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Advances and directions of research in Symbolic Data Analysis

Advances and directions of research in Symbolic Data Analysis. E. Diday CEREMADE. Paris–Dauphine University. June 14, 2014 SDA Workshop – Tutorial Academica Sinica. OUTLINE. PART 1 BUILDING SYMBOLIC DATA PART 2 OPEN DIRECTION OF RESEARH . PART 3

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Advances and directions of research in Symbolic Data Analysis

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  1. Advances and directions of research in Symbolic Data Analysis E. Diday CEREMADE. Paris–Dauphine University June 14, 2014 SDA Workshop – Tutorial AcademicaSinica

  2. OUTLINE • PART 1 BUILDING SYMBOLIC DATA • PART 2 OPEN DIRECTION OF RESEARH. • PART 3 AN ILLUSTRATIVE EXAMPLE : TRACHOMA STUDY

  3. PART 1 Building Symbolic data: . Some principles . Ten kinds of Symbolic Variables

  4. Someprinciples • Symbolic Data are not given or foundlike standard or complex data. • They are buildfrom classes of individuals in case of standard data or from classes of severalkinds of individualsin case of complex data. • Symbolic data are not only distributions.

  5. Ten examples of Symbolic variables

  6. PART 2: OPEN DIRECTION OF RESEARH • Building Symbolic Data. • Extending methods to Symbolic Data • Four theorems of convergence needed to be proved on any extended method to Symbolic Data • Models of models • Law of parameters of laws and Laws of vectors of laws. • Copulas needing. • Optimisation in non supervisedlearning (hierarchical and pyramidal clustering).

  7. BUILDING SYMBOLIC DATA The discretization of the initial classical variables has to be donne in order to optimize at least three kinds of aims: 1) The quality of the obtained distribution It can be measured by model selection criteria BIC, MDL, AIC, MML like or other criterion of this kind based on the likelihood estimation. Flat distributions are not interesting so criterion of “information” like (Sum of pi Log(pi)) can be used. 2) The level of discrimination between the obtained symbolic description. It can be measured by the sum of their dissimilarities two by two. 3) The correlation between the bins associated to the different symbolic variables (metabins).

  8. EXTENDING METHODS ON SYMBOLIC DATA: MUCH REMAINS TO BE DONE • Graphical visualisation of Symbolic Data • Correlation, Mean, Mean Square, distribution of a symbolic variables. • Dissimilarities between symbolic descriptions, K-nearest neighbourg • Clustering, spatial hierarchies and pyramids of symbolic descriptions, S-KohonenMappings • S-Decision Trees • S-Principal Component, Discriminant FactorialAnalysis • S- Canonical Analysis, Regression • S- Bayesiantrees, Multilevelanalysis, Variance Analysis, Vector Support Machine, Mixture decomposition, MultilevelAnalysis, Learnong machine by groups. • Etc...

  9. FOUR THEOREM TO BE PROVED ON ANY EXTENDED METHOD TO SYMBOLIC DATA M(n, k) issupposed to be a SDA methodwhere k is the number of classes obtained on n initial individuals. THEOREME 1 : If the k classes are fixed and n tends towardsinfinity, then M(n, k) converges towards a stable position. THEOREME 2 : If k increasesuntilgetting a single individual by class, then M(n, k) converges towards a standard method. THEOREME 3 : If k and n increasesimulataneouslytowardsinfinity, then M(n, k) converges towards a stable position. THEOREME 4 If the k lawsassociated to the k classes are considered as a sample of a law of laws, then M(n, k) applied to thissample converges to M(n, k) applied to thislaw. Exemples : Théorème 1: il a été démontré dans Diday, Emilion (CRAS, Choquet 1998), pour les treillis de Galois: à mesure que la taille de la population augmente les classes (décrites par des vecteurs de distributions), s’organisent dans un treillis de Galois qui converge. Emilion (CRAS, 2002) donne aussi un théorème dans le cas de mélanges de lois de lois utilisant les martingales et un modèle de Dirichlet. Théorème 2: Par ex, l’ACP classique MO est un cas particulier de l’ACP notée M(n, k) construite sur les vecteurs d’intervalles. Théorème 3: c’est le cadre de données qui arrivent séquentiellement (de type « Data Stream ») et des algorithmes de type one pass (voir par ex Diday, Murty (2005)). Théorème 4: Dans le cas d'une classification hiérarchique ou pyramidale 2D, 3D etc. la convergence signifie que les grands paliers et leur structure se stabilisent. Dans le cas d’une ACP la convergence signifie que les axes factoriels se stabilisent.

