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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 NC
1 1.5
ST
0.8
C 1
ST
0.6
0.4
C + E
0.5 NR NR
0.2
0 0
0 10 20 30 40 0 20 40 60 80 100
NC
1.5
N
d = ∑ c s +e
1
D D ij k=1 ik kj ij
0.5 NR
0
Bilinearity!
0 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
5
pure spectrà non-negative concentration
AI
normalization
to unit area 4
PC1
3 1,2,3,4,5
AII 2 experimental
values
II 1
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
•0.4 Tmax
•0.3
•0.2
Tmin
How •0.1
to •0
•0 •5 •10 •15 •20 •25 •30 •35 •40 •45 •50
calculate Tmax
•1.5
Tmax
and •1
Tmin? •0.5
•0
Tmin
•0 •5 •10 •15 •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, 617-
633
•O.S.Borgen and B.R.Kowalski, Anal. Chim. Acta, 1985, 174, 1-
26
•K.Kasaki, S.Kawata, S.Minami, Appl. Opt., 1983 (22), 3599-
3603
•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) subject to ge(T) = 0
T 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
This gives the
c i si ∑c sij ij relative signal
contribution of
fi (T) = f (T ) =
or i
j
CS T
∑c s
i,j
ij ij
species i respect
the global
measured signal !
f(T) is scalar value between 0 and 1!
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 …)
1 0 ... 0
0 1 ... 0
Tini = eye(N) = ... ... ... ...
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 Find T max which gives a maximum
of f(T) of f(T)
under constraints gi(T)<0, ge(T)=0 under constraints gi(T)<0. ge(T)=0
Built minimum band Built maximum band
c min = cALS / T min cmax = cALS / T max
s min = sALS / T min smax=sALS / T max
Experimental data system under
study
1.5
1
0.5
D
0
0 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 MCR-
0.8
ALS to the
Conc. relativa
C
0.6
experimental data
0.4 pK2 matrix D
pK1
0.2
Applied constraints in ALS
0
2 3 4 5 6 7 8 9 were:
pH
a) non-negative spectra
spectra profiles
0.8 b) non-negative
concentrations
0.6
c) closure in concentrations
Absorbànica /a.u.
0.4
ST
Initial estimates were obtained
0.2
from purest variables
0
0 10 20 30 40 50 60 70 80 90
Nº canal /nm
•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 Proposed species:
pK1 3.6660 poly(I)-poly(C+)
poly(I)-poly(C)-poly(C+) + H
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
ambiguities 0.2
were present 0
5 10 15 20 25 30 35
when constraints
applied were 3
only closure 2.5
non-negativity!!! 2
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 0.6
ambiguities 0.4
nearly 0.2
dissappear 0
5 10 15 20 25 30 35
when selectivity
constraint was 0.8
applied!!! 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 Experimental
Data Data
Montecarlo Noise Jackknife
Simulation Addition
Building theoretical data
Experimental
Data, Dexp
MCR-ALS
ST Experimental
C error
E
Theoretical
Data, D
Montecarlo Simulations
M0.1 = D +N0.1 M1 = D +N1
Concentration
M2= D +N2 M5= D +N5 profiles C
Theoretical
Data MCR-ALS
New Data Matrix
D
ST
Pure Spectra
Random
Error
250 times each noise level!
N0.1, N1, N2 and N5
1000 simulations!
MATLAB function randomn with zero
mean and relative sd 0.1%, 1%, 2% and
5% of maximum signal in D
Noise Addition Simulations
D0.1 = Dexp +N0.1 D1 = Dexp +N1
D2= Dexp +N2 D5= Dexp +N5 Concentration
C
profiles
Experimental MCR-ALS
New Data Matrix
Data
ST
Dexp Pure Spectra
Random
Error
250 times each noise level!
N0.1, N1, N2 and N5 1000 simulations!
