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Measuring inequality through a non-compensatory approach

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Abstract

The aggregation of variables is one of the most critical procedures in the building of Composite Indicators. Most of the debate is about the issue of compensation between variables with poor and above-average performance. Researchers understand that non-compensatory aggregation is more appropriate than compensatory aggregation for measuring multidimensional phenomena for several reasons. Among them are the impossibility of substituting one element of the phenomenon for the other, the maintenance of weights as a measure of relative importance, and the emphasis on poorest performances. Despite its desirable properties, the literature review on Composite Indicators of inequality reveals a broad preference for compensatory aggregation of variables over non-compensatory aggregation. The absence of Composite Indicators of inequality built by non-compensatory aggregation leads to the following questions: to what extent does non-compensatory aggregation favor the representation of multidimensional social phenomena in geography? What are the shortcomings of non-compensatory aggregation, and how to resolve them? The results show that the Ordered Weighted Averaging (OWA) operator makes it possible to overcome problems associated with non-compensatory aggregation and obtain a statistically consistent Composite Indicator to represent inequality in a Brazilian city. The advantage of using non-compensatory aggregation is demonstrated by comparing the Composite Indicators constructed through OWA, Simple Additive Weighting, Principal Component Analysis, and Doubt Benefit. Another novelty of this research lies in demonstrating how the regulation of compensation levels between variables and the direction of bias can help in maximizing the Composite Indicator's capability to capture the most relevant variable of the multidimensional phenomenon.

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  1. The Capes Journal Portal is a virtual library that houses international scientific production and has a collection of more than 45,000 journals, 130 reference bases, and 12 patent bases: https://www.periodicos.capes.gov.br/.

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Funding

This work was carried out with the support of: (i) The Coordination for the Improvement of Higher Education Personnel—Brazil (CAPES)—Financing Code 001; and (ii) The National Council for Scientific and Technological Development of Brazil (CNPq) : a) Process 423443/2016-0-“Mapping and analysis of territorial inequalities in medium-sized cities in the interior of Paraná”—Edital Universal 001/2016; and b) Productivity Grant, Grant 311032/2016–8.

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Appendices

Appendix I—Composite Indicators of inequality

The literature review on Composite Indicators of inequality was carried out on the Capes Journal Portal,Footnote 1 Web of Science and Social Indicators Research Journal. The prescribers used were Inequality AND Index OR Indicator. Table

Table 3 Inequality Composite Indicators

3 summarizes the Composite Indicators on inequality found in the literature, revealing its volume and diversification.

See Table 3.

Some conclusions can be extracted from the 44 articles in Table 3. First, data-driven methods are used in 47% of the papers (Chen, 2015; Yoon & Klasen, 2018). Second, participatory methods are used in 33% of the papers (Parada et al., 2019; Rubio et al., 2018). Third, 16% of the papers use equal weighting (Botha, 2016; Soares & Delgado, 2016). Fourth, the other 5% use mixed methods that combine data-driven and participatory methods (You et al., 2020). Fifth, none of the 44 reviewed papers use non-compensatory methods. Sixth, 37% of the papers use maps to represent Composite Indicators.

Appendix II—OWA operator

O operador de agregação OWA é um método com diversas aplicações na solução de problemas de tomada de decisões multiobjetivo. The mathematical formulation of the OWA operator is given as follows:

$${\text{OWA}}\left( {v_{1} ,v_{2} , \ldots ,v_{q} } \right) = \mathop \sum \limits_{i = 1}^{q} w_{i} b_{i} ,$$
(4)

where \(b_{i} ,\) corresponds to the i-th largest value between \(v_{1} ,v_{2} , \ldots ,v_{q}\) and the weights \(w_{i} { }\) satisfy the conditions \(w_{i} \in \left[ {0,1} \right]\) e \(\mathop \sum \limits_{i = 1}^{q} w_{i} = 1\).

The OWA operator is performed through three steps.

  • Organize variables \(v_{1} ,v_{2} , \ldots ,v_{q}\) in descending order

  • Define the weights \(w_{1} ,w_{2} , \ldots ,w_{q}\) associated with the OWA operator

  • Employ the OWA (4) operator to aggregate the variables.

The weights \(w_{i}\) are defined according to the \(i\) position of the ordered variable \(a_{i}\). The adjustment of these weights is made through operators that allow regulating the compensation levels between variables. For example, the \(max\) operator reproduces an optimistic approach in that there is full compensation between the variables.

$$w_{q} = 0, w_{i} = 0, i = 2, \ldots ,q$$
(5)

In contrast, the \(min\) operator reproduces a pessimistic approach in that the worst performance of the variable prevails, and there is no compensation between the variables.

$$w_{q} = 1, w_{i} = 0, i = 1, \ldots ,q - 1$$
(6)

The weights of the OWA operator can also be regulated through so-called linguistic fuzzy quantifiers. The linguistic quantifiers represent a fuzzy set \(Q_{\left( x \right)}\) of the portion of \(x \left[ {0,1} \right]\) variables (Ekel et al., 2020; Pedrycz et al., 2011). The solution will be defined if there are at least \(x\) variables that outperform the variable according to the fuzzy set \(Q\) that must satisfy the conditions:

  • \(Q\left( 0 \right) = 0\);

  • \(Q\left( 1 \right) = 1\);

  • if \(x_{1} > x_{2}\), then \(Q\left( {x_{1} } \right) > Q\left( {x_{2} } \right)\).