  10. MODELS OF MODELS ARE NEEDED Teams X’1 X’j C1 Ci Ck Table 1 Table 2 A symbolic data (age of Messi team) A number (age of Messi) • Xj is a standard random numerical variable • X’j is a random variable with histogram value • Question: if the law of Xj is given what is the law of X’j? (Dirichlet models useful).

  11. Law of parameters of laws Find the law of the parameters for eachsymbolic variable Yj and the law of the associatedvector of parameterslaws. Estimated parameters of the law Xijof the class Ci Example: Parij = ( ij , ϭij) Example: If f is the density of the parameters of the uniformlaw of intervals and g the law of intervalsthen: g(y) = 6 p f(x) / j = 1,p(x jmax -x jmin) (Diday à SFC 2011 Orléans).

  12. Copulas needing in Symbolic Data Analysis • In eachll of the symbolic data table, we supose to have a densityfunction f(i,j) • f(i, j, j’) is the joint probability of the variables j and j’ for the individual i. • In case of independency , we have • f(i, j, j’) = f(i, j’). f(i, j’), • If thereis no independancy: • f(i, j, j’) = Copula(f(i, j’). f(i, j’)) • Aim of Copula model in SDA: • find the Copula whichminimise the differenceswith the joint. • In order to avoid the restriction to independencyhypotheses and to reduce the cost of f(i, j, j’) computing. • In thatway we canobtain a Copular PCA, Regression, Canonical, Analysis, ….

  13. Bi-plot of histogram variables Y1 Y2 C1 • The joint probability can be inferred by a copula model Ci Copula Ck

  14. x1 x2 x5 x3 x4 x2 x3 x4 x1 x5 Optimisation in clustering d is the given dissimilarity Ultrametric dissimilarity = U Hierarchies W = |d - U | Each class is described by symbolic data Pyramides Robinsonian dissimilarity = R 3D Spatial Pyramid S1 W = |d - R | S2 Yadidean dissimilarity = Y C3 C2 A 1B1 C1 W = |d - Y |

  15. PART 3 ILLUSTRATIVE EXAMPLE ON TRACHOMA • Trachoma, caused by repeated ocular infections with Chlamydia tra- chomatis whose vector is a fly, is an important cause of blindness in the world. This study was conducted in Mali. • The first aim was to choose among three antibiotic strategies those with the best cost-effectiveness ratio. • The second aim was to find the demographic and environmental parameters on which we could try to intervene.

  16. Symbolic Table of Degradation The classes 0x0, 0x1, 1x0 and 1x1 of degradation (0 = healthy , 1 = ill at the (beginning x end) of the one year study. These classes are directly issued from the given data and not from a clustering process. INTERPRETATION: The THIRD STRATEGY is the most frequent in the worth class (0x1). Nevertheless we cannot conclude that it is the worth strategy as the degradation can come from the environmental of this class 0X1.

  17. The third strategy remains the worse in three homogeneous environmental conditions obtained by clustering

  18. PCA OF THE SYMBOLIC DATA TABLE OF DEGRADATION A Standard PCA is applied on the categories of the symbolic fariables (considered as numerical variables) of the “degradation symbolic data table” on which the piecharts of the strategies are projected.

