Chemometrics in Spectroscopy (eBook)

Front Cover 1
Chemometrics in Spectroscopy 4
Copyright page 5
Table of Contents 8
Preface 12
Note to Readers 14
Chapter 1. A New Beginning . . . 16
THE MULTIVARIATE NORMAL DISTRIBUTION 18
REFERENCES 22
Chapter 2. Elementary Matrix Algebra: Part 1 24
MATRIX OPERATIONS 25
ELEMENTARY OPERATIONS FOR LINEAR EQUATIONS 27
SUMMARY 30
REFERENCES 30
Chapter 3. Elementary Matrix Algebra: Part 2 32
ELEMENTARY MATRIX OPERATIONS 32
CALCULATING THE INVERSE OF A MATRIX 34
SUMMARY 35
REFERENCES 36
Chapter 4. Matrix Algebra and Multiple Linear Regression: Part 1 38
QUASI-ALGEBRAIC OPERATIONS 40
MULTIPLE LINEAR REGRESSION 43
THE LEAST SQUARES METHOD 44
REFERENCES 47
Chapter 5. Matrix Algebra and Multiple Linear Regression: Part 2 48
THE POWER OF MATRIX MATHEMATICS 53
REFERENCES 56
Chapter 6. Matrix Algebra and Multiple Linear Regression: Part 3 – The Concept of Determinants 58
REFERENCES 60
Chapter 7. Matrix Algebra and Multiple Linear Regression: Part 4 – Concluding Remarks 62
A WORD OF CAUTION 63
REFERENCE 64
Chapter 8. Experimental Designs: Part 1 66
REFERENCES 70
Chapter 9. Experimental Designs: Part 2 72
REFERENCES 77
Chapter 10. Experimental Designs: Part 3 78
Chapter 11. Analytic Geometry: Part 1 – The Basics in Two and Three Dimensions 86
THE DISTANCE FORMULA 86
DIRECTION NOTATION 87
THE COSINE FUNCTION 87
DIRECTION IN 3-D SPACE 89
DEFINING SLOPE IN TWO DIMENSIONS 90
RECOMMENDED READING 91
Chapter 12. Analytic Geometry: Part 2 – Geometric Representation of Vectors and Algebraic Operations 92
VECTOR MULTIPLICATION (SCALAR × VECTOR) 92
VECTOR DIVISION (VECTOR ÷ SCALAR) 93
VECTOR ADDITION (VECTOR + VECTOR) 93
VECTOR SUBTRACTION (VECTOR VECTOR) 94
Chapter 13. Analytic Geometry: Part 3 – Reducing Dimensionality 96
REDUCING DIMENSIONALITY 96
3-D TO 2-D BY PROJECTION 96
2-D INTO 1-D BY ROTATION 99
Chapter 14. Analytic Geometry: Part 4 – The Geometry of Vectors and Matrices 100
ROW VECTORS IN COLUMN SPACE 100
COLUMN VECTORS IN ROW SPACE 100
PRINCIPAL COMPONENTS FOR REGRESSION VECTORS 101
RECOMMENDED READING 103
Chapter 15. Experimental Designs: Part 4 – Varying Parameters to Expand the Design 104
REFERENCES 105
Chapter 16. Experimental Designs: Part 5 – One-at-a-time Designs 106
REFERENCES 107
Chapter 17. Experimental Designs: Part 6 – Sequential Designs 108
REFERENCES 110
Chapter 18. Experimental Designs: Part 7 – ß, the Power of a Test 112
REFERENCES 114
Chapter 19. Experimental Designs: Part 8 – ß, the Power of a Test (Continued) 116
REFERENCES 117
Chapter 20. Experimental Designs: Part 9 – Sequential Designs Concluded 118
REFERENCES 120
Chapter 21. Calculating the Solution for Regression Techniques: Part 1 – Multivariate Regression Made Simple 122
REFERENCES 123
Chapter 22. Calculating the Solution for Regression Techniques: Part 2 – Principal Component(s) Regression Made Simple 124
REFERENCES 126
Chapter 23. Calculating the Solution for Regression Techniques: Part 3 – Partial Least Squares Regression Made Simple 128
REFERENCES 131
Chapter 24. Looking Behind and Ahead: Interlude 132
Chapter 25. A Simple Question: The Meaning of Chemometrics Pondered 134
REFERENCES 140
Chapter 26. Calculating the Solution for Regression Techniques: Part 4 – Singular Value Decomposition 142
REFERENCES 144
Chapter 27. Linearity in Calibration 146
REFERENCE 149
Chapter 28. Challenges: Unsolved Problems in Chemometrics 150
REFERENCE 154
Chapter 29. Linearity in Calibration: Act II Scene I 156
