1. An Introduction to Model-Free Chemical Analysis Hamid Abdollahi IASBS, Zanjan e-mail: abd@iasbs.ac.ir Lecture 2

2. Position of a known profile in corresponding space: dxdx T v1 T v2 v1v1 v2v2 T v1 is the length of projection of d x on v 1 vector T v1 = d x. v 1 T v2 is the length of projection of d x on v 1 vector T v2 = d x. v 2 T v1 T v2 Coordinates of d x point:

3. Position of real spectral profiles in V space Real spectrum 1 Real spectrum 2

4. Position of real spectral profiles in V space Real spectrum 1 Real spectrum 2

5. Position of real spectral profiles in V space Real spectrum 1 Real spectrum 2

6. Heuristic Evolving Latent Projection (HELP) The main contributions of HELP have been offering a sophisticated graphical tool to visually detect potential selective zones in the score plot of the data matrix and a statistical method to confirm the presence of the selectivity in the concentration and/or spectral windows graphically chosen.

7. Free Discussion Heuristic Evolving Latent Projection (HELP)

8. Vspace.m file Visualizing the points in V space

18. ? Use the Vspace.m file and find the points which define the similar spectral shapes.

19. Solution of a soft-modeling method D = USV = CA C ≠ USA ≠ V D = US (T -1 T) V = CA C = US T -1 A =T V t 11 t 12 t 21 t 22 T= ti 11 ti 12 ti 21 ti 22 T -1 = Two component systems:

20. a 1 = t 11 v 1 + t 12 v 2 a 2 = t 21 v 1 + t 22 v 2 A = C = [c 1 = ti 11 s 11 u 1 + ti 21 s 22 u 2 c 2 = ti 12 s 11 u 1 + ti 22 s 22 u 2 ] Solution of a soft-modeling method The elements of T matrix are the coordinates of real spectral profiles in V space The elements of ST -1 matrix are the coordinates of real concentration profiles in U space

21. V_U_space.m file Visualizing the points in V and U spaces

23. Real spectrum 1 Real spectrum 2

40. Intensity ambiguity in V space

41. Intensity ambiguity in U space

70. Rotational ambiguity in V space

71. Rotational ambiguity in U space

72. Rotational ambiguity The one major problem with all model-free methods is the fact that often there is no unique solution for the task of decomposing the data matrix into the product of two physically meaningful matrices. Where there is rotational ambiguity, the solution of soft-modeling methods is one particular point within the range of possibilities.

73. ? Use the V_U_space.m file and investigate the effect of overlapping in concentration and spectral profiles on the possible solutions

74. A first order kinetic as a closed system

80. ? Use the V_U_space.m file and investigate the possible solutions for a first order kinetic data

81. Intensity ambiguity in V space v1v1 v2v2 a k1ak1a k2ak2a T 11 T 12 k 1 T 11 k 1 T 12 k 2 T 11 k 2 T 12

82. Normalization a = T 1 v 1 + T 2 v 2 k 1 a = k 1 T 1 v 1 + k 1 T 2 v 2 k 2 a = k 2 T 1 v 1 + k 2 T 2 v 2 k n a = k n T 1 v 1 + k n T 2 v 2 … a’ = v 1 + T v 2

83. v1v1 v2v2 Normalization 1 1 2 4 3 5 a = T 1 v 1 + T 2 v 2 a’ = v 1 + T v 2 1’2’3’ 4’ 5’

84. n_V_U_space.m file Visualizing the normalized points in V and U spaces