pH Emission Spectrum Emission(3 λ) λ1 λ2 λ3 A λ λ1λ2λ3λ1λ2λ3 A Ex 1 Emission(3 λ) λ1λ2λ3λ1λ2λ3 A Ex 2 Emission(3 λ) λ1λ2λ3λ1λ2λ3 A Ex 3 λ1λ2λ3λ1λ2λ3 Second order (matrix) Ex 1 Ex 2 Ex 3 λ1 λ2 λ3 A λ A

λ1λ2λ3λ1λ2λ3 Ex 1 Ex 2 Ex 3 pH Ex λ pH Emission(3 λ) λ1λ2λ3λ1λ2λ3 Ex 1 Emission(3 λ) λ1λ2λ3λ1λ2λ3 Ex 2 Emission(3 λ) λ1λ2λ3λ1λ2λ3 Ex 3 pH Ex 1 Ex 2 Ex 3 3-way Methods: PARAFAC, …

λ em. λ ex M pH λ em. pH

M Sample pH Sample pH λ λ

Soft modeling parallel factor analysis method attempts to decompose a three- way data into the product of three significantly smaller matrices. K I = + B I D K C K I E J A P P P

o In 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. o However, appropriate selection of the initial parameters and restrictions (e.g. non-negativity) still make PARAFAC useful in this regard.

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 PARAFAC algorithm begins with an initial guess of the two loading modes 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

 Suppose initial estimates of B and C loading modes are given = K J I Matricizing I JK I N N A = XZ A +

X (I×J×K)X (J×IK) B =X Z B + = J IK N J N Matricizing X (J×IK) = B (J×N) (C  A) T = B Z B T

X (I×J×K)X (K×IJ) = K IJ N K N Matricizing C =X Z C + X (K×IJ) = C (K×N) (B  A) T = C Z C T

5. Go to step 1 until relative change in fit is small Reconstructing Three-way Array from obtained A and B and C profiles 4-2. Calculating the norm of residual array

Initialize B and C 2 A = X (I×JK ) Z A (Z A Z A ) −1 3 B = X (J×IK ) Z B (Z B Z B ) −1 4 C = X (K×JI ) Z C (Z C Z C ) −1 Given: X of size I × J × K Go to step 1 until relative change in fit is small 5 ZA=CBZA=CB ZB=CAZB=CA ZC=BAZC=BA

Chemical Model HA A - +H + K I = + B I D K C K I E J A Hard constraint for two components Nonlinear fitting constraint A A A ALS A FIT

Initialize B and C 2 A = X (I×JK ) Z A (Z A Z A ) −1 3 B = X (J×IK ) Z B (Z B Z B ) −1 4 C = X (K×JI ) Z C (Z C Z C ) −1 Given: X of size I × J × K Go to step 1 until relative change in fit is small 5 ZA=CBZA=CB ZB=CAZB=CA ZC=BAZC=BA 2-1

HA A - + H + λ em. λ ex M pH λ em. pH

×10 -23

pK a =5

HA A - + H + HB B - + H + Closure Rank Deficiency λ em. λ ex M pH λ em. pH

= a [HA] + [A - ] = a([HB] + [B - ]) [HB] + [B - ] [HB] = C t2 [H + ] [H + ] + K a2 CK [B - ] = t2a2 [H + ] + K a2 HB B - + H + [HA] = C t1 [H + ] [H + ] + K a1 CK [A - ] = t1a1 [H + ] + K a1 HA A - + H + [HA] + [A - ]

Case IV em HA = em HB Case III b em HA = em A - Case III a Case II.b Case II.a Description of Three-way Data em HA = em A- em HA = em HB Closure Rank Deficiency HA A - +H + HB B - +H + HA A - +H + HB B - +H + HA A - +H + HB B - +H + HA A - +H + HB B - +H + HA A - +H + HB B - +H +

HA A - +H + HB B - +H + B - is not Spectroscopic active em HA =em A- Hard constraints has been applied on rank overlap components UniqueTime(s) Iter. Methods Free Noise Data Noisy Data Time(s) pK a 5.00 Rank overlap problem.

ex pH = ex pH T V U T-T- M. Vosough et. al. J. Chemom. 2006; 20:

Calculation of excitation profiles as a function of (t 12,t 21 ) Calculation of pH profiles as a function of t 12 and t 21

Hard constraints have been applied only on one the rank overlap components. HA A - +H + HB B - +H + B - is not Spectroscopic active em HA =em HB UniqueTime(s) Iter. Methods Free Noise Data Noisy Data Time(s) pK a 5.00 Rank overlap problem ?

ex pH = ex pH T V U T-T- M. Vosough et. al. J. Chemom. 2006; 20:

Calculation of excitation profiles as a function of (t 12,t 21 ) Calculation of pH profiles as a function of t 12 and t 21

Hard constraints have been applied on rank overlap components. HA A - +H + HB B - +H + All components are Spectroscopic active em HA =em A- UniqueTime(s) Iter. Methods Free Noise Data Noisy Data Time(s) pK a 4.00 Rank overlap and closure rank deficiency

HA A - +H + HB B - +H + All components are Spectroscopic active em HA =em HB UniqueTime(s) Iter. Methods Free Noise Data Noisy Data Time(s) pK a 5.00 Rank overlap and closure rank deficiency ? Hard constraints has been applied on one of the rank overlap components.

There were not good initialization for rank overlap species HA A - +H + HB B - +H + All components are Spectroscopic active em HA =em A- em HB =em B- UniqueTime(s) Iter. Methods Free Noise Data Noisy Data Time(s) pK a Rank overlap and two closure rank deficiency

H 2 A HA - + H + HA - A H + HA - A H +

λ em. λ ex λ em. λ ex λ em. λ ex λ em. λ ex pH=2.2 pH=4.8 pH=7.8pH=12

Reported pK a1 and pK a2 of PY are 4.8 and 9.2 respectively. Ghasemi J, Abbasi B, Kubista M. J. Korean Chem. Soc. 2005; 49:

M Sample pH Sample pH λ λ Rank Overlap Closure Rank Deficiency

CtCt CtCt2 CtCt CtCt

Reported pK a 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

Please analyze each data with these algorithms to find the advantages of HSPARAFAC rather than PARAFAC !!! All the mentioned simulated three-way data, the GUI program of PARAFAC, the HSPARAFAC for monoporotic acids are available on the web.