GeoJournal
https://doi.org/10.1007/s10708-023-10889-4
A new method for multispace analysis of multidimensional
social exclusion
Matheus Pereira Libório · Hamidreza Rabiei‑Dastjerdi
Sandro Laudares · Chris Brunsdon Christopher ·
Rodrigo Correia Teixeira · Patrícia Bernardes
·
Accepted: 25 April 2023
© The Author(s), under exclusive licence to Springer Nature B.V. 2023
Abstract Social phenomena are multidimensional
and dependent on geographic space. Numerous methods are capable of representing multidimensional
social phenomena through a composite indicator.
Among these methods, principal component analysis (PCA) is the most used when considering the
geographical perspective. However, the composite
H. Rabiei‑Dastjerdi
School of Architecture, Planning, and Environmental
Policy & CeADAR, University College Dublin,
Dublin D04 V1W8, Ireland
e-mail: hamid.rabiei@ucd.ie
indicators built by the method are sensitive to outliers
and dependent on the input data, implying informational loss and specific eigenvectors that make multispace–time comparisons impossible. This research
proposes a new method to overcome these problems:
the Robust Multispace PCA. The method incorporates
the following innovations. The sub-indicators are
weighted according to their conceptual importance in
the multidimensional phenomenon. The non-compensatory aggregation of these sub-indicators guarantees
the function of the weights as of relative importance.
Aggregating indicators in dimensions balances the
weight structure of dimensions in the composite indicator. A new scale transformation function that eliminates outliers and allows multispatial comparison
reduces by 1.52 times the informational loss of the
composite indicator of social exclusion in eight cities’
urban areas. The Robust Multispace-PCA has a high
potential for appropriation by researchers and policymakers, as it is easy to follow, offers more informative and accurate representations of multidimensional
social phenomena, and favors the development of policies at multiple geographic scales.
H. Rabiei‑Dastjerdi
Social Determinants of Health Research Center, Isfahan
University of Medical Sciences, 81746‑73461 Isfahan ,
Iran
Keywords Social exclusion · Multidimensional
analysis · Composite indicators · Spatial analysis ·
Principal component analysis
M. P. Libório (*) · S. Laudares · R. C. Teixeira ·
P. Bernardes
Pontifical Catholic University of Minas Gerais,
Belo Horizonte 30535‑012, Brazil
e-mail: m4th32s@gmail.com
S. Laudares
e-mail: sandrolaudares@gmail.com
R. C. Teixeira
e-mail: rteixeira@pucminas.br
P. Bernardes
e-mail: patriciabernardes@pucminas.br
C. B. Christopher
National Centre for Geocomputation, Maynooth
University, Maynooth, Ireland
e-mail: christopher.brunsdon@mu.ie
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Introduction
Social phenomena are dynamic events of behavioral
origin that operate within a specific time and historical context through groups of people and multifaceted
processes (Bourdieu et al., 1991). Within this definition are phenomena such as well-being (Artelaris,
2017), inequality (Arretche, 2018), social exclusion
(Schwartzman, 2004), poverty (Marlier & Atkinson,
2010), and social vulnerability (Mavhura et al., 2017).
Social phenomena have five characteristics in
common. First, space matters in explaining social
phenomena (Dangschat, 2009; Fayard, 2012; Musterd & Murie, 2006; Shome, 2003; Stretesky et al.,
2004). Decisive features in explaining the social phenomenon of one location may be irrelevant in other
locations. Second, although income plays a relevant
role in understanding social phenomena, this variable offers a very limited understanding of these
phenomena as it ignores their multidimensional
nature (Artelaris, 2017; Mavhura et al., 2017; Murphy & Scott, 2014). Third, they are formed by constructs or dimensions resulting from the aggregation
of multiple sub-indicators that represent and connect
the concept of the social phenomenon to a quantitative measure. Fourth, a wide variety of sub-indicators
provide empirical content and operability to the social
phenomena concept (Marticuni & Libório, 2022).
Fifth, researchers constantly disagree about the relevant sub-indicators for representing social phenomena, resulting in different ways of measuring the same
phenomenon (Libório et al., 2022).
Among these characteristics, the multidimensional
nature of social phenomena creates several operational challenges for their representation (Libório
et al., 2023). These particular challenges have been
widely explored in the literature on composite indicators through studies such as that by Nardo et al.
(2005), Munda (2012), Becker et al. (2017), and KucCzarnecka et al. (2020).
Composite indicators are one-dimensional measures that offer a more straightforward interpretation of multidimensional phenomena (Terzi et al.,
2021). Recent advances have contributed to reducing challenges and limitations associated with building composite indicators (Dialga & Giang, 2017).
However, these advances are insufficient to build a
perfect composite indicator (Mazziotta & Pareto,
2017; Greco et al., 2019). Basically, a composite
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indicator is built by aggregating the normalized subindicators, weighted or not (Machado et al., 2022).
Naturally, each process of building a composite indicator involves challenges and limitations. The works
by Nardo et al. (2005), Munda (2012), Becker et al.
(2017), and Cinelli et al. (2021) provide a detailed
explanation of the challenges and limitations of the
scale transformation, weighting, and aggregation of
sub-indicators. This research is particularly interested
in overcoming challenges and limitations of the preferred method by geographers in building composite
indicators: Principal component analysis (PCA, see
literature review in appendix).
