You may be interested in the set of papers that make up these inter-related projects

https://www.researchgate.net/project/Modelling-segregation-applying-the-new-methdologies

https://www.researchgate.net/project/Modelling-segregation-methodological-developments

Thank you very much, Mr. Jones!

To Milan B. Vemić:

Thank you for your response, Mr. Vemić! By the way, when I collect data on that indices for all the municipalities in the region, which measures shall I use to determine, how evenly is income (population, literacy etc.) dispersed in region? I'm planning to use Gini index, but maybe there are some better measurements you happen to know?

To Milan B. Vemić:

Thanks a lot, Mr. Vemić, for your answer!

Unfortunately there are a lot of issues with using an index approach (like Gini) to analysing fine-grained spatial data :

Article Modelling residential segregation as unevenness and clusteri...

" Methodologically, the finer the spatial scale and the fewer the number of people living in an area, the more likely are chance results. Many traditional segregation indices (INCLUDING GINI )assume that there are no stochastic variations and provide upwardly-biased estimates with apparent segregation when there is none (Carrington and Troske,1997). It is thus necessary to adopt inferential modelling to assess segregation net of stochastic variation as pioneered by Leckie et al. (2012). Segregation is characterised as an estimated variance around a mean. If all places have the same share of a subgroup the variance estimate will be zero; any apparent differences are merely due to chance. However, an estimated variance whose uncertainty intervals do not include zero indicates genuine segregation in the form of unevenness. Importantly they show that several well-known indices [INCLUDING THE GINI] are an increasing monotonic function of the variance. Consequently, the same conclusions will result whichever are used to compare different groups or groups over time. They also provide a method of transforming the variance and accompanying uncertainty into these indices if that helps comparison.

Mobility research finds that movers choose (or are forced to choose) a broad zone of a city and then decide at a micro scale where to locate within that zone (Johnston, Forrest et al. 2016). There can be multiple processes operating at multiple scales and a group can be concentrated in zones and within certain subareas within zones. Methodologically, the scale at which segregation analysis is conducted is critical and can determine the outcomes found. Previous research has resulted in the stylised fact that the most marked segregation is at the finest scale and declines at higher spatial scales. However, there has been much misunderstanding about analysis at multiple scales. Typically (e.g. Lee et al. 2008) the analysis moves through a sequence of ever-growing scales finding the maximum segregation at the finest scale. These analyses are problematic as the finest scale implicitly includes segregation at any relevant higher scale (Duncan et al., 1961) and the apparent decline is likely to be an artefact of the aggregation process whereby data are smoothed as they are combined into larger units.

Fortunately, the modelling approach can be extended to analyse multiple scales simultaneously where unevenness is characterised by the variance at one scale net of the variation at other scales (and net of stochastic variation). Several studies using this approach have found that segregation is greater at the macro scale, as for example for London (Johnston, Jones et al., 2016; Jones et al. 2015).

Processes of structured segregation are likely to produce not just unevenness of distribution but also clustering whereby similar groups are spatially concentrated in certain parts of the city and separated from dissimilar groups. Methodologically, however, most segregation analyses are inherently aspatial merely analysing the frequency distribution of minority rates and ignoring location. What is needed is an assessment of the degree of spatial dependence of the rates – the degree to which similar rates are adjacent to each other. The developing modelling approach in partitioning the variance at different levels implicitly models the dependency at each scale below the topmost one (Jones et al. 2015); larger variances indicate both greater differences between and equivalently greater similarity within areas (Bullen et al., 1997). But this crude approach has no explicit parameter for the degree of spatial dependence; the variance measures only unevenness and there is a need for a modelling approach that additionally assesses spatial clustering as well as aspatial unevenness. Initially, the modelling approach was applied to educational segregation which analyses data for preexisting higher-level level entities (at its simplest, children in schools). For residential segregation, this is not the case and the areas of interest must be defined, thereby confronting the MAUP There are two aspects to this: the scale problem is that results may depend on the number of units studied; while the zonation problem is that different results can be found for a constant number of geographical units but with differing spatial arrangements.

To date the model-based approach has taken a strictly hierarchical approach where the scales and zoning are taken as given and fixed. Thus, a model-based study of segregation in London (Johnston, Jones et al 2016) used census Output Areas at the finest scale which are exactly nested in Middle Layer Super Output Areas which are in turn nested in Boroughs. It may be that this specific mesh is determining the results so we need to research the MAUP to gauge potential sensitivity of results to different spatial architectures. This problem is closely bound up with spatial dependence for if the rates are truly without map pattern then differential aggregations and zonings are unlikely to determine the results (Wong 2009, 114). Views on MAUP range from an ‘essentially unpredictable’ intractable problem (Fotheringham and Wong, 1991,1025) to a ‘very powerful analytical device’ (Openshaw, 1984,7). Another striking feature is that much research simply concludes that results differ and gives little insight as to why. There are several reasons for this. Some studies use real data and then create zones from these – the million correlation coefficients of Openshaw and Taylor (1979) are based on re-grouping areal voting data for Iowa. It is then difficult to know what the ‘true’ relationship is and to isolate what is producing the results. Other studies use simulated data of known properties but are unrealistic in not considering different forms of spatial configuration in determining the results (Amrhein, 1995). A more pervasive problem is that studies have not been set in a general modelling framework (Wrigley, 1995) resulting in no goodness-of-fit measure for determining what is the ‘best’ zonal arrangement. Here we see the MAUP as an opportunity to vary scale and zonation to bring the patterns into focus and thereby gain insight to the underlying processes producing segregation. However, we lack a statistic to provide guidance when this focus has been achieved. This is especially difficult in real data as there may be multiple and unknown configurations where spatial processes are operating; working at a single scale may miss what is going on.

Three other issues are methodologically important for us. Firstly, in assessing change there is a need to incorporate statistical uncertainty. A decreasing population per unit area will result in smaller numbers which will emphasize stochastic variation upwardly biasing segregation estimates. Such analysis of change must be multiscalar as countervailing tendencies (polarization at one scale alongside dispersal at another) may cloud the picture of what is happening (Haggett, 1965). Secondly, ethnicity in the 21st city is not a binary variable, with multiple ethnicities characterising contemporary society. This diversity requires simultaneous modelling of multiple groups which results in smaller numbers as ethnicity is disaggregated, making a modelling approach particularly valuable. The third aspect is that there may be multiple influences on location decision-making so that residential segregation may be differently driven by ethnicity and class. It is vital to have a methodology for assessing which are the key influences on segregation and to do so for multiple groups at multiple scales. We have addressed all three in previous work (Johnston, Jones et al., 2016; Jones et al. 2017) and here we concentrate on spatializing the modelling approach to measure clustering while examining sensitivity to the MAUP for a single time cross-section for just two ethnic groups.

Our agenda should now be clear. A modelling approach is required that can handle and separate stochastic variation from ‘true’ pattern. It must work at multiple scales simultaneously distinguishing unevenness and spatial clustering. It should allow sensitivity analysis to assess which alternative spatial arrangements bring the patterns into focus at one scale net of others. This needs to be set in an inferential framework where the uncertainty of parameters is evaluated and where appropriate measures of goodness of fit allow the preferment of some spatial arrangement over others. The method of comparison must consider model complexity in coming to a parsimonious judgement on models with many potential parameters. The method must also be computationally feasible on realistically-sized data sets. "