Estimation of error propagation and
prediction intervals in Multivariate
Curve Resolution Alternating Least
Squares using resampling methods
Joaquim Jaumot, Raimundo Gargallo
and Romà Tauler*
Department of Analytical Chemistry
University of Barcelona
Outline:
•Introduction
•Rotational ambiguities and feasible bands
•Error propagation and resampling methods
•Results
•Conclusions
Multivariate (Soft) Self Modeling
Curve Resolution
NC
1
1.5
C
0.8
0.6
0.4
ST
ST
1
NC
C
0.5
+
NR
NR
E
0.2
0
0
10
20
30
40
0
0
20
40
60
80 100
NC
1.5
1
D
D
NR
0.5
0
0
N
d = ∑ c s +e
ij k=1 ik kj ij
Bilinearity!
10
20
30
40
50
60
70
80
90
Multivariate (Soft) Self Modeling
Curve Resolution
• Multivariate Curve Resolution (MCR) methods have
been shown to be powerful self-soft-modeling tools able to
investigate complex chemical systems with a minimum
number of assumptions.
• Alternating Least Squares (ALS) has become a
popular method for Multivariate Curve Resolution (MCR)
due to its flexibility in constraint implementation during the
optimization of resolved profiles.
Multivariate (Soft) Self Modeling
Curve Resolution
• What are the reliability of MCR-ALS
estimations?
•
Do the MCR-ALS solutions have rotational and
scale freedom?
•
Are they unique solutions or exist instead a band
of feasible solutions?
•
How errors and noise are propagated from
experimental data to ALS estimations?
Goals of this study
• Find the reliability of ALS resolved profiles in
multivariate curve resolution.
• Estimate prediction error intervals for ALS profiles
• Estimate prediction error intervals for parameters
calculated from MCR-ALS resolved profiles
• Investigate the interaction between propagation of
errors and rotational ambiguities (noise effects on
rotational ambiguities and constraints).
Outline:
•Introduction
•Rotational ambiguities and calculation of
feasible bands
•Error propagation and resampling methods
•Results
•Conclusions
Lawton and Sylvestre feasible bands
non-negative spectra
I
pure spectrà
normalization
to unit area
AI
5
non-negative concentration
4
3
AII
2
II
1
1,2,3,4,5
experimental
values
PC1
non-negative concentration
PC2
non-negative spectra
W.H. Lawton and EA Sylvestre, Technometrics 1971, 13, 617-33
Rotational Ambiguities
Factor Analysis (PCA) Data Matrix Decomposition
D = U VT + E
‘True’ Data Matrix Decomposition
D = C ST + E
D = U T T-1 VT + E = C ST + E
C = U T; ST = T-1 VT
How to find the rotation matrix T?
Matrix decomposition is not unique!
T(N,N) is any non-singular matrix
There is rotational freedom for T
Rotational Ambiguities
Because of rotational ambiguities instead of
unique solutions, a set of feasible solutions are
obtained
Feasible solutions are different solutions that fit
equally well the data under a set of constraints
For a particular system under a set of constraints,
feasible solutions are defined from a set of
possible T values.
Rotational Ambiguities
• T values define the band of feasible
solutions or feasible bands
• How to define the boundaries of these
feasible bands?
• How to represent graphically these
boundaries?
Is it possible to define band boundaries
(Tmax and Tmin)?
•0.5
Tmax
•0.4
•0.3
How
to
calculate
Tmax
and
Tmin?
Tmin
•0.2
•0.1
•0
•0
•5
•10
•15
•20
•25
•30
•35
•40
•45
•50
Tmax
•1.5
•1
•0.5
•0
•0
•5
•10
•15
Tmin
•20
•25
•30
•35
•40
How to define and find the band
boundaries?
• What are the T values giving the maximum/outer
and minimum/ inner boundaries of the feasible
bands under a set of constraints?
D*= Cinic STinic =
= Cinic Tmin T-1min STinic = CminSTmin =
= CinicTmax T-1max STinic = CmaxSTmax
where: D(NR,NC), C(NR,N), ST(N,NC), T(N,N)
How to define and evaluate Tmax and Tmin?
