Visualization

There exist alternatives to the membership function

to capture categorization. Fuzzy representations and

uncertainty visualization are suitable solutions like the

Disk diagrams (Park and Park, 2010a; Park and Park,

2010b) proposed by Yeseul Park, to visualize fuzzy

set operations. It can describe the complexity of fuzzy

sets by showing the degree among sets with the layout

of star coordinates, etc. (Zhu et al., 2018) extended

the circular disk diagram layouts to improve sets mem-

bership analysis, by using color opacity and optimized

layout to convey fuzzy sets membership and reveal the

uncertain owner-member relationship. Besides, they

also designed a computational framework by combin-

ing physical simulation and geometric interpolation.

However, those works focus on communicating cate-

gories rather than supporting their creation.

2.3 Uncertainty Visualization

Research on uncertainty visualization also offers some

solutions for the presentation of ambiguity while cate-

gorization. In this ﬁeld, Works related to uncertainty

visualization provide guidance on advanced properties

in the data, i. e. incorrectness, incompleteness, and

ambiguity (Dressel and Nori, 2014). Brodlie (Brodlie

et al., 2012) reviewed state of the art in uncertainty

visualization according to different data, for example,

point data, scalar data, multi-ﬁeld scalar data, and vec-

tor data. At the same time, they also concluded the

uncertainty visualization in the various dimensional

datasets. Skeels proposed a classiﬁcation of uncer-

tainty (Skeels et al., 2010) for information visualiza-

tion. (Dong and Hayes, 2012) designed an interactive

tool to help users recognize the situations and com-

prehend the ambiguity, which is able to evaluate the

domain-dependent decision support system (DSS).

3 DEFINING CRISP AND FUZZY

MEMBERSHIP FUNCTIONS

Table 1: Dataset with quantities and categories.

Name Age Group Profession

Lisa 8 Children Student

Barney 39 Adult Engineer

Smithers 33 Adolescent Professor

Mr. Burns 90 Senior Retired

The challenge we address is the explicit mapping be-

tween quantities and categories. While most visualiza-

tion techniques and tools usually address it during the

visual encoding of data, it remains internal—or at best

using a legend—without providing a fully explicit set

of properties for this mapping.

To introduce our approach, we progressively intro-

duce the deﬁnitions by ﬁrst stating our challenge as

ﬁnding the relationship between Q (Quantitative) and

C (Categories) as a mapping function:

Quantity → Category (1)

Quantities are the measures of counts or values,

which is able to be expressed by numbers and also can

be compared in terms of ”more”, ”less”, or ”equal”,

like attribute Age showed in Table. 1. On the contrary,

categories are measures of type and can be expressed

by a symbol, name, or label, which is categories, like

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Figure 1: Manipulation of the interactive membership function. By adjusting the parameter core, the membership function

is changed along with the new value of core, which is shown as the red dot line; Another interaction is deﬁning the name of

categories by inputting action and it is shown as the green dot line.

attribute Age group in Table. 1. Mapping function con-

nect quantities (

) and categories (

), e. g., the quan-

tity ”age” is separated into ordinal range, showed as

[0,:] →< YOUNG,ADULT,OLD >

. Thus the trans-

formation from quantities to categories is deﬁned as

Eq. (2):

x →







Y OUNG i f age(x) ≤ 20

ADULT i f 20 < age(x) ≤ 60

OLD i f 60 < age(x).

(2)

We implemented a prototype

of the mapping func-

tion above from quantitative scales to a domain of

user-deﬁned categories. Fig. 2 illustrates the user in-

terface that represents the mapping of each value to

categories. A user can change the value parameter

adjusting the slider, which is able to change the num-

ber of bars of the mapping function. A user can also

drag the vertical bars using circles so he can adjust the

categories. A table below indicates the categorization

result for each category.

This mapping relates to the classical sets theory,

where the categorization is a crisp process that splits

quantities into categories with a binary function: ac-

cepting or rejecting the object belonging to a cate-

gory (Massad et al., 2009). An element

either be-

longs to a category or not. Those are strict perimeters

and each element only belongs to a single category.

Such a mapping function can be combined using in-

tervals that often converts a whole range of values. If

there is a set

that is not empty and a set

S ⊂ W

, the

characteristic function of S will be shown as follows:

(x) =



1 i f x ∈ S

0 i f x /∈ S

(3)

https://observablehq.com/d/4560d69baca4663c

Figure 2: Categorization using crisp membership function,

there are three crisp categories generated from the crisp mem-

bership function showed in (a) and the detail information of

categories in (b).