PCA can be easily operationalized through online
tools (Metsalu & Vilo, 2015) and modules integrated
into Geographic Information Systems (GIS) such as
QGIS (Ramdani et al., 2018). Besides, it is possible to
assign local weights to the sub-indicators by considering the spatial weights matrix and building a composite indicator with the properties of spatial dependence and heterogeneity (Cartone & Postiglione, 2020;
Sarra & Nissi, 2020; Tsutsumida et al., 2017).
However, PCA is sensitive to outliers, small samples, and poorly correlated data (Nardo et al., 2005).
The presence of this element results in composite
indicators with extracted variance and general consistency of the data below acceptable thresholds indicated below:
• Variance extracted from the sub-indicators in the
principal component must be greater than 0.50
(Nardo et al., 2005; Libório et al., 2020);
• The general consistency of the data measured
by the Kaiser–Meyer–Olkin (KMO) test must be
greater than 0.60 (Kaiser, 1974) for the sample to
be considered adequate;
• Bartlett’s (1937) sphericity test must be less than
0.05 to ensure that the correlation matrix is an
identity matrix.
PCA attributes greater weights to sub-indicators
that are more correlated with each other regardless
of their contextual importance (Mazziotta & Pareto,
2019; Libório et al., 2021; Terzi et al., 2021). This
problem is intensified when sub-indicators are not
previously aggregated in their respective dimensions.
The weight of dimensions will be strongly impacted
by the correlation of their underlying sub-indicators,
unbalancing the weight structure of the dimensions
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(Nardo et al., 2005). In addition, the weights of the
sub-indicators, obtained from standardization by
mean and standard deviation, vary with the input
data. This variation builds composite indicators with
sub-indicators weighted by different values, preventing multi-space and time comparisons (Libório et al.,
2022).
This research develops the Robust MultispacePCA to solve three of the PCA problems. First is
the incomparability between composite indicators
in different space–time due to sub-indicators with
different weight structures. Second, the unbalanced
structure of weights of dimensions that contain a
different number of sub-indicators. Third, the high
loss of information arising from an extracted variance and overall data consistency below acceptable
limits.
The Robust Multispace-PCA includes three innovations. First, the relative importance of the subindicators in the concept of the multidimensional
phenomenon is considered in the weighting process. Second, the balance of the weight structure of
the dimensions is ensured by aggregating the subindicators by the geometric mean in their respective
dimensions. Third, outliers are eliminated through
a new scaling normalization function that enables
multi-space–time comparison of composite indicators. The method is applied to represent intra-urban
multidimensional social exclusion, offering comparable information on the conditions of social exclusion in the urban census tracts of eight Brazilian
cities.
The Robust Multispace-PCA has a high degree
of applicability in problems of a multidimensional
nature, enabling comparisons of complex phenomena from multiple cities in different states or neighborhoods in different cities. This new method offers
highly relevant information for understanding phenomena at different geographic scales and monitoring
the impact of public actions on these phenomena over
time.
This research is organized into five sections. In
addition to this introduction, Sect. “Robust multispace-PCA” presents the Robust Multispace-PCA.
The multidimensional social exclusion phenomenon, the information about the study area, and the
data used in building the composite indicator are
presented in Sect. “Multispace analysis of multidimensional social exclusion in medium-sized cities
in Paraná, Brazil”. Sect. “Results and discussions” is
divided into results from the application of Principal
Components Analysis and results from the application of Robust Multispace-PCA. Sect. “Conclusions”
presents final considerations, potential contributions,
limitations of the method, and suggestions for future
work.
Robust multispace‑PCA
PCA is one of the factorial family methods. These
methods are based on the variability between possibly
correlated variables and include the PCA (Pearson,
1901), Factor Analysis (Spearman, 1904), Cronbach
Alpha Coefficient (Cronbach, 1951), Structural Equation Modeling (Jöreskog, 1970) and Correspondence
Analysis (Greenacre, 1984).
PCA is based on the analysis of an Xij data matrix,
where i = 1, 2, ..., n denotes the sub-indicators and
j = 1, 2, ..., m denotes the geographic regions. The
variance–covariance matrix Σ can be decomposed in its
auto-structure (Jolliffe, 2002) as follows:
Σ = AΛAt
(1)
where Λ is the diagonal matrix of eigenvalues of Σ;
A is the corresponding matrix of eigenvectors of Σ;
superscript t indicates the transposed matrix. The
eigenvalues in Λ represent the variance of the principal component Yz defined as:
Yz = XAZ
(2)
where AZ is the Z -th column of the matrix of eigenvectors Λ of Σ, which represents the contribution of
each sub-indicator in X to the z-th principal component Yz . The Yz entries are defined as the scores for
the principal component representing the composite
indicator.
The operationalization of (1) and (2) results in
unique eigenvectors dependent on the input data. Different input data result in different sub-indicator weight
structures, making the multispatial comparison of composite indicators impossible.
