Abstract
Multi-objective evaluation is a necessary aspect when managing complex systems, as the intrinsic complexity of a system is generally closely linked to the potential number of optimization objectives. However, an evaluation makes no sense without its robustness being given (in the sense of its reliability). Statistical robustness computation methods are highly dependent of underlying statistical models. We propose a formulation of a model-independent framework in the case of integrated aggregated indicators (multi-attribute evaluation), that allows to define a relative measure of robustness taking into account data structure and indicator values. We implement and apply it to a synthetic case of urban systems based on Paris districts geography, and to real data for evaluation of income segregation for Greater Paris metropolitan area. First numerical results show the potentialities of this new method. Furthermore, its relative independence to system type and model may position it as an alternative to classical statistical robustness methods.
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Notes
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We design by Multi-Objective Evaluation all practices including the computation of multiple indicators of a system (it can be multi-objective optimization for system design, multi-objective evaluation of an existing system, multi-attribute evaluation; our particular framework corresponds to the last case).
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The discrepancy is defined as the L2-norm of local discrepancy which is for normalized data points \(\mathbf {X}=(x_{ij})\in \left[ 0,1\right] ^d\), a function of \(\mathbf {t}\in \left[ 0,1\right] ^d\) comparing the number of points falling in the corresponding hypercube with its volume, by \(disc(\mathbf {t}) = \frac{1}{n}\sum _i \mathbbm {1}_{\prod _j x_{ij}<t_j} - \prod _j t_j\). It is a measure of how the point cloud covers the space.
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References
Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004)
Newman, M.E.J.: Complex systems: a survey (2011). arXiv:1112.1440
Haken, H., Portugali, J.: The face of the city is its information. J. Env. Psychol. 23(4), 385–408 (2003)
Picon, A.: Smart cities: théorie et critique d’un idéal auto-réalisateur. B2 (2013)
Souami, T.: Ecoquartiers: secrets de fabrication. Scrineo (2012)
Bavoux, J.-J., Beaucire, F., Chapelon, L., Zembri, P.: Géographie des transports. Paris (2005)
Carver, J.S.: Integrating multi-criteria evaluation with geographical information systems. Int. J. Geogr. Inf. Syst. 5(3), 321–339 (1991)
Jégou, A., Augiseau, V., Guyot, C., Judéaux, C., Monaco, F.-X., Pech, P. et al.: L’évaluation par indicateurs: un outil nécessaire d’aménagement urbain durable?. réflexions à partir de la démarche parisienne pour le géographe et l’aménageur. Cybergeo: European Journal of Geography (2012)
Launer, R.L., Wilkinson, G.N.: Robustness in statistics. Academic Press (2014)
Deb, K., Gupta, H.: Introducing robustness in multi-objective optimization. Evol. Comput. 14(4), 463–494 (2006)
Barrico, C., Antunes, C.H.: Robustness analysis in multi-objective optimization using a degree of robustness concept. In: Evolutionary Computation, 2006. CEC 2006. IEEE Congress on, pp. 1887–1892 (2006)
Brunsdon, C., Fotheringham, S., Charlton, M.: Geographically weighted regression. J. R. Stat. Soc. Ser. D (The Statistician) 47(3), 431–443 (1998)
Dobbie, M.J., Dail, D.: Robustness and sensitivity of weighting and aggregation in constructing composite indices. Ecol. Indic. 29, 270–277 (2013)
Ali, A., Carneiro, I., Dussarps, L., Guédel, F., Lamy, E., Raimbault, J., Viger, L., Cohen, V., Aw, T., Sadeghian, S.: Les eco-quartiers lus par la mobilité : vers une évaluation intégrée. Technical report, Ecole des Ponts ParisTech (2014). June
Mangin, D., Panerai, P.: Projet urbain. Parenthèses (1999)
Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi–Monte Carlo Integration. Cambridge University Press (2010)
Livet, P., Muller, J.-P., Phan, D., Sanders, L.: Ontology, a mediator for agent-based modeling in social science. J. Artif. Soc. Soc. Simul. 13(1), 3 (2010)
Wang, J.-J., Jing, Y.-Y., Zhang, C.-F., Zhao, J.-H.: Review on multi-criteria decision analysis aid in sustainable energy decision-making. Renew. Sustain. Energy Rev. 13(9), 2263–2278 (2009)
Varet, S.: Développement de méthodes statistiques pour la prédiction d’un gabarit de signature infrarouge. Ph.D. thesis, Université Paul Sabatier-Toulouse III (2010)
Niederreiter, H.: Discrepancy and convex programming. Annali di matematica pura ed applicata 93(1), 89–97 (1972)
DT QGis. Quantum gis geographic information system. Open Source Geospatial Foundation Project (2011)
R Core Team: R language definition (2000)
Jessica, F., Delphine, D., Olivier, R., Astrid, J.: Dicedesign-package. Designs of Computer Experiments, pp. 2 (2009)
Ram, K.: Git can facilitate greater reproducibility and increased transparency in science. Source Code Biol. Med. 8(1), 7 (2013)
Bennett, J.: OpenStreetMap. Packt Publishing Ltd. (2010)
Louf, R., Barthelemy, M.: Patterns of residential segregation (2015). arXiv:1511.04268
Gilli, F., Offner, J.-M.: Paris, métropole hors les murs: aménager et gouverner un Grand Paris. Sciences Po, les presses (2009)
Guérois, M., Le Goix, R.: La dynamique spatio-temporelle des prix immobiliers à différentes échelles: le cas des appartements anciens à paris (1990–2003). Cybergeo: European Journal of Geography (2009)
Tivadar, M., Schaeffer, Y., Torre, A., Bray, F.: Oasis–un outil d’analyse de la ségrégation et des inégalités spatiales. Cybergeo: European Journal of Geography (2014)
Kwan, M.-P.: Space-time and integral measures of individual accessibility: a comparative analysis using a point-based framework. Geogr. Anal. 30(3), 191–216 (1998)
Banos, A.: Pour des pratiques de modélisation et de simulation libérées en géographie et shs. Thèse d’Habilitation à Diriger des Recherches, UMR CNRS 8504 Géographie-Cités, ISCPIF, Décembre (2013)
Dragomir, S.S.: The ostrowski’s integral inequality for lipschitzian mappings and applications. Comput. Math. Appl. 38(11), 33–37 (1999)
Acknowledgements
The author would like to thank Julien Keutchayan (Ecole Polytechnique de Montréal) for suggesting the original idea of using discrepancy, and anonymous reviewers for the useful comments and insights.
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Raimbault, J. (2017). A Discrepancy-Based Framework to Compare Robustness Between Multi-attribute Evaluations. In: Fanmuy, G., Goubault, E., Krob, D., Stephan, F. (eds) Complex Systems Design & Management. CSDM 2016. Springer, Cham. https://doi.org/10.1007/978-3-319-49103-5_11
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DOI: https://doi.org/10.1007/978-3-319-49103-5_11
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