(x)

is the function, the domain of which is

. The

value of

(x)

is included in set

{0,1}

. With

(x)

= 1, it means element

belongs to set

; if

(x)

0, it illustrates element

does not belong to set

that this mapping function

(x) → {0,1}

are able to

completely represent the relationship between element

x and set S.

However, previous mapping function has a limi-

tation on splitting ambiguous sets. For example, it

can not represent the children, adolescent and adult,

because adolescent normally has overlap with children

and adult. Thus, it is not any more suitable for sepa-

rating quantitative values when there is ambiguity or

there are more than one quantitative scale. In order to

solve the problem that does not have sharp boundaries

while categorizing, Zadeh proposed the membership

degree that means an element can partially belong to a

set (Zadeh et al., 1996).

In order to implement the fuzzy membership func-

Categorizing Quantities using an Interactive Fuzzy Membership Function

197

tion into interactive categorization, we implemented

the second prototype

to map quantities to categories

based on fuzzy logic. Fig. 3(a) shows the interface

for a user to adjust parameters. In this ﬁgure, x-axis

represents quantitative values and y-axis shows the dis-

tribution of membership degree(

). Compared to the

ﬁrst prototype, this one is able to be adjusted among

fuzzy logic parameters core and n by manipulating

the sliders. Besides, a user also can adjust core by

dragging small black circles on the fuzzy membership

function and make them different from other fuzzy

categories. Finally, he can also deﬁne the name of

generated categories by inputting texts so that a user

can generate any fuzzy categories that they want with

membership values and speciﬁc names. The member-

ship values and names are illustrated in Fig. 2(b).

Fuzzy category is deﬁned with relaxing belonging

constraints, which proposes an intermediate member-

ship value as interval

[0,1]

. This value represent the

degree how many possibility element x belongs to set

. Thus, the membership function can be described as

(x) → [0,1]

. The value of

(x)

means the member-

ship degree of membership function. With

(x) = 1

, it

represents the complete belongingness while

(x) = 0

shows the complete non-belongingness. Let

X =

,..}

denote the set

of elements

. A fuzzy

category is then a pair

(X,µ)

. So fuzzy category is

denoted as

{µ(x

)/x

,...,µ(x

)/x

}

. The core and

support respectively represent all elements that belong

to a set are denoted

Core(A) = {x ∈ X|A(x) = 1}

and

all the elements which membership degree is

> 0

such

as Supp(A) = {x ∈ X |A(x) > 0}.

Any function

(x) → [0,1]

is membership func-

tion framework, instead of accurate representation.

The real membership function depends on its shapes.

Those membership functions with speciﬁc shapes are

composed of speciﬁc properties and they are chosen

based on different data and applied ﬁelds, e. g., trian-

gular, trapezoidal, Gaussian and sigmoidal functions.

In this paper, we chose the trapezoidal membership

function and it is given by:

f (x) =











0 i f x < a

(x − a)/(b − a) i f a ≤ x < b

1 i f b ≤ x < c

(d − x)/(d − c) i f c ≤ x < c

0 i f x > c

(4)

Where,

a,b,c,d

are the parameters of trapezoidal

membership function. The membership functions can

be displayed over a line chart in which x-axis is the

quantitative value and the y-axis is the membership

degrees. Each line is a category, and their membership

degrees are from 0 or 1.

https://observablehq.com/d/b887da8c1dfdb975

Figure 3: The mapping function based on fuzzy logic theory

generates both crisp categories and fuzzy categories in (a)

and all those categories and its information are in (b).

4 INTERACTIVE MEMBERSHIP

FUNCTION

4.1 Interaction

We introduce how to use the interactive membership

function in this section. There are two general param-

eters of membership function:

and

core

are shown

as the sliders in Fig. 1. In this visual tool, a user can

adjust the two parameters to generate different shapes

of membership functions. By adjusting parameters in

the control panel, all categories generated would be

changed with the same value of core.

If a user would like to operate it more speciﬁcally,

he can build a new shape of membership function by

dragging those little circles showed in Fig. 4(a), for

example, A user can change the parameter

core

of spe-

ciﬁc categories by dragging black circle showed in the

left of Fig. 4(a). And then, the membership function

is able to be changed as the right of Fig. 4(a), the ﬁrst

category in this membership function is changed. Fi-

nally, this membership function also offers the naming

method. By inputting texts in the input ﬁelds shown in

Fig. 4(b), the categories are able to be related to input

labels and appear as new categories in the underlying

dataset.

