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Evolving Factor Analysis The evolution of a chemical system is gradually known by recording a new response vector at each stage of the process under study. EFA performs subsequent PCA on gradually increasing submatrices in the process direction, enlarged by adding one new row at a time. This procedure is performed from top to bottom of the data set (forward EFA) and from bottom to top (backward EFA) to investigate the emergence and the decay of the process contribution, respectively. The forward and backward EFA plots are built by representating the singular values of each PCA analysis vs. the process variable related to the last row included in the window analyzd.

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1.4327 0.0024 Singular values (0-2 sec)

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2.2664 0.0083 0.0000 Singular values (0-4 sec)

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3.2730 0.0231 0.0000 0.0000 Singular values (0-6 sec)

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4.4044 0.0563 0.0001 0.0000 Singular values (0-8 sec)

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5.5834 0.1245 0.0004 0.0000 Singular values (0-10 sec)

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6.7299 0.2517 0.0012 0.0000 Singular values (0-12 sec)

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7.7864 0.4668 0.0036 0.0000 Singular values (0-14 sec)

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8.7323 0.7956 0.0099 0.0000 Singular values (0-16 sec)

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9.5808 1.2484 0.0244 0.0000 Singular values (0-18 sec)

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10.3637 1.8119 0.0552 0.0000 Singular values (0-20 sec)

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11.1136 2.4512 0.1133 0.0000 Singular values (0-22 sec)

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11.8506 3.1232 0.2110 0.0000 Singular values (0-24 sec)

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12.5772 3.7923 0.3561 0.0000 Singular values (0-26 sec)

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13.2808 4.4360 0.5455 0.0000 Singular values (0-28 sec)

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13.9413 5.0402 0.7623 0.0000 Singular values (0-30 sec)

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14.5360 5.5893 0.9812 0.0000 Singular values (0-32 sec)

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15.0430 6.0633 1.1776 0.0000 Singular values (0-34 sec)

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15.4449 6.4435 1.3359 0.0000 Singular values (0- 36 sec)

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15.7348 6.7216 1.4512 0.0000 Singular values (0- 38 sec)

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15.9215 6.9040 1.5268 0.0000 Singular values (0- 40 sec)

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16.0273 7.0098 1.5713 0.0000 Singular values (0- 42 sec)

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16.0794 7.0634 1.5942 0.0000 Singular values (0- 44 sec)

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16.1015 7.0868 1.6044 0.0000 Singular values (0- 46 sec)

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16.1096 7.0955 1.6083 0.0000 Singular values (0-48 sec)

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16.1122 7.0983 1.6096 0.0000 Singular values (0-50 sec)

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0.7231 0.0017 0.000 0.000 Singular values (50-48 sec)

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1.2648 0.0060 0.0000 0 Singular values (50-46 sec)

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2.0245 0.0164 0.0000 0.0000 Singular values (50-44 sec)

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3.0143 0.0395 0.0000 0.0000 Singular values (50-42 sec)

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4.2074 0.0865 0.0001 0.0000 Singular values (50-40 sec)

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5.5408 0.1738 0.0003 0.0000 Singular values (50-38 sec)

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6.9305 0.3215 0.0009 0.0000 Singular values (50-36 sec)

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8.2934 0.5483 0.0029 0.0000 Singular values (50-34 sec)

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9.5650 0.8639 0.0082 0.0000 Singular values (50-32 sec)

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10.7064 1.2627 0.0213 0.0000 Singular values (50-30 sec)

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11.7008 1.7245 0.0504 0.0000 Singular values (50-28 sec)

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12.5455 2.2228 0.1080 0.0000 Singular values (50-26 sec)

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13.2478 2.7381 0.2091 0.0000 Singular values (50-24 sec)

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13.8235 3.2656 0.3639 0.0000 Singular values (50-22 sec)

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14.2956 3.8130 0.5684 0.0000 Singular values (50-20 sec)

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14.6900 4.3880 0.8003 0.0000 Singular values (50-18 sec)

