Report

pH Emission Spectrum A A λ1 Ex1 Emission(3λ) λ2 λ λ1 λ2 λ3 λ3 A Ex2 A Emission(3λ) λ1 λ2 λ3 λ λ2 λ2 λ3 λ3 A λ1 Emission(3λ) λ1 λ1 A Ex3 Ex1 Ex2 Ex3 λ2 λ3 Second order (matrix) Ex1 Ex2 Ex3 pH Emission(3λ) Ex1 λ1 λ2 λ3 pH Ex1 Ex2 Ex3 pH Emission(3λ) Ex2 pH λ1 λ1 λ2 λ2 λ3 λ3 λ Ex1 Ex2 Ex3 Ex1 Ex2 Ex3 pH Emission(3λ) Ex3 λ1 λ2 λ3 3-way Methods: PARAFAC, … Ex Second-order, i.e., matrix data for a given sample can be produced in a variety of ways: one of them is pH λex pH λem. M λem. λex analyte acid–base properties Sample pH-Spectral λ Sample pH M pH λ Soft modeling parallel factor analysis method attempts to decompose a threeway data into the product of three significantly smaller matrices. K K P D A = C P P K B J E + I I I P vecD x p y p z p vecE p 1 oIn some of three-way data array, some factors are strictly proportional in one mode of a three-way array and the PARAFAC may lead to false minima. oHowever, appropriate selection of the initial parameters and restrictions (e.g. non-negativity) still make PARAFAC useful in this regard . o The and constraints are optional selection in the PARAFAC rutine belonging to N-way toolbox o There are not any reports on the applying hard modeling on some species to in the PARAFAC solution. Incorporating hard constraint for some or all of the concentration profiles in the soft modeling PARAFAC algorithm that is called model (HSPARAFAC) An interesting discussion is whether the chemical model imposed as a constraint on the data for some of the species, lead to ensure unique profiles in some or all of modes for corresponding species while the Alternating least squares PARAFAC algorithm Algorithms for fitting the PARAFAC model are usually based on alternating least squares. This is advantageous because the algorithm is simple to implement, simple to incorporate constraints in, and because it guarantees convergence. However, it is also sometimes slow. The solution to the PARAFAC model can be found by alternating least squares (ALS) by successively assuming the loadings in two modes known and then estimating the unknown set of parameters of the last mode. Determining the rank of three-way array The PARAFAC algorithm begins with an initial guess of the two loading modes Suppose initial estimates of B and C loading modes are given Step1. Determining of A profile K JK J Matricizing I I N = N I JK A = XZA+ Matricizing X (I×J×K) IK N N = J X (J×IK) IK J X(J×IK) = B(J×N)(CA)T = B ZBT B =X ZB+ X (I×J×K) Matricizing IJ N = K X (K×IJ) N IJ K X(K×IJ) = C(K×N)(BA)T = C ZCT C =X ZC+ 4-1. Reconstructing Three-way Array X from obtained A and B and C profiles 4-2. Calculating the norm of residual array I J K Rss (x ijk x ijk ) 2 i 1 j 1 k 1 % fit (1 Rss I J K 2 x ijk ) 100 i 1 j 1 k 1 5. Go to step 1 until relative change in fit is small. Given: X of size I × J × K Initialize B and C 2 ZA=CB A = X(I×JK ) ZA(ZAZA)−1 3 ZB=CA B = X(J×IK ) ZB(ZBZB)−1 4 5 ZC=BA C = X(K×JI ) ZC(ZCZC)−1 Go to step 1 until relative change in fit is small K K C K B = D A + I I J E I Hard constraint for two components A Nonlinear fitting constraint A Chemical Model HA A- +H+ AALS AFIT Given: X of size I × J × K Initialize B and C 2 ZA=CB A = X(I×JK ) ZA(ZAZA)−1 2-1 3 4 5 ZB=CA ZC=BA B = X(J×IK ) ZB(ZBZB)−1 C = X(K×JI ) ZC(ZCZC)−1 Go to step 1 until relative change in fit is small A- + H+ HA λex pH pH λem. M λem. λex ×10-23 pKa =5 Data Case I Noise free Noisy PARAFAC No. of Time iter. 2 0.02 4 0.023 HPARAFAC No. of Time pKa iter. 13 0.33 5.00 12 0.25 5.00 HA A- + H+ HB B- + H+ λex pH pH λem. M λem. Closure Rank Deficiency λex HA Ct1 [H+] [HA] = [H+] + Ka1 A - + H+ [A-] Ct1 Ka1 = [H+] + Ka1 [HA] + [A-] HB B- + H+ Ct2 [H+] [HB] = [H+] + Ka2 [B -] Ct2 Ka2 = [H+] + Ka2 [HB] + [B-] =a [HA] + [A-] = a([HB] + [B-]) Closure Rank Deficiency Description of Three-way Data Case II.a HA A- +H+ HB B- +H+ Case II.b HA A- +H+ HB B- +H+ Case III a HA A- +H+ HB B- +H+ emHA = emA- Case III b HA A- +H+ HB B- +H+ emHA = emHB Case IV HA A- +H+ HB B- +H+ emHA = emAemHA = emHB HA A- +H+ HB B- +H+ B- is not Spectroscopic active emHA=emA- Rank overlap problem Free Noise Data Noisy Data Unique Iter. Time(s) pKa Iter. Time(s) pKa Methods PARAFAC HSPARAFAC . 