ph emission spectrum emission(3 λ) λ1 λ2 λ3 a λ λ1λ2λ3λ1λ2λ3 a ex 1 emission(3 λ)...
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pH Emission Spectrum
Emission(3λ)
λ1 λ2 λ3
A
λ
λ1
λ2
λ3
A
Ex1
Emission(3λ)λ1
λ2
λ3
A
Ex2
Emission(3λ)
λ1
λ2
λ3
A
Ex3
λ1
λ2
λ3
Second order
(matrix)
Ex1 Ex2 Ex3
Second order data
λ1 λ2 λ3
A
λ
A
1
λ1
λ2
λ3Ex1 Ex2 Ex3
pH
Exλ
pH
Emission(3λ)λ1
λ2
λ3Ex1
Emission(3λ)
λ1
λ2
λ3Ex2
Emission(3λ)λ1
λ2
λ3Ex3
pH
pH
pH
Ex1 Ex2 Ex3
Ex1 Ex2 Ex3
Ex1 Ex2 Ex3
3-way Methods: PARAFAC, …
2
λem.
λex
M
pH λex
λem.
pH
Second-order, i.e., matrix data for a given sample can be
produced in a variety of ways:
one of them is Fluorescence spectroscopy
3
M
Sample
pH
Sample
pH
λ
λ
Another attractive possibility is to take advantage of the analyte acid–base properties, i.e. obtaining pH-
Spectral data matrix for each sample
4
Soft modeling parallel factor analysis method attempts to
decompose a three-way data into the product of three
significantly smaller matrices.
EzyxD vecvecP
pppp
1
K
I
= +
B
ID
KC K
I
E
JA
PPP
Parallel Factor Analysis
5
The popularity of PARAFAC model in resolving multi-way data is due to its unique properties
The reason for the importance of uniqueness of solutions is clear: obtaining unique solutions allows for an unambiguous interpretation of the estimated parameters (spectra, chromatograms, etc.) in terms of the chemical system. In short, uniqueness generates fundamental insight into the chemistry of the studied system.
6
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 .
7
o The non-negativity, unimodality and orthogonality 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 reduce rotational ambiguity in the PARAFAC solution.
8
Incorporating hard constraint for some or all of the
concentration profiles in the soft modeling PARAFAC
algorithm that is called hard-soft PARAFAC 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
traditional PARAFAC didn't provide unique solution. 9
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.10
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
11
Step1. Determining of A profile
Suppose initial estimates of B and C loading modes are given
=
K
J
I
Matricizing
I
JK
IJK
N
N
A = XZA+
12
5. Go to step 1 until relative change in fit is small.
4-1. Reconstructing Three-way Array from obtained A and B and C profiles
4-2. Calculating the norm of residual array
2
1 1 1
( )I J K
ijk ijki j k
Rss x x
X
100)1(%
1 1 1
2
I
i
J
j
K
kijkx
Rssfit
15
Initialize B and C
2 A = X(I×JK ) ZA(ZAZA)−1
3 B = X(J×IK ) ZB(ZBZB)−1
4 C = X(K×JI ) ZC(ZCZC)−1
Given: X of size I × J × K
Go to step 1 until relative change in fit is small
1
5
ZA=CB
ZB=CA
ZC=BA
16
Chemical Model
HA A- +H+
K
I
= +B
ID
K C K
IE
JA
Hard constraint for two components
Nonlinear fitting constraint
AA
AALSAFIT
18
Initialize B and C
2 A = X(I×JK ) ZA(ZAZA)−1
3 B = X(J×IK ) ZB(ZBZB)−1
4 C = X(K×JI ) ZC(ZCZC)−1
Given: X of size I × J × K
Go to step 1 until relative change in fit is small
1
5
ZA=CB
ZB=CA
ZC=BA
Hard Constraint 2-1
19
HPARAFAC PARAFACData Case
IpKa Time No. of iter.
Time No. of iter.
5.00 0.33 13 0.02 2 Noise free
5.00 0.25 12 0.023 4 Noisy
26
Ct1 = a Ct2[HA] + [A-] = a([HB] + [B-])
[HB] + [B-] = Ct2
[HB] = Ct2 [H+]
[H+] + Ka2
C K[B-] = t2 a2
[H+] + Ka2
HB B- + H+
[HA] = Ct1 [H+]
[H+] + Ka1
C K[A-] = t1 a1
[H+] + Ka1
HA A- + H+
[HA] + [A-] = Ct1
28
Case IV
emHA = emHBCase III b
emHA = emA-Case III a
B- is not spectroscopic active# emHA = emHBCase II.b
B- is not spectroscopic active, emHA
= emA-Case II.a
Description of Three-way Data
emHA = emA-
emHA = emHB
Closure Rank Deficiency
Five Different Simulation Three-way Data
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+
29
Results of PARAFAC and HSPARAFAC for Case IIa Three-way Data
HA A- +H+
HB B- +H+
B- is not Spectroscopic active
emHA=emA-
Hard constraints has been applied on rank overlap components
Unique Time(s)Iter. Iter.Methods
Free Noise Data Noisy Data
5.00
839 7.47
552 4.94
170
87
2.57
1.41
Time(s) pKapKa
5.00
Rank overlap problem
.PARAFAC
HSPARAFAC
30
The unique Span spanned by excitation and pH profiles of rank overlapped species
HA and A-
11 12
21 22
t t=
t t
T 12
21
1 t=
t 1
T
1t
t1
tt1
1
21
12
2112
1T
ex
pH=
ex
pH
T V
UT-
Normalizing
M. Vosough et. al. J. Chemom. 2006; 20: 302-310. 31
Calculation of excitation profiles as a function of (t12,t21)
Calculation of pH profiles as a function of t12
and t21
T T T T12T T1 1 12 2 1
T T T T21 2 21 1 2 2
1 t +t= = = =
t 1 t +
v v v sTV S
v v v s
12-11 2
21 12 21
1 21 2 2 12 1 1 212 21
1 -t 1= =
-t 1 1-t t
1= -t -t = =
1-t t
UT u u
u u u u c c C 32
Is hard constraint equivalent to unify t elements?
