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Uncertainty in Hard, Soft and Hard-Soft Modeling

Uncertainty in Hard, Soft and Hard-Soft Modeling. Uncertainty in Calculated Model Parameters using Hard- Modeling Method. Model Based Analyses.

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Uncertainty in Hard, Soft and Hard-Soft Modeling

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  1. Uncertainty in Hard, Soft and Hard-Soft Modeling

  2. Uncertainty in Calculated Model Parameters using Hard- Modeling Method

  3. Model Based Analyses The very rigid constraints of a chemical model form a framework within which the fit is confined and which results in a robust analysis, in model-free analysis, this framework is dramatically wider and looser and these methods suffer gradually from a sever lack of robustness. It must be remembered, however, that the choice of the wrong model necessarily results in the rung analysis and wrong resulting parameters.

  4. Complex Formation Equilibrium M + L ML [ [ ML ] ] C = [L] + [ ML ] L K = f [M] [L] = [M] + [ ] C ML M CL = [L] + KF [M] [L] CM = [M] + KF [M] [L]

  5. Data.m Spectrophotometric monitoring of complex formation titration

  6. Calculation of Model Parameter E R A = C + 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 R = A – C E = A – C C+ A = f( A, model, K) The quality of the fit is represented by the matrix of residuals. Assuming white noise, the sum of the squares, ssq, of all elements ri,j is statistically the best measure to be minimized ssq = ΣΣ r2I,j

  7. Calculation of Model Parameter

  8. How we can calculate the precision of model parameter?

  9. Distribution of Fitted Model Parameters log (Kf) (mean) = 3.5004 Standard Deviation of log (Kf)= 0.0021

  10. Main_ML_S.m Search for K in a certain range

  11. ? Based on repeatation procedure, calculate the standard deviation of fitted parameter in different level of noise

  12. Error Propagation y = f (x) var (y) = (df/ dx)2var (x) y = f (x1, x2) var (y) = (df/dx1)2var (x1) + (df/dx2)2var(x2) + (df/d(x1) d(f)/d(x2) 2cov(x1, x2)

  13. General Error Propagation y = f (x1, x2, x3, …) var (y) = JT [Var (x)]J var(x1), cov(x1, x2), … , cov(x1,xn) cov(x2, x1), var(x1), … , cov(x2, xn) Var(x) = … … … cov(xn, x1), cov(xn, x2), … , var(xn) JT= [ df/dx1, df/dx2, …, df/dxn]

  14. Uncertainty of fitted model parameters A = C E + R R = A – C E = A – C C+ A = f( A, model, p) var (R) =JT [Var(p)] J Var(p)=(JT J)-1var (R) var(p1), cov(p1, p2), … , cov(p1,pn) cov(p2, p1), var(p2), … , cov(p2, pn) Var(p) = … … … cov(pn, p1), cov(pn, p2), … , var(pn) var (R) = (Ri,j)2/(nm-np) = ssq/df

  15. Jacobian Matrix R(p1+dp1) – R(p1-dp1) J1= dR/dp1= 2dp1 J1 J2 Jn … J = … J = J2 J1 Jn

  16. Hessian Matrix J1TJ1 J1TJ2 … J1TJn J2TJ1 J2TJ2 … J2TJn JT J = … … … JnTJ1 JnTJ2 … JnTJn The inverted Hessian matrix H-1, is the variance-covariance matrix of the fitted parameters. The diagonal elements contain information on the parameter variances and the off-diagonal elements the covariances.

  17. Newton-Gauss-Levenberg-Marquardt Algorithm ssqold< = > ssq mp=0 guess parameters, p=pstart initial value for mp Calculate residuals, r(p) and sum of squares, ssq < = yes End, display results no > mp ×10 mp / 3 mp=0 Calculate Jacobian, J Calculate shift vector Dp, and p = p + Dp

  18. Main_ML.m NGLM algorithm for Fitting

  19. ? Use Main_ML m-file for fitting the three parameters (K, CM and CL) with different initial estimates

  20. ? Check the uncertainty calculated for K when the initial concentrations are fixed or fitted

  21. Correlation between Fitted Parameters When two parameters are fitted, is there any relation between calculated parameters? Is there any relation between the estimated uncertainties on K and C0? K C0 ?????????????????????????????????????????

  22. Main_ML_corr.m Correlation between fitted parameters

  23. ? What are the relations between the shapes and values of Jacobian with variance and covariance of parameters?

  24. ? Using the J matrix and calculate the corelation between parameters

  25. Propagation of Uncertainty from Initial Concentration to Equilibrium Constant K = f(residual, C0) var(K) = (df/d(residual))2var(residual) + (dK/dC0)2var (C0) (dK/dC0)2 Sensitivity of K to C0 Kopt(C0+dC0) – Kopt(C0-dC0) dK/dC0 = 2dC0

  26. Main_ML_C0 Propagation of uncertainty from C) to K

  27. ? What is the effect of noise on measured signal in uncertainty of K due to C0?

  28. ? Modify the Main_ML_cO m-file for considering the uncertainty in C0M and C0L

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