Evaluation of boundaries of feasible bands:
Previous studies
• W.H.Lawton and E.A.Sylvestre, Technometrics, 1971, 13, 617633
•O.S.Borgen and B.R.Kowalski, Anal. Chim. Acta, 1985, 174, 126
•K.Kasaki, S.Kawata, S.Minami, Appl. Opt., 1983 (22), 35993603
•R.C.Henry and B.M.Kim (Chemomet. and Intell. Lab. Syst.,
1990, 8, 205-216)
•P.D.Wentzell, J-H. Wang, L.F.Loucks and K.M.Miller
(Can.J.Chem. 76, 1144-1155 (1998))
•P. Gemperline (Analytical Chemistry, 1999, 71, 5398-5404)
•R.Tauler (J.of Chemometrics 2001, 15, 627-46)
•M.Legger and P.D.Wentzell, Chemomet and Intell. Lab. Syst.,
2002, 171-188
Definition of band boundaries
The whole measured signal is:
D = ∑ Di = ∑ ci siT
The contribution of each species to the whole signal is:
Di = cisiT
Solving the Optimization Problem:
max/outer boundary: Find Tmax that makes ci siT maximum
min/inner boundary: Find Tmin that makes ci siT minimum
Constrained Non-Linear Optimization
Problem (NCP)
Find T which makes:
min/max f(T)
T
subject to ge(T) = 0
and to
gi(T) ≤ 0
where T is the matrix of variables, f(T) is a
non-linear scalar function of T and g(T) is the
vector of constraints (non-linear function of T)
Matlab Optimizarion Toolbox fmincon function
1) What are the variables of the problem?
T (rotation matrix),
D = C T T-1 ST
2) What is the objective function f(T) to be
optimized?
For each species i = 1,..,ns
fi (T) =
ij ij
c i si
CS
∑c s
f (T ) =
∑c s
T
or i
j
ij ij
i,j
f(T) is scalar value between 0 and 1!
This gives the
relative signal
contribution of
species i respect
the global
measured signal !
3) What are the constraints g(T)?
The following constraints may be considered:
normalization/closure
gnorm/gclos
non-negativity
gcneg/gsneg
known values/selectivity
gknown/gsel
unimodality
gunim
trilinearity (three-way data)
gtril
Are they equality or inequality constraints?
4) What are the initial estimates of C, ST?
•Initial estimates of C and ST are obtained by MCR-ALS
•Initial estimates are feasible solutions fulfilling the
constraints of the system (non-negativity, unimodality,
closure, selectivity, local rank,…)
5) What are the initial values of T?
•NCP depends on initial estimates of T! (local minima,
convergence, speed …)
Tini = eye(N) =
1

0
 ...

0

0
...
1
...
...
...
0
...
0

0
... 

1 
Optimization algorithm
•R.Tauler (J.of Chemometrics 2001, 15, 627-46)
Initial estimations of CALS and S ALS
profiles are obtained by MCR-ALS
T=eye(number of species)
For each species define objective function
f(T)=norm(c(T)s(T))=norm(cALS T sALS / T)
Select constraints g(T):
equality ge: normalization/closure, known values,
inequality gi: non-negartivity, selectivity, unimodality, trilinearity,
Find T min which gives a minimum
of f(T)
under constraints gi(T)<0, ge(T)=0
Find T max which gives a maximum
of f(T)
under constraints gi(T)<0. ge(T)=0
Built minimum band
c min = cALS / T min
s min = sALS / T min
Built maximum band
cmax = cALS / T max
smax=sALS / T max
Experimental data system under
study
1.5
1
0.5
0
0
D
10
20
30
40
50
60
70
80
90
Acid-base spectrophotometric titration of the double
stranded heteropolynucleotide polyinosinic-polycytidylic
acid. Spectral region between 240-320 nm and pH region
between pH 2 and pH 9
concentration profiles
1
Application of MCRALS to the
experimental data
matrix D
Conc. relativa
0.8
C
0.6
0.4
0.2
0
pK2
pK1
2
3
4
5
6
7
8
9
pH
spectra profiles
Absorbànica /a.u.
0.8
0.6
ST
0.4
Initial estimates were obtained
from purest variables
0.2
0
0
10
20
30
40
50
Nº canal /nm
60
70
Applied constraints in ALS
were:
a) non-negative spectra
b) non-negative
concentrations
c) closure in concentrations
80
90
•This system has selectivity! local rank resolution conditions!
