Estimation of error propagation and prediction intervals in Multivariate Curve Resolution Alternating Least Squares using resampling methods

Transcription

1 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

2 Outline: Introduction Rotational ambiguities and feasible bands Error propagation and resampling methods Results Conclusions

3 Multivariate (Soft) Self Modeling Curve Resolution NC NC C.5.5 S T C NR S T + NR E NC.5 D NR D N d = c s e ij = + k ik kj ij Bilinearity!

4 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.

5 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?

6 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).

7 Outline: Introduction Rotational ambiguities and calculation of feasible bands Error propagation and resampling methods Results Conclusions

8 Lawton and Sylvestre feasible bands non-negative spectra pure spectrà normalization to unit area A I A II I 2 II non-negative concentration,2,3,4,5 experimental values PC non-negative concentration PC2 non-negative spectra W.H. Lawton and EA Sylvestre, Technometrics 97, 3, 67-33

9 Rotational Ambiguities Factor Analysis (PCA) Data Matrix Decomposition D = U V T + E True Data Matrix Decomposition D = C S T + E D = U T T - V T + E = C S T + E C = U T; S T = T - V T 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

10 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.

11 Rotational Ambiguities T values define the band of feasible solutions or feasible bands How to define the boundaries of these feasible bands? Howto represent graphically these boundaries?

12 Is it possible to define band boundaries (T max and T min )?.5.4 T max.3 How to calculate T max and T min? T max T min T min

13 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 * = C inic S T inic = = C inic T min T - min S T inic = C min S T min = = C inic T max T - max S T inic = C max S T max where: D(NR,NC), C(NR,N), S T (N,NC), T(N,N) How to define and evaluate T max and T min?

14 Evaluation of boundaries of feasible bands: Previous studies W.H.Lawton and E.A.Sylvestre, Technometrics, 97, 3, O.S.Borgen and B.R.Kowalski, Anal. Chim. Acta, 985, 74, - 26 K.Kasaki, S.Kawata, S.Minami, Appl. Opt., 983 (22), R.C.Henry and B.M.Kim (Chemomet. and Intell. Lab. Syst., 99, 8, 25-26) P.D.Wentzell, J-H. Wang, L.F.Loucks and K.M.Miller (Can.J.Chem. 76, (998)) P. Gemperline (Analytical Chemistry, 999, 7, ) R.Tauler (J.of Chemometrics 2, 5, ) M.Legger and P.D.Wentzell, Chemomet and Intell. Lab. Syst., 22, 7-88

15 Definition of band boundaries The whole measured signal is: D = D i = c i s i T The contribution of each species to the whole signal is: D i =c i s i T Solving the Optimization Problem: max/outer boundary: Find T max that makes c i s it maximum min/inner boundary: Find T min that makes c i s it minimum

16 Constrained Non-Linear Optimization Problem (NCP) Find T which makes: min/max f(t) subject to g e (T) = T and to g i (T) 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

17 ) What are the variables of the problem? T (rotation matrix), D = C T T - S T 2) What is the objective function f(t) to be optimized? For each species i =,..,ns T cs = T = C S i i f( i ) or f( T i ) j i,j cs ij cs f(t) is scalar value between and! ij ij ij This gives the relative signal contribution of species i respect the global measured signal!

18 3) What are the constraints g(t)? The following constraints may be considered: normalization/closure g norm /g clos non-negativity g cneg /g sneg known values/selectivity g known /g sel unimodality g unim trilinearity (three-way data) g tril Are they equality or inequality constraints?

19 4) What are the initial estimates of C, S T? Initial estimates of C and S T 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 ) T ini = eye(n) =

20 Optimization algorithm R.Tauler (J.of Chemometrics 2, 5, ) Initial estimations of CALS and SALS 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 Tmin which gives a minimum of f(t) under constraints gi(t)<, ge(t)= Find Tmax which gives a maximum of f(t) under constraints gi(t)<. ge(t)= Built minimum band cmin = cals / Tmin smin = sals / Tmin Built maximum band cmax = cals / Tmax smax=sals / Tmax

21 Experimental data system under study.5.5 D Acid-base spectrophotometric titration of the double stranded heteropolynucleotide polyinosinic-polycytidylic acid. Spectral region between nm and ph region between ph 2 and ph 9

22 Conc. relativa Absorbànica /a.u concentration profiles ph pk pk 2 spectra profiles C S T Application of MCR- ALS to the experimental data matrix D Applied constraints in ALS were: a) non-negative spectra b) non-negative concentrations c) closure in concentrations Initial estimates were obtained from purest variables 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

23 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 pk Proposed species: poly(i)-poly(c+) pk poly(i)-poly(c)-poly(c+) + H poly(i)-poly(c) + H

24 Estimation of band boundaries (max/min contribution of each species) of feasible solutions.8 Large Rotational ambiguities were present when constraints applied were only closure non-negativity!!!

25 Estimation of band boundaries (max/min contribution of each species) of feasible solutions.8 Rotational ambiguities nearly dissappear when selectivity constraint was applied!!!

26 Outline: Introduction Rotational ambiguities and feasible bands Error propagation and resampling methods Results Conclusions

27 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

28 Resampling Methods Resampling Methods Theoretical Data Experimental Data Montecarlo Simulation Noise Addition Jackknife

29 Building theoretical data Experimental Data, D exp MCR-ALS C S T Theoretical Data, D Experimental error E

30 Montecarlo Simulations M. = D +N. M = D +N M = D +N M = D +N Concentration profiles C Theoretical Data D New Data Matrix MCR-ALS Random Error N., N, N 2 and N 5 MATLAB function randomn with zero mean and relative sd.%, %, 2% and 5% of maximum signal in D Pure Spectra S T 25 times each noise level! simulations!

