Correspondence Analysis for Symbolic Multi–Valued Variables

 

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Correspondence Analysis for Symbolic Multi–Valued Variables

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17/03/2012 university of costa rica classic correspondence factor analysis single-selection questionnaires · what is your political party 1 2 3 4 5 left green social-democratic christian-socialism right university of costa rica 1

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17/03/2012 classic correspondence factor analysis single-selection questionnaires · what is your educational level 1 2 3 4 primary secondary college-degree university-degree university of costa rica 799 people were interviewed required a single selection · contingency table is xty university of costa rica 2

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17/03/2012 classic correspondence factor analysis university of costa rica multiple-choice questionnaires correspondence factor analysis university of costa rica 3

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17/03/2012 un poco de historia del ads · l ads ou encore l analyse des données appliquée à des concepts a été introduite par edwin diday [diday 1987a [diday 1987b [diday 1989 university of costa rica university of costa rica 4

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17/03/2012 objetos simbólicos · real-life objects are too complex to be represented by points in a vectorial space [bock&diday 2000 · symbolic objects overcome this limitation by representing concepts rather than individuals [bock&diday 2000 · knowledge extraction from large data bases is our main aim as in data mining [diday 1998 university of costa rica symbolic data table symbolic object university of costa rica 5

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17/03/2012 creando tablas simbólicas classical description of schools schools jaurès condorcet chevreul st hélène st sernin st hilaire town paris paris lyon lyon toulouse toulouse nb of pupils 320 450 200 380 290 210 kind public public public private public private level 1 3 2 3 1 2 symbolic description of the towns by the schools variables town paris lyon toulouse nb of pupils [320 450 [200 380 [210 290 kind 100 public 50 public 50 private 50 public 50 private level {1 3 {2 3 {1 2 university of costa rica ¿cómo se construyen las tablas simbólicas millones id-trx 3457 causal 36 28 39 35 35 27 34 sucursal curridabat san pedro grecia san pedro alajuela alajuela heredia monto 2,500.00 1,750.00 2,400.00 1,900.00 1,850.00 1,900.00 1,600.00 tarjeta 1000 1001 1000 1001 1001 1002 1002 análisis análisis multivariado clásico numérico datos 1251 3245 7635 3245 5367 6486 cientos análisis multivariado simbólico conceptos tarjeta 1000 1001 1002 causal 361/2 391/2 281/3 352/3 271/2 341/2 sucursal {curr-50 gre-50 {sp-66 al-33 {al-50 her-50 monto [2.4,2.5 [1.75,1.9 [1.6,1.9 salario 255.4 122,2 534,5 university of costa rica 6

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17/03/2012 from a relational database to a symbolic data table vsog university of costa rica rsda assumptions on symbolic data analysis assumption 1 assumptions 2 input symbolic output symbolic university of costa rica 7

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17/03/2012 · in this presentation we will extend the correspondence analysis to the case of symbolic multi-valued variables and then we will apply it to process multiple­choice multiple­ questionnaire questionnaire · but the first question is what is a symbolic multi-valued variable multiuniversity of costa rica eyes-color hair-color · a symbolic variable y is called multi-valued if its multivalues yk are all finite subsets ­ yk green,blue · so there is imprecise information in the input data that will be represented in the two dimensional plane as a rectangle instead of a point university of costa rica 8

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17/03/2012 relationship between two symbolic multi-valued variables · as it is very well know in classic correspondence analysis a contingency table associated with two qualitative variables is build · example 1 if there are two qualitative variables ­ x =eyes-color with 3 modalities green blue and brown and ­ y hair-color with 2 modalities blond and black university of costa rica · if we have 5 individuals the following disjunctive complete tables could be obtained · then the crossed or contingency table between the variables x and y is university of costa rica 9

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17/03/2012 symbolic case · example 1 ­ if x1 eyes-color1 green or x1 eyes-color1 blue ­ if y3 hair-color3 blond or y3 hair-color3 black · in this case there are two possible disjunctive complete tables for the variable x and two for y university of costa rica · using this information we have 4 possible contingency tables between the variable x and y university of costa rica 10

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17/03/2012 · taking the minimum and the maximum of the components of these 4 matrices an interval contingency data table is obtained university of costa rica there is a problem · the construction of the matrix k requires too many calculations · in fact it requires pmnm matrix products · the following theorem reduces the calculation to only two matrix multiplications · before presenting the theorem the following definition must be given university of costa rica 11

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17/03/2012 definition · the minimum possibilities matrix meet matrix is · the maximum possibilities matrix join matrix is university of costa rica example 2 using the same variables x and y from example 1 we have the following result university of costa rica 12

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17/03/2012 university of costa rica then university of costa rica 13

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17/03/2012 correspondence analysis between two multi-valued variable then we can start with the following matrix the idea of the method is to transform the matrix k to the matrix kc and then to apply a classical ca to kc in order to project there the interval profiles university of costa rica university of costa rica 14

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17/03/2012 · the following two theorems will allow us to project as a rectangles the interval column profiles and the interval row profiles university of costa rica university of costa rica 15

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