INTRODUCTION TO SYMBOLIC DATA ANALYSIS

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INTRODUCTION TO SYMBOLIC DATA ANALYSIS. E. Diday CEREMADE. Paris–Dauphine University. TUTORIAL: 13 June 2014 Activity Center, Academia Sinica, Taipei, Taiwan. OUTLINE. PART 1: BUILDING SYMBOLIC DATA FROM STANDARD OR COMPLEX DATA PART 2: SYMBOLIC DATA ANALYSIS

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### INTRODUCTION TOSYMBOLIC DATA ANALYSIS

E. Diday

TUTORIAL: 13 June 2014

Activity Center, Academia Sinica, Taipei, Taiwan

OUTLINE

PART 1: BUILDING SYMBOLIC DATA FROM STANDARD OR COMPLEX DATA

PART 2: SYMBOLIC DATA ANALYSIS

Is Symbolic Data Analysis a new paradigm?

.PART 3: OPEN DIRECTION OF RESEARH

PART 4: SDA SOFTWARES: SODAS, SYR and R

PART 5: INDUSTRIAL APPLICATIONS

### PART 1

BUILDING SYMBOLIC DATA FROM STANDARD OR COMPLEX DATA

What is a standard Data Table?
• It is a set of individuals (i.e. observations) described by a set of
• Numerical variables (as age, weight,..) or
• Categorical variables (as Nationality, club name,…).
• Example:
What are Complex Data?

Any data which cannot be considered as a

“standard observations x standard variables”

data table.

Example

The individuals are Towers of nuclear power plants described by

Table 1) Observations: Cracks .

Variables: Cracks description.

Table 2) Observations: corrosions.

Variables: corrosion description .

Table 3) Observations: vertices of a grid.

Variables: Gap depression from the ground.

.

Why considering classes of individuals as new individuals?

• Example:
• if wewish to know whatmakes a playerwins, we are interested by a standard data table where the individuals are the players (in rows) described (in columns) by their standard caracteristic variables.
• If ourwishisnow to know whatmakes a team wins, we are interested by a data table where the teams (in rows) are descibed by caracteristic variables of the teams taking care on the variability of the playersinsideeach team.
• The teams canbenowconsidered as new individuals of higherleveldescribed by symbolic variables taking care on the variability of the individualsinsideeach class.

X’1

X’j

C1

Ci

Ck

From standard data tables to symbolic data tables

Symbolic Data Table describing Teams (i.e. classes of individuals)

Standard data table describing Football players (individuals).

in each cell a number

(age) or

a category

(Nationality)

A symbolic data in each cell

(Bar chart age of the Messi Team)

NationalitiesBar chart

Weightinterval

Age Bar chart

Some columns are contigency tables

Here the variation (of weight, nationality, …) concerns the players of each team.

Therefore each cell can contain:

A number, an interval, a sequence of categorical values, a sequence of weighted values as a barchart, a distribution, …

THIS NEW KIND OF VARIABLES ARE CALLED « SYMBOLIC » BECAUSE THEY ARE NOT PURELY NUMERICAL IN ORDER TO EXPRESS THE INTERNAL VARIATION INSIDE EACH CLASS.

The failure is that in the actual practice

• Only the “individual” kind of observations is considered.
• Therefore these individual observations are only described by standard numerical and categorical variables.
The SDA paradigm shift

It is the transition

• from “individual observations” described by standard variables of numerical or categorical values.
• To “classes of individuals” (considered as “higher level observations”)
• Described by “symbolic variables”, of “symbolic values” (intervals, probability distributions, sets of categories or numbers, random variables,…)
• taking care on the variability inside the classes
• “symbolic values” can not be treated as numbers.

Building Symbolic Data needs three steps

First Step: we have a standard data table TAB1, where individuals are described by numerical or categorical random variables Yj .

Second step : we have a Table 2: where classes of individuals are described by random variables Y’j with random variables Yijvalue.

• Thirdstep: we have a symbolic data table Table3:where the random variables Yij are represented by:
• Probability distributions, histograms, bar charts, percentiles,…
• Intervals Min, Max, interquartilintervaletc.
• Set of numbers or categories
• Functions as Time Series.
VARIABLES
• Standard variables value:
• numerical (income, profit,…),
• categorical (Countries, Stock-Exchange places,..)
• Symbolic variables value:
• interval,
• bar chart,
• Histogram, etc.
How to build symbolic data from standard or complex data?
• How to categorize the numerical, ordinal, nominal ground variables, in order that the obtained symbolic histograms or barchart variables for each class?
• First: find the discretisation which discriminates as well as possible these classes.
• Second or simultaneously: Maximize the correlation between the bins.