  19. ANY PIECHART oF SYMBOLIC VARIABLE CAN BE SEEN: Borehole well

  20. CORRELATION CIRCLE OF ALL THE CATEGORIES ( ie BINS) OF THE SYMBOLIC VARIABLESON THE FIRST AXIS.

  21. SYMBOLIC VARIABLES PROJECTION IN HYPERCUBE QUADRANT SYMBOLIC VARIABLES PROJECTION

  22. THE SDA STRATEGY • The classes are generally not obtained from a clustering process. The classes 0x0, 0x1, 1x0 and 1x1 of degradation are directly issued from the given data. • the clustering strategy in SDA is not much used to build the classes to be studied, it is mainly used in order to show dependencies or independencies between groups of symbolic variables. Here the environmental conditions

  23. CONCLUSION Classical, Complex and Big Data are GIVEN. Symbolic data are BUILD. Complex and Big Data data can be simplified and reduced in Symbolic Data. The quality the obtained Symbolic Data can be improved by optimization of several criteria. The number of papers for building Symbolic Data remains few. Much remains to do in this direction. Symbolic data are not only distributions. SYMBOLIC DATA ARE THE NUMBERS OF THE FUTURE.

  24. Basic books and papers • Bock H.H., Diday E. (editors and co-authors) ( 2000): Analysis of Symbolic Data.Exploratory methods for extracting statistical information from complex data. Springer Verlag, Heidelberg, 425 pages, ISBN 3-540-66619-2. • L. Billard, E. Diday (2003) "From the statistics of data to the statistic of knowledge: Symbolic Data Analysis". JASA . Journal of the American Statistical Association. Juin, Vol. 98, N° 462. • Billard, L. and Diday, E. (2006). Symbolic Data Analysis: Conceptual Statistics and Data Mining. 321 pages. Wiley series in computational statistics. Wiley, Chichester, ISBN 0-470-09016-2. • E. Diday, M. Noirhomme (eds and co-authors) (2008) “Symbolic Data Analysis and the SODAS software”. 457 pages. Wiley. ISBN 978-0-470-01883-5. • Noirhomme-Fraiture, M. and Brito, P. (2012) Far beyond the classical data models: symbolic data analysis. Statistical Analysis and Data Mining 4 (2), 157-170. • Lazare N. (2013) "Symbolic Data Analysis". CHANCE magazine. Editor’s Letter – Vol. 26, No. 3.

  25. In Building SymbolicData • Stéphan V., Hébrail G.,Lechevallier Y. (2000) « Generation of symbolicobjectsfromrelationnal data base ».Chapter in book : Analysis of Symbolic Data: Exploratory Methods for Extracting Statistical Information from Complex Data (eds. H.-H.Bock and E. Diday). Springer-Verlag, Berlin, 103-124. • Chiun-How, K., Chih-Wen, O., Yin-Jing, T., Chuan-kai, Yang, Chun-houh, Chen (2012) “A Symbolic Database for TIMSS”. Arroyo J., Maté C., Brito P. Noihomme M. eds, 3rd Workshop in Symbolic Data Analysis. Universidad Compiutense de Madrid. http://www.sda-workshop.org/. • E. Diday, F. Afonso, R. Haddad (2013) : “The symbolic data analysis paradigm, discriminate discretization and financial application”. In Advances in Theory and Applications of High Dimensional and Symbolic Data Analysis, HDSDA 2013. Revue des Nouvelles Technologies de l'Information vol. RNTI-E-25, pp. 1-14