•Initial estimates from pure variable detection methods provide good
initial estimates that produce solutions close to the true profiles
Parameter estimation
Mass-action law is only assumed at the site level
and not for the whole polynucleotide molecule
Evaluation of constants
from intersection profiles
pK1
3.6660
Proposed species:
poly(I)-poly(C+) Ù
poly(I)-poly(C)-poly(C+) + H Ù
pK2
4.9244
poly(I)-poly(C) + H
Estimation of band boundaries
(max/min contribution of each species)
of feasible solutions
1
0.8
0.6
Large Rotational 0.4
0.2
ambiguities
0
were present
when constraints
3
applied were
2.5
only closure
non-negativity!!! 2
5
10
15
20
25
30
35
1.5
1
0.5
0
0
10
20
30
40
50
60
70
80
Estimation of band boundaries
(max/min contribution of each species)
of feasible solutions
1
0.8
Rotational
ambiguities
nearly
dissappear
when selectivity
constraint was
applied!!!
0.6
0.4
0.2
0
5
10
15
20
25
30
35
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
Outline:
•Introduction
•Rotational ambiguities and feasible bands
•Error propagation and resampling methods
•Results
•Conclusions
Error propagation and resampling
methods
•How experimental error/noise in the input data
matrices affects MCR-ALS results?
•For ALS calculations there is no known
analytical formula to calculate error estimations.
(i.e. like in linear lesast-squares regressions)
•Bootstrap estimations using resampling methods
is attempted
Resampling Methods
Resampling Methods
Theoretical
Data
Montecarlo
Simulation
Experimental
Data
Noise
Addition
Jackknife
Building theoretical data
Experimental
Data, Dexp
MCR-ALS
ST
C
Theoretical
Data, D
Experimental
error
E
Montecarlo Simulations
M0.1 = D +N0.1 M1 = D +N1
M2= D +N2
M5= D +N5
Theoretical
Data
D
New Data Matrix
Random
Error
N0.1, N1, N2 and N5
MATLAB function randomn with zero
mean and relative sd 0.1%, 1%, 2% and
5% of maximum signal in D
Concentration
profiles
C
MCR-ALS
Pure Spectra
ST
250 times each noise level!
1000 simulations!
Noise Addition Simulations
D0.1 = Dexp +N0.1 D1 = Dexp +N1
D2= Dexp +N2 D5= Dexp +N5
Experimental
Data
Dexp
New Data Matrix
Random
Error
N0.1, N1, N2 and N5
MATLAB function randomn with zero
mean and relative sd 0.1%, 1%, 2% and
5% of maximum signal in D
Concentration
profiles
C
MCR-ALS
Pure Spectra
ST
250 times each noise level!
1000 simulations!
Experimental
Data
N0.1
N1
N2
and
N5
Jackknife Simulations
Dexp (36,81)
J1
+
J2
Random Error
=
New Data
Matrix
Dnoise (36,81)
D0.1, D1, D2, D5
Reduced Matrix (1,10,19,28) (32,81)
Reduced Matrix (2,11,20,29) (32,81)
J3
Reduced Matrix (3,12,21,30)
J4
Reduced Matrix (4,13,22,31)
J5
Reduced Matrix (5,14,23,32)
J6
Reduced Matrix (6,15,24,33)
J7
Reduced Matrix (7,16,25,34)
J8
Reduced Matrix (8,17,26,35)
J9
Reduced Matrix (9,18,27,36)
(32,81)
(32,81)
(32,81)
(32,81)
(32,81)
(32,81)
(32,81)
Jackknife Simulations
Jack Knife
Reduced
Data Matrix
JN
N = 1,..., 9
Concentration
profiles
MCR-ALS
C
Pure Spectra
ST
Outline:
•Introduction
•Rotational ambiguities and feasible bands
•Error propagation and resampling methods
•Results
•Conclusions
Presentation of Results
1. Calculation of species profiles error bands:
Mean profile, maximum and minimum profiles,
standard deviation profiles and confidence range
profiles
2. pKa (parameter) error estimations
3. Rotational ambiguity effects on error estimates
from resampling methods. Calculation of
boundaries of feasible bands from mean
species profiles error bands
Mean, bands and confidence range of the concentration profiles
1
0.9
0.8
Noise 0%
Rel. concentration
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
3
4
5
6
7
8
9
Monte Carlo Simulations
Concentration profiles:
Mean max and min profiles
Confidence range profiles
pH
Mean, bands and confidence range of concentratios profiles
1
Mean, bands and confidence range of the concentration profiles
1
0.8
Noise 0.1%
0.8
0.7
0.6
Rel. conc.
