1. Rank Annihilation Based Methods

2. p n X The rank of matrix X is equal to the number of linearly independent vectors from which all p columns of X can be constructed as their linear combination Geometrically, the rank of pattern of p point can be seen as the minimum number of dimension that is required to represent the p point in the pattern together with origin of space rank(P p ) = rank(P n ) = rank(X) < min (n, p) Rank

3. xPxP yPyP Variance P Q R yQyQ yRyR xQxQ xRxR x y O x P y P x Q y Q x R y R OP 2 = x P 2 + y P 2 OQ 2 = x Q 2 + y Q 2 OR 2 = x R 2 + y R 2 OP 2 + OQ 2 + OR 2 = x P 2 + y P 2 + x Q 2 + y Q 2 + x R 2 + y R 2 Sum squared of all elements of a matrix is a criterion for variance in that matrix

4. Eigenvectors and Eigenvalues For a symmetric, real matrix, R, an eigenvector v is obtained from: Rv = v is an unknown scalar-the eigenvalue Rv – v  = 0 (R – I  v  = 0 The vector v is orthogonal to all of the row vector of matrix (R- I) Rv = v0v - RI =

5. Variance and Eigenvalue 2 4 3 6 D = 1 2 Rank (D) = 1 Variance (D) = (1) 2 + (2) 2 + (3) 2 + (2) 2 + (4) 2 + (6) 2 = 70 Eigenvalues (D) = [70 0] 4 2 3 6 D = 1 2 Rank (D) = 2 Variance (D) = (1) 2 + (4) 2 + (3) 2 + (2) 2 + (2) 2 + (6) 2 = 70 Eigenvalues (D) = [64.4 5.6]

6. Free Discussion The relationship between eigenvalue and variance

7. 10 uni-components samples Eigenvalues (D) = 204.7 0 Without noise 204.6 0.0012 0.0011 0.0009 0.0008 0.0006 0.0005 Eigenvalues (D) = with noise

8. 10 bi-components samples Eigenvalues (D) = 262.65 18.94 0 Without noise Eigenvalues (D) = with noise 262.64 18.93 0.0011 0.0009 0.0008 0.0007 0.0006 0.0005

9. Bilinearity As a convenient definition, the matrices of bilinear data can be written as a product of two usually much smaller matrices. D = C E + R =

10. Bilinearity in mono component absorbing systems

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26. Bilinearity in multi-component absorbing systems A B C k1k1 k2k2 D = C S T D = c A s A T + c B s B T + c C s C T

27. sATsAT cAcA cA sATcA sAT Bilinearity

28. sBTsBT cBcB cB sBTcB sBT

29. sCTsCT cCcC cC sCTcC sCT

30. cA sATcA sAT cB sBTcB sBT cC sCTcC sCT ++

31. Spectrofluorimetric spectrum Excitation-Emission Matrix (EEM) is a good example of bilinear data matrix Excitation wavelength Emission wavelength EEM

32. One component EEM

33. Two component EEM

34. Free Discussion Why the EEM is bilinear?

35. C=1.0 C=0.8 C=0.6 C=0.4 C=0.2 Trilinearity

36. Quantitative Determination by Rank Annihilation Factor Analysis

37. Two components mixture of x and y C x =1.0 and C y =2.0

38. Two components mixture of x and y

52. - = C x =1.0 C y =2.0 C x =0.4 C y =0.0 Residual Two components mixture of x and y

53. C x =1.0 C y =2.0 C x =1.5 C y =0.0 Residual - = Two components mixture of x and y

54. C x =1.0 C y =2.0 C x =1.0 C y =0.0 Residual - = Two components mixture of x and y

55. Mixture StandardResidual - = M – S = R Rank(M) = nRank(S) = 1 Rank(R) = n n-1 Rank Annihilation

56. The optimal solutions can be reached by decomposing matrix R to the extent that the residual standard deviation (RSD) of the residual matrix obtained after the extraction of n PCs reaches the minimum  j=n+1 c j n (c– 1) ( ) RSD(n) = 1/2

58. RAFA1.m file Rank Annihilation Factor Analysis

59. Saving the measured data from unknown and standard samples

60. Calling the rank annihilation factor analysis program

62. Rank estimation of the mixture data matrix

68. ? Use RAFA1.m file for determination one analyte in a ternary mixture using spectrofluorimetry

69. Simulation of EEM for a sample

80. ? Investigate the effects of extent of spectral overlapping on the results of RAFA