  26. IN SYMBOLIC DATA ANALYSIS • In Pricipal Component Analysis Cazes P., Chouakria A., Diday E., Schektman Y. (1997). Extension de l’analyse en composantes principales à des données de type intervalle, Rev. Statistique Appliquées, Vol. XLV Num. 3, pp. 5-24, France. 29. Cazes P. (2002) Analyse factorielle d’un tableau de lois de probabilité. Revue de statistique appliquée, tome 50, n0 3. Diday E. (2013) "Principal Component Analysis for bar charts and Metabins tables". Statistical Analysis and Data Mining. Article first published online: 20 May 2013. DOI: 10.1002/sam.11188. 2013 Wiley. Statistical Analysis and Data Mining,6,5, 403-430. Ichino, M. (2011). The quantile method for symbolic principal component analysis. Statistical Analysis and Data Mining, Wiley. 184-198. Makosso-Kallyth S. and Diday E. (2012) Adaptation of interval PCA to symbolic histogram variables. Advances in Data Analysis and Classification (ADAC). July, Volume 6, Issue 2, pp 147-159. Rademacher, J., Billard , L., (2012) Principal component analysis for interval data. Wiley interdisciplinary Reviews: Computational Statistics .Volume 4, Issue 6, pp. 535–540. Shimizu N., Nakano J. (2012) Histograms Principal Component Analysis. Arroyo J., Maté C., Brito P. Noihomme M. eds, 3rd Workshop in Symbolic Data Analysis. Universidad Compiutense de Madrid. http://www.sda-workshop.org/ Wang H., Guan R., Wu J. (2012a). CIPCA: Complete-Information-based Principal Component Analysis for interval-valued data, Neurocomputing, Volume 86, Pages 158-169.

  27. In SymbolicForecasting Arroyo, J. and Maté, C. (2009). Forecasting histogram time series with k-nearest neighbors' methods. International Journal of Forecasting 25, 192–207. García-Ascanio, C.; Maté, C. (2010). Electric power demand forecasting using interval time series: A comparison between VAR and iMLP. Energy Policy 38, 715-725 Han, A., Hong, Y., Lai, K.K., Wang, S. (2008). Interval time series analysis with an application to the sterling-dollar exchange rate. Journal of Systems Science and Complexity, 21 (4), 550-565. He, L.T. and C. Hu (2009). Impacts of Interval Computing on Stock Market Variability Forecasting. Computational Economics 33, 263-276. • In Symbolic rule extraction Afonso, F. et Diday, E. (2005). Extension de l’algorithme Apriori et des regles d’association aux cas des donnees symboliques diagrammes et intervalles. Revue RNTI, Extraction et Gestion des Connaissances (EGC 2005), Vol. 1, pp 205-210, Cepadues, 2005.

  28. In Symbolic Decision Tree Ciampi, A., Diday, E., Lebbe, J., Perinel, E. et Vignes, R. (2000). Growing a tree classifier with imprecise data. Pattern Recognition letters 21: 787-803. Mballo C., Diday E. (2006)  The criterion of Smirnov-Kolmogorov for binary decision tree : application to interval valued variables. Intelligent Data Analysis. Volume 10, Number 4 . pp 325 – 341 Winsberg S., Diday E., Limam M. (2006). A tree structured classifier for symbolic class description. Compstat 2006. Physica-Verlag. Bravo, M. et Garcia-Santesmases, J. (2000). Symbolic Object Description of Strata by Segmentation Trees, Computational Statistics, 15:13-24, Physica-Verlag.

  29. In Clustering • De Carvalho F., Souza R., Chavent M., and Lechevallier Y. (2006) Adaptive Hausdorff distances and dynamic clustering of symbolic interval data. Pattern Recognition Letters Volume 27, Issue 3, February 2006, Pages 167-179. • De Souza R.M.C.R, De Carvalho F.A.T. (2004). Clustering of interval data based on City-Block distances. Pattern Recognition Letters, 25, 353–365. • Diday E. (2008) Spatial classification. DAM (Discrete Applied Mathematics) Volume 156, Issue 8, Pages 1271-1294. • Diday, E., Murty, N. (2005) "Symbolic Data Clustering" in Encyclopedia of Data Warehousing and Mining . John Wong editor . Idea Group Reference Publisher. • Irpino, A. and Verde, R. (2008): Dynamic clustering of interval data using a Wasserstein-based distance. Pattern Recognition Letters 29, 1648-1658. • In Multidimensional Scaling • Terada, Y., Yadohisa, H. (2011) Multidimensional scaling with hyperbox model for percentile dissimilarities, In: Watada, J., Phillips-Wren, G., Jain, L. C., and Howlett, R. J. (Eds.): Intelligent Decision Technologies Springer Verlag, 779–788 • Groenen, P.J.F.,Winsberg, S., Rodriguez, O., Diday, E. (2006). I-Scal: Multidimensional scaling of interval dissimilarities. Computational Statistics and Data Analysis 51, 360–378.

