1. Y + = (Y T Y) -1 Y T Y T Y is non-singular and squared ? (Full rank) Inversion is possible if: =Y Y+c=Y Y+c

2. Y’*Y is (3  3) but: rank(Y’*Y)=2 ! rank(Y) = 2 =min(#r,#c) => Y is full rank

3. Y should have: #rows > #col.s 1

4. Y should not be: Rank deficient 2 Column are linearly dependent

5. !

6. 3 compon.s, 4 samples 4 wavel.s, 4 samples rank(x)=min(r(c),r(s))=3 rank(x) < min(# r, #c) =4 => x is rank deficient

7. pinv can be performed when x is rank deficient..

9. pinv ?X= I (not square and singular X) svd & estimation of X using significant factors ?U *  * V *T =I V *  *-1 U* T U*  *V* T =I pseudo inverse pinv(X)= X + = V *  *-1 U *T U* T U*=I  *-1  *=I V * V* T =I ?

10. ksks kCkC X = CC + X || -X || Criterion for fitting ks Hard Model Projection of X onto space of C X = C S classic 1. # samp.s ≥ # compon.s 2. C : full rank (rank(C)= #compon.s) (lin indep conc profiles)

11. ksks kCkC X Hard Model Projection of X onto space of C C = X Z inverse = XX + C || -C || Criterion for fitting ks 1. # samp.s ≥ # wavel.s 2. X: full rank (rank(X)= # wavel.s) - variab. Select. - Factor based methods ! !

12. X is usually near to singular…  # samples < # wavel.s  # wavel.s > # compon.s

13. XX + =U*  *V* T V *  *-1 U* T (signif factors) =U*  *  *-1 U* T =TT +

14. ksks kCkC X Hard Model Projection of C onto space of T C = T R Z inverse = TT + C || -C || Criterion for fitting ks 1. # samp.s ≥ # PCs 2. T: full rank (lin indep col.s) SVD T

15. ksks kCkC X Hard Model Projection of T onto space of C T = C R classic 1. # samp.s ≥ # compon.s 2. C : full rank (lin indep. conc prof.s) T SVD = CC + T || -T || Criterion for fitting k

16. = CC + X = XX + C = CC + T = TT + C pcrC (Target Transform) ccrX (classical curve resolution) pcrT ccrC T J Thurston, R G Brereton Analyst 127, 2002, 659.

17. The considered kinetic system: Second order consecutive A+B  CDA+B  CD Spectral meas. In 101 wavel.s each 30 sec (41 times) r(C)=3 # indep react.s +1

18. ccrC: X1=[X(:,50) X(:,70) X(:,90)] =X1*inv(X1‘*X1)*X1'*C =X*inv(X‘*X)*X'*C  1 =X*pinv(X)*C  X (41x101) 41 samples r(X)=3 101 wavel.s 1 # samp.s ≥ # wavel.s 2 rank(X)= # wavel.s Information content !

19. ccrX: =C*inv(C’*C)*C’*X  C1=C(:,2:4) =C1*inv(C1’*C1)*C1’*X =C*pinv(C)*X 1. # samp.s ≥ # compon.s 2. rank(C)= #compon.s C (41x4) 41 samples r(X)=3 4 compon.s

20. pcrT: =C*inv(C’*C)*C’*T  C1=C(:,2:4) =C1*inv(C1’*C1)*C1’*T =C*pinv(C)*T

21. pcrC: =T*T’*C 1. # samp.s ≥ # PCs 2. rank(T)= # col.s (always it is so…)

22. Overlap effect

24. +Rand noise

26. Spectral overlap (in the presence of some noise) results in some deviation in the results from ***C methods

27. Results from application of ***C and ***X methods are different … One way to obtain more similar results from ***C and ***X methods are application of constraints

28. Presence of heteroscedastic noise

29. + a heterosced. noise 41 reaction times &101 wavelengths

30. Inaccurate results from ccrX !

31. weights weighted regression… ||W ( -X) ||

32. 1/SD1 1/SD2 … 1/SDn W =W =

33. Accurate results from weighted ccrX ! n=50

34. Recognition of the presence of heterosc. noise FSMWFA

35. Non-random sampling error A more serious source of error

36. Square, symmetric, But not diagonal W matrix: J Chemometr 2002, 16, 378. R. Bro, N.D. Sidiropoulos, A.K. Smilde Maximum likelihood fitting

37. Presence of non-random sampling error nS=0.005 || -X ||||W ( -X) || Weighted regression ccrX J Chemom 2002, 16,387. R.Bro et al

38. Presence of unknown interference

39. Changing interference, drift, or shift rank(Data)=4

40. pcrT pcrC ccrX ccrC

41. Presence of shift or drift (a changing interference) results in serious deviations in ***X Methods (but not in ***C methods) Why?

42. = CC + X = XX + C = CC + T In the presence of shift, drift or changing interferences: T or X space includes 1. the concentration changes according to the model 2. variations from shift, drift or changing interference C space includes only the concentration changes according to the model Projection of a larger space to a smaller one Projection of a smaller space to a larger one  = TT + C

43. in the presence of unknown interference, drift or shift. Target Transform (pcrC) is the most preferred method

44. Constant interference rank(Data)=3 ! A+B  CDA+B  CD

45. ccrC ccrX pcrTpcrC

46. A constant interference does not show any significant effect the accuracy of ***X and ***C methods.

47. Target test fitting From: J Chemometr. 2001, 15, 511. P.Jandanklang, M. Maeder, A. C. whitson

48. Differential pulse Voltammetry Each voltammog. depends only on its own E 1/2

49. Successive complexation:

50. Analyst, 2001, 126, 371-377 Each concn. profile includes  1,…,  n

51. X

52. X=CS X=U  V T =TV = VV T s = UU T c = TT T c voltammogr concn.

53. For estimation of concn. profiles  1,…,  n (n parameters) should be optimized simultaneously  1,…,  n are dependent parameters

54. Simultaneous optimization of n dependent nonlinear parameters: Simplex method. Levenberg-Marquardt …

55. estimation of (E 1/2 ) 1, …, (E 1/2 ) n values for voltammograms (E 1/2 ) 1, …, (E 1/2 ) n are independent parameters

60. r = || - s||  0 (E1/2) M

65. r = || - s||  0 (E1/2) M (E1/2) ML

71. r = || - s||  0 (E1/2) M (E1/2) ML2 (E1/2) ML

74. r = || - s||  0 (E1/2) M (E1/2) ML (E1/2) ML2 (E1/2) ML3

77. r = || - s||  0 (E1/2) M (E1/2) ML (E1/2) ML2 (E1/2) ML3 (E1/2) L

80. r = || - s||  0 (E1/2) M (E1/2) ML (E1/2) ML2 (E1/2) ML3 (E1/2) L (E1/2) I

82. Optimum values for n independent parameters can be estimated by grid search of one parameter.

83.  A difficult aspect of hard modeling is determination of correct model

84. Thanks.

85. Thanks to: Miss Maryam Khoshkam and Mr Yaser Beyad for a number of m-files and slides.