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Y + = ( Y T Y ) -1 Y T

= Y Y + c. Y + = ( Y T Y ) -1 Y T. Inversion is possible if:. ?. . Y T Y is non-singular and squared. (Full rank). rank( Y ) = 2 =min(#r,#c) => Y is full rank. Y ’* Y is (3 3) but: rank( Y ’* Y )=2 !. 1. Y should have: #rows > #col.s. Column are linearly dependent. 2.

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Y + = ( Y T Y ) -1 Y T

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  1. =YY+c Y+= (YTY)-1 YT Inversion is possible if: ?  YTY isnon-singularandsquared (Full rank)

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

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

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

  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. pinvcan be performed when x is rank deficient..

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

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

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

  11. X is usually near to singular… • # samples < # wavel.s • # wavel.s > # compon.s

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

  13. || -C|| = TT+C C = T RZinverse X SVD Hard Model T ks kC Criterion for fitting ks  1. # samp.s ≥ # PCs 2. T: full rank (lin indep col.s)   Projection of C onto space of T

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

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

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

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

  18. C(41x4) 41 samples r(X)=3 4 compon.s 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

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

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

  21. Overlap effect

  22. +Rand noise

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

  24. 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

  25. Presence of heteroscedastic noise

  26. 41 reaction times &101 wavelengths + a heterosced. noise

  27. Inaccurate results from ccrX !

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

  29. W=

  30. n=50 Accurate results from weighted ccrX !

  31. FSMWFA Recognition of the presence of heterosc. noise

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

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

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

  35. Presence of unknown interference

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

  37. ccrX ccrC pcrC pcrT

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

  39. = CC+X = XX+C = CC+T = TT+C 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 smaller space to a larger one Projection of a larger space to a smaller one

  40. = TT+C in the presence of unknown interference, drift or shift. Target Transform (pcrC) is the most preferred method

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

  42. ccrC ccrX pcrT pcrC

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

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

  45. Differential pulse Voltammetry Each voltammog. depends only on its own E1/2

  46. Successive complexation:

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

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