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)(CA)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)(BA)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=CB
A = X(I×JK ) ZA(ZAZA)−1
3
ZB=CA
B = X(J×IK ) ZB(ZBZB)−1
4
5
ZC=BA
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=CB
A = X(I×JK ) ZA(ZAZA)−1
2-1
3
4
5
ZB=CA
ZC=BA
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 !!!