4.2 Implementation Details

This membership function is implemented in Observ-

able, using library D3 (Bostock et al., ) and is bundled

as a JavaScript ES6 module compliant with modern

Web standards. An online prototype

is available

along with the datasets described in this paper. Addi-

https://observablehq.com/d/12575cfbd09fe8e8

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Table 2: Summary of case studies illustrating interesting conﬁgurations constructed in the three case studies datasets.

Data FUZZYCUT # of sets Name Interval

Age 5 Children, Teenager, Adult, Less old, Old [0.99, 90.90]

Temperature 3 Cold, Warm, Hot [-12.98, 68.58]

Speed 5 Low, Low-Middle, Middle, Mid-High, High [0,104.04]

Figure 4: The top circles on membership function are able to

be dragged to change the parameter core of a speciﬁc cate-

gory in (a); The labels of categories are deﬁned by inputting

texts in input ﬁeld in (b).

tional datasets can be added using a JSON ﬁle spec-

iﬁcation that captures the conﬁgurations that a user-

deﬁned. The parameters of conﬁgurations include the

parameters of speciﬁc membership function and cate-

gories’ names are showed as follows:

{

"title": "Temperature",

"attribute": "0",

"parameters": [{

"n": "3",

"Core": "13.6",

"Support": "27.2",

"names": [

"Cold",

"Warm",

"Hot"

]}]}

Such conﬁguration enables the rapid export and

sharing of the parameters, so they can be used in a

different tool and can further be analyzed to understand

the categorization process of multiple users.

5 CASE STUDIES

In this section, we introduce three typical quantitative

datasets and the categorization we generated using

our interactive visual tool. The datasets we used in-

clude age datasets, temperature datasets, and taxi speed

datasets.

5.1 Case Study with Age Dataset

A typical age dataset consists of a semi-bounded in-

terval and rounded values, there exists a minimum

value but no ﬁxed maximum value. The dataset we

used, in this case, is The Simpsons, a popular U.S.

TV Series. The dataset includes 24 characters with an

age attribute. Age range from 0 to 90. The dataset

is shown in Fig. 5 where ages are represented on the

x-axis. The y-axis shows the membership degree for

every category so that a user can easily check cate-

gories and their membership degrees. The table list

below shows the categories and their properties. Mem-

bership values and the range of categories are shown

in the table shown in Fig. 5(b), which is corresponding

to the membership function and has 5 categories. We

can see that the membership degrees of category 1 and

category 3 are equal to 0.5, and other categories are

equal to 1. The name of every category is shown on the

table where each character now has an age category.

5.2 Case Study with Temperature

Dataset

We use a temperature dataset of the United State in this

second case study. As such temperature value does not

have a clear maximum value and minimum value, the

data is non-bounded and continuous. The dataset we

used consists of temperatures from 357 divisions in

the USA for 12 months. The categorization is shown

in Fig. 6. Fig. 6(a) represents the fuzzy membership

function and Fig. 6(b) is the table list of categories.

In Fig. 6(a), temperature degrees are distributed along

with the x-axis and the y-axis are membership values

for each category. In this prototype, temperature values

Categorizing Quantities using an Interactive Fuzzy Membership Function

199

Figure 5: The age data are from 0 to 90, there are three

main categories with name Chidren, Adult and Old. Mean-

while, two fuzzy categories created and they are named with

Teenager and Less old.

are separated into three categories, which are from -

12.98 to 68.56. The name of those three categories

are Cold, Warm and Hot. But the Warm category is

derived from Cold and Hot, membership degree of

which is 0.5.

Figure 6: Temperature data are separated with two main

categories Cold and Hot. The intermediate category Warm

is created.

5.3 Case Study with Taxi Speed Dataset

We conduct the third case using a taxi speed dataset.

The speed data are semi-rounded and continuous

values types. Trafﬁc analysts usually are interested

in characterizing taxi drivers’ characteristics, which

is helpful for them to identify driving behaviors. Ob-

viously, speed is a very essential parameter to deﬁne

categories, as it reﬂects many characteristics of drivers.

The prototype of taxi speed is showed in Fig. 7(a)

and the table list is showed in Fig. 7(b). In this case,

speed is separated into ﬁve categories and their main

categories are Low, Middle and High. Besides, there

are two fuzzy categories with membership degrees less

than 1, named Low-Middle and Mid-High. Other cate-

gories, more domain-speciﬁc, could have been used,

e.g. Slow, Fast by editing the label input ﬁeld

in the prototype.

