One of the limits when creating segregation measures for ordinal vari-ables is computational performance. For example, exposure and clustering may be easy to code for a few (ideally two) categories, but become di cult to adapt in the case of a continuous ordinal variable. One strategy could be to discretise the variable into many categories and use a categorical algorithm, however this is likely to prove di cult for the computer. As a matter of fact, the operation of sampling the ordinal variable is added to the operation of sampling the two natural spatial dimensions. Even the global measures such as the NSI, CGI, OITI and OVRI may require intensive computations if cus-tom neighbourhoods are created and the performance declines steeply when the scale is decreased. In contrast, the moment strategy in the multifractal methodology ensures that complex operations inside each neighbourhood are never required. The process remains manageable at all scales.
Chapter V has proved that multifractal analysis is not sensitive to the modi able areal unit problem contrary to the global segregation measures.
Section 2.1 in particular shows that it can yield more coherent and resilient results. Reardon and O'Sullivan (2004); Reardon et al. (2006) additionally suggest that the measure should be insensitive to order preserving change in the distribution. Sections 2.1 and 2.2 of chapter IV demonstrate that the multifractal spectrum is largely dominated by the spatial repartition of the variable (as identi ed in gure IV.18, IV.20 and IV.23). However, we argue that its ability to pick up dierences in the value distribution of the variable when appropriate, such as when the distribution is randomized ( gures IV.20 and IV.23), is an advantage.
It was also required that the measure would have scale interpretability, meaning that it reaches 0 when group proportions are uniform and the max-imum value only when there is no proximity between any two members of each group. Instead, our measure reaches its maximum value (the fractal dimension of its support) when the spatial distribution is uniform, and its minimum value (0) when groups are isolated. This situation is equivalent, up to an easy inversion. Other criteria that were devised for multi-categorical segregation, but that can have an interpretation for ordinal segregation are brie y discussed in relation to our method in appendix B. In conclusion,
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the multifractal approach fares well against the generally agreed criteria in the eld, while oering a computationally e cient improvement over clas-sical measures, particularly since it has an important and unique capacity to quantify the exposure and clustering dimensions of segregation, even for ordinal variables, as identi ed in chapter V.
This ful ls our rst and primary goal, which was to improve the charac-terisation of the spatial patterns that emerge from inequality over the current existing methods. We also aimed at consolidating the nascent literature that suggests that the multifractal framework is well adapted to study cities (and may be more truthful to reality than monofractal analysis). We have proved in chapter IV that cities often do present the scaling properties that make the multifractal framework relevant. Indeed, we have identi ed particularly con-vincing multifractal patterns for real estate measures in several cities spread across the globe: London, Paris, New York and Kyoto. We have also shown that income and energy consumption are good candidates for multifractal analysis. However, all measures are not suitable, as section 3.1 of chapter IV has illustrated for the diversity of land use measure.
We now recall the main practical results of chapters IV and V. Sections 2.1 and 2.2 of chapter IV have enlightened a decrease in multifractality with modernisation that can be understood as an arguably positive reduction of inequality, but also as a negative loss of diversity that can imply situations where all properties become unaordable. These two sections have also pre-sented convincing similarities in the independent evolution of the spatial repartition of real estate measures across three dierent continents during the 20th century. Section 3 of chapter IV has revealed discrepancies be-tween the spectra of income and the spectra of the other measures. Finally, Section 2.2 of chapter IV, complemented by section 2.2 of chapter V, led to an observation that in New York the rich tend to live clustered and iso-lated from the poor, contrary to London which presents a dierent pro le composed of clustered, but balanced in terms of economic class exposure, neighbourhoods. As a matter of fact, there has been an active strategy to orient London's recent development in that direction. More than just observ-ing these situations, multifractal analysis has allowed to obtain a detailed quanti cation containing a full range of values.
Outside from the academic analysis of current and past situations, the fact that the information contained inside the spectra is delivered over this range of values (rather than through a unique global parameter) makes it particularly useful for practical urban planning. We can not only observe if inequality is evolving through time or space, but we can also observe how it is evolving and observe which end of the distribution is most aected by
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the evolution. Ideally, we would want to go further and be able to predict how a development scenario will impact inequality before it is actually im-plemented. A natural extension to this thesis would therefore be to couple the methodology with urban modelling tools. We make a step in that direc-tion in appendix A where we present an agent-based model that emphasises the role of economic segregation in the process of choosing a domestic prop-erty to buy. Contrary to other locational models that are oriented towards relating the population distribution to economic e ciency, we focus on build-ing a model that correctly represents the evolution of economic segregation following a new urban development.
Another natural extension of this work could consist in nding other ap-plications to the identi ed multifractal urban patterns. One could think for example of testing if industrial energy consumption measures are multifrac-tal, and if the spectrum can be associated with economic e ciency.
In conclusion, we have devised a new computationally e cient method-ology to study the spatial patterns emerging from inequality in the urban environment that solves many of the issues usually encountered in this eld.
It provides richer insights due to its results being presented as a continuous spectrum instead of one global parameter. We have taken particular care to develop an interpretative framework for our results focused on studying inequality and have presented the analysis in a way that aims at emphasiz-ing the advantages of the multifractal methodology for comparemphasiz-ing diverse situations and their evolutions through time. This is done to invite planners and policy makers to incorporate the impact on inequality induced by their future projects.
APPENDIX A