0.7
Rel. concentration
Noise 2%
0.9
0.9
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0.1
3
0
2
3
4
5
6
7
8
5
6
7
8
pH
pH
Mean, bands and confidence range of concentration profiles
Mean, bands and confidence range of concentration profiles
1.2
1
0.9
1
Noise 1%
0.8
0.7
0.8
Rel. conc.
0.6
Rel. conc.
4
9
0.5
0.4
0.3
0.6
0.4
Noise 5%
0.2
0.2
0
0.1
0
3
4
5
6
pH
7
8
-0.2
2
3
4
5
6
pH
7
8
9
Mean, bands and confidence range of concentration profiles
0.8
0.7
Noise 0%
Absorbance /a.u.
0.6
0.5
0.4
0.3
0.2
0.1
0
240
250
260
270
280
Wavelength /nm
290
300
310
320
Monte Carlo Simulations
Spectra profiles:
Mean max and min profiles
Confidence range profiles
Mean, bands and confidence range of the spectra
0.8
Mean, bands and confidence range of the spectra
0.8
0.7
Noise 0.1%
Noise 2%
0.7
0.6
0.5
0.5
Absorbance /a.u.
Absorbance /a.u.
0.6
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
240
-0.1
250
260
270
280
290
300
310
320
240
250
260
270
280
Wavelength /nm
290
300
310
320
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
240
Mean, bands and confidence range of the spectra
1.6
Noise 1%
1.4
1.2
1
Absorbance /a.u.
Absorbance /a.u.
Wavelength /nm
Mean, bands and confidence range of spectra
0.8
0.6
Noise 5%
0.4
0.2
0
250
260 Wavelength
270 280
/nm
290
300 310
320
-0.2
240
250
260
270
280
Wavelength /nm
290
300
310
320
Monte Carlo Simulations
pKa error estimations
pK2
pK1
Noise
added
Value
Std. dev
Value
Std. dev
0%
3.6660
4e-15
4.9244
9e-15
0.1 %
3.6662
6e-4
4.9243
0.0012
1%
3.6696
0.0065
4.9262
0.0128
2%
3.6761
0.0127
4.9173
0.0245
5%
3.9762
0.4349
5.0745
0.7595
Calculation of band boundaries from mean species
profiles error bands (under non-negativity and
closure constraints)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
5
10
15
20
25
30
35
2.5
0
Noise 0.1%
1.5
10
15
20
25
30
35
1.5
1
0.5
0.5
0
10
20
30
40
50
60
70
80
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
5
10
15
20
2.5
25
30
35
1.5
0
0
10
5
20
30
10
40
15
50
20
60
25
2
Noise 1%
2
Noise 2%
2
1
0
5
2.5
2
0
0
70
30
80
35
Noise 5%
1
1
0.5
0
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Calculation of band boundaries from mean profile
error bands (under non-negativity, closure and
selectivity constraints)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
5
10
15
20
25
30
35
0.8
5
10
15
20
25
30
35
0.8
Noise 0.1%
0.6
0.4
Noise 2%
0.6
0.4
0.2
0
0
0.2
0
10
20
30
40
50
60
70
80
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
5
10
15
20
25
30
35
0.8
0
0
10
5
20
30
10
40
15
50
20
60
25
70
30
80
35
1
Noise 1%
0.6
0.4
Noise 5%
0.8
0.6
0.4
0.2
0
0.2
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Mean, bands and confidence range of the concentration profiles
1
0.9
0.8
Rel. concentration
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
3
4
5
6
7
8
9
pH
Noise Addition Simulations
Concentration profiles:
Mean max and min profiles
Confidence range profiles
Mean, bands and confidence range of the concentration profiles
Mean, bands and confidence range of concentration profiles
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Rel. conc
Rel. conc.