Hard constraints have been applied only on one the rank overlap components.
Results of PARAFAC and HSPARAFAC for Case IIb Three-way Data
HA A- +H+
HB B- +H+
B- is not Spectroscopic active
emHA=emHB
Unique Time(s)Iter. Iter.Methods
Free Noise Data Noisy Data
5.0065 0.96 48 0.80
Time(s) pKapKa
5.00
Rank overlap problem
PARAFAC
HSPARAFAC
2483 22.08 417 3.76
?
33
The unique Span spanned by excitation and pH profiles of rank overlapped species
HA and HB
11 12
21 22
t t=
t t
T 12
21
1 t=
t 1
T
1t
t1
tt1
1
21
12
2112
1T
ex
pH=
ex
pH
T V
UT-
Normalizing
M. Vosough et. al. J. Chemom. 2006; 20: 302-310. 34
Calculation of excitation profiles as a function of (t12,t21)
Calculation of pH profiles as a function of t12
and t21
T T T T12T T1 1 12 2 1
T T T T21 2 21 1 2 2
1 t +t= = = =
t 1 t +
v v v sTV S
v v v s
12-11 2
21 12 21
1 21 2 2 12 1 1 212 21
1 -t 1= =
-t 1 1-t t
1= -t -t = =
1-t t
UT u u
u u u u c c C 35
Hard constraint was applied on only one of the rank overlapped pH profiles so….
Hard constraints have been applied on rank overlap components.
Results of PARAFAC and HSPARAFAC for Case IIIa Three-way Data
HA A- +H+
HB B- +H+
All components are Spectroscopic active
emHA=emA-
Unique Time(s)Iter. Iter.Methods
Free Noise Data Noisy Data
5.00839 12.09 421 6.15
Time(s) pKapKa
4.00
Rank overlap and closure rank deficiency
PARAFAC
HSPARAFAC .7508 62.78 2650 21.89
36
Results of PARAFAC and HSPARAFAC for Case IIIb Three-way Data
HA A- +H+
HB B- +H+
All components are Spectroscopic active
emHA=emHB
Unique Time(s)Iter. Iter.Methods
Free Noise Data Noisy Data
5.001398 19.21 991 16.01
Time(s) pKapKa
5.00
Rank overlap and closure rank deficiency
PARAFAC
HSPARAFAC ?
Hard constraints has been applied on one of the rank overlap components. 37
There were not good initialization for rank overlap species
Results of PARAFAC, HSPARAFAC and HPARAFAC for Case IIIc Three-way Data
HA A- +H+
HB B- +H+
All components are Spectroscopic activeemHA=emA-
emHB=emB-
Unique Time(s)Iter. Iter.Methods
Free Noise Data Noisy Data
Time(s) pKapKa
Rank overlap and twoclosure rank deficiency
PARAFAC&
HSPARAFAC
HPARAFAC 5.00763 13.62 371 7.135.00.
38
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300
350400
450
200
400
600
800
250
300
350400
450
200
400
600
250
300
350400
450
200
400
600
250
300
350400
450
200400600
800
λem. λex λem. λex
λem. λex λem.λex
pH=2.2 pH=4.8
pH=7.8 pH=12
EEM at Different pH for PY Solution (2.40×10-6M)
40
pKa2 pKa1 Time No. of iter. Method
7.58 1410 PARAFAC
9.09 4.81 1.03 68 HSPARAFAC
9.05 4.81 0.32 13 HPARAFAC
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. 44
Spectrophotometric Study of
Tarterazin and Sunset Yellow in
Mixtures
M
Sample
pH
Sample
pH
λ
λ
Rank OverlapClosure Rank Deficiency 46
250 300 350 400 450 500 550 6000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Abs. Wavelength (nm)
Ab
s.
1 2 3 4 5 6 7 80.1
0.2
0.3
0.4
0.5
0.6
0.7
Sample number
con
cen
tra
tion
(M
icro
mo
lar)
7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 120
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
pH
Re
lativ
e c
on
cen
tra
tion
48
1 2 3 4 5 6 7 8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Sample number
con
cen
tra
tion
(M
icro
mo
lar)
250 300 350 400 450 500 550 600
0
0.02
0.04
0.06
0.08
0.1
0.12
Abs. Wavelength (nm)
Ab
s.
7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
pH
Re
lativ
e c
on
cen
tra
tion
49
Method No. of iter. Time pKa1 pKa2
PARAFAC 6548 560.17 - -
HPARAFAC 429 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.
50
Conclusion HSPARAFAC method can decompose data
uniquely when the equilibrium model of rank overlap species are incorporated in the HSPARAFAC algorithm even in the presence of unknown interference.
The parameters of the models, i.e. pKas, were calculated.
Compared with PARAFAC, HSPARAFAC take much less time and the number of iterations greatly reduced.
51
54
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.
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