Rank Annihilation Based Methods

1. Rank Annihilation Based Methods

2. p X n rank(Pp) = rank(Pn) = rank(X) < min (n, p) Rank 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

3. y P Q R O xP x Variance xP yP xQ yQ xR yR OP2= xP2 + yP2 yP OQ2= xQ2 + yQ2 yQ OR2= xR2 + yR2 yR OP2 + OQ2 + OR2= xP2 + yP2 + xQ2 + yQ2 + xR2 + yR2 xQ xR Sum squared of all elements of a matrix is a criterion for variance in that matrix

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

5. Variance and Eigenvalue 1 2 D = 2 4 3 6 1 2 D = 4 2 3 6 Rank (D) = 1 Variance (D) = (1)2 + (2)2 + (3)2 + (2)2 + (4)2 + (6)2 = 70 Eigenvalues (D) = [70 0] 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 204.7 0 0 0 0 0 0 0 0 0 204.6 0.0012 0.0012 0.0011 0.0009 0.0009 0.0008 0.0006 0.0006 0.0005 Eigenvalues (D) = Eigenvalues (D) = Without noise with noise

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

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 k2 k1 A B C D = C ST D = cA sAT + cB sBT + cC sCT

27. Bilinearity cA cA sAT sAT

28. Bilinearity cB cB sBT sBT

29. Bilinearity cC cC sCT sCT

30. cA sAT cB sBT cC sCT + +

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

32. One component EEM

33. Two component EEM

34. Free Discussion Why the EEM is bilinear?

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

36. Quantitative Determination by Rank Annihilation Factor Analysis

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

38. Two components mixture of x and y

39. Two components mixture of x and y

40. Two components mixture of x and y

41. Two components mixture of x and y

42. Two components mixture of x and y

43. Two components mixture of x and y

44. Two components mixture of x and y

45. Two components mixture of x and y

46. Two components mixture of x and y

47. Two components mixture of x and y

48. Two components mixture of x and y

49. Two components mixture of x and y

50. Two components mixture of x and y