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An Introduction to Model-Free Chemical Analysis Hamid Abdollahi IASBS, Zanjan e-mail: abd@iasbs.ac.ir Lecture 3
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? Use the n_V_U_space.m file and find the feasible band for a two component system
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Rank Deficiency
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v 1 vector u 1 vector
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Augmentation =
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Real Spectrum 1 Real Spectrum 2
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Augmentation and Normalization
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? Investigate the number of Augmented samples on ranges of possible solutions
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How can we determine that some target spectra belong to a particular space?
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The row space (V space) of measured data matrix.
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Projections of targets in V space
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Comparison of projections with targets (Target Testing)
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Defining a criteria
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Target Factor Analysis (TFA)
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TFA.m file Target Factor Analysis (TFA)
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? Modify TFA.m file for using the correlation coefficient as criteria for target testing
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Using TFA for determination of chemical model parameters
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What is the pK a of a monoprotic acid?
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The column space (U space) of measured data matrix.
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Simulated targets
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Projections of targets in U space
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Defining a criteria
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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.
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Using ITTFA for calculating the concentration profiles from HPLC-DAD data
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ITTFA U Space
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ITTFA Initial estimate
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ITTFA U Space
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ITTFA Output
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ITTFA Constrained Output
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ITTFA U Space
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ITTFA Output
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ITTFA Constrained Output
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ITTFA U Space
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ITTFA Output
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ITTFA Constrained Output
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ITTFA U Space
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ITTFA Constrained Output
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ITTFA.m file Iterative Target Transformation Factor Analysis
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? Use ITTFA.m file for finding another concentration profile.
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? Use ITTFA.m file and investigate the effect of initial estimate
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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
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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
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Measured data matrix, D
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Rows of data matrix
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Columns of data matrix
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Initial estimate of T matrix
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Calculated T -1 based on initial estimate of T
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Calculated C matrix based on T -1
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Shifted T matrix
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Calculated A matrix based on new T
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T -1 corresponding to new T
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Calculated C matrix
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Residual
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Shifted T matrix
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Calculated A matrix
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T -1 matrix
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Calculated C matrix
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Residual
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Converged T matrix after 10 iteration
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Solution for A after 10 iteration
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T -1 matrix after 10 iteration
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Solution for C after 10 iteration
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Residual
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RFA.m file Visulizing the RFA method
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? Use RFA.m file and investigate the effect of initial estimate of T matrix
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