1. Equilibrium systems Chromatography systems Number of PCs original Mean centered Number of PCs original Mean centered 21 21 21 2 1 22 32

2. The rank of a product of two matrices X and Y is equal or smaller to the smallest of the rank of X and Y: Rank (X Y) ≤ min (rank (X), rank (Y)) A = C S

3. HA A - + H + [HA] = C t [H + ] [H + ] + K a [A - ] = C t K a [H + ] + K a [Int] =p [HA] + [A - ] = C t C t =  p =  [Int] [HA] + [A - ] =  [Int] [HA] + [A - ] -  [Int] = 0

4. ? Determine the rank of data matrix of following hypothesis system (rank.mat file)

5. A Rank Deficiency problem and Its Solution

6. Rank deficient system = = Full rank system

7. Augmentation =

8. A rd C E = AsAs C E = E = AsAs C C Rank (A rd ) = 2 Rank (A s ) = 1 Rank (A rd ; A s ) = 3 Augmentation

10. 123123 321321 0.5 123123 321321 ++ (-8) 000000 = Rank deficiency in C matrix

11. 123123 321321 0.5 246246 642642 000000 Augmentation 123123 321321 0.5 246246 642642 000000

12. Rank deficiency in a system with two independent chemical processes HA A - + H + HB B - + H + [HA] [A-][A-] [HB] [B-][B-] HA A-A- HB B-B-

13. HAM.m file Spectrophotometric monitoring of pH- metric titration of mixture of two acids

19. HA A - + H + [HA] = C t1 [H + ] [H + ] + K a1 [A - ] = C t1 K a1 [H + ] + K a1 [HA] + [A - ] =  ([HB] + [B - ]) C t1 =  C t2 HB B - + H + [H + ] + K a2 [HB] = C t2 [H + ] [A - ] = C t2 K a2 [H + ] + K a2 [HB] + [B - ] = C t2

20. 123123 321321 246246 642642 123123 321321 + + 000000 = 246246 642642 + 123123 321321 + 444444 = 246246 642642 + 888888 = 444444 888888 =(1/2) (-1/2)

21. Augmentation =

23. 123123 321321 246246 642642 123123 321321 000000 000000 123123 321321 246246 642642 123123 321321 000000 000000

24. 123123 321321 123123 321321 + 444444 444444 = 246246 642642 000000 000000 + 888888 000000 =

25. ? Is it possible to solve rank deficiency in systems by exact same pK a values?

26. Non-linear data

27. Principal Component Analysis

28. Samples in two dimensional wavelength space

29. Non-homogeneous deviation in wavelength space

30. Principal Component Analysis

31. Samples in two dimensional wavelength space

32. ? Use NLC.m file and create data matrix for two component system and investigate the effect of non- linearity on numbers of PCs

33. NLC.m file Non-linear calibration absorbance data matrix

37. Number of significant eigenvalues and evolutionary chemical processes

62. 1.59150.00230.00000.0000 Singular values (0-6 sec)

63. 2.45180.00690.00000.0000 Singular values (2-8 sec)

64. 3.4849 0.0193 0.0001 0.0000 Singular values (4-10 sec)

65. 4.5878 0.0488 0.0002 0.0000 Singular values (6-12 sec)

66. 5.6211 0.1120 0.0008 0.0000 Singular values (8-14 sec)

67. 6.4493 0.2323 0.0023 0.0000 Singular values (10-16 sec)

68. 6.9869 0.4338 0.0060 0.0000 Singular values (12-18 sec)

69. 7.2375 0.7233 0.0138 0.0000 Singular values (14-20 sec)

70. 7.3157 1.0627 0.0279 0.0000 Singular values (16-22 sec)

71. 7.4354 1.3499 0.0495 0.0000 Singular values (18-24 sec)

72. 7.8152 1.4677 0.0752 0.0000 Singular values (20-26 sec)

73. 8.5105 1.4042 0.0951 0.0000 Singular values (22-28 sec)

74. 9.3713 1.2588 0.0963 0.0000 Singular values (24-30 sec)

75. 10.1608 1.1141 0.0761 0.0000 Singular values (26-32 sec)

76. 10.6552 0.9779 0.0476 0.0000 Singular values (28-34 sec)

77. 10.6909 0.8265 0.0245 0.0000 Singular values (30-36 sec)

78. 10.1926 0.6522 0.0108 0.0000 Singular values (32-38 sec)

79. 9.1870 0.4713 0.0042 0.0000 Singular values (34-40 sec)

80. 7.7941 0.3089 0.0014 0.0000 Singular values (36-42 sec)

81. 6.1977 0.1830 0.0004 0.0000 Singular values (38-44 sec)

82. 4.5999 0.0980 0.0001 0.0000 Singular values (40-46 sec)

83. 3.1735 0.0475 0.0000 0.0000 Singular values (42-148 sec)

84. 2.0275 0.0209 0.0000 0.0000 Singular values (44-50 sec)

85. FSW.m file Eigen analysis on moving fixed size window

90. ? Use FSW.m file and investigate on pH-window of each component for H2A system (H2A.m file)

91. ? Investigate the effects of selected window size on accuracy of determining the concentration window for each species

117. 1.4327 0.0024 Singular values (0-2 sec)

118. 2.2664 0.0083 0.0000 Singular values (0-4 sec)