MATLAB function randomn with zero
mean and relative sd 0.1%, 1%, 2% and
5% of maximum signal in D
Experimental Jackknife Simulations
Data
Dexp (36,81) J1
Reduced Matrix (1,10,19,28) (32,81)
J2
N0.1 + Reduced Matrix (2,11,20,29) (32,81)
N1 J3
Random Error Reduced Matrix (3,12,21,30) (32,81)
N2 J4
and = Reduced Matrix (4,13,22,31) (32,81)
N5 J5
Reduced Matrix (5,14,23,32) (32,81)
New Data J6
Matrix Reduced Matrix (6,15,24,33) (32,81)
J7
Dnoise (36,81) Reduced Matrix (7,16,25,34) (32,81)
J8
Reduced Matrix (8,17,26,35) (32,81)
D0.1, D1, D2, D5 J9
Reduced Matrix (9,18,27,36) (32,81)
Jackknife Simulations
Concentration
profiles
Jack Knife
MCR-ALS C
Reduced
Data Matrix
Pure Spectra
JN
ST
N = 1,..., 9
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
0.7
Noise 0% Monte Carlo Simulations
Concentration profiles:
Rel. concentration
0.6
0.5
0.4
0.3
0.2
Mean max and min profiles
Confidence range profiles
0.1
0
2 3 4 5 6 7 8 9
pH
Mean, bands and confidence range of concentratios profiles
1
Mean, bands and confidence range of the concentration profiles
0.9
1
0.9
0.8
Noise 2%
0.8
0.7
Noise 0.1% 0.7
0.6
Rel. conc.
Rel. concentration
0.6 0.5
0.5 0.4
0.4 0.3
0.3 0.2
0.2 0.1
0
0.1
3 4 5 6 7 8
0 pH
2 3 4 5 6 7 8 9
pH
Mean, bands and confidence range of concentration profiles Mean, bands and confidence range of concentration profiles
1 1.2
0.9
1
Noise 1%
0.8
0.7 0.8
0.6
0.6
Rel. conc.
Rel. conc.
0.5
0.4 0.4
0.3
Noise 5%
0.2
0.2
0.1 0
0
-0.2
3 4 5 6 7 8 2 3 4 5 6 7 8 9
pH pH
Mean, bands and confidence range of concentration profiles
0.8
0.7
0.6
Noise 0% Monte Carlo Simulations
Spectra profiles:
0.5
Absorbance /a.u.
0.4
Mean max and min profiles
0.3
0.2
Confidence range profiles
0.1
0
240 250 260 270 280 290 300 310 320
Wavelength /nm
Mean, bands and confidence range of the spectra
0.8
Mean, bands and confidence range of the spectra
0.8
0.7
0.6 Noise 0.1% 0.7
0.6 Noise 2%
0.5 0.5
Absorbance /a.u.
Absorbance /a.u.
0.4 0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0 -0.1
240 250 260 270 280 290 300 310 320
240 250 260 270 280 290 300 310 320 Wavelength /nm
Wavelength /nm
Mean, bands and confidence range of spectra
0.8
Absorbance /a.u.
Mean, bands and confidence range of the spectra
1.6
0.7
Noise 1%
1.4
0.6 1.2
0.5 1
0.4
Absorbance /a.u.
0.8
0.3 0.6
0.2
Noise 5%
0.4
0.2
0.1 0
0 -0.2
240 250 270 280
260 Wavelength /nm 290 300 310 320 240 250 260 270 280
Wavelength /nm
290 300 310 320
Monte Carlo Simulations
pKa error estimations
pK1 pK2
Noise
Value Std. dev Value Std. dev
added
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 0
5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
2.5 2.5
2
1.5
Noise 0.1% 2
1.5
Noise 2%
1 1
0.5 0.5
0 0
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 5 10 15 20 25 30 35 5 10 15 20 25 30 35
2.5 2
2
1.5
Noise 1% Noise 5%
1
1
0.5
0 0
0 10 20 30 40 50 60 70 80 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 0
5 10 15 20 25 30 35 5 10 15 20 25 30 35
0.8 0.8
0.6
0.4
Noise 0.1% 0.6
0.4
Noise 2%
0.2 0.2
0 0
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
5 10 15 20 25 30 35 5 10 15 20 25 30 35
0.8 1
Noise 1% Noise 5%
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 0
0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80
Mean, bands and confidence range of the concentration profiles
1
0.9
Noise Addition Simulations
0.8
0.7
Rel. concentration
0.6
0.5
0.4
Concentration profiles:
Mean max and min profiles
0.3
0.2
0.1
0
2 3 4 5
pH
6 7 8 9
Confidence range profiles
Mean, bands and confidence range of the concentration profiles Mean, bands and confidence range of concentration profiles
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
Rel. conc.