1
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
3
4
5
6
7
8
3
4
5
pH
6
7
8
6
7
8
pH
Mean, bands and confidence range of concentration profiles
Mean, bands and confidence range of concentration profiles
1
1
0.9
0.8
0.8
0.7
Rel. conc.
Rel. conc
0.6
0.5
0.4
0.3
0.6
0.4
0.2
0.2
0.1
0
0
3
4
5
6
pH
7
8
3
4
5
pH
Mean, bands and confidence range of the spectra
0.8
0.7
0.6
Absorbance /a.u.
0.5
0.4
0.3
0.2
0.1
0
240
250
260
270
280
290
300
310
320
Wavelength /nm
Noise Addition Simulations
Spectra profiles:
Mean, max and min profiles
Confidence range profiles
Mean, bands and confidence range of spectra
Mean, bands and confidence range of spectra
0.8
0.9
0.7
0.8
0.7
0.6
Absorbance /a.u.
Absorbance /a.u.
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
240
0
250
260
270
280
Wavelength /nm
290
300
310
320
240
250
Mean, bands and confidence range of spectra
270
280
Wavelength /nm
290
300
310
320
290
300
310
320
Mean, bands and confidence range of spectra
0.8
1.6
0.7
1.4
1.2
0.6
1
0.5
Absorbance /a.u.
Absorbance /a.u.
260
0.4
0.3
0.2
0.8
0.6
0.4
0.2
0.1
0
0
240
250
260
270
280
Wavelength /nm
290
300
310
320
-0.2
240
250
260
270
280
Wavelength /nm
Noise Addition Simulations
pKa error estimations
pK2
pK1
Noise
added
Value
Std. dev
Value
Std. dev
0%
3.6539
2e-14
4.9238
2e-14
0.1 %
3.6540
6e-4
4.9226
0.0022
1%
3.6592
0.0061
4.9134
0.0264
2%
3.6656
0.0101
4.9100
0.0409
5%
4.0754
0.4873
5.3308
1.1217
Calculation of band boundaries from mean profile
error bands (under non-negativity and closure
constraints) at 1% error noise addition
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
5
10
15
20
25
30
35
40
3
2.5
2
1.5
1
0.5
0
-0.5
0
10
20
30
40
50
60
70
80
90
Jackknife Simulations at 1% noise; Concentration profiles:
Mean max and min profiles and confidence range profiles
M e a n , b a n d s a n d c o n fid e n c e ra n g e p f c o n c e n t ra tio n p ro file s
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
R e l. c o n c e n tra t io n
2
3
4
5
6
7
8
0
2
3
4
5
6
7
8
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
2
3
4
5
6
7
8
3
4
5
6
7
8
1
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
3
4
5
6
7
8
3
4
5
6
7
8
2
3
4
5
6
7
8
2
3
4
5
6
7
8
0
2
0.9
2
2
0
2
3
4
5
pH
pH
6
7
8
Jackknife Simulations at 1% noise; spectra profiles:
Mean max and min profiles and confidence range profiles
M ean, bands and c onfidenc e range of s pec tra
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
A bs orbanc e /a.u.
-0. 1
240
250
260
270
280
290
300
310
320
-0.1
240
0
250
260
270
280
290
300
310
320
240
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
240
250
260
270
280
290
300
310
320
240
260
270
280
290
300
310
320
240
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
240
0
250
260
270
280
290
300
310
320
240
260
270
280
290
300
310
320
250
260
270
280
290
300
310
320
250
260
270
280
290
300
310
320
0
250
0.8
0
250
0
250
260
270
280
290
300
310
320
240
Jackknife Simulations
pKa error estimations
at 1% noise level
Nº exp
pK1
1
2
3
4
5
6
7
8
9
3.6629 ± 0.0066
3.6601 ± 0.0074
3.6590 ± 0.0059
3.6580 ± 0.0056
3.6333 ± 0.0130
3.6882 ± 0.0198
3.6591 ± 0.0064
3.6592 ± 0.0059
3.6582 ± 0.0065
pK2
4.9135 ± 0.0277
4.8989 ± 0.0221
4.9122 ± 0.0261
4.9221 ± 0.0189
4.9018 ± 0.0236
4.9144 ± 0.0267
4.9144 ± 0.0256
4.9144 ± 0.0253
4.9233 ± 0.0239
Parameter Estimation
Summary of results
Real
Theoretical Value
MonteCarlo
Simulations
0.1 %
1%
2%
5%
pk1
pk2
pk1
pk2
pk1
pk2
pk1
pk2
pk1
pk2
Value
3.6660
4.9244
-
-
-
-
-
-
-
-
Value
-
-
3.6662
4.9244
3.6696
4.9262
3.6761
4.9173
3.9762
5.0745
Stand.
dev.