The Robust Multispace-PCA employs the normalization function proposed by Mazziotta and Pareto
(2022) to obtain the eigenvectors from all input datasets. The function is applied on a database that
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aggregates the multiple input data, being operationalized by the following expression:
yijc =
xijc − Minijc
Maxijc − Minic
(3)
where xijc is the value of the sub-indicator j in the
census tract i from the city c; Minijc and Maxijc are the
minimum and maximum values of the sub-indicator j
from all census tracts i from all cities c.
The application of (3) makes it possible to build comparable composite indicators as the normalization function’s maximum and minimum values are fixed. However,
this normalization function does not solve the problem of
high information loss in the input data since the PCA is
sensitive to the presence of outliers (Nardo et al., 2005).
Therefore, the application of (1), (2), and (3) is still insufficient to build a robust composite indicator.
Robust Multispace-PCA introduces one function to
eliminate outliers and build robust composite indicators. In short, the function applies the 3-sigma rule as
a capping method (Zheng et al., 2022) to define the
maximum and minimum values used in the scale normalization function.
First, the function that transforms outliers with values of three standard deviations below the mean into
numbers equivalent to the 3-sigma rule is applied as
follows:
( )
Lwijc yijc = 𝜇ijc − 𝜎 ijc × 3
(4)
where 𝜇ijc is the mean of the scores of the normalized
sub-indicator j in all census tracts i of all cities c, 𝜎ijc
is the standard deviation of the normalized sub-indicator j for all census tracts i of all cities c.
Second, the function that transforms outliers with
values of three standard deviations above the mean
into numbers equivalent to the 3-sigma rule is applied
as follows:
( )
Hgijc yijc = 𝜇ijc + 𝜎ijc × 3
(5)
From (3), (4), and (5), it is possible to obtain the new
scale normalization function for the construction of
robust and multi-space comparable composite indicators.
The following expression gives this new function:
zijc =
yijc − Lwic
Hgic − Lwic
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(6)
( )
Lwic yijc < Minijc
Note
that
if
than
( )
( )
Lwic yijc = Minijc, and if Hgic yijc < Maxijc than
( )
Hgic yijc = Maxijc.
The Robust Multispace-PCA aggregates the
sub-indicators in their respective dimensions to
ensure the balance of the weight structure of the
dimensions. The sub-indicators are weighted
according to expert opinion. This weighting
scheme aims to prevent the sub-indicator weights
from differing from the relative importance of
these sub-indicators concerning the multidimensional phenomenon (Mazziotta & Pareto, 2019;
Terzi et al., 2021).
Then the weighted sub-indicators are aggregated
using the geometric mean. A geometric mean is
an aggregation approach that avoids compensation
between sub-indicators with poor and above-average performance (Greco et al., 2019). This noncompensatory aggregation property ensures that
sub-indicator weights are maintained as coefficients
of importance, not as a substitute for performance
(Becker et al., 2017; Libório et al., 2021).
The following expression operationalizes the
weighted aggregation of the sub-indicators by the
geometric mean:
Dr =
n
∏
j=1
w
zr z
(7)
where z is the sub-indicator z of dimension r and wz
is the weight defined by the group of experts for the
sub-indicator z.
In short, Robust Multispace-PCA involves the
following steps and solutions:
1. Grouping of input data into a single database,
allowing multispatial comparisons;
2. Application of the new scale transformation function to eliminate outliers, reducing the informational loss of input data;
3. Weighting of sub-indicators based on the opinion
of specialists, matching the weights of the subindicators to their conceptual importance in the
multidimensional phenomenon;
4. Non-compensatory aggregation of the weighted
sub-indicators in their respective dimensions,
ensuring that the weights of the sub-indicators
maintain their relative importance function and
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balancing the weight structure of the dimensions;
5. Application of the PCA to build robust and multispatial comparable composite indicators.
Multispace analysis of multidimensional social
exclusion in medium‑sized cities in Paraná, Brazil
Social exclusion theoretical framework
Poverty, inequality, social exclusion, social vulnerability, and well-being are expressions of social phenomena that portray the living conditions of different
societies (Martinuci & Libório, 2022). These phenomena are often confused and used as synonyms,
although they have very different definitions. Poverty,
for example, reflects a situation of deprivation resulting from a lack of resources. Inequality refers to the
distribution of social wealth (between people, cities,
or countries). Social exclusion corresponds to an individual or group’s partial or non-integral participation in existing social systems (Libório et al., 2022).
Social vulnerability is associated with the degree of
risk an individual or society is subject to and their
ability to overcome adverse events (Mavhura et al.,
2017). Finally, well-being is associated with the feelings and emotions arising from the enjoyment of
material and immaterial goods (Sarra & Nissi, 2020).
The complex nature of these social phenomena,
which involve many sub-indicators in multiple dimensions, makes their measurement by geographers,
historians, economists, and sociologists quite challenging (Artelaris, 2017; Mazziotta & Pareto, 2017;
Murphy & Scott, 2014). Social phenomena have the
typical characteristics of multidimensional phenomena on which composite indicators are applied to support public interest decisions (Levitas et al., 2007;
Rabiei-Dastjerdi et al., 2018).
