Multilingual textual data an approach through multiple factor analysis Belchin Kostov

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Multilingual textual data: an approach through multiple factor

analysis

Belchin Kostov

Department of Statistics and Operational Research, Universitat Politècnica de

Catalunya, C/ Jordi Girona 1-3, 08034 Barcelona, Spain

Ramón Alvarez-Esteban1

Department of Economics and Statistics, Universidad de León, Campus de Veg-

azana s/n, 24071 León, Spain

Mónica Bécue-Bertaut

Department of Statistics and Operational Research, Universitat Politècnica de

Catalunya, C/ Jordi Girona 1-3, 08034 Barcelona, Spain

François Husson

Institut Agro, Univ Rennes1, CNRS, IRMAR, 35000, Rennes, France

Abstract This paper focuses on the analysis of open-ended questions answered in dif-

ferent languages. Closed-ended questions, called contextual variables, are asked to all

respondents in order to understand the relationships between the free and the closed re-

sponses among the different samples since the latter assumably affect the word choices.

We have developed "Multiple Factor Analysis on Generalized Aggregated Lexical Tables"

(MFA-GALT) to jointly study the open-ended responses in different languages through

the relationships between the choice of words and the variables that drive this choice.

MFA-GALT studies if variability among words is structured in the same way by variabil-

ity among variables, and inversely, from one sample to another. An application on an

international satisfaction survey shows the easy-to-interpret results that are proposed.

Keywords: Correspondence analysis, Lexical tables, Textual and contextual data, Multi-

ple factor analysis, Generalized aggregated lexical table

1. INTRODUCTION

Socio-economic surveys beneﬁt from introducing open-ended questions in

addition to closed-ended questions because they enrich each other. Closed-ended

questions inform the interpretation of open-ended questions because the meaning

of words is related to the characteristics or opinions of those who speak. For

1Ramón Alvarez-Esteban, ramon.alvarez@unileon.es. ORCID 0000-0002-4751-2797

arXiv:2210.06527v1 [cs.CL] 12 Oct 2022

instance, in a satisfaction survey, customers are asked to rate certain aspects of the

product and then freely give their opinion on which aspects could be improved,

which is clearly linked to the ratings. In a survey that includes the question "What

does health mean to you?" closed-ended questions such as gender, age, education

and health status will greatly assist in exploring how the deﬁnitions of health

vary with these variables. In the case of international surveys, our framework,

these open-ended questions raise the issue of analyzing the responses expressed

in different languages by several samples.

In the case of a single language, textual statistics (Benzécri, 1981; Lebart

et al., 1998) offer multidimensional tools for processing free responses. Sepa-

rately for each sample, the free responses are encoded in the form of a frequency

table respondents ×words, called a lexical table (LT). A standard methodology

is to apply correspondence analysis to this LT (CA-LT; direct analysis) and to use

the closed information as a complement. It is also usual to group the responses of

the categories of a closed question (such as age crossed with gender or education

level, called contextual variable) and to build the frequency table of the words ×

categories, called aggregated lexical table (ALT) which can also be analyzed by

CA (CA-ALT).

This approach is extended to several quantitative or qualitative contextual

variables by using linearly constrained CA methods (Takane et al., 1991). Thus,

Balbi and Giordano (2001) address textual data including external information,

Balbi and Misuraca (2010) propose a double projection strategy by involving ex-

ternal information both on documents and words while Spano and Triunfo (2012)

apply canonical correspondence analysis (CCA; ter Braak (1986, 1987)) to tex-

tual data. In line with these works, Bécue-Bertaut et al. (2014) and Bécue-Bertaut

and Pagès (2015) propose the method called CA on a generalized aggregated lex-

ical table (CA-GALT). First, a table words ×variables, called "Generalized Ag-

gregated Lexical Table" (GALT), is developed by positioning the words on the

variable-columns based on the values taken by the respondents who use them.

Afterwards, this GALT is analyzed by means of a CCA adapted to textual data.

