A B C k1k1 k2k2 Consecutive Reaction d[A] dt = -k 1 [A] d[B] dt = k 1 [A] - k 2 [B] d[C] dt = k 2 [B] [A] = [A] 0 exp (-k 1 t) [B] = [A] 0 (k 1 /(k 2 -k 1 )) (exp (-k 1 t) - exp (-k 2 t)) [C] = [A] 0 – [A] – [B] A three component system
k 1 =5 k 2 =0.5
Kinfit4.m
k 1 =5 k 2 =0.5 k 1 =0.5 k 2 =5 Multiple Solutions
Non-negativity constraints for eliminations of multiple solutions in fitting of multivariate kinetic models to spectroscopic data Joaquim Jaumot, Paul J. Gemperline and Alexandra stang J. Chemometrics, 2005, 19,
Model-based non-linear fitting The task of model-based data fitting for a given matrix A, is to determine the best parameters defining matrix C, as well as the best pure responses collected in matrix E. A = C E + R AC E R =+ The quality of the fit is represented by the matrix of residuals. Assuming white noise, the sum of the squares, ssq, of all elements r i,j is statistically the best measure to be minimized ssq = ΣΣ r 2 I,j
Linear and non_linear parameters There are two fundamentally different kinds of parameters, a smal number of model constants, which are nonlinear parameters, and the large number of response coefficients, which are linear parameters. E = C + A = (C T C) -1 C T A E = C \ A (MATLAB notation) R = A – C E = A – C C + A = f( A, model, k)
A B C k1k1 k2k2 Newton-Gauss Algorithm d[A] dt = -k 1 [A] d[B] dt = k 1 [A] - k 2 [B] d[C] dt = k 2 [B] [A] = [A] 0 exp (-k 1 t) [B] = [A] 0 (k 1 /(k 2 -k 1 ))(exp (-k 1 t) - exp (-k 2 t)) [C] = [A] 0 – [A] – [B] A = C E + R A Model (consecutive reaction) Measured data (A) Non-linear parameters (k 1, k 2 ) Linear parameters (E)
Newton-Gauss Algorithm i) Estimate of non-linear parameters (k 1, k 2 ) k 1 =0.3 k 2 =0.15
Newton-Gauss Algorithm ii) Calculation the residuals and ssq according to model and estimated parameters [A] = [A] 0 exp (-k 1 t) [B] = [A] 0 (k 1 /(k 2 -k 1 ))(exp (-k 1 t) - exp (-k 2 t)) [C] = [A] 0 – [A] – [B] Model k 1 =0.2 & k 2 =0.15 E=C\A
Newton-Gauss Algorithm ii) Calculation the residuals and ssq according to model and estimated parameters Calculation of residual matrix R = A – C (C\A) ---= ssq = ΣΣ r 2 I,j = 1.66
Newton-Gauss Algorithm iii) Calculation the shift parameters vector The Taylor series expansion R = f (parameters)ssq = f (parameters) A shift in nonlinear parameters should be calculated in such a way that the ssq moves towards its minimum value. R (p) = A – C (C\A) R (p + p ) minimum p = ? f (x + x) = f (x) + ( ) ( x) + ( ) ( x) 2 df (x) d x 2! 1 d n f (x) d n x d 2 f (x) d2xd2x + … + ( ) ( x) n n! 1 f (x + x) = f (x) + ( ) ( x) df (x) d x
Newton-Gauss Algorithm iii) Calculation the shift parameters vector The Taylor series expansion R (p + p ) = R(p) + ( ) p dR (p) d p The goal is to determine the vector of parameter shifts that moves R(p 0 + p) towards zero. R(p 0 ) = - ( ) p dR (p 0 ) d p R(p 0 ) = - p 1 - p 2 - … - p np dR (p 0 ) d p 1 dR (p 0 ) d p 2 dR (p 0 ) d p np
Newton-Gauss Algorithm iii) Calculation the shift parameters vector Calculation the partial derivative It is always possible to approximate the partial derivative numerically by the method of finite differences. In the limit as p 1 approaches zero, the derivative of R(p 0 ) with respect to p 1 can be approximated: dR(p 0 ) d p i R(p 0 + p i ) – R (p 0 ) p i = ~ R([0.3, 0.15]) R([0.3 + , 0.15]) dR([k 1,k 2 ])/dk 1 = p i
Newton-Gauss Algorithm iii) Calculation the shift parameters vector R(p 0 ) = - p 1 - p 2 - … - p np dR (p 0 ) d p 1 dR (p 0 ) d p 2 dR (p 0 ) d p np R (p 0 )dR(p 0 )/dR(p 1 )dR(p 0 )/dR(p 2 )=- - p1p1 p2p2 - … - How p vector can be calculated? One solution to this problem is to vectorise (unfold into long column vectors) the residuals and partial derivative matrices.