Social exclusion is a good example of applying composite indicators for three reasons. First, the
concept of social exclusion is a relatively recent and
controversial (Marlier & Atkinson, 2010). The lack
of a clearly defined and definitive theoretical framework for social exclusion makes two measurements
challenging. Second, the concept of social exclusion
is not limited to objective measures such as income
and the distribution of resources and includes subjective measures that are difficult to measure, such
as citizenship (Schwartzman, 2004) and social relations (Levitas et al., 2007). Any attempt to measure
social exclusion will be associated with informational
loss. Third, the social exclusion concept is sufficiently
broad and comprehensive for different sub-indicator
structures to be considered when measuring it in the
same geographical area (IBGE, 2017; Rodrigues,
2005). Sub-indicators selected to measure social
exclusion in one city may not make much sense to
others, especially when these cities are from different
countries.
The theoretical framework adopted in this research
is based on a set of studies developed in the project
"Mapping and analysis of territorial inequalities in
medium-sized cities in the interior of Paraná" and
consolidated in the book "Intraurban Inequalities:
methods for the production and analysis of composite
indicators" (anonymized citation for review).
Dimensions and sub‑indicators
Anonymized citation for review (year) develops
a five-dimensional social exclusion construct that
aggregates fifteen sub-indicators. Although the construct includes the main dimensions and sub-indicators associated with social phenomena, it has two
important limitations.
First, the household dimension includes sub-indicators that reflect housing and public infrastructure
conditions. Although housing and public infrastructure conditions are good proxies for social exclusion,
these dimensions must be worked on separately, as
they impact social phenomena in different intensities (Libório et al., 2020). Furthermore, improving
the conditions of homes and public infrastructure
requires specific measurements because they produce
unique and specific information for urban planning.
Second, the sub-indicator vegetation cover of the
environmental dimension is unreliable. The vegetation cover sub-indicator is obtained from satellite
image processing using the normalized difference
vegetation index method. The results of applying this
method are very sensitive to climatological changes
(Huang et al., 2021). Rainfall does not occur uniformly in space, changing the vegetation cover of
cities differently. This research includes the neighborhood infrastructure dimension and suppresses
the environmental dimension to overcome these
limitations.
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The economic dimension aggregates the most
relevant sub-indicators for representing social phenomena (Levitas et al., 2007). Income, for example,
directly influences sub-indicators of the dimensions of educational characteristics, demographic
aspects, and household conditions (Libório et al.,
2020). Furthermore, sub-indicators of the economic dimension are more easily obtained and have
greater stratification of geographic scales (Arretche,
2018).
The dimension of educational characteristics
included extremely relevant sub-indicators for
actions aimed at reducing two social problems. The
sub-indicators of this dimension are strongly associated with social mobility due to the direct and
positive correlation between individual income and
individual education level (Becker & Chiswick,
1966; Menezes Filho & Kirschbaum, 2019).
Demographics such as infant mortality and fertility rates are important sub-indicators in measuring social phenomena (Boing et al., 2020). These
sub-indicators have a direct and negative correlation
with the life expectancy of populations (Libório
et al., 2022). Socially more vulnerable populations
have a lower life expectancy at birth.
The conditions of the homes portray social phenomena based on the characteristics of the properties as building material (Gutberlet & Hunter, 2008;
Lago & Cardoso, 2017).
Finally, the neighborhood infrastructure dimension plays an important role in measuring and
understanding social phenomena in cities, as they
connect the geographic space with the other dimensions of social exclusion (Libório et al., 2022).
Selection and data collection
Data from the sub-indicators of the five dimensions of
social exclusion were selected based on accessibility,
geographic scale, reliability, and theoretical framework. Based on these criteria, 14 sub-indicators of the
Brazilian Institute of Geography and Statistics were
selected for the scale of urban census tracts (IBGE,
2010).
The sub-indicator data are from the last demographic census carried out in 2010. The polarity of
the sub-indicator with the concept of social exclusion
shown in Fig. 1 was defined based on its correlation
with the individual income indicator. This definition
of the sub-indicator polarity allows the correct choice
Fig. 1 The dimensions and sub-indicators in the composite indicator of social exclusion. Note sub-indicators’ relative importance in
the social exclusion concept was obtained in the work of (anonymized citation for review)
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Fig. 2 Location map of
cities
of the scale transformation function (Terzi et al.,
2021).
Study area
Three criteria were used to select the cities shown in
Fig. 2. The first criterion is the location of the cities.
Only cities in the interior of the State of Paraná were
included in the analysis. The second criterion is the
influence of the city. Cities that exert some regional
influence were selected (IBGE, 2008). The third criterion is demographic size. Municipalities with over
100,000 inhabitants in the last demographic census of
2010 were selected (IBGE, 2010).
The geographical unit of reference of the composite indicator of social exclusion is the census
tracts of cities. Census tracts are the smallest units
for aggregating demographic census data and comprise between 200 and 250 households (IBGE, 2010).
The number of census tracts varies with the population’s size in each of the eight cities: Toledo 109;
Apucarana 113; Guarapuava 182; Foz do Iguaçu 320;
Ponta Grossa 398; Cascavel 418; Maringá 483; and
Londrina 770.
