Betweenness relation orientated by Guttman effect in critical - PowerPoint PPT Presentation

Betweenness relation orientated by Guttman effect in critical edition M. Le Pouliquen 1 M. Csernel 2 1 Telecom Bretagne, Labsticc UMR 3192 , BP 832, 29285 Brest Cedex - France 2 AXIS Inria-Rocqencourt, BP-105- 78180 Le Chesnay Cedex - France CARME 2011 The 10th International Conference on CORRESPONDENCE ANALYSIS AND RELATED METHODS 1 / 32

Plan of the presentation Introduction 1 Different characterizations for betweenness 2 Textual tradition De Nuptiis and first result 3 Application to seriation 4 Conclusions 5 2 / 32

Plan Introduction 1 Different characterizations for betweenness 2 Textual tradition De Nuptiis and first result 3 Application to seriation 4 Conclusions 5 3 / 32

Critical edition Critical edition project consists in different steps : Inventory of the manuscripts called witnesses in the corpus. Codicologic and paleographic studies of the manuscripts in order to carry out a first classification of the manuscripts. Text collation. Development of the stemma codicum in order to explain the text history. Reading selection. Critical edition. 4 / 32

Example of stemma sodicum F IGURE : Stemma codicum established by Danuta Shanzer : Each letter is a manuscript ; the Greek letters indicate lost or supposed manuscripts. 5 / 32

b b b b b Idea of Don Quentin Don Quentin came up with the idea of using the betweenness in order to draw up a stemma. In fact, he restored small chains of three manuscripts where one is between the others , he assembled these small chains in order to infer the complete tree. I Q G F P 6 / 32

b b b b b Idea of Don Quentin Don Quentin came up with the idea of using the betweenness in order to draw up a stemma. In fact, he restored small chains of three manuscripts where one is between the others , he assembled these small chains in order to infer the complete tree. I The mns Q is between I and G. That is written : ( I , Q , G ) Q G F P 7 / 32

b b b b b Idea of Don Quentin Don Quentin came up with the idea of using the betweenness in order to draw up a stemma. In fact, he restored small chains of three manuscripts where one is between the others , he assembled these small chains in order to infer the complete tree. I - ( I , Q , G ) - ( I , Q , F ) Q ... G F P 8 / 32

Idea of Don Quentin Method : If B is between A and C then : (i) The readings of the middle manuscript B agree alternately with these of A or C (ii) The readings A et C never agree against these of B. 9 / 32

Idea of Don Quentin Method : If B is between A and C then : (i) The readings of the middle manuscript B agree alternately with these of A or C (ii) The readings A et C never agree against these of B. As an example, consider the three following sentences corresponding to the same sentence of manuscripts copied one from the other. A = This is a sentence invented for the example B = Here is a sentence invented for the example C = Here is a sentence built for the example 10 / 32

Idea of Don Quentin Method : If B is between A and C then : (i) The readings of the middle manuscript B agree alternately with these of A or C (ii) The readings A et C never agree against these of B. As an example, consider the three following sentences corresponding to the same sentence of manuscripts copied one from the other. A = This is a sentence invented for the example B = Here is a sentence invented for the example C = Here is a sentence built for the example 11 / 32

Idea of Don Quentin Method : If B is between A and C then : (i) The readings of the middle manuscript B agree alternately with these of A or C (ii) The readings A et C never agree against these of B. As an example, consider the three following sentences corresponding to the same sentence of manuscripts copied one from the other. B = Here is a sentence invented for the example A = This is a sentence invented for the example C = Here is a sentence built for the example 12 / 32

Plan Introduction 1 Different characterizations for betweenness 2 Textual tradition De Nuptiis and first result 3 Application to seriation 4 Conclusions 5 13 / 32

Metric betweenness Among many geometrical characterizations of betweenness, Menger has introduced a definition under the name of metric betweenness in the following way : Definition A ternary relation ( , , ) on a set E is a metric betweenness relation if there is a metric d on E such that : ( a , b , c ) ⇔ d ( a , b ) + d ( b , c ) = d ( a , c ) 14 / 32

Metric betweenness The metric d between two strings is the number of operations (substitution deletion insertion) required to transform one of them into the other (a kind of word edit distance). In our example, calculations of d give : A This is a sentence invented for the example B Here is a sentence invented for the example C Here is a sentence built for the example d ( A , B ) = 1 d ( B , C ) = 1 d ( A , C ) = 2 It is noted that ( A , B , C ) because d ( A , C ) = d ( A , B ) + d ( B , C ) and not ( A , C , B ) . 15 / 32

