Evolving Factor AnalysisThe 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.

1.4327 0.0024

Singular values (0-2 sec)

2.2664 0.0083 0.0000

Singular values (0-4 sec)

3.2730 0.0231 0.0000 0.0000

Singular values (0-6 sec)

4.4044 0.0563 0.0001 0.0000

Singular values (0-8 sec)

5.5834 0.1245 0.0004 0.0000

Singular values (0-10 sec)

6.7299 0.2517 0.0012 0.0000

Singular values (0-12 sec)

7.7864 0.4668 0.0036 0.0000

Singular values (0-14 sec)

8.7323 0.7956 0.0099 0.0000

Singular values (0-16 sec)

9.5808 1.2484 0.0244 0.0000

Singular values (0-18 sec)

10.3637 1.8119 0.0552 0.0000

Singular values (0-20 sec)

11.1136 2.4512 0.1133 0.0000

Singular values (0-22 sec)

11.8506 3.1232 0.2110 0.0000

Singular values (0-24 sec)

12.5772 3.7923 0.3561 0.0000

Singular values (0-26 sec)

13.2808 4.4360 0.5455 0.0000

Singular values (0-28 sec)

13.9413 5.0402 0.7623 0.0000

Singular values (0-30 sec)

14.5360 5.5893 0.9812 0.0000

Singular values (0-32 sec)

15.0430 6.0633 1.1776 0.0000

Singular values (0-34 sec)

15.4449 6.4435 1.3359 0.0000

Singular values (0- 36 sec)

15.7348 6.7216 1.4512 0.0000

Singular values (0- 38 sec)

15.9215 6.9040 1.5268 0.0000

Singular values (0- 40 sec)

16.0273 7.0098 1.5713 0.0000

Singular values (0- 42 sec)

16.0794 7.0634 1.5942 0.0000

Singular values (0- 44 sec)

16.1015 7.0868 1.6044 0.0000

Singular values (0- 46 sec)

16.1096 7.0955 1.6083 0.0000

Singular values (0-48 sec)

16.1122 7.0983 1.6096 0.0000

Singular values (0-50 sec)

0.7231 0.0017 0.000 0.000

Singular values (50-48 sec)

1.2648 0.0060 0.0000 0

Singular values (50-46 sec)

2.0245 0.0164 0.0000 0.0000

Singular values (50-44 sec)

3.0143 0.0395 0.0000 0.0000

Singular values (50-42 sec)

4.2074 0.0865 0.0001 0.0000

Singular values (50-40 sec)

5.5408 0.1738 0.0003 0.0000

Singular values (50-38 sec)

6.9305 0.3215 0.0009 0.0000

Singular values (50-36 sec)

8.2934 0.5483 0.0029 0.0000

Singular values (50-34 sec)

9.5650 0.8639 0.0082 0.0000

Singular values (50-32 sec)

10.7064 1.2627 0.0213 0.0000

Singular values (50-30 sec)

11.7008 1.7245 0.0504 0.0000

Singular values (50-28 sec)

12.5455 2.2228 0.1080 0.0000

Singular values (50-26 sec)

13.2478 2.7381 0.2091 0.0000

Singular values (50-24 sec)

13.8235 3.2656 0.3639 0.0000

Singular values (50-22 sec)

14.2956 3.8130 0.5684 0.0000

Singular values (50-20 sec)

14.6900 4.3880 0.8003 0.0000

Singular values (50-18 sec)

15.0288 4.9811 1.0266 0.0000

Singular values (50-16 sec)

15.3247 5.5579 1.2200 0.0000

Singular values (50-14 sec)

15.5782 6.0693 1.3680 0.0000

Singular values (50-12 sec)

15.7824 6.4753 1.4711 0.0000

Singular values (50-10 sec)

15.9307 6.7613 1.5372 0.0000

Singular values (50-8 sec)

16.0254 6.9387 1.5759 0.0000

Singular values (50-6 sec)

16.0776 7.0349 1.5963 0.0000

Singular values (50- 4 sec)

16.1022 7.0801 1.6058 0.0000

Singular values (50-2 sec)

16.1122 7.0983 1.6096 0.0000

Singular values (50-0 sec)

Using MATLAB for evolving factor analysis

hplc.m file

Creating HPLC-DAD data

HPLC-DAD data for three components system

EFA.m file

Evolving Factor Analysis

Ret

enti

on T

ime

Wavelength

D

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

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

zero

An example for zeros command in MATLAB

Plot the results of forward analysis

Change in order of columns of the matrix

Comparison of real and estimated profiles

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

Transformation the concentration windows calculated with EFA to concentration profiles

Retention Time

C = S T

=

Con

cen

trat

ion

mat

rix

Scor

e m

atri

x

Transformation matrix

c1 = S t1

=

Con

cen

trat

ion

vec

tor

Scor

e m

atri

x

Transformation vector

=

c0 = S0 t1

0= t11 s1 + t21 s2 + t31 s3

HPLC-DAD data for three components system

Results from EFA

Retention Time

From row number 35 to 61

concEFA.m file for calculation the concentration

profiles according to results of EFA

Comparison the results with true values

?

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

Application of EFA in chemical equilibria study

Stepwise dissociation of triprotic acid H3A

H3A.m file

for simulating the spectrophotometric monitoring

of pH-meteric titration

Evolving Factor Analysis (EFA)

Evolving Factor Analysis (EFA)

?

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

Application of EFA in chemical Linetics study

Consecutive reaction

consecutive.m file

for simulating the spectrophotometric monitoring of consecutive A B C reaction

Evolving Factor Analysis (EFA)

Evolving Factor Analysis (EFA)

?

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

Fixed concentration of interference and EFA

EFA

HPLC-DAD data after column mean centering

Results of forward and backward eigen analysis

Results of applying EFA on mean centered data

Score plot without mean centering

Score plot after mean centering

Distribution of objects of a two component system

O A2

A1

Mean centering

O A1

A2

Mean centering and then PCA

O

PC1PC2

Distribution of objects of a two component system

O A1

A2

Mean centering on window data

O A1

A2

Before appearance the analyte the variance is equal to zero

Mean centering on window data and then PCA

O PC1

PC2

Before appearance the analyte the variance is equal to zero

Mean centering on window data and then PCA

O PC1

PC2

O PC1

PC2

Before appearance the analyte the variance is equal to zero

Mean centering on window data and then PCA

Before appearance the analyte the variance is equal to zero

Mean centering on window data and then PCA

O PC1

PC2

Mean centering on window data

O A1

A2

Mean centering and then PCA on window data

O

PC1PC2

Mean centering on window data

O A1

A2

Mean centering and then PCA on window data

O

PC1PC2

Mean centering on window data

O A1

A2

Mean centering and then PCA on window data

O

PC1PC2

IEFA.m

Evolving factor analysis in the presence of fixed concentration

interferent

Results of applying IEFA.m file

Results of applying IEFA.m file

Comparison between results of IEFA and real values of analyte

?

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

Titration of H3A in the presence of an inert species

Titration of H3A in the presence of an inert species

EFA results

EFA results in the absence of interference

?

WHY?