Results and discussions
The first part of this section presents the results associated with the composite indicators of social exclusion constructed through the PCA. Figure 3 shows
that the extracted variance and KMO from Principal Components Analysis change from one city to
another. In particular, the extracted variance did not
reach the 0.50 acceptance threshold in any cities. The
variance extracted in the first principal component
was 0.23 in Londrina, 0.27 in Maringa, 0.30 in Ponta
Grossa and Toledo, 0.33 in Apucarana and Cascavel,
and 0.40 in Guarapuava.
The general consistency of the data measured by
the KMO has exceeded the limit value of 0.60 (Kaiser, 1974). However, the composite indicators built
by the PCA showed an information loss greater than
50%.
The results are not robust, and the comparability between cities is compromised as the informational loss and the weight of the sub-indicators vary
with the city. “Heads of household illiterate” is the
most important sub-indicator in the first principal
component in six cities. However, the weight of this
indicator is 1.4 times greater in Londrina compared
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Fig. 3 The variance extracted from the first principal component in the eight cities
Table 1 Dimension weight in the composite indicator
Demography
Economy
Education
Households
Neighborhood
Apucarana
Cascavel
Foz do Iguaçu
Guarapuava
Londrina
Maringá
Ponta Grossa
Toledo
0.26
0.30
0.16
0.14
0.15
0.34
0.25
0.15
0.13
0.13
0.35
0.25
0.17
0.18
0.06
0.33
0.24
0.14
0.20
0.08
0.34
0.29
0.19
0.13
0.05
0.27
0.32
0.15
0.11
0.15
0.30
0.31
0.10
0.21
0.08
0.30
0.30
0.15
0.19
0.06
to Guarapuava. Besides, Table 1 shows a significant
imbalance in the weight structure of dimensions
across cities.
The Demography dimension’s weight in the composite indicator of social exclusion is, on average, 3.3
times greater than the weight of the neighborhood
dimension. This result shows that the direct aggregation of the sub-indicators creates a very unbalanced
dimension weighting structure. This imbalance is
especially significant in the city of Foz do Iguacu,
where the contribution of the Demography dimension to the composite indicator is 5.7 times greater
than that of the neighborhood dimension. This result
supports the previous studies that highlight the limitations of PCA in building composite indicators
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(Libório et al., 2020; Mazziotta & Pareto, 2019; Terzi
et al., 2021).
The second part of this section presents the results
associated with the composite indicators of social
exclusion built through the Robust Multispace-PCA,
which involves the following solutions to overcome the deficiencies of the Principal Components
Analysis:
• Running the PCA from a database that groups
information from all cities allows for obtaining
sub-indicators with equal weights (eigenvectors)
for all cities, making the scores of the composite
indicators fully comparable.
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• The sub-indicator normalization function that
limits the highest and lowest value to up to 3
standard deviations from the mean transforms
atypical data (outliers) into typical data, generating robust results with informational loss within
the acceptance thresholds.
• Sub-indicators’ aggregation in dimensions
avoids the imbalance of the composite indicator’s weighting structure, making the multidimensional phenomenon’s theoretical and operational framework compatible.
• The aggregation of sub-indicators in their respective dimensions by the geometric mean prevents
sub-indicators of poor performance from being
offset by sub-indicators of above-average performance.
• The weighting of the sub-indicators based on specialists’ opinions allows for considering each subindicator’s importance in the multidimensional
phenomenon concept.
These solutions made it possible to increase the
information retained in the composite indicator
by 1.93 times. The variance extracted in the first
Fig. 4 Variance extracted from principal components and contribution of dimensions in the first principal component. Note
Overall KMO = 0.70 (Demography and Economy KMO = 0.70;
Table 2 Contribution
(weight) of the dimensions
of social exclusion in the
first component (composite
indicator) obtained by the
Robust Multispace-PCA in
the eight cities
Minimum
Maximum
Maximum-Minimum
Standard deviation
component increased from 0.27 in the Principal
Components Analysis to 0.53 in the Robust Multispace-PCA, surpassing the acceptance threshold of
this test.
Figure 4 shows that the Robust Multispace-PCA
also makes it possible to balance the contribution of
dimensions in the composite indicator.
The contribution (weight) of each of the five
dimensions of social exclusion in the first Principal Component (composite indicator) is the same as
shown in Table 2.
The contribution (weight) of each of the five
dimensions of social exclusion in the first Principal Component (composite indicator) does not vary
across cities in the A Robust Multispace-PCA. In
other words, a dimension of social exclusion contributes equally to the composite indicator of the
eight cities. Table 2 brings statistics that prove the
inexistence of differences in the contribution of the
dimensions of social exclusion in the first Principal
Component.
In turn, Table 3 shows that the contribution of
the dimensions of social exclusion varies between
Education KMO = 0.87; Households KMO = 0.68; Neighborhood KMO = 0.66). Bartlett test: chi-square = 144 and
p-value < 0.001
Demography
Economy
Education
Households
Neighborhood
0.20
0.20
0.00
0.00
0.22
0.22
0.00
0.00
0.16
0.16
0.00
0.00
0.22
0.22
0.00
0.00
0.20
0.20
0.00
0.00
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Table 3 Contribution of
the dimensions of social
exclusion in the first
component (composite
indicator) obtained by the
PCA in the eight cities
Minimum
Maximum
Maximum-Minimum
Standard deviation
Demography
Economy
Education
Households
Neighborhood
0.26
0.35
0.09
0.04
0.24
0.32
0.08
0.03
0.10
0.19
0.09
0.03
0.11
0.21
0.10
0.04
0.05
0.15
0.10
0.04
cities when the PCA is used to build the composite
indicator.