Rel. conc
0.5 0.5
0.4
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 6 7 8
pH 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
0.6 0.6
Rel. conc.
Rel. conc
0.5
0.4
0.4
0.3
0.2
0.2
0.1 0
0
3 4 5 6 7 8 3 4 5 6 7 8
pH pH
Mean, bands and confidence range of the spectra
0.8
0.7
0.6
0.5
Noise Addition Simulations
Spectra profiles:
Absorbance /a.u.
0.4
0.3
0.2
0.1
Mean, max and min profiles
0
240 250 260 270 280
Wavelength /nm
290 300 310 320
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
0.6
0.5
Absorbance /a.u.
Absorbance /a.u.
0.5
0.4
0.4
0.3
0.3
0.2 0.2
0.1 0.1
0
0
240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320
Wavelength /nm Wavelength /nm
Mean, bands and confidence range of spectra 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.
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0 -0.2
240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320
Wavelength /nm Wavelength /nm
Noise Addition Simulations
pKa error estimations
pK1 pK2
Noise
Value Std. dev Value Std. dev
added
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 0
2 3 4 5 6 7 8 2 3 4 5 6 7 8 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
R e l. c o n c e n tra t io n
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 0
2 3 4 5 6 7 8 2 3 4 5 6 7 8 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 0
2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8
pH
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 0.8
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
0
-0. 1 -0.1
240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320
0.8 0.8 0.8
0.7 0.7 0.7
0.6 0.6 0.6
0.5 0.5 0.5
A bs orbanc e /a.u.
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 0
240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320
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 0
240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320 240 250 260 270 280 290 300 310 320
Jackknife Simulations
pKa error estimations
at 1% noise level
Nº exp pK1 pK2
1 3.6629 ± 0.0066 4.9135 ± 0.0277
2 3.6601 ± 0.0074 4.8989 ± 0.0221
3 3.6590 ± 0.0059 4.9122 ± 0.0261
4 3.6580 ± 0.0056 4.9221 ± 0.0189
5 3.6333 ± 0.0130 4.9018 ± 0.0236
6 3.6882 ± 0.0198 4.9144 ± 0.0267
7 3.6591 ± 0.0064 4.9144 ± 0.0256
8 3.6592 ± 0.0059 4.9144 ± 0.0253
9 3.6582 ± 0.0065 4.9233 ± 0.0239
Parameter Estimation
Summary of results
Real 0.1 % 1% 2% 5%
pk1 pk2 pk1 pk2 pk1 pk2 pk1 pk2 pk1 pk2
Theoretical Value Value 3.6660 4.9244 - - - - - - - -
Value - - 3.6662 4.9244 3.6696 4.9262 3.6761 4.9173 3.9762 5.0745
MonteCarlo
Simulations
Stand.
- - 0.0006 0.0012 0.0065 0.0128 0.0127 0.0245 0.4349 0.7595
dev.
Value - - 3.6540 4.9226 3.6592 4.9134 3.6656 4.9100 4.0754 5.3308
Noise Addition
Stand.
- - 0.0006 0.0022 0.0061 0.0264 0.0101 0.0409 0.4873 1.1217
dev.
Value - - 3.6546 4.9199 3.6598 4.9128 3.6673 4.9131 4.0822 5.3292
JackKnife
Stand.
- - 0.0038 0.0032 0.0086 0.0244 0.0124 0.0471 0.5145 1.0906
dev.
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 three-
component 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 MICROCONTA-
MINANTS 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
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