Metric betweenness The metric d between two strings is the number of operations (substitution deletion insertion) required to transform one of them into the other (a kind of word edit distance). In our example, calculations of d give : A This is a sentence invented for the example B Here is a sentence invented for the example C Here is a sentence built for the example d ( A , B ) = 1 d ( B , C ) = 1 d ( A , C ) = 2 For application, we use the index I M = d ( A , B )+ d ( B , C ) − d ( A , C ) null if d ( A , C ) ( A , B , C ) . 16 / 32

Betweenness defined by a score built with Don Quentin’s conditions We want to build an index to detect if a manuscript is between two others. If B is between A and C then : Let n reading’s number in B Let n 1 reading’s number in B which do not belong either in A , or in C . Put down I Q 1 = n 1 n . Therefore I Q 1 ∈ [ 0 , 1 ] and if I Q 1 = 0 the first condition of Don Quentin is verified. Let n 2 The number of common readings in A and C which do not belong in B Put down I Q 2 = n 2 n . Therefore I Q 2 ∈ [ 0 , 1 ] and if I Q 2 = 0 the second condition of Don Quentin is verified. So, I Q = 0 . 8 ∗ I Q 2 + 0 . 2 ∗ I Q 1 = 0 if both Don Quentin’s conditions are verified. 17 / 32

Betweenness defined by a score built with Don Quentin’s conditions With the preceding example : A This is a sentence invented for the example B Here is a sentence invented for the example C Here is a sentence built for the example n = 2, n 1 = 0, n 2 = 0 ⇒ I Q = 0 Therefore B is between A and C A This is a sentence invented for the example C Here is a sentence built for the example B Here is a sentence invented for the example n = 2, n 1 = 1, n 2 = 1 ⇒ I Q = 1 therefor C isn’t betweenn A and B 18 / 32

Betweenness relation based on set theory Definition One set B is between two other sets A and C for Restle if and only if : (i) A ∩ B ∩ C = ∅ (ii) A ∩ B ∩ C = ∅ B A ∩ B ∩ C B C A C A A ∩ B ∩ C 19 / 32

Betweenness relation based on set theory Definition One set B is between two other sets A and C for Restle if and only if : (i) A ∩ B ∩ C = ∅ (ii) A ∩ B ∩ C = ∅ We use the index I E = Card ( A ∩ ¯ B ∩ C )+ Card (¯ A ∩ B ∩ ¯ C ) which is null if Card ( B ) ( A , B , C ) . 20 / 32

Betweenness relation based on set theory Using the preceding example again : The set A contains variants { This, invented }, B { Here, invented } and C { Here, built }. On a diagram : This invented Here built 21 / 32

Betweenness relation based on set theory Using the preceding example again : The set A contains variants { This, invented }, B { Here, invented } and C { Here, built }. On a diagram : This invented Here built A ∩ B ∩ C = ∅ 22 / 32

Betweenness relation based on set theory Using the preceding example again : The set A contains variants { This, invented }, B { Here, invented } and C { Here, built }. On a diagram : This invented Here built A ∩ B ∩ C = ∅ A ∩ B ∩ C = ∅ 23 / 32

Plan Introduction 1 Different characterizations for betweenness 2 Textual tradition De Nuptiis and first result 3 Application to seriation 4 Conclusions 5 24 / 32

Corpus De Nuptiis Use of the textual tradition engendered by the poem which opens the book IX Nuptiis Philologiae et Mercurii Tertullianus of Martianus Capella (5th century ap. J.-C). Collected by Jean-Baptiste Guillaumin Stemma, certainly incomplete, draw up by Danuta Shanzer The corpus is made up of 18 poems indicated by the following letters : Co = ( A , B , C , D , E , F , H , K , L , M , O , P , R , S , U , V , W , Z ) The collation table is consisted of 234 readings 25 / 32

Results on the corpus De Nuptiis Betweenness index I Q ordered for some triplets : Triplets SFV MFV KFV UFV LFV PFV RFV HF I Q 0 , 086 0 , 087 0 , 087 0 , 087 0 , 087 0 , 087 0 , 088 0 , 0 No intermediate manuscript in the real corpus. Manuscripts E and F are present in the first 70 triplets. So E and F are between the textual sub-tradition CEFV and the rest of the corpus ? We must relaxe the definitions and define the notion of weak-betweenness : Definition Let I ( = I M ,I E or I Q ) a between index, B is weak-between A and C if : (i) I ( A , B , C ) ≤ I ( B , A , C ) et I ( A , B , C ) ≤ I ( A , C , B ) (ii) There is a threshold index I S such as I ( A , B , C ) ≤ I S 26 / 32