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An Introduction to Model-Free Chemical Analysis

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

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An Introduction to Model-Free Chemical Analysis

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

  2. Model-based vs. model-free analysis There are no generally applicable tools available to guide the researcher towards finding the model that correctly describes the chemical process under investigation. Model fitting is much easier than model finding. Information obtained from model-free analysis can guide the researcher toward the correct model In many instances there is no model or mathematical function at all that could be used to quantitatively describe the process under investigation.

  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 s11 v1 = d2,: u21 s11 v1 = … … dp,: up1 s11 v1 =

  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. ? Is the pattern of points depend on selected variables?

  16. ? How is the dependency of pattern to overlapping of concentration and spectral profiles?

  17. 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 s11 v1 + u12 s22 v2 = u21 s11 v1 + u22 s22 v2 d2,: = … … … up1 s11 v1 + up2 s22 v2 dp,: = D = USV = u1s11 v1 + … + ursrr vr Row vectors:

  18. Singular Value Decomposition For r=2 D = u1s11 v1 + u2s22 v2 [ d:,1 d:,2 … d:,q ] = u1s11 [v11 v12 … v1q] + u1s22 [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:

  19. 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

  20. 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

  21. Visualizing the rows and columns of data matrix

  22. ? How is the dependency of pattern in one space to other space?

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