Fuzzy quantifiers \(Q\) can be represented in natural language through linguistic terms such as: "As much as possible," "Majority," "Average," and "At least half" (see Zadeh, 1983). This research employs fuzzy linguistic quantifiers "More than J" (7) e "At least J" (8)

$$Q_{\left( x \right)} = \left\{ {\begin{array}{*{20}l} {0 if 0 \le x \le \alpha } \\ {\frac{x}{1 - \alpha } if \alpha < x \le 1} \\ \end{array} } \right.$$
(7)
$$Q_{\left( x \right)} = \left\{ {\begin{array}{*{20}c} {\frac{x}{\alpha } if 0 \le x \le \alpha } \\ {1 \quad if \alpha < x \le 1} \\ \end{array} } \right.$$
(8)

where \(\alpha = \frac{j}{q}\).

The fuzzy linguistic quantifier "More than J" has a smaller compensation level than "At least J." The first considers the poorest performances of the variables, while the second allows a more significant influence of the above-average performances of the variable, being more compensatory. The OWA operator allows estimating the degree of optimism or pessimism expressed by the weight values through the quantitative measure known as Orness Degree (Yager, 1988) applying the expression:

$$\theta = \frac{1}{q - 1}\mathop \sum \limits_{i = 1}^{q} \left( {q - i} \right)w_{i} ,$$
(9)

Onde \(\theta\) assume valores entre 0 e 1, sendo 0.50 uma abordagem neutra, 0.00 uma abordagem totalmente pessimista e 1.00 uma abordagem totalmente otimista (Ekel et al., 2020; Liu & Han, 2008).

Appendix III—Urban landscape analysis

The landscape is a form, an element of space, the result of past moments and spatial contexts (Santos, 2008). Despite being eminently appearance, it is a starting point for understanding space complexity because its analysis can reveal inequality and indicate processes of space production that generated them (Santos, 2013). Although the urban landscape analysis is an activity associated with a field visit, resource limitations, difficulty of access, and even the circulation restrictions imposed by COVID have stimulated new ways to analyze spatial patterns in the urban environment (Libório et al., 2021b). Several works based on Big Data have shown the efficiency of the urban landscape analysis from Google Street View images (Badland et al., 2010).

For example, Hilal et al. (2018) analyze the vegetation present in the urban landscape from neighborhoods with a high and low density. Li et al. (2015) evaluate urban vegetation at street level using Google Street View as a tool for cities' urban environmental planning. Yin and Wang (2016) analyze the characteristics of the urban landscape to verify the sidewalks' conditions.

In the present study, the urban landscape analysis was performed using images from Google Street View. The following elements were checked: paving, trees, garbage accumulated on the street, open sewers, public lighting, and the sidewalks' conditions and the front of households. The analysis is based on the comparison of pairs of census sectors that allow obtaining a Correctness rate. Four steps are performed to calculate the Correctness rate of the OWA and SAW-EW Composite Indicators.

First, the census tracts with atypical scores are selected. The box-plot method (Ross, 2014) was applied to select the census tracts with atypical differences between the OWA and SAW-EW Composite Indicators. These atypical differences represent census tracts where SAW-EW scores are very different from OWA scores. Considering that the OWA scores are lower than the SAW-EW, these atypical differences indicate the census tracts for which the SAW-EW may have underestimated the level of inequality.

Second, census tracts with similar scores are selected. Then, the census tracts that had similar OWA and SAW-EW scores and at the same time had scores close to a census tract identified in the first step are selected. The census tracts selected in this step represent spatial units where SAW-EW scores are similar to OWA scores.

Third, the urban landscape of 28 census tracts is analyzed through Google Street View. Then, the urban conditions observed in the census tracts selected in the first and second steps are compared. The responses to these comparisons are divided into two. One, "A > B," when the urban conditions between the census tracts are not similar. That is, the SAW-EW underestimated the level of inequality in the census tract selected in the first step. Two, "A = B" when the census tracts' urban conditions chosen in the first and second steps are similar. That is, the OWA overestimated the inequality of the census tract selected in the first step.

Fourth, the responses of the 14 comparisons are computed by assigning "1" (one) to "A > B" and "0" (zero) to "A = B." Then, the OWA Correctness rate was calculated by dividing the sum of the values assigned to each answer by the total comparisons. The SAW-EW Correctness rate was calculated by subtracting the OWA's Correctness rate by 1. Figure 

Fig. 9
figure 9

Urban landscape analysis

9 illustrates the four steps used in the OWA and SAW-EW correctness calculation.

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Libório, M.P., Martinuci, O.d., Ekel, P.I. et al. Measuring inequality through a non-compensatory approach. GeoJournal 87, 4689–4706 (2022). https://doi.org/10.1007/s10708-021-10519-x

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