  30. Some Symbolic Data Analysis references • In Self Organizing map • Hajjar C., Hamdan H. (2011). Self-organizing map based on L2 distance for interval-valued data. In SACI 2011, 6th IEEE International Symposium on Applied Computational Intelligence and Informatics (Timisoara, Romania), pp. 317–322.P. • In Dissimilarities between Symbolic Data • Kim, J. and Billard, L. (2013): Dissimilarity measures for histogram-valued observations, Communications in Statistics-Theory and Method, 42, 283-303. • Verde, R., Irpino, A. (2010). Ordinary Least Squares for Histogram Data Based on Wasserstein Distance, in: Proc. COMPSTAT’2010, Y. Lechevallier and G.Saporta (Eds).PP.581-589. PhysicaVerlag Heidelberg.

  31. In Regression and Canonical analysis extended to Symbolic Data • Dias, S., Brito, P., (2011). A New Linear Regression Model for Histogram-Valued Variables. In Proceedings of the 58th ISI World Statistics Congress (Dublin, Ireland). • Lauro, C., Verde, R. , Irpino, A. (2008). Generalized canonical analysis, in: Symbolic Data Analysis and the Sodas Software, E. Diday and M. Noirhomme. Fraiture (Eds.), 313-330, Wiley, Chichester. • Tenenhaus A., Diday E., Emilion R., Afonso F. (2013) Regularized General Canonical Correlation Analysis Extended To Symbolic Data. ADAC (publication on the way). • Neto, E.A, De Carvalho F.A.T. (2010). Constrained linear regression models for symbolic interval-valued variables. Computational Statistics and Data Analysis 54, 333-347. • Wang H., Guan R., Wu J. (2012c). Linear regression of interval-valued data based on complete information in hypercubes, Journal of Systems Science and Systems Engineering, Volume 21, Issue 4, Page 422-442.

  32. In Symbolic Data Models referencies • P. Bertrand, F. Goupil (2000) “ Descriptive Statistics for symbolic data“ . In H.H. Bock, E. Diday (Eds) “Analysis of Symbolic Data “. Springer-Verlag, pp. 106-124.  • Brito, P. and Duarte Silva, A.P. (2012). Modelling interval data with Normal and Skew-Normal distributions. Journal of Applied Statistics, 39 (1), 3-20. • E. Diday, M. Vrac (2005) "Mixture decomposition of distributions by Copulas in the symbolic data analysis framework". Discrete Applied Mathematics (DAM). Volume 147, Issue1, 1 April, pp. 27-41. • E. Diday (2011) Modélisation de données symboliques et application au cas des intervalles. Journées Nationales de la Société Francophone de Classification. Orléans • E. Diday (2002) “From Schweizer to Dempster: mixture decomposition of distributions by copulas in the symbolic data analysis framework” IPMU 2002, July, Annecy, France • Diday E., Emilion R. (1997) "Treillis de Galois Maximaux et Capacités de Choquet" . C.R. Acad. Sc. t.325, Série 1, p 261-266. Présenté par G. Choquet en Analyse Mathématiques • Diday E., R. Emilion (2003) Maximal and stochastic Galois lattices. Discrete appliedMath. Journal. Vol. 27 (2), pp. 271-284. • Emilion R., Classification et mélanges de processus. C.R. Acad. Sci. Paris, 335, série I, 189-193 (2002). • Emilion R., Unsupervised Classification and Analysis of objectsdescribed by nonparametricprobability distributions. StatisticalAnalysis and Data Mining (SAM), Vol 5, 5, 388-398 (2012). • J. Le-Rademacher, L. Billard (2011) “Likelihood functions and some maximum likelihood estimators for symbolic data”. Journal of Statistical Planning and Inference 141 1593–1602. Elsevier. • T. Soubdhan, R. Emilion, R. Calif (2009) “Classification of daily solar radiation distributions”. SolarEnergy 83 (2009) 1056–1063. Elsevier.

