Visualizing the Microscopic Structure of Bilinear Data: Two components chemical systems

1. Visualizing the Microscopic Structure of Bilinear Data: Two components chemical systems

2. Y D R X Factorization: In many chemical studies, the measured or calculated properties of the system can be considered to be the linear sum of the term representing the fundamental effects in that system times appropriate weighing factors. A matrix can be decomposed into the product of two significantly smaller matrices. D = X Y + R + =

3. A simple one component system

4. Observing the rows of data in wavelength space

5. Observing the columns of data in time space

6. Singular Value Decomposition = S V U D d1,: u11 v1 s11 d2,: u21 = … … up1 dp,: D = USV = u1s11 v1 + … + ursrr vr For r=1 D = u1s11 v1 Row vectors: d1,: u11 s11v1 = d2,: u21 s11v1 = … … dp,: up1 s11v1 =

7. Singular Value Decomposition S V U D d:,1 u1 s11 v11 = d:,2 u1 s11 v12 = … … d:,q u1 s11 v1q = D = USV = u1s11 v1 + … + ursrr vr = For r=1 D = u1s11 v1 Column vectors: [ d:,1 d:,2 … d:,q ] = u1s11 [v11 v12 … v1q]

8. Rows of measured data matrix in row space: up1s11v1 u11s11v1 v1 p points (rows of data matrix) in rows space have the following coordinates: u11s11 u21s11 … up1s11

9. Columns of measured data matrix in column space: q points (columns of data matrix) in columnss space have the following coordinates: v11s11 v12s11 … v1qs11 u1 v11 s11u1 v1q s11u1

10. Visualizing the rows and columns of data matrix

11. Solutions v1js11 Pure spectrum ui1s11 Pure conc. profile

12. Two component systems

13. Visualizing data in three selected wavelengths

14. Visualizing data in three selected Times

15. Measured Data Matrix

16. Singular Value Decomposition For r=2 D = u1s11 v1 + u2s22 v2 d1,: u11 v1 u12 v2 s11 s22 d2,: u21 u22 = + … … … up1 up2 dp,: d1,: u11 s11v1 + u12 s22 v2 = u21 s11v1 + u22 s22 v2 d2,: = … … … up1 s11v1 + up2 s22 v2 dp,: = D = USV = u1s11 v1 + … + ursrr vr Row vectors:

17. Singular Value Decomposition For r=2 D = u1s11 v1 + u2s22 v2 [ d:,1 d:,2 … d:,q ] = u1s11 [v11 v12 … v1q] + u2s22 [v21 v22 … v2q] d:,1 s11 v11 u1 + s22 v21 u2 = d:,2 s11 v12 u1 + s22 v22 u2 = … … s11 v1q u1 + s22 v2q u2 d:,q = D = USV = u1s11 v1 + … + ursrr vr Column vectors:

18. Rows of measured data matrix in row space: v2 Coordinates of rows u11s11 u12s22 u21s11 u22s22 up2s22 … … up1s11 up2s22 u22s22 d2,: d1,: v1 up1s11 u21s11 dp,: … u12s22 u11s11

19. Columns of measured data matrix in column space: u2 Coordinates of columns v11s11 v12s11 . . .v1qs11 v21s11 v22s11 . . .v2qs11 v21s22 d:, 1 d:, 2 v22s22 … v2qs22 d:, q u1 v11s11 v12s11 v1qs11

20. Two component chromatographic system

21. Two component Kinetic system

22. Two component multivariate calibration

23. Position of a known profile in corresponding space: v2 Tv2 dx Tv1 Coordinates of dx point: v1 Tv1 Tv2 Tv1 is the length of projection of dx on v1 vector Tv1 = dx . v1 Tv2 is the length of projection of dx on v1 vector Tv2 = dx. v2

24. Position of real profiles in V and U spaces

25. Position of real profiles in V and U spaces

26. Position of real profiles in V and U spaces

27. Duality in Measured Data Matrix

28. xj p xiT n up vn xi xj u1 v1 uj vi Geometrical interpretation of an n x p matrix X xij Pn Sp Sn Pp xij xij

29. X U Duality based relation between column and row spaces R= C ST = U D VT = X VT RT= S CT = VD UT = Y UT X = U D = RV = U YT V Y= V D = RT U = V XT U YT = V

30. Non-negativity constraint and the system of inequalities: U z ≥ 0 V z ≥ 0 X = U D U= X D-1 Y = V D V= Y D-1 U-space Y Points U z = X D-1z ≥ 0 Hyperplanes V-space X Points V z = Y D-1z ≥ 0 Hyperplanes

31. Duality based relation between column and row spaces The ith point in V-space: xi xiD-1= [Ui,1 Ui,2 … Ui,N] The ithhyperplane in U-space: Ui,1zi,1 + Ui,2zi,2 … Ui,Nzi,N≥ 0 The coordinates of each point in one space defines the coefficient of related hyper plane in dual space Point x in V-space Hyper plane (D-1x) z in U-space For two-component systems: xi = [xi,1 xi,2] A point in V-space: A half-plane in V-space: Ui,1z1 + Ui,2z2 ≥ 0

32. Half-plane calculation in two-component systems: Ui,1z1 + Ui,2z2 ≥ 0 z2 ≥ (-Ui,2/Ui,1)z1 General half-plane General border line can be defined for all points that the ith element of the profile is equal to zero z2= (-Ui,2/Ui,1)z1 General border line z2 ith border line 0 ith half-plane z1

33. Chromatographic data

34. Chromatographic data

35. Kinetics data

36. Kinetics data

37. Calibration data

38. Calibration data

39. Intensity ambiguity in V space v2 k2T12 k2a k1T12 k1a T12 v1 k1T11 T11 k2T11 a

40. Normalization to unit length v2 k2T12 k2a k1T12 k1a T12 v1 k1T11 T11 k2T11 an = (1/||a||) a a an

41. Normalization to first eigenvector v2 k2T12 k2a k1T12 k1a T12 v1 k1T11 T11 k2T11 an = (1/(v.a))a an a 1

42. Normalization to unit length v2 1’ 2’ 3’ v1 3 4 4’ 5 2 1 5’

43. Normalization to first eigenvector v2 4’ a’ = v1 + T v2 1’ 2’ 3’ 5’ v1 a = T1 v1 + T2v2 3 4 5 2 1 1

44. Chromatographic data- Normalized to unit length

45. Chromatographic data- Normalized to 1th eigenvector

46. Kinetics data- Normalized to unit length

47. Kinetics data- Normalized to 1th eigenvector

48. Multivariate calibration data- Normalized to unit length

49. Multivariate calibration data- Normalized to 1th eigenvector

50. Lawton-Sylvester Plot • The normalized abstract space of two component systems is one dimensional One dimensional normalized space Data points region • There are 4 critical points in normalized abstract space of two-component systems: Second inner point First inner point First outer point Second outer point • The 4 critical points can be calculated very easily and so the complete resolving of two component systems is very simple 12 Second feasible region First feasible region