6 DISCUSSION AND

PERSPECTIVES

In this paper, we designed an interactive tool to catego-

rize quantities into categories. By using Fuzzy Logic,

it enables splitting quantitative values into categories

with a degree called membership degrees. We are able

to present uncertainties with membership degrees to

better capture user knowledge and demonstrated its ap-

plication using three datasets included in an interactive

prototype.

The summary of case studies is shown in Table 2.

The results suggest that, in most situations, users are

likely to categorize quantities into 3 - 5 categories

with particular names for each category. First, naming

generated categories is usually based on social basic

knowledge, e. g., the name of age categories, which

includes Children, Adult and Old. But when the cat-

egories are complicated, the name of categories tend

to be Very ... or Super ... There is also another naming

method, which combines the basic categories, e. g., the

name Mid-High is derived from Middle and High in

speed data. Our tool supports an extensive number of

categories, up to several dozens. Beyond this number,

it may however become difﬁcult to read the category’s

names and interact with them.

We identiﬁed the following limits in our current

work.

It does not consider the distribution of quan-

titative values.

The trapezoid membership function

can not consider the real distribution of the dataset,

instead of building a membership function based on

the maximum value, minimum value, and the num-

ber of categories. The distribution is not considered

in this method.

Only categorize dataset with clear

Figure 7: Taxi speed are separated with three main categories

and two fuzzy categories. Those two categories are named

Low-Middle and Mid-High.

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maximum value and minimum value.

Currently, we

can only use the maximum value existed in the dataset

or dataset with a clear boundary. For example, in the

age dataset, the maximum value is

, so the maxi-

mum value in this prototype is 90, it is totally based on

the range of the dataset.

Limitation in categorizing

quantities with rounded values

: When we catego-

rize age dataset, the results of categories are not integer,

which leads to the abnormal categories e. g., category

Children is from 0.99 years old to 17.97 years old.

Our main perspective is to validate our approach

using a formal evaluation process to assess how the

fuzzy membership function helps a user categorizing

quantities. The evaluation includes three steps:

manual categories creation; 2) using fuzzy mem-

bership function and 3) post-study questionnaire.

In manual categories creation, investigating How

does a user create categories for datasets such as the

characters in The Simpsons would be a major question

to be answered as in this paper we only reported on

typically used manually created to illustrate our ap-

proach. In this evaluation step, a user will be asked

by questionnaire to focus on the number of categories,

the intervals, and the name of categories, and we will

log the results for remote analysis.

Our study protocol will be as follows (using the

prototype shown in Fig. 3(a)):

Step 1:

we introduce

FUZZYCUT to participants and tell them how to use

and operate FUZZYCUT which will last 10 minutes.

In this part, the user is able to know how to change

parameters of fuzzy membership function and inter-

act with FUZZYCUT to generate and adjust categories.

Step 2:

those parameters, categories, and their inter-

vals are saved remotely by recording every interaction.

Step 3:

all those information collected from a user

are organized and analyzed to support a formalized

function, which is very important for future work to de-

velop speciﬁc mapping function because the collected

information will be the major proof of mapping spe-

ciﬁc quantities. A post-study questionnaire will collect

feedback, the content of which aims to illustrate if this

tool inﬂuences participants’ original intent, and how

those categories change after using this tool.

7 CONCLUSION

This paper introduced an interactive visualization tech-

nique to assist a user in categorizing quantities into

categories. We relied upon a well-known function, the

fuzzy membership function from fuzzy logic theory,

which we implemented as an interactive prototype. We

illustrated its use for 3 case studies: age, temperature,

and taxi speed data. As the prototype enables an ex-

plicit mapping of the categorization function, we plan

to use it to trace the process a user follows when cre-

ating categories. In particular to understand if there

is consensus between groups of users regarding the

choice of categories values interval, labeling, and con-

ﬁdence. We also plan to use the interactive function

to communicate this mapping, e. g., as an interactive

legend (Park and Park, 2010b) for both visual commu-

nication and exploration of datasets.

ACKNOWLEDGMENTS

This work was partially supported by Chinese Schol-

arship Council (CSC). This work was also partially

supported by the M2I project on Urban Mobility

funded by the French Agency for Durable Develop-

ment (ADEME).

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