SOME ADVANTAGES of SYMBOLIC DATA:

• Work at the needed level of generality without loosing variability.
• Reduce simple or complex huge data.
• Reduce number of observations and number of variables.
• Reduce missing data.
• Ability to extract simplified knowledge and decision from complex data.
• Solve confidentiality (classes are not confidential as individuals).
• Facilitate interpretation of results: decision trees, factorial analysis new graphic kinds.
• Extent Data Mining and Statistics to new kinds of data with much industrial applications.

SYMBOLIC DATA ANALYSIS TOOLS HAVE BEEN DEVELOPPED

• Graphical visualisation of Symbolic Data
• Correlation, Mean, Mean Square, distribution of a symbolic variables.
• Dissimilarities between symbolic descriptions
• Clustering of symbolic descriptions
• S-Kohonen Mappings
• S-Decision Trees
• S-Principal Component Analysis
• S-Discriminant Factorial Analysis
• S-Regression
• Etc...

Y2

x

x

x

x

Y1

• Observations data are uniformly distributed in the circle:
• no correlation between Y1 and Y2 for intial observations data.
• A correlation appears between the two variables for the centers of a given partition in 4 classes.

Symbolic Data Table

Transformation in classical data

Concern:

The initial variables are lost and the variation is lost!

Divisive Clustering or Decision tree

Symbolic Analysis

Classical Analysis

Weight

Max Weight

Any symbolic variable (set of bins variables) can be projected. Here the species variable.

* SYROKKO Company afonso@syrokko.com

The Symbolic Variables contributions are inside the smallest hyper cube containing the correlation sphere of the bins

Numerical versus symbolical space of representation

a1

a2

C1

Ci

Ck

a2i

a1i

b1

b1i

a1i

b2i

b1i

a2i

b2i

b2

Y1

Y2

C1

Ci

(Y1(Ci ), Y2(Ci )) = ([a1i , b1i ], ([a2i , b2i ])

Y2

Ck

x

Numerical representation of interval variables

Bi-plot of interval variables

b1

Ci

Y1

b2

Ci

x

x

a1

a2

Bi-plot of histogram variables

Y1

Y2

C1

• The joint probability can be inferred by a copula model

Ci

Copula

Ck

PART 3: OPEN DIRECTION OF RESEARH
• Models of models
• Law of parameters of laws
• Laws of vectors of laws.
• Copulas needed.
• Four general convergence theorem.
• Optimisation in non supervised learning (hierarchical and pyramidal clustering).
From lower level of individual observation to higher level observation of classes: higher level 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).

Why using copula models in Symbolic Data Analysis?

• 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 dépendancy:
• f(i, j, j’) = Copula(f(i, j’). f(i, j’))
• Aim of Copula model in SDA:
• find the Copula which minimises the differencewith the joint.
• In order to avoid the restriction to independencyhypotheses and to reduce the cost of f(i, j, j’) computing.

FOUR THEOREM TO BE PROVED FOR 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 one.

THEOREME 3 : I k and n increasessimulataneouslytowardsinfinity, then M(n, k) converges towards a stableposition.

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.

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 |

### PART 4: SDA SOFTWARES:

SODAS

RSDA

SYR

SoftwareTo build symbolic data from standard or complex data and analyze symbolic data, different software packages exist today.SODAS - academic free package, though registration required and a code needed for installation, http://www.info.fundp.ac.be/asso/sodaslink.htmMuch Symbolic data data bases can be found at http://www.ceremade.dauphine.fr/SODAS/RSDA: academic free packages are available on CRAN: oldemar.rodriguez@gmail.comSYR: professional package, see : afonso@syrokko.com

SODAS SOFTWARE

CARTE DE KOHONEN DE CONCEPTS

ANALYSE FACTORIELLE: ACP de variables à valeur intervalle

Superposition de deux deux étoîles associées à deux classes de la pyramides

Arbre de décision sur variables à valeur histogramme ou intervalle

The objective of SCLUST is the clustering of symbolic objects by a dynamic algorithm based on symbolic data tables. The aim is to build a partition of SO´s into a predefined number of classes. Each class has a prototype in the form of a SO. The optimality criterion used is based on the sum of proximities between the individuals and the prototypes of the clusters.

Pyramide classifiante

FROM DATA BASE TO SYMBOLIC DATA IN SODAS

Individuals

Classes

Relational Data Base

QUERY

Description of individuals

Columns: symbolic variables

Classes

Class description

Symbolic Data Table

Cells contain Symbolic Data

SYR SOFTWARE

Produce a Symbolic Data Table from complex data.

Manage Symbolic Data Tables: sort rows and columns by discriminant power

Analyse Symbolic data tables: SPCA,Sclustering…

Produce network, rules and decision trees.

SYR: SYMBOLIC DATA TABLE MANAGEMENT

SYMBOLIC DATA TABLE

• Sorting rows by min, max of intervals or frequencies of barchart is possible.
• Sorting variables by discriminate power of the concepts is also possible.