These results show the limitations of the PCA in
terms of comparability between the census tracts of
different cities while revealing the importance of the
Robust Multispace-PCA in comparative studies.
Another advantage of Robust Multispace-PCA
over PCA is associated with the number of Principal
Components needed to reach an extracted average
variance above the threshold of 0.50. Table 4 shows
that one Principal Component is enough to reach the
0.50 threshold in the Robust Multispace-PCA. In
turn, three Principal Components are needed to reach
an extracted average variance of 0.50 in the PCA.
In short, the composite indicator built by Robust
Multispace-PCA presents a structure in which the
weights of sub-indicators and dimensions do not vary
by city. This fixed weight structure makes it possible
to compare the mapping of social exclusion in the
eight cities shown in Fig. 5.
Visually, it is possible to observe that Guarapuava
and Foz do Iguaçu concentrate the largest number of
census tracts classified as high social exclusion. This
finding is confirmed in Tables 5 and 6, which also
point out Londrina and Maringá as the cities with the
highest number and proportion of census tracts classified as nonsocial exclusion.
Another advantage of the Robust Multispace-PCA
is to allow the geographical analysis of the different
dimensions. This analysis allows us to infer which
Table 4 The number of principal components (PCs) needed to
reach the 0.50 threshold
PCA
Robust MultispacePCA
PC-1
PC-2
PC-3
0.32
0.52
0.44
0.71
0.54
0.86
Three principal components are necessary to reach an average
variance extracted above the threshold of 0.50, except in the
city of Guarapuava, where the PC-2 was 0.51
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dimensions contribute positively and negatively to
social exclusion. Figure 6 shows that the poor performance of the Economy and Neighborhood dimensions impacts social exclusion in Apucarana.
These maps are especially useful because they
allow the definition of public policies with the most
significant impact on reducing social exclusion at different geographic scales. In Brazil, investment in education is the responsibility of the State and Municipalities. The comparability between census tracts of
different cities allows the State to invest in education
in the census tracts of cities with poorer performance.
Conclusions
Social phenomena such as poverty, social vulnerability, well-being, and social exclusion are multidimensional and dependent on geographic space.
This research reveals that researchers prefer the
PCA method to measure this multidimensional phenomenon. However, the PCA presents several shortcomings that result in composite indicators with
low informational power and do not allow comparisons between different areas of study. The Robust
Multispace-PCA allows for overcoming these
shortcomings.
The Robust Multispace-PCA groups data from different areas in a single database. Then, the sub-indicators are normalized by the highest and lowest limits
defined by the value of three standard deviations from
the mean. Then, these sub-indicators are weighted
according to their relative importance in the social
exclusion concept and aggregated into five dimensions of social exclusion, preventing sub-indicators
of poor performance from being fully compensated
by sub-indicators of above-average performance.
Finally, the composite indicator of social exclusion is
built through the PCA based on the five dimensions
of social exclusion.
GeoJournal
Fig. 5 Map of social exclusion in the eight cities. Note social exclusion classes were defined using the quartile method
Table 5 The number of census tracts by the level of social exclusion in the eight cities
Social exclusion
Apucarana
Cascavel
Foz do Iguaçu
Guarapuava
Londrina
Maringá
Ponta Grossa
Toledo
High
Medium
Low
None
0
32
65
16
6
89
211
112
11
84
176
49
12
39
96
35
7
81
390
292
0
65
230
188
7
88
208
95
1
26
65
17
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GeoJournal
Table 6 Percentage of census tracts by the level of social exclusion in the eight cities
Social exclusion
Apucarana
Cascavel
Foz do Iguaçu
Guarapuava
Londrina
Maringá
Ponta Grossa
Toledo
High
Medium
Low
None
0.00
0.28
0.58
0.14
0.01
0.21
0.50
0.27
0.03
0.26
0.55
0.15
0.07
0.21
0.53
0.19
0.01
0.11
0.51
0.38
0.00
0.13
0.48
0.39
0.02
0.22
0.52
0.24
0.01
0.24
0.60
0.16
Fig. 6 Map of the dimensions and social exclusion of Apucarana. Note social exclusion classes were defined using the quartile
method
Robust Multispace-PCA offers two important
contributions to the field of multidimensional social
phenomena. These contributions are evidenced
through the case of social exclusion in eight Brazilian cities. The first contribution is associated
with the increase in the informational power of
the results. The weighted aggregation of the subindicators in their respective dimensions allows
for identifying which dimensions most influence
social exclusion in each area of the city, allowing the formulation of specific public policies for
each area. The second contribution of this research
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is associated with the possibility of applying the
Robust Multispace-PCA to analyze composite
social indicators over time. This possibility occurs
as the grouping of data from different periods prevents the weights of the sub-indicators of the composite indicator from being different in time.
