An Introduction to Model-Free Chemical Analysis Hamid Abdollahi IASBS, Zanjan Lecture 3

? Use the n_V_U_space.m file and find the feasible band for a two component system

Rank Deficiency

v 1 vector u 1 vector

Augmentation =

Real Spectrum 1 Real Spectrum 2

Augmentation and Normalization

? Investigate the number of Augmented samples on ranges of possible solutions

How can we determine that some target spectra belong to a particular space?

The row space (V space) of measured data matrix.

Projections of targets in V space

Comparison of projections with targets (Target Testing)

Defining a criteria

Target Factor Analysis (TFA)

TFA.m file Target Factor Analysis (TFA)

? Modify TFA.m file for using the correlation coefficient as criteria for target testing

Using TFA for determination of chemical model parameters

What is the pK a of a monoprotic acid?

The column space (U space) of measured data matrix.

Simulated targets

Projections of targets in U space

Defining a criteria

Iterative Target Transformation Factor Analysis (ITTFA) Algorithm: 1.Calculation of the score matrix by PCA. 2. Use of the estimated concentration profile as initial target. 3. Projection of the target onto the score space. 4. Constraint of the target projected. 5. Projection of the constrained target. 6. Return to step 4 until convergence is achieved.

Using ITTFA for calculating the concentration profiles from HPLC-DAD data

ITTFA U Space

ITTFA Initial estimate

ITTFA U Space

ITTFA Output

ITTFA Constrained Output

ITTFA U Space

ITTFA Output

ITTFA Constrained Output

ITTFA U Space

ITTFA Output

ITTFA Constrained Output

ITTFA U Space

ITTFA Constrained Output

ITTFA.m file Iterative Target Transformation Factor Analysis

? Use ITTFA.m file for finding another concentration profile.

? Use ITTFA.m file and investigate the effect of initial estimate

Resolving Factor Analysis (RFA) RFA is the combination of nonlinear parameter fitting and free-model analysis. RFA combine the advantages of the small number of parameters of model-based analyses with the lack of model constraints of the model-free methods. D = USV = C A D = (UST -1 ) (TV) = C A C = UST -1 A = TV

Resolving Factor Analysis (RFA) Algorithm: 1.Initial Guess of the Elements of T. 2. Calculation of the Matrices C and A. C = UST -1 A=TV 3. Using Constraint for C and A. 4. Residuals and Sum of Squares. D calc = C A R = D – D calc 5. Calculation of Parameter Shifts. 6. Return to Step 2 until Convergence. ssq = ΣΣ r 2 I,j

Measured data matrix, D

Rows of data matrix

Columns of data matrix

Initial estimate of T matrix

Calculated T -1 based on initial estimate of T

Calculated C matrix based on T -1

Shifted T matrix

Calculated A matrix based on new T

T -1 corresponding to new T

Calculated C matrix

Residual

Shifted T matrix

Calculated A matrix

T -1 matrix

Calculated C matrix

Residual

Converged T matrix after 10 iteration

Solution for A after 10 iteration

T -1 matrix after 10 iteration

Solution for C after 10 iteration

Residual

RFA.m file Visulizing the RFA method

? Use RFA.m file and investigate the effect of initial estimate of T matrix