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Principal Component Analysis of an XI- XV century silver coins’ XRF dataset and verification by additional multilinear methods Anita Rácz a, János Elek.

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Presentation on theme: "Principal Component Analysis of an XI- XV century silver coins’ XRF dataset and verification by additional multilinear methods Anita Rácz a, János Elek."— Presentation transcript:

1 Principal Component Analysis of an XI- XV century silver coins’ XRF dataset and verification by additional multilinear methods Anita Rácz a, János Elek a,b, Károly Héberger c, Róbert Rajkó d, Adrienn Lengyel a,b a University of Debrecen Department of Inorganic and Analytical Chemistry b ANALAB Ltd. c Chemical research Center, Academy of Hungarian Sciences d University of Szeged, Faculty of Engineering 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

2 Scope of the study − − Does any sort of grouping occur? − − Do we need the whole XRF spectra, or is the elemental composition information sufficient for the classification? − − Is there coherence between the groups and the historical periods? − − Check up of ranking by additional chemometrical methods (LDA, CART, PLS) 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

3 Theory of X-ray fluorescent elemental analysis 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

4 Emission of charactersitic X- ray radiation 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

5 Energy dispersive X-ray spectrum 2048 excitation energies chemometric data evaluation Typical composition (%) 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

6 248 coins 0: Unknown origin 1: I.István - Kálmán 2: II.István - Imre 3: II.András - III.András 4: Anjou 5: Bohemian 6: Luxembourg 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

7 Principal component analysis − − Unsupervised method. − − The X data matrix was constructed by using 248 objects and 2048 variables (excitation channels). − − Then the data matrix was resolved as the product of T „scores” and P „loadings” matrices. X = TP T objective: finding n „virtual properties” (PC n ) instead of 2048 „real properties” (excitation voltages) that reveals the differences amongst the coins. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

8 c PC 1 (35,93%) PC 5 (8,39%)

9 Kiértékelés: Főkomponensek értéke egymás függvényében I. - Elemösszetétel szerinti vizsgálat, I2 indikátor változó csoportosítással - értéktengelyen: átskálázás; várható érték =0, szórás =1 CC%=76% PC 5 (8,39%) PC 1 (35,93%) Unknown I. István – Imre II. András - Luxemburg

10 Főkomponensek értéke egymás függvényében ( spektrum) I2 indikátor változó csoportosítással CC%= 72% PC 1 (25,91%) PC 5 (2,05%) Unknown I. István – Imre II. András - Luxemburg

11 Linear Discriminant Analysis (LDA) − − LDA finds a linear combination of features which characterize or separate two or more classes of objects. Supervised method. − − An ellipse (hyper-ellipsoid) is constructed around each group. − − The discriminant equation: 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

12 Kiértékelés LDA Osztályozási mátrix Ábrázolás kanonikus változók függvényében I2b csoportosítás létrehozása CC%=77% Unknown I. István - Imre II. András - Lux.

13 Classification and regression tree(CART) − − Recursive classification method, that creates subdivisions by repeated binary splitting of the original dataset: each node of the tree contains a binary „yes/no” question. − − The algorithm finds the question about some feature which splits the data maximizing the purity of the two partitions. − − „Leaves” represent class labels and „branches” represent conjunctions of features that lead to those class labels. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

14 Kiértékelés CART CC%=78% group 2 group 3

15 Parcial Least Squares regression (PLS) L: rank of matrix X M: number of variables N: number of samples U=Q T V T Q T (PLS components) V T (base vectors) Instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable variables to a new space. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

16 CC%=77% Unknown I. István – Imre II. András - Luxemburg

17 Summary − − With using PCA two historically adequate groups were found. The accuracy of ranking is not affected by thy chioce of the dataset (CC spec :72%, CC ec :76%). − − During LDA, PLS and CART analysis using compositional information, the coins were classified to the two historical predefined groups. The efficiency of the classification was found to be 76-78% in all cases. − − Most of the published research focused on the analysis of the main components (Au, Ag, Cu). The fingerprint-type examination of the low concentration alloying constituents can provide useful information for numismatic experts. − − Our work shows, that ranking of these pieces becomes possible by selection of the appropriate test method and constructing a didactic set of minted coins with known origin and an analyzing algorithm. − − Though many of the unknown coins can be classified as a member of the two groups, the rest seem to form an independent set. (Correct dating?) 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

18 Multivariate Curve Resolution – Alternating Least Squares (MCR-ALS) Compulsory assumptions: - non negativity (C,S) - unimodality (C) - closure (C) - Hard-modeling (C) © Roma Tauler 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

19 MCR-ALS components Var 1: Cu Var 2: Pb, Cu, Sn Var 3: Sn, Cu, Zn, Fe, Pb Var 4: Ag 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

20 MCR-ALS scoreplot For further separation with dimensional reduction LDA was preformed on the score matrix. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

21 Statistics of gLDA 6 groups 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

22 Statistics of gLDA 2 groups 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

23 Target Rotation Supervised non orthogonal „PCA”. (c) Prof. Olav Christie 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

24 Summary II. − − Use of non Euclidian models results in better classification of the unknown origin coins. − − All the unknowns can be date to the period I. István Imre reign (verification with independent methods?) − − If the above statements are true, the use of multistep chemomertical data analysis should be considered. − − With creative approach, classical methods of chemometrics can be employed for obtaining useful scientific information without inventing new sophisticated algorithms and data processing tools. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino

25 Acknowledgement − − László Szolnoki, János Dani, Róbert Ujszászi, Ferenc Benus − − Olav Christie, Peter Filzmoser − − ANALAB Kft. 8th Winter Symposium on Chemometrics 2012 Russia, Drakino


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