The PCA is a method sensitive to the correlation
between sub-indicators. The average correlation
between the original sub-indicators of this research
is only 0.19. Therefore, the Robust MultispacePCA may fail for correlations between sub-indicators below this threshold. This limitation opens
GeoJournal
an opportunity to improve the Robust MultispacePCA. This improvement may include procedures
to increase the correlation between sub-indicators,
such as asymmetry correction techniques (e.g.,
Box-Cox transformation) and sub-indicator selection (e.g., Hierarchical clustering).
• Indirect strategy methods: present flexible structure of normalization, weighting, and aggregation
of sub-indicators. This category includes methods such as Simple Additive Weighting (Equal
Weights, Data-Driven Weights, and Participatory
Weights) and Utility and Distance functions.
Funding This work was carried out with the support of the
National Council for Scientific and Technological Development of Brazil (CNPq) under grant 151518/2022–0.
The works by Nardo et al. (2005), El Gibari et al.
(2019), and Terzi et al. (2021) offer essential information for understanding these methods. The main
attribute of Benefit-of-the-Doubt is the individual
weighting of sub-indicators by geographic area. This
weighting approach considers the performance of
each geographic area against a benchmark performance. Higher weights are assigned to the highestperforming sub-indicators in each area (Fusco et al.,
2018, 2020). In short, Benefit-of-the-doubt is a
method that considers spatial heterogeneity but makes
areas not fully comparable. Factor family methods
assign weights to sub-indicators to maximize the variance/correlation of sub-indicators in the first Principal
Component/Factor. The weights of the sub-indicators
do not vary with the geographic area, allowing comparison between the areas. Simple Additive Weighting (SAW) is a flexible method of building composite
indicators. The SAW allows different combinations of
normalization function, weighting approach, and subindicator aggregation scheme. For example, it is possible to weigh sub-indicators using Equal Weights,
Data-driven, and Participatory weighting schemes.
Utility functions and Distance functions are methods of building composite indicators that require the
participation of specialists, which may or may not
involve the prior normalization of the sub-indicators.
Experts should evaluate the alternatives (geographical areas) and the sub-indicator weights. The main
methods in this category are the Multi-Utility Theory
(MAUT) and the Technique for Order Preferences by
Similarity to Ideal Solutions (TOPSIS).
The main works in the literature on composite
indicators (Cherchye et al., 2007; Nardo et al., 2005;
Saltelli, 2007) ignore entirely a method widely used
in the construction of composite indicators in the
field of geography: remote sensing. Literature review
studies also do not include remote sensing among
the construction methods of composite indicators (El
Gibari et al., 2019; Fernandez et al., 2020). However,
the review of publications on composite indicators in
Data availability Martinuci, O. S., Machado, A. M. C.,
Libório, M. P. (2021). Data for: Time-in-space analysis of multidimensional phenomena, Mendeley Data, V4, https://​doi.​org/​
10.​17632/​m3y4j​ncvch.4.
Declarations
Conflict of interest The authors declare they have no financial interests.
Human and/or animals rights No Human Participants and/
or Animals are involved in this research.
Appendix—composite indicators in Geography
Until August 2022, journals titled with the terms
“Geo*” or “Spatial” and indexed in the Scopus and
Web of Science databases had published 200 articles
with the term “composite ind*” in the abstract. One
hundred and three articles have been published in the
last five years, showing how composite indicators
have become popular among geographers.
Of the 200 articles identified, a content analysis of
60 randomly selected articles was performed. Content
analysis identified the method used to build the composite indicator and which composite indicator was
built. Two of these 60 articles are literature reviews
and do not build a composite indicator (see Miller
et al., 2013; Torres-Delgado & Saarinen, 2014). The
methods used in the building of the composite indicators were identified based on the works of Miller et al.
(2013) and El Gibari et al. (2019) as follows:
• Direct strategy methods: have their particular
normalization, weighting, and aggregation structure of the sub-indicators. This category includes
methods such as Benefit-of-the-doubt, Data Envelopment Analysis, Factor Analysis, and PCA.