839 7.47 170 2.57 552 4.94 5.00 87 1.41 Hard constraints has been applied on rank overlap components 5.00 ex T = pH T- V ex U pH t11 t12 T= t 21 t 22 T 1 1 1 t 12 t 21 1 T= t 21 1 t 21 t 12 1 M. Vosough et. al. J. Chemom. 2006; 20: 302-310. t12 1 Calculation of excitation profiles as a function of (t12,t21) T T T T 1 t v1 +t12 v 2 s1 T 12 v1 T TV = = =S T = T T T t 21 1 v 2 t 21v1 +v 2 s2 Calculation of pH profiles as a function of t12 and t21 UT = u1 -1 = u1 -t 21u 2 1 -t12 1 u2 = -t 21 1 1-t12 t 21 1 u 2 -t12u1 = c1 c2 = C 1-t12 t 21 HA HB B- +H+ A- B- is not Spectroscopic active +H+ emHA=emHB Rank overlap problem Methods Noisy Data Unique Iter. Time(s) pKa Iter. Time(s) pKa PARAFAC HSPARAFAC Free Noise Data ? 2483 22.08 65 0.96 417 3.76 5.00 48 0.80 5.00 Hard constraints have been applied only on one the rank overlap components. ex T = pH T- V ex U pH t11 t12 T= t 21 t 22 T 1 1 1 t 12 t 21 1 T= t 21 1 t 21 t 12 1 M. Vosough et. al. J. Chemom. 2006; 20: 302-310. t12 1 Calculation of excitation profiles as a function of (t12,t21) T T T T 1 t v1 +t12 v 2 s1 T 12 v1 T TV = = =S T = T T T t 21 1 v 2 t 21v1 +v 2 s2 Calculation of pH profiles as a function of t12 and t21 UT = u1 -1 = u1 -t 21u 2 1 -t12 1 u2 = -t 21 1 1-t12 t 21 1 u 2 -t12u1 = c1 c2 = C 1-t12 t 21 HA A- +H+ HB B- +H+ All components are Spectroscopic active emHA=emA- Rank overlap and closure rank deficiency Methods Noisy Data Unique Iter. Time(s) pKa Iter. Time(s) pKa PARAFAC HSPARAFAC Free Noise Data 7508 . 839 62.78 2650 21.89 12.09 4.00 421 6.15 Hard constraints have been applied on rank overlap components. 5.00 HA A- +H+ HB B- +H+ All components are Spectroscopic active emHA=emHB Rank overlap and closure rank deficiency Methods Free Noise Data Noisy Data Unique Iter. Time(s) pKa Iter. Time(s) pKa PARAFAC HSPARAFAC ? 1398 19.21 5.00 991 16.01 Hard constraints has been applied on one of the rank overlap components. 5.00 HA A- +H+ HB B- +H+ All components are Spectroscopic active emHA=emAemHB=emB- Rank overlap and two closure rank deficiency Methods Free Noise Data Noisy Data Unique Iter. Time(s) pKa Iter. Time(s) pKa PARAFAC & HSPARAFAC HPARAFAC . 763 13.62 5.00 371 7.13 5.00 There were not good initialization for rank overlap species H2A HA- + H+ HA- A- 2 + H+ pH=2.2 pH=4.8 800 600 400 200 600 400 200 450 400 λem. 350 250 λex 300 pH=7.8 450 400 λem. 350 250 λex 300 pH=12 800 600 400 200 600 400 200 450 400 λem. 350 250 λex 300 450 400 λem. 350 250 λex 300 Method No. of iter. Time pKa1 pKa2 PARAFAC 1410 7.58 HSPARAFAC 68 1.03 4.81 9.09 HPARAFAC 13 0.32 4.81 9.05 Reported pKa1 and pKa2 of PY are 4.8 and 9.2 respectively. Ghasemi J, Abbasi B, Kubista M. J. Korean Chem. Soc. 2005; 49: 269-277. Sample λ Sample pH M pH Rank Overlap Closure Rank Deficiency λ [A ] CCt t K a 2 [H ] Ka Ct [ H ] [HA] [H ] K a Ct Ct 0.14 0.7 concentration (Micro molar) 0.12 0.08 0.06 0.04 0.02 0 250 0.5 0.4 0.3 0.2 0.1 300 350 400 450 500 550 600 1 2 3 Abs. Wavelength (nm) 4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 7 5 Sample number 0.4 Relative concentration Abs. 0.1 0.6 7.5 8 8.5 9 9.5 pH 10 10.5 11 11.5 12 6 7 8 n (Micro mol ar) 0.7 0.12 0.1 0.6 0.5 concentratio 0.4 0.06 0.3 0.2 0.04 0.1 1 0.02 0 250 300 350 400 450 500 550 2 3 600 4 5 Sample num gth (nm) Abs. W avelen 0.4 Relative concentration Abs. 0.08 0.35 0.3 0.25 0.2 0.15 0.1 0.05 7 7.5 8 8.5 9 9.5 pH 10 10.5 11 11.5 12 6 ber 7 8 Method No. of iter. PARAFAC HPARAFAC 6548 429 Time pKa1 pKa2 560.17 28.15 10.61 9.51 Reported pKa of TA ans SY are 9.6 and 10.4 respectively. Pérez-Urquiza M, Beltrán JL. Journal of Chromatography A. 2001;917:331-6. Thanks to Mr. Javad Vallipour from Tabriz University All the mentioned simulated three-way data, the GUI program of PARAFAC, the HSPARAFAC for monoporotic acids are available on the web. Please analyze each data with these algorithms to find the advantages of HSPARAFAC rather than PARAFAC !!!