  33. In SDA Industrial Applications • Afonso F., Diday E., Badez N., Genest Y. (2010) Symbolic Data Analysis of Complex Data: Application to nuclear power plant. COMPSTAT’2010 , Paris. • Bezerra B., Carvalho F. (2011) Symbolic data analysis tools for recommendation systems. Knowl. Inf. Syst 01/2011; 26:385-418. DOI:10.1007/s10115-009-0282-3. • Bouteiller V., Toque C., A., Cherrier J-F., Diday E., Cremona C. (2011) Non-destructive electrochemical characterizations of reinforced concrete corrosion: basic and symbolic data analysis. Corros Rev . Walter de Gruyter • Berlin • Boston. DOI 10.1515/corrrev-2011-002. • Courtois, A., Genest, G., Afonso, F., Diday, E., Orcesi, A., (2012) In service inspection of reinforced concrete cooling towers – EDF’s feedback ,IALCCE 2012, Vienna, Austria • Cury, A., Crémona, C., Diday, E. (2010). Application of symbolic data analysis for structural modification assessment. Engineering Structures Journal. Vol 32, pp 762-775. • ChristelleFablet, Edwin Diday, Stephanie Bougeard, Carole Toque, Lynne Billard (2010). Classification of Hierarchical-Structured Data with Symbolic Analysis. Application to Veterinary Epidemiology. COMPSTAT’2010 , Paris. • Haddad R., Afonso F., Diday E., (2011) Approche symbolique pour l'extraction de thématiques: Application à un corpus issu d'appels téléphoniques. In actes des XVIIIèmes Rencontres de la Sociéte francophone de Classification. Universitéd'Orléans • Laaksonen, S. (2008). People’s Life Values and Trust Components in Europe - Symbolic Data Analysis for 20-22 Countries. In. Edwin Diday and Monique Noirhomme-Fraiture, “Symbolic Data Analysis and the SODAS Software", Chapter 22, pp. 405-419. Wiley and Sons: Chichester, UK. • Quantin C., Billard L., Touati M., Andreu N., Cottin Y., Zeller M., Afonso F., Battaglia G., Seck D., Le Teuff G., and Diday E.. (2011) Classification and Regression Trees on Aggregate Data Modeling: An Application in Acute Myocardial Infarction. Journal of Probability and Statistics Volume 2011 (2011), 19 pages. • Terraza V, Toque C. (2013) Mutual Fund Rating: A Symbolic Data Approach. In "Understanding Investment Funds Insights from Performance and Risk Analysis". Edited by VirginieTerraza and HeryRazafitombo . Economics & Finance Collection 2013. The Palgrave Macmilan editor. UK. • He, L.T. and C. Hu (2009). Impacts of Interval Computing on Stock Market Variability Forecasting. Computational Economics 33, 263-276. • E. Diday, F. Afonso, R. Haddad (2013) : The symbolic data analysis paradigm, discriminate discretization and financial application, in Advances in Theory and Applications of High Dimensional and Symbolic Data Analysis, HDSDA 2013. Revue des Nouvelles Technologies de l'Information vol. RNTI-E-25, pp. 1-14 • Han, A., Hong, Y., Lai, K.K., Wang, S. (2008). Interval time series analysis with an application to the sterling-dollar exchange rate. Journal of Systems Science and Complexity, 21 (4), 550-565.

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