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G eneralized R ank A nnihilation F actor A nalysis

G eneralized R ank A nnihilation F actor A nalysis. Anal Chem 58( 1986 )496. E Sanchez B R Kowalski. Bilinear data. One component  Rank =1. f (Emiss.). e (Excit.). Conc. . . X 1 (Fluoresc.). =. Two components. E. Conc. F. . . X 2 (Fluoresc.). =.

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G eneralized R ank A nnihilation F actor A nalysis

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  1. GeneralizedRank Annihilation FactorAnalysis Anal Chem 58(1986)496. E Sanchez B R Kowalski

  2. Bilinear data One component  Rank =1 f (Emiss.) e (Excit.) Conc.   X1 (Fluoresc.) =

  3. Two components E Conc. F   X2 (Fluoresc.) =

  4. X2- 0.5 X1 = E Rank=2 Rank=1 Lorber, 1984 Two components X2 = Rank = 2 one component (calibration matrix) X1 = Quantification of one component. Rank = 1

  5. Two components(sample) X2 = Rank = 2 Two components(calibration) X3 = Rank = 2 What about quantific. of more than one component?

  6. Generalized RAFA [Anal Chem 1986, 58, 496-499. B.R. Kowalski] 1. Non-iterative [Lorber, 1984]. 2. Simultaneous detn. of analytes using Just one bilinear calibration spectrum from one mixture of standards. • Bilinear spectrum of each analyte • b. Relative conc.s

  7. Common F and E Trilinearity Theory sample :  E = X2(FT)+ -1 E FT =X2 Calibration : E FT =X1  E = X1(FT)+ -1 X1(FT)+ -1 = X2(FT)+ -1 X1Z= USVT Z -1  Z = VS-1Z* (definition)

  8. I I X1VS-1Z*= USVT VS-1Z*-1  UTX1VS-1Z*= Z*-1  R V = V (eigenvector analysis) FT= (VS-1Z*)+ ? E =UZ* -1 -1  = 

  9. H2A  HA  A H2B  HB  B using pH-metric titration Simult. detn. of two acids in a sample

  10. C0A ? H2A  HA  A sample C0B ? H2B  HB  A C0A =0.02 M H2A  HA  A calibr. C0B =0.04 M H2B  HB  A Data matrices

  11. sample calibration

  12. Only HA- and HB- are optically active.

  13. sample calibration

  14. [Usm,Ssm,Vsm] = svd(Xsm') [Zstar,λ]=eig(Usm‘ * Xcl‘ * Vsm* inv(Ssm))

  15. (CoA)cl (CoA)sm (CoB)cl (CoB)sm λ= , (CoB)cl=0.04 M => (CoB)sm=0.02 M , (CoA)cl=0.02 M => (CoA)sm=0.03 M β= 15

  16. Conc. profiles F = pinv( Vsm * inv( Ssm ) * Zstar)

  17. spectral profiles E = Usm * Zstar * inv(β)

  18. Example: HPLC-DAD chromatogram for A,B, and C (as CL), for ?,?,and ? (as SM) What if: The calibration sample includes some components that are not present in unknown sample, And there be some components in unknown sample not present in the calibration sample. The General Condition

  19. Xcl CAcl= 1 mM CBcl= 3 mM CCcl= 2 mM

  20. Xsm ?, ?, and ?, ..

  21. The total space, rank =4 (includes A, B, C ,and D) Xtot = Xcl + Xsm [Utot,Stot,Vtot] = svd(Xtot') [Zstar,λ]=eig(Utot‘ * Xsm‘ * Vtot* inv(Stot))

  22. C?sm C?sm+C?cl =0.9999 0.0003 0.5000 0.3334 B A D C?sm= C?cl C 2C?sm= C?cl C?cl=0 Only in sm C?sm=0 Only in cl CBsm= 3 mM CCsm= 1 mM λ= β/ ( β+ ξ)

  23. Conc. profiles F = pinv( Vtot * inv( Stot ) * Zstar)

  24. spectral profiles E = Utot * Zstar

  25. Non-bilinear RA Rank for the pure component >1 Analyte detn. ..in the presence of unaccounted spectral interference..

  26. H2A  HA  A

  27. Xcl One compon, but Rank=… 3

  28. H2A and H2B

  29. Interference H2A and H2B Rank(Xsm)=5

  30. Conc. Prof.s Spect. Prof.s

  31. λ of H2B

  32. DECRA DirectExponentialCurveResolutionAlgorithm J. Chemom. 14 (2000) 213-227.

  33. shift 162/54= 3 54/18= 3 18/6= 3 6/2= 3  Model base:an exponential decay x2 x1 x

  34. C1 e –k t l ===e –kt +k(t+S)= e –k S C2 e –k (t+S) x1 : C1 = e –k t x2 : C2 = e –k (t+S)  k = ln(l) / S Shift

  35. Shift=7 x1 x2 1st Ord Data From 1 sample

  36. k = ln(2.013) / 7 =0.1

  37. sPT sQT sRT cP cQ cR X Expon. Decay 2st Ord Data From 1 sample = + +

  38. Stacking N E 1 F 1+S λ M-S M Gives k1 and k2 X X E (MN) ((M-S) N 2) 2-way 3-way Trilinear structure X1 = + X2

  39. A A’ B B’ C C’ … 1st order reactions Decomposition of a number of colorants to colorless products..

  40. svd(X)= 6279.5 294.0 34.4 0.7 0.6 0.6 … Three components

  41. Shift = 10 min

  42. Estimated F

  43. estmated E

  44.   λ= k = ln(λ) / shift

  45. A consecutive reaction:

  46. k1 k2 A B  D No Expon. Decaying concn.

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