119. 3.2730 0.0231 0.0000 0.0000 Singular values (0-6 sec)

120. 4.4044 0.0563 0.0001 0.0000 Singular values (0-8 sec)

121. 5.5834 0.1245 0.0004 0.0000 Singular values (0-10 sec)

122. 6.7299 0.2517 0.0012 0.0000 Singular values (0-12 sec)

123. 7.7864 0.4668 0.0036 0.0000 Singular values (0-14 sec)

124. 8.7323 0.7956 0.0099 0.0000 Singular values (0-16 sec)

125. 9.5808 1.2484 0.0244 0.0000 Singular values (0-18 sec)

126. 10.3637 1.8119 0.0552 0.0000 Singular values (0-20 sec)

127. 11.1136 2.4512 0.1133 0.0000 Singular values (0-22 sec)

128. 11.8506 3.1232 0.2110 0.0000 Singular values (0-24 sec)

129. 12.5772 3.7923 0.3561 0.0000 Singular values (0-26 sec)

130. 13.2808 4.4360 0.5455 0.0000 Singular values (0-28 sec)

131. 13.9413 5.0402 0.7623 0.0000 Singular values (0-30 sec)

132. 14.5360 5.5893 0.9812 0.0000 Singular values (0-32 sec)

133. 15.0430 6.0633 1.1776 0.0000 Singular values (0-34 sec)

134. 15.4449 6.4435 1.3359 0.0000 Singular values (0- 36 sec)

135. 15.7348 6.7216 1.4512 0.0000 Singular values (0- 38 sec)

136. 15.9215 6.9040 1.5268 0.0000 Singular values (0- 40 sec)

137. 16.0273 7.0098 1.5713 0.0000 Singular values (0- 42 sec)

138. 16.0794 7.0634 1.5942 0.0000 Singular values (0- 44 sec)

139. 16.1015 7.0868 1.6044 0.0000 Singular values (0- 46 sec)

140. 16.1096 7.0955 1.6083 0.0000 Singular values (0-48 sec)

141. 16.1122 7.0983 1.6096 0.0000 Singular values (0-50 sec)

142. 0.7231 0.0017 0.000 0.000 Singular values (50-48 sec)

143. 1.2648 0.0060 0.0000 0 Singular values (50-46 sec)

144. 2.0245 0.0164 0.0000 0.0000 Singular values (50-44 sec)

145. 3.0143 0.0395 0.0000 0.0000 Singular values (50-42 sec)

146. 4.2074 0.0865 0.0001 0.0000 Singular values (50-40 sec)

147. 5.5408 0.1738 0.0003 0.0000 Singular values (50-38 sec)

148. 6.9305 0.3215 0.0009 0.0000 Singular values (50-36 sec)

149. 8.2934 0.5483 0.0029 0.0000 Singular values (50-34 sec)

150. 9.5650 0.8639 0.0082 0.0000 Singular values (50-32 sec)

151. 10.7064 1.2627 0.0213 0.0000 Singular values (50-30 sec)

152. 11.7008 1.7245 0.0504 0.0000 Singular values (50-28 sec)

153. 12.5455 2.2228 0.1080 0.0000 Singular values (50-26 sec)

154. 13.2478 2.7381 0.2091 0.0000 Singular values (50-24 sec)

155. 13.8235 3.2656 0.3639 0.0000 Singular values (50-22 sec)

156. 14.2956 3.8130 0.5684 0.0000 Singular values (50-20 sec)

157. 14.6900 4.3880 0.8003 0.0000 Singular values (50-18 sec)

158. 15.0288 4.9811 1.0266 0.0000 Singular values (50-16 sec)

159. 15.3247 5.5579 1.2200 0.0000 Singular values (50-14 sec)

160. 15.5782 6.0693 1.3680 0.0000 Singular values (50-12 sec)

161. 15.7824 6.4753 1.4711 0.0000 Singular values (50-10 sec)

162. 15.9307 6.7613 1.5372 0.0000 Singular values (50-8 sec)

163. 16.0254 6.9387 1.5759 0.0000 Singular values (50-6 sec)

164. 16.0776 7.0349 1.5963 0.0000 Singular values (50- 4 sec)

165. 16.1022 7.0801 1.6058 0.0000 Singular values (50-2 sec)

166. 16.1122 7.0983 1.6096 0.0000 Singular values (50-0 sec)

167. EFA.m file Forward and backward eigen analysis

175. ? Investigate the variation of eigenvalues in an evolutionary chemical process in the presence of noise

176. ? Show that the eigenvalue analysis can be used for estimating the selective wavelength range for each chemical species

183. Principal Component Regression (PCR) PCR is simply PCA followed by a regression step A= C E = S L AC E = S L = A= C E = (S R) (R -1 L) C = S R CS R = S r = c1c1

184. A data matrix can be represented by its score matrix A regression of score matrix against one or several dependent variables is possible, provided that scores corresponding to small eigenvalues are omitted This regression gives no matrix inversion problem PCR has the full-spectrum advantages of the CLS method PCR has the ILS advantage of being able to perform the analysis one chemical components at a time while avoiding the ILS wavelength selection problem

185. c = S b Calibration and Prediction Steps in PCR = c1c1 n 1 S n r b r 1 b = ( S T S) -1 S T c Calibration Step AxAx m p L p r r m SxSx = Prediction Step S x = A x L c x = S x b