In CA-GALT, like in any CA, both the variability of vocabulary through the vari-

ability of variables, and the variability of variables through the variability of vo-

cabulary are explained. This ﬁts perfectly with the perspective we have chosen to

take here.

In the case of multilingual surveys, we propose to analyze simultaneously

the different GALTs, one for each sample, by means of a multiple factor analy-

sis (MFA; (Escoﬁer and Pagès, 2016; Pagès, 2014)), tailored to process a multi-

ple GALT. This leads to the Multiple Factor Analysis for Generalized Aggregate

Lexical Tables (MFA-GALT). This work outlines how to adapt MFA reasoning in

order to handle a multiple GALT, and details its properties and graphical repre-

sentations.

The aim of MFA-GALT is to jointly study the open-ended responses given

by several samples in different languages through the relationships between the

choice of words and the variables that motivate this choice. These relationships

may or may not have similar structures. In other words, MFA-GALT studies

whether variability among words is structured in the same way by variability

among variables, and inversely, across samples.

The paper is organized as follows: Section 2 presents the data structure and

the notation. Section 3 recalls the principles of CA-GALT and MFA, the methods

that form the basis of our approach; Section 4 is devoted to MFA adapted to mul-

tiple GALTs (MFA-GALT) and Section 5 exposes the properties of the method.

Enventually, the application of MFA-GALT to a full-scale application (Section 6)

shows its capabilities. The main conclusions are presented in Section 7.

2. DATA STRUCTURE AND NOTATION

Lsamples have answered a questionnaire with closed questions, either quan-

titative or categorical, all of identical type, which constitute the contextual data.

They have also answered an open-ended question in different languages conform-

ing the textual data. The lsample has Ilrespondents who all together use Jl

different words in the llanguage. From these responses, we construct the (Il×Jl)

table Yl, respondents ×words; Nlis the grand total for this table.

The responses to the closed-ended questions, common to all samples, are en-

coded in the (Il×K)table Xl, whose columns correspond either to quantitative

variables or to dummy variables encoding the categories of one or more cate-

gorical variables. Whatever the type, kand Kdenote, respectively, the column

variable kand the total number of column variables. In what follows, the term

variable will be used for both types. From Yl, we compute the (Il×Jl) proportion

table Pl=Yl/Nl.

If we consider only the sample l, the weights of the respondents are obtained

from the margin of the rows of Pl, thus proportional to the length of their free

answers, and stored in the (Il×Il) diagonal matrix Dl. The total weight of the

respondents belonging to the same sample is equal to 1. In the same way, the

weights of the words are obtained from the margin of the columns of Pl, thus

proportional to their counts, and stored in the (Jl×Jl) diagonal matrix Ml. The

total weight of the words used by the same sample is equal to 1. Xlis centered

and possibly normalized in the case of quantitative variables, for the weighting

system Dl. The data structure including the relations between words and variables

is the (Jl×K) table Ql=YT

lXl

Nl=PT

lXl.Qlis called generalized aggregated lexical

table.

Remark. The name Generalized Aggregated Lexical Table and the acronym

GALT are used to emphasize the great similarity between this table and the clas-

sic aggregated lexical table (ALT) developed in the case of a single categorical

variable (Lebart et al., 1998).

In fact, the calculation is exactly the same in both cases. What changes is only

the expression of the matrix Xitself. In the case of an ALT, this table is composed

of the dummy variables corresponding to the categories of a single categorical

variable.

words

respondent

Figure 1: Sequence of Lcoupled tables

In the global analysis of the Lsamples, we have to deal with I=∑lIlrespon-

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Multilingualtextualdata:anapproachthroughmultiplefactoranalysisBelchinKostovDepartmentofStatisticsandOperationalResearch,UniversitatPolitècnicadeCatalunya,C/JordiGirona1-3,08034Barcelona,SpainRamónAlvarez-Esteban1DepartmentofEconomicsandStatistics,UniversidaddeLeón,CampusdeVeg-azanas/n,24071León,Spa...

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Multilingual textual data an approach through multiple factor analysis Belchin Kostov

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