* SYROKKO Company eliezer@syrokko.com

each cell contains until 800.000 values.

Each cell is transformed in HISTOGRAM from a PROJECTION or from WAVELETS

Time Series Data table:Anomaly detection on a bridge

LCPC (Laboratoire Central Des Ponts et Chaussées) and SNCF Data

Sensor 1 Sensor 2 Sensor 3 …. Sensor N

Trains

HIERARCHICAL DATA*

Symbolic procedure

From numerical description of pigs to symbolic description of Farms

• Numerical variables

and

• Categorical variables

are transformed in Bar Chart of the frequencies based on 30 animals,

Or in interval value variables

19 variables

Description of pig respiratory diseases

125 farms x 30 animals

Median score (continuous var.)

Animal frequencies (categorical var.)

64 variables

Description of pig respiratory diseases

125 farms

*C. Fablet, S. Bougeard (AFSSA)

Step 1: Symbolic Description of Farms*

* SYROKKO Company afonso@syrokko.com

### Nuclear Power PlantFindCorrelationsBetween3 Standard Data Tables of Different observation units and different Variables

Craks

Cartography of the towel by a grid

Inspection machine

NUCLEAR POWER PLANT

Nuclear thermal power station

Inspection :

PB: FIND CORRELATIONS BETWEEN 3 CLASSICAL DATA TABLES OF DIFFERENT UNITS AND VARIABLES:

Table 1) Observations: Cracks . Variables: Cracks description.

Table 2) Observations: vertices of a grid. Variables: Gap deviation at different periods compared to the initial model position.

Table 3) Observations: vertices of a grid. Variables: Gap depression from the ground.

ARE Transformed in ONE Symbolic Data Table where the classes the towers. On this new table SDA can be applied.

Towers on PCA first axes
• PCA on chooosen symbolic variables
• Three clusters.visualisation
• Interval and bar chart variables can be seen..
• A network of the strongest links can be represented.

NETSYR results (SYR software)

### Telephone calls text mining in order to discover “themes” without using semantic

INITIAL DATA: 2 814 446 rows

Each calling session is called a document.

We start after lemmatisation with a table of

• 31454 documents
• 2258 words

Correspondence between documents and words.

First Steps:building overlapping clusters of documents and words: CLUSTSYR

70 x 2258

2 814 446 rows:

Correspondence documents, words

31454 documents x2258 words

70 Overlapping Clusters of Documents described by the tf-idf of 2258 words.

80 x 70

2258 x 70

80 overlapping clusters of words described by their tf-idf in the 70 clusters of Docs.

2258 Words described by their tf-idf on the 70 clusters of Docs.

Next step: STATSYREach cluster of documents is described by the 80 clusters of words called “themes”

Themes

Classes of documents

WORDS in Each Theme

GRAPHICAL REPRESENTATIONby NETSYR from SYR software

GRAPHICAL REPRESENTATION of themes ,

document classes, by Pie Charts

And their Bar chart description.

Overlapping

Clusters

SOCIAL NEWORK

Based on dissimilarities

ANNOTATION :

of Themes and Document classes

Moving, Zooming…

We obtain finally a clear representation of the main themes , their classes and their links : “failures”, “budget”,”addresses”, “vacation” etc..

A Survey on Security
• A sample of people of three regions (Vex, Val, Plai) have answered to three questions:
• Gender: M or W,
• Security: priority to
• Fight Against Unemployment (FAU),
• Juvenile Delinquency (JD)
• Drug addict (D)),
• Death penalty (Yes or No).

Gender, Security , D. Penalty are «  barchart value variables »

M, W, FAU, JD…are « bins »

Frombarchartsymbolic variables to Metabin latent variables

Table 1

Initial bar chart data table

Table 2

Metabin latent variables

CONCLUSION

If you have standard units described by numerical and (or) categorical variables, these variables induce “classes” described by symbolic variables taking care of their internal variation. Then SDA can be applied on these new units in order to get complementary and enhancing results by extending standard analysis to symbolic analysis.

Symbolic data have to be build from given standard or complex data.

Symbolic data cannot be reduced to standard data.

Complex data can be simplified in symbolic data.

Big Data bases can be reduced in symbolic data

Symbolic data are not only distributions, they are the numbers of the future.

Références

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.
• 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.
• 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.
• 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.
Building Symbolic Data and representationReferencies
• 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
SOME SYMBOLIC DATA ANALYSIS REFERENCIES
• 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.

Symbolic Data Analysis references
• 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.

### Symbolic Data Analysis referencies

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.

Symbolic Data Analysis references

• 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.

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.
Some Symbolic Data Analysis references
• 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.
Some 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.

Some SDA Industrial Applications

• AfonsoF., 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.