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15.0288 4.9811 1.0266 0.0000 Singular values (50-16 sec)

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15.3247 5.5579 1.2200 0.0000 Singular values (50-14 sec)

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15.5782 6.0693 1.3680 0.0000 Singular values (50-12 sec)

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15.7824 6.4753 1.4711 0.0000 Singular values (50-10 sec)

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15.9307 6.7613 1.5372 0.0000 Singular values (50-8 sec)

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16.0254 6.9387 1.5759 0.0000 Singular values (50-6 sec)

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16.0776 7.0349 1.5963 0.0000 Singular values (50- 4 sec)

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16.1022 7.0801 1.6058 0.0000 Singular values (50-2 sec)

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16.1122 7.0983 1.6096 0.0000 Singular values (50-0 sec)

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Using MATLAB for evolving factor analysis

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hplc.m file Creating HPLC-DAD data

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HPLC-DAD data for three components system

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EFA.m file Evolving Factor Analysis

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Retention Time Wavelength D

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Delete the SVF and SVB variables from the memory in work space

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Creating the SVF matrix with (m m- 1) dimensions and all elements equal to zero

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An example for zeros command in MATLAB

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Plot the results of forward analysis

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Change in order of columns of the matrix

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Comparison of real and estimated profiles

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? Employ the EFA in wavelength direction of data matrix and interpret the results

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Transformation the concentration windows calculated with EFA to concentration profiles Retention Time

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C= S T = Concentration matrix Score matrix Transformation matrix c 1 = S t 1 = Concentration vector Score matrix Transformation vector = c 0 = S 0 t 1 0= t 11 s 1 + t 21 s 2 + t 31 s 3

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HPLC-DAD data for three components system

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Results from EFA Retention Time From row number 35 to 61

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concEFA.m file for calculation the concentration profiles according to results of EFA

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Comparison the results with true values

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? Use the concEFA.m file and calculate the concentration profile for third component

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Application of EFA in chemical equilibria study Stepwise dissociation of triprotic acid H 3 A

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H3A.m file for simulating the spectrophotometric monitoring of pH-meteric titration

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Evolving Factor Analysis (EFA)

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? Use the H3A.m file and investigate the effects of pKas on results of EFA.

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Application of EFA in chemical Linetics study Consecutive reaction

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consecutive.m file for simulating the spectrophotometric monitoring of consecutive A B C reaction

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Evolving Factor Analysis (EFA)

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? Use the consecutive.m file and investigate the effects of rate constants on results of EFA.

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Fixed concentration of interference and EFA

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EFA

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HPLC-DAD data after column mean centering

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Results of forward and backward eigen analysis

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Results of applying EFA on mean centered data

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Score plot without mean centering

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Score plot after mean centering

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Distribution of objects of a two component system O A2 A1

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Mean centering O A1 A2

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Mean centering and then PCA O PC1 PC2

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Distribution of objects of a two component system O A1 A2

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Mean centering on window data O A1 A2

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Before appearance the analyte the variance is equal to zero Mean centering on window data and then PCA O PC1 PC2

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Before appearance the analyte the variance is equal to zero Mean centering on window data and then PCA O PC1 PC2

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O PC1 PC2 Before appearance the analyte the variance is equal to zero Mean centering on window data and then PCA

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Before appearance the analyte the variance is equal to zero Mean centering on window data and then PCA O PC1 PC2

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Mean centering on window data O A1 A2

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Mean centering and then PCA on window data O PC1 PC2

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Mean centering on window data O A1 A2

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Mean centering and then PCA on window data O PC1 PC2

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Mean centering on window data O A1 A2

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Mean centering and then PCA on window data O PC1 PC2

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IEFA.m Evolving factor analysis in the presence of fixed concentration interferent

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Results of applying IEFA.m file

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Comparison between results of IEFA and real values of analyte

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? Use IEFA.m file and analyze the three co-eluting components system with fix concentration of one of them

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Titration of H3A in the presence of an inert species

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EFA results

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EFA results in the absence of interference

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? WHY?

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