بنام خدا

1. بنام خدا

2. Institute for Advanced Studies in Basic Sciences, Zanjan - Iran Chemometrics workshop, 1384 Rank Annihilation Factor Analysis Mohsen Kompany-Zareh

3. Rank Annihilation Factor Analysis (RAFA) for two-way (2nd Order) data

4. Transformation:of abstract factors into real (basic) factors (Chemically meaningful) Factor Analysis Data matrix + • Preparation : Variab. Select. / mean cent. /scaling • Reprodution: • svd signif No. of abstract (principal) factors • Combination: => complete models of real factors • Prediction

5. Transformation abstract factors into basic factors 1. Abstract rotation: varimax, quartimax, oblimin, .. 2. Target testing: of potentially real factors.. r?=μT 3. Special Meth.s: EFA, PLS, KSFA, RAFA Use of chemical constraints (No proper target vect.)

7. Rank of matrix X the number of linearly independent columns or rows of X: 2 3 5 Random Numbers : 5 1 4 3 4 2 1 6 3 Rank=min(r,c) Rank = 3 Mathem.: Full Rank

8. 2 6 3 4 12 2 3 9 4 1 3 1 3 Rank the number of linearly independent columns or rows of X: Rank =2 <min(r,c) Mathem.: Rank deficient 2 linearly dependent col.s

9. 2 7 3 3 6 3 5 4 3 6 3 3 x1 + x2 -3(x3) =0 linear dependent col.s ? Rank the number of linearly independent columns or rows of X: Rank =2 <min(r,c) Mathem.: Rank deficient

10. Evolutionary data - Chromatography - Reaction kinetics -Titrations (pH, L, …) -Ex.-Em. Fluorersc. Sp. - … Evolutionary methods: -Modeling meth.s (Hard models) -Self-modeling meth.s (Soft models)

11. 0.5 1.5 One component A X1= excitation emission fluorescence spectrum Rank = 1 Mathem.: Rank defic. Chemical : Full rank

12.   = One component A X1 f (Emiss.) e (Excit.) Conc. Bilinear data xij=keikkfkj

14. Two components (A and B) E Conc. F   X2 (Fluoresc.) =

15. Completely eliminating the contribution of A = Two components (A and B) X2 = Rank = 2 Rank Annihilation

17. Rank Annihilation Requires a second set of data,... …the calibration set

18. Two components (A and B) Conc.s =1, 3 X2 = Rank = 2 Conc. =2 Calibration set (A) X1 = Rank = 1 What about subtraction of X1 from X2 ?

19. [A]X2 [A]X1 Rank (X2–2.0 X1) 2 2 (X2 –1.5 X1) (X2 - X1) 2 (X2 – 0.5X1) 1 Annihil. (X2 – 0.2X1) 2 =   -1= λ Iterative

20. A GRAY system (A and …) X3 = Rank =5 one component (A) X1 = Rank = 1 Can RAFA be applied ?

21. [A]X3 [A]X1 For the GRAY system (A and …) (X3 – 0.5X1)=E Rank =5 Rank =4 =   -1 =λ Quantification of a component in a complex mixture without concern for other components.

23. Direct Calculation of λ A. Lorber, Anal Chim Acta 1984,164,293-297 λ-1 X2 –X1=E Rank=n Rank=n-1 Mathematics: Reducing the rank of matrix is equivalent to saying that the determinant of the matrix is zero. | λ-1X2 –X1|=0 Or, as a generalized eigenvalue/eigenvector problem X1z = λ-1X2z

24. X1z = λ-1X2z 1. MatricesX1andX2are usually rectangular. !! 2. X2is not an Identity matrix.!! Solution: U1S1V1’z = λ-1U2S2V2’z U2’U1S1V1’ V2S2-1z = λ-1U2’U2S2V2’ V2S2-1z U2’U1S1V1’ V2S2-1z = λ-1z

25. U2’U1S1V1’ V2S2-1z = λ-1z Eigenvalue problem ab’S2-1z = λ-1z a b’ S2-1

26. sum ab’S2-1 = 2.000= λ-1

28. Rank Annihilation with incomplete information

29. Rank Annihilation Requires a second set of data,... …the calibration set

30. the calibration set (component A) X1 Bilinear data f (Emiss.) e (Excit.) Conc.   = Twovectors for one component is the complete information

31. Incomplete Information only one of the vectors is known for a particular component. Elution profile Spectrum

32. Two components …

36. Pure spectrum of component B The only information (s)

37. A soft method…

38. Rank=1 Rank=2 The pseudo-inverse estimate of c

39. Results from pseudoinverse solution Negative

40. Transformation matrix t-1 t X = m×n m×p p×1 1×p p×n s' p signif. scores and loadings

41. 1×p p×p p×1 1×p p×1 1×n n×p p×1

42. t a1 a2 t1 t2 1×p p×1 Necessary condition for t 1 = a1 t1 + a2 t2