Preface |
|
ix | |
Symbols |
|
xiii | |
|
|
1 | (6) |
|
|
7 | (22) |
|
Matrices, Vectors, Scalars |
|
|
8 | (18) |
|
Elementary Matrix Operations |
|
|
10 | (1) |
|
|
10 | (2) |
|
|
12 | (4) |
|
|
16 | (5) |
|
|
21 | (1) |
|
|
21 | (1) |
|
|
22 | (1) |
|
|
22 | (1) |
|
|
23 | (1) |
|
|
24 | (1) |
|
Orthogonal and Orthonormal Matrices |
|
|
25 | (1) |
|
Solving Systems of Linear Equations |
|
|
26 | (3) |
|
|
29 | (72) |
|
|
33 | (3) |
|
Chromatography / Gaussian Curves |
|
|
36 | (4) |
|
Titrations, Equilibria, the Law of Mass Action |
|
|
40 | (36) |
|
A Simple Case: Fe3+ + SCN- |
|
|
40 | (3) |
|
The General Case, Definitions |
|
|
43 | (2) |
|
A Chemical Example, Cu2+, Ethylenediamine, Protons |
|
|
45 | (3) |
|
Solving Complex Equilibria |
|
|
48 | (1) |
|
The Newton-Raphson Algorithm |
|
|
48 | (8) |
|
Example: General 3-Component Titration |
|
|
56 | (2) |
|
Example: pH Titration of Acetic Acid |
|
|
58 | (2) |
|
|
60 | (2) |
|
Complex Equilibria Including Activity Coefficients |
|
|
62 | (2) |
|
Special Case: Explicit Calculation for Polyprotic Acids |
|
|
64 | (5) |
|
Solving Non-Linear Equations |
|
|
69 | (1) |
|
One Equation, One Parameter |
|
|
69 | (2) |
|
Systems of Non-Linear Equations |
|
|
71 | (5) |
|
Kinetics, Mechanisms, Rate Laws |
|
|
76 | (25) |
|
|
77 | (1) |
|
Rate Laws with Explicit Solutions |
|
|
77 | (3) |
|
Complex Mechanisms that Require Numerical Integration |
|
|
80 | (1) |
|
|
80 | (2) |
|
Fourth Order Runge-Kutta Method in Excel |
|
|
82 | (4) |
|
Interesting Kinetic Examples |
|
|
86 | (1) |
|
|
87 | (2) |
|
|
89 | (2) |
|
The Steady-State Approximation |
|
|
91 | (1) |
|
Lotka-Volterra / Predator-Prey Systems |
|
|
92 | (3) |
|
The Belousov-Zhabotinsky (BZ) Reaction |
|
|
95 | (2) |
|
Chaos, the Lorenz Attractor |
|
|
97 | (4) |
|
|
101 | (112) |
|
Background to Least-Squares Methods |
|
|
102 | (7) |
|
The Residuals and the Sum of Squares |
|
|
103 | (1) |
|
Linear Example: Straight Line |
|
|
103 | (2) |
|
Non-Linear Example: Exponential Decay |
|
|
105 | (4) |
|
|
109 | (39) |
|
Straight Line Fit - Classical Derivation |
|
|
109 | (4) |
|
|
113 | (1) |
|
Generalised Matrix Notation |
|
|
114 | (1) |
|
|
115 | (2) |
|
|
117 | (2) |
|
Linear Dependence, Rank of a Matrix |
|
|
119 | (1) |
|
|
120 | (1) |
|
Errors in the Fitted Parameters |
|
|
121 | (4) |
|
|
125 | (2) |
|
Applications of Linear Least-Squares Fitting |
|
|
127 | (1) |
|
Linearisation of Non-Linear Problems |
|
|
127 | (3) |
|
Polynomials, the Savitzky-Golay Digital Filter |
|
|
130 | (1) |
|
|
131 | (4) |
|
Calculation of the Derivative of a Curve |
|
|
135 | (3) |
|
|
138 | (1) |
|
Linear Regression with Multivariate Data |
|
|
139 | (4) |
|
|
143 | (1) |
|
Computation of Component Spectra, Known Concentrations |
|
|
144 | (1) |
|
Computation of Component Concentrations, Known Spectra |
|
|
145 | (1) |