-
-
0.0006
0.0012
0.0065
0.0128
0.0127
0.0245
0.4349
0.7595
Value
-
-
3.6540
4.9226
3.6592
4.9134
3.6656
4.9100
4.0754
5.3308
Stand.
dev.
-
-
0.0006
0.0022
0.0061
0.0264
0.0101
0.0409
0.4873
1.1217
Value
-
-
3.6546
4.9199
3.6598
4.9128
3.6673
4.9131
4.0822
5.3292
Stand.
dev.
-
-
0.0038
0.0032
0.0086
0.0244
0.0124
0.0471
0.5145
1.0906
Noise Addition
JackKnife
Outline:
•Introduction
•Rotational ambiguities and feasible bands
•Error propagation and resampling methods
•Experimental system and simulations
•Results
•Conclusions
Summary
•Different approaches for calculation of error propagation
and prediction intervals of estimations have been
compared including: Monte Carlo simulations, Noise
addition resampling approaches and Jackknife based
methods.
•The obtained results allowed a preliminary investigation
of the noise effects on MCR-ALS resolved profiles and on
parameters from them estimated, and allowed also a
preliminary investigation of noise effects on rotational
ambiguities.
•The study has been shown for the resolution of a threecomponent equilibrium system with overlapping
concentration and spectra profiles
Conclusions
-Rotational ambiguity effects on species profiles depend
on the structure and constraints of the data system.
-Rotational ambiguities effects at low noise levels in a
system with low selectivity are more important than error
propagation effects
-However, at high noise levels (≥ 5%), error propagation
effects became larger than rotational ambiguities effects
and they are both mixed and undistinguishable
- Obviously the best is to have a system with enough
selectivity (low rotational ambiguities) and with low noise
levels (low error propagation)
Poster presentations of the Chemometrics
group from the University of Barcelona at
CAC2002
ELUCIDATION OF THE STRUCTURE OF A PROTEIN
FOLDING INTERMEDIATE (MOLTEN GLOBULE STATE)
USING MULTIVARIATE CURVE RESOLUTION
ALTERNATING LEAST SQUARES (MCR-ALS)
Susana Navea, Anna de Juan and Romà Tauler
MULTIVARIATE CURVE RESOLUTION ALTERNATING
LEAST SQUARES ANALYSIS OF THE CONFORMATIONAL
EQUILIBRIA OF THE OLIGONUCLEOTIDE
d<TGCTCGCT>
Joaquim Jaumot, Núria Escaja, Raimundo Gargallo, Enrique
Pedroso and Romà Tauler
HARD AND SOFT MODELLING OF ACID-BASE CHEMICAL
EQUILIBRIA OF BIOMOLECULES USING 1H-NMR
Joaquim Jaumot, Montserrat Vives, Raimundo Gargallo and Romà
Tauler
IDENTIFICATION AND DISTRIBUTION OF MICROCONTAMINANTS SOURCES OF NONIONIC SURFACTANTS, THEIR
DEGRADATION PRODUCTS AND LINEAR ALKYLBENZENE
SULFONATES IN COASTAL WATERS AND SEDIMENTS IN
SPAIN BY MEANS OF CHEMOMETRIC METHODS
Emma Peré-Trepat, Mira Petrovic, Damià Barceló and Romà Tauler
MULTIWAY DATA ANALYSIS OF ENVIRONMENTAL
CONTAMINATION SOURCES IN SURFACE NATURAL
WATERS OF CATALONIA (SPAIN)
Emma Peré-Trepat, Mónica Flo, Montserrat Muñoz, Manel Vilanova,
Josep Caixach, Antoni Ginebreda, Romà Tauler