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GeoJournal
Table 7 Composite indicators and methods used in Geography
Method
Composite indicator
Benefit-of-the-doubt
Socioeconomic (Ogneva-Himmelberger et al., 2013) and social exclusion (Libório et al.,
2022a, 2022b, 2022c)
Factor analysis
Intra-urban deprivation (Oyebanji, 1984), travel attitudes and reasons for location choice
(Ettema & Nieuwenhuis, 2017), assessment of oil and gas geopolitical influence (Gu &
Wang, 2015), urban spheres of influence (Wang et al., 2011), environments of disadvantage (Pacione, 2004a), target-regional and target-country cultural variation (Slangen,
2016), social capital (Kemeny & Cooke, 2017), marginalization of youth in the labor
market (Bauder & Sharpe, 2000), disadvantaged live in rural areas (Pacione, 2004b), and
Vulnerability to COVID-19 (Fall et al., 2022)
PCA
Neighborhood socio-demographic characteristics (Wang & Lindsey, 2019), risk and resilience (González et al., 2018), vulnerability to dengue fever (Hagenlocher et al., 2013),
intraregional agricultural characteristics (Su et al., 2020), agricultural development (Halder, 2021), material well-being (Sinha & Basu, 2022), settlement development (Kallingal
& Mohammed Firoz, 2022), family living conditions (Das et al., 2021a, 2021b), urban
spatial inequality (Rabiei‐Dastjerdi & Matthews, 2021), and access to services and basic
urban amenities (Das et al., 2021a, 2021b)
Remote sensing
Off-road vehicle recreation across a landscape (Westcott & Andrew, 2015), vulnerability
to climate change (Lawal & Adesope, 2019), evapotranspiration of planting winter wheat
(Wu et al., 2019), night light urban index (Zheng et al., 2016), land surface temperature
(Zhu et al., 2018), functional land-use (Lang et al., 2014), and environmental benefits in
urban land use (Rahman & Szabó, 2022)
SAW with equal weights
Levels of living (Barke, 1989; Oyebanji, 1986), serious violence (Harries, 2004), the
geography of corporate directors (O’Hagan & Rice, 2012), community livelihood vulnerability (Lin & Polsky, 2016), sustainability of urban growth and form (Ogle et al., 2017),
multiple deprivations (Raheem, 2017) and, vulnerability to motor fuel price increases
(Mattioli et al., 2019). It is also possible to avoid compensation between sub-indicators
by aggregating sub-indicators by the geometric and harmonic mean to obtain composite
indicators of environmental stress indicator and social relevance (Fernández & Wu, 2018)
and social exclusion (Libório et al., 2022)
SAW with data-driven weights
Affordable housing demand (Baker & Beer, 2007), spatial risks for allocating automated
external defibrillators (Lin et al., 2016), information and communication technologies
development (Song et al., 2014), global energy security (Wang et al., 2019) and inequality (Libório et al., 2022a, 2022b, 2022c), and neighborhood social change (Kitchen &
Williams, 2009)
SAW with participatory weights
Transportation disadvantage impedance (Duvarci et al., 2015), suitability location mapping
for entrepreneurs (Ahamed et al., 2020), wetland health (Zhou et al., 2020), and intraurban inequality (Libório et al., 2021a, 2021b; Libório et al., 2021)
Utility functions and distance functions Airport dependency (Koo et al., 2016), development of aviation markets (Jankiewicz &
Huderek-Glapska, 2016), river basin vulnerability (Varis et al., 2012), environmental
quality (Montero et al., 2010), potential spatial access to urban health services (Apparicio
et al., 2017), urban liveability (Higgs et al., 2019), gender-based vulnerability (Nelson
et al., 2020), tourism seasonality (Martín et al., 2018), and employment protection legislation (Percoco, 2016)
Geography magazines consolidated in Table 7 reveals
a reasonable number of composite indicators constructed through remote sensing.
The literature review summarized in Table 7 shows
a certain balance between the strategies preferred by
geographers in building composite indicators. Direct
strategy methods are used in 43% of composite indicators, while the remaining 57% use indirect strategy
Vol:. (1234567890)
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methods (Miller et al., 2013). The preference for
methods such as Factor Analysis, PCA, and Simple
Additive Weighting is much greater in articles published in Geography journals than in literature. Such
methods are used in 65% of the articles published in
geographic journals. In contrast, these methods are
used in only 12% of the articles in the literature on
composite indicators (El Gibari et al., 2019).
GeoJournal
The max–min and z-scores normalization functions are used in 70% of the composite indicators
analyzed. Like most researchers, geographers prefer
to weigh sub-indicators statistically from the data.
However, among the 28 articles that weigh the subindicators from the data, only two use the Benefitof-the-Doubt. In short, the sub-indicator weighting
approach most used in the literature on composite
indicators is rarely used in Geography. Most articles
in Geography reviewed apply factorial family methods to weigh the sub-indicators. Factor Analysis and
PCA are used in 64% of the articles. These percentages indicate that geographers privilege the comparison between geographic areas to the consideration of
spatial heterogeneity.
Compensatory aggregation approaches are used in
72% of the articles reviewed. This means that most
composite indicators published in geography journals
allow sub-indicators of poor performance to be offset by sub-indicators of above-average performance.
Besides, many geographic composite indicators used
in practice have a very simplified structure. Some
composite indicators aggregate only two or three
sub-indicators.
Despite this, there are several composite indicators
built from satellite images. The normalized difference vegetation index (NDVI) by Tucker (1979) is the
most famous among them. Satellite image data has
also been useful in building social composite indicators in the urban environment. First, building social
composite indicators from the processing of raster
data (Di Bella et al., 2018; Duque et al., 2015; Niu
et al., 2020). Second, building sub-indicators to be
aggregated with other sub-indicators in a composite
indicator that combines raster and conventional data
(Rabiei-Dastjerdi & Matthews, 2021). Third, validating composite indicators built using conventional
data (Libório et al., 2020; Su et al., 2020; Zhou et al.,
2020).
Finally, the literature review shows that geographers have used composite indicators in several fields.
The relevance of composite indicators in geography
grows yearly, with the number of articles published
on the topic doubling in the last five years. The preferred method of geographers for building composite
indicators is PCA. The preference for this method
does not correspond to the preference of researchers
in the literature of composite indicators that use Data
Envelopment Analysis and TOPSIS in 16% and 14%
of the studies, respectively.
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