|
The Pseudo-Inverse in Excel |
|
|
146 | (2) |
|
|
148 | (50) |
|
The Newton-Gauss-Levenberg/Marquardt Algorithm |
|
|
148 | (1) |
|
A First, Minimal Algorithm |
|
|
149 | (4) |
|
Termination Criterion, Numerical Derivatives |
|
|
153 | (2) |
|
The Levenberg/Marquardt Extension |
|
|
155 | (6) |
|
Standard Errors of the Parameters |
|
|
161 | (1) |
|
Multivariate Data, Separation of the Linear and Non-Linear Parameters |
|
|
162 | (6) |
|
Constraint: Positive Component Spectra |
|
|
168 | (1) |
|
Structures, Fixing Parameters |
|
|
169 | (6) |
|
Known Spectra, Uncoloured Species |
|
|
175 | (5) |
|
Reduced Eigenvector Space |
|
|
180 | (3) |
|
|
183 | (6) |
|
Non-White Noise, Χ2-Fitting |
|
|
189 | (1) |
|
|
190 | (5) |
|
|
195 | (2) |
|
Finding the Correct Model |
|
|
197 | (1) |
|
|
198 | (15) |
|
The Newton-Gauss Algorithm |
|
|
198 | (6) |
|
|
204 | (3) |
|
Optimisation in Excel, the Solver |
|
|
207 | (4) |
|
|
211 | (2) |
|
|
213 | (100) |
|
|
213 | (33) |
|
The Singular Value Decomposition, SVD |
|
|
214 | (3) |
|
|
217 | (2) |
|
Magnitude of the Singular Values |
|
|
219 | (2) |
|
The Structure of the Eigenvectors |
|
|
221 | (1) |
|
The Structure of the Residuals |
|
|
222 | (1) |
|
The Standard Deviation of the Residuals |
|
|
223 | (1) |
|
Geometrical Interpretations |
|
|
224 | (1) |
|
|
224 | (4) |
|
Reduction in the Number of Dimensions |
|
|
228 | (3) |
|
|
231 | (4) |
|
Three and More Components |
|
|
235 | (4) |
|
|
239 | (2) |
|
|
241 | (2) |
|
|
243 | (3) |
|
Target Factor Analyses, TFA |
|
|
246 | (13) |
|
|
250 | (1) |
|
Iterative Target Transform Factor Analysis, ITTFA |
|
|
251 | (2) |
|
Target Transform Search/Fit |
|
|
253 | (4) |
|
Parameter Fitting via Target Testing |
|
|
257 | (2) |
|
Evolving Factor Analyses, EFA |
|
|
259 | (21) |
|
Evolving Factor Analysis, Classical EFA |
|
|
260 | (8) |
|
Fixed-Size Window EFA, FSW-EFA |
|
|
268 | (3) |
|
Secondary Analyses Based on Window Information |
|
|
271 | (1) |
|
Iterative Refinement of the Concentration Profiles |
|
|
271 | (5) |
|
Explicit Computation of the Concentration Profiles |
|
|
276 | (4) |
|
Alternating Least-Squares, ALS |
|
|
280 | (10) |
|
Initial Guesses for Concentrations or Spectra |
|
|
281 | (1) |
|
Alternating Least-Squares and Constraints |
|
|
282 | (6) |
|
|
288 | (2) |
|
Resolving Factor Analysis, RFA |
|
|
290 | (5) |
|
Principle Component Regression and Partial Least Squares, PCR and PLS |
|
|
295 | (18) |
|
Principal Component Regression, PCR |
|
|
296 | (1) |
|
Mean-Centring, Normalisation |
|
|
297 | (1) |
|
|
298 | (2) |
|
|
300 | (3) |
|
|
303 | (3) |
|
Partial Least Squares, PLS |
|
|
306 | (2) |
|
|
308 | (1) |
|
PLS Prediction / Cross Validation |
|
|
309 | (1) |
|
|
310 | (3) |
Further Reading |
|
313 | (4) |
List of Matlab Files |
|
317 | (4) |
List of Excel Sheets |
|
321 | (1) |
Index |
|
322 | |