31 Noise Addition Simulations D. = D exp +N. D = D exp +N D 2 = D exp +N 2 D 5 = D exp +N 5 Concentration profiles C Experimental Data New Data Matrix MCR-ALS D exp Random Error N., N, N 2 and N 5 MATLAB function randomn with zero mean and relative sd.%, %, 2% and 5% of maximum signal in D Pure Spectra S T 25 times each noise level! simulations!

32 Experimental Data Jackknife Simulations D exp (36,8) J Reduced Matrix (,,9,28) (32,8) J2 N. + Reduced Matrix (2,,2,29) (32,8) N J3 Random Error Reduced Matrix (3,2,2,3) (32,8) N 2 J4 and = Reduced Matrix (4,3,22,3) (32,8) N 5 J5 Reduced Matrix (5,4,23,32) (32,8) New Data J6 Matrix Reduced Matrix (6,5,24,33) (32,8) J7 D noise (36,8) Reduced Matrix (7,6,25,34) (32,8) J8 Reduced Matrix (8,7,26,35) (32,8) D., D, D 2, D 5 J9 Reduced Matrix (9,8,27,36) (32,8)

33 Jackknife Simulations Jack Knife Reduced Data Matrix J N N =,..., 9 MCR-ALS Concentration profiles C Pure Spectra S T

34 Outline: Introduction Rotational ambiguities and feasible bands Error propagation and resampling methods Results Conclusions

35 Presentation of Results. 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

36 Rel. concentration Mean, bands and confidence range of the concentration profiles Noise % ph Monte Carlo Simulations Concentration profiles: Mean max and min profiles Confidence range profiles Mean, bands and confidence range of the concentration profiles Noise.% Mean, bands and confidence range of concentratios profiles Noise 2% Rel. concentration Rel. conc ph Mean, bands and confidence range of concentration profiles ph Mean, bands and confidence range of concentration profiles Noise %.8 Rel. conc Rel. conc Noise 5% ph ph

37 Absorbance /a.u Mean, bands and confidence range of concentration profiles Noise % Wavelength /nm Monte Carlo Simulations Spectra profiles: Mean max and min profiles Confidence range profiles Absorbance /a.u Mean, bands and confidence range of the spectra Mean, bands and confidence range of the spectra.8.7 Noise.%.6 Noise 2% Absorbance /a.u Wavelength /nm Absorbance /a.u. Wavelength /nm Mean, bands and confidence range of spectra Noise % Wavelength /nm Absorbance /a.u Mean, bands and confidence range of the spectra Wavelength /nm Noise 5%

38 Monte Carlo Simulations pka error estimations pk pk 2 Noise added Value Std. dev Value Std. dev % e e-5. % e % % %

39 Calculation of band boundaries from mean species profiles error bands (under non-negativity and closure constraints) Noise.% Noise 2% Noise % 2 Noise 5%

40 Calculation of band boundaries from mean profile error bands (under non-negativity, closure and selectivity constraints) Noise.%.6 Noise 2% Noise % Noise 5%

41 Rel. concentration Mean, bands and confidence range of the concentration profiles 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 Rel. conc..5.4 Rel. conc ph ph Mean, bands and confidence range of concentration profiles Mean, bands and confidence range of concentration profiles Rel. conc Rel. conc ph ph

42 Absorbance /a.u Mean, bands and confidence range of the spectra Wavelength /nm Noise Addition Simulations Spectra profiles: Mean, max and min profiles Confidence range profiles.8 Mean, bands and confidence range of spectra.9 Mean, bands and confidence range of spectra Absorbance /a.u Absorbance /a.u Wavelength /nm Wavelength /nm.8 Mean, bands and confidence range of spectra.6 Mean, bands and confidence range of spectra Absorbance /a.u Absorbance /a.u Wavelength /nm Wavelength /nm

43 Noise Addition Simulations pka error estimations pk pk 2 Noise added Value Std. dev Value Std. dev % e e-4. % e % % %

44 Calculation of band boundaries from mean profile error bands (under non-negativity and closure constraints) at % error noise addition

45 ph Jackknife Simulations at % noise; Concentration profiles: Mean max and min profiles and confidence range profiles Mean, bands and confidence range pf concentration profiles Rel. concentration

46 Jackknife Simulations at % noise; spectra profiles: Mean max and min profiles and confidence range profiles.8.8 Mean, bands and confidence range of spectra Absorbance /a.u

47 Jackknife Simulations pka error estimations at % noise level Nº exp pk pk ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.239

48 Parameter Estimation Summary of results Real. % Stand. dev Value JackKnife Stand. dev Value Noise Addition Stand. dev Value MonteCarlo Simulations Value Theoretical Value pk2 pk pk2 pk pk2 pk pk2 pk pk2 pk 5 % 2 % %

49 Outline: Introduction Rotational ambiguities and feasible bands Error propagation and resampling methods Experimental system and simulations Results Conclusions

50 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

51 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)

52 Poster presentations of the Chemometrics group from the University of Barcelona at CAC22 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

53 HARD AND SOFT MODELLING OF ACID-BASE CHEMICAL EQUILIBRIA OF BIOMOLECULES USING H-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