REFERENCES 159
Chapter 30. Linearity in Calibration: Act II Scene II – Reader’s Comments . . . 160
REFERENCES 163
Chapter 31. Linearity in Calibration: Act II Scene III 164
REFERENCES 172
Chapter 32. Linearity in Calibration: Act II Scene IV 174
REFERENCES 177
Chapter 33. Linearity in Calibration: Act II Scene V 178
REFERENCES 181
Chapter 34. Collaborative Laboratory Studies: Part 1 – A Blueprint 182
EXPERIMENTAL DESIGN 183
ANALYTICAL METHODS 188
METHOD A and B analysis 188
RESULTS AND DATA ANALYSIS 188
REFERENCES 192
Chapter 35. Collaborative Laboratory Studies: Part 2 – using ANOVA 194
ANOVA TEST COMPARISONS FOR LABORATORIES AND METHODS (ANOVA_s4 WORKSHEET) 194
ANOVA test comparisons (using ANOVA_s2 worksheet) 195
REFERENCES 196
Chapter 36. Collaborative Laboratory Studies: Part 3 – Testing for Systematic Error 198
TESTING FOR SYSTEMATIC ERROR IN A METHOD: COMPARISON TEST FOR A SET OF MEASUREMENTS VERSUS TRUE VALUE – SPIKED RECOVERY METHOD (COMPARET WORKSHEET) 198
REFERENCES 199
Chapter 37. Collaborative Laboratory Studies: Part 4 – Ranking Test 200
RANKING TEST FOR LABORATORIES AND METHODS (MANUAL COMPUTATIONS) 200
REFERENCE 201
Chapter 38. Collaborative Laboratory Studies: Part 5 – Efficient Comparison of Two Methods 202
COMPUTATIONS FOR EFFICIENT COMPARISON OF TWO METHODS (COMP_METH WORKSHEET) 202
SUMMARY 207
ACKNOWLEDGEMENT 207
REFERENCES 207
Chapter 39. Collaborative Laboratory Studies: Part 6 – MathCad Worksheet Text 208
REFERENCES 237
Chapter 40. Is Noise Brought by the Stork? Analysis of Noise: Part 1 238
REFERENCES 241
Chapter 41. Analysis of Noise: Part 2 242
APPENDIX 247
REFERENCES 248
Chapter 42. Analysis of Noise: Part 3 250
REFERENCES 257
Chapter 43. Analysis of Noise: Part 4 258
REFERENCES 267
Chapter 44. Analysis of Noise: Part 5 268
ALTERNATE ANALYSIS 273
ABSORBANCE NOISE IN THE “HIGH NOISE” REGIME 281
REFERENCES 283
Chapter 45. Analysis of Noise: Part 6 286
REFERENCES 291
Chapter 46. Analysis of Noise: Part 7 292
EFFECT OF NOISE ON COMPUTED TRANSMITTANCE 294
COMPUTED TRANSMITTANCE NOISE 296
REFERENCES 298
Chapter 47. Analysis of Noise: Part 8 300
REFERENCES 307
Chapter 48. Analysis of Noise: Part 9 308
REFERENCES 313
Chapter 49. Analysis of Noise: Part 10 314
DISCUSSION 323
REFERENCES 326
Chapter 50. Analysis of Noise: Part 11 328
REFERENCES 330
Chapter 51. Analysis of Noise: Part 12 332
REFERENCES 336
Chapter 52. Analysis of Noise: Part 13 338
REFERENCES 342
Chapter 53. Analysis of Noise: Part 14 344
PRELIMINARY STEPS 347
EVALUATION OF THE FUNCTION 350
REFERENCES 352
Chapter 54. Derivatives in Spectroscopy: Part 1 – The Behavior of the Derivative 354
THE BEHAVIOR OF THEORETICAL DERIVATIVES 354
THE BEHAVIOR OF COMPUTED DERIVATIVES 359
REFERENCES 365
Chapter 55. Derivatives in Spectroscopy: Part 2 – The “True” Derivative 366
BETTER DERIVATIVE APPROXIMATIONS 367
REFERENCES 372
Chapter 56. Derivatives in Spectroscopy: Part 3 – Computing the Derivative 374
METHODS OF COMPUTING THE DERIVATIVE 375
LIMITATIONS OF THE SAVITZKY–GOLAY METHOD 378
EXTENSIONS TO THE SAVITZKY–GOLAY METHOD 380
REFERENCES 384
Chapter 57. Derivatives in Spectroscopy: Part 4 – Calibrating with Derivatives 386
ACKNOWLEDGEMENT 393
REFERENCES 393
Chapter 58. Comparison of Goodness of Fit Statistics for Linear Regression: Part 1 – Introduction 394
REFERENCE 399
Chapter 59. Comparison of Goodness of Fit Statistics for Linear Regression: Part 2 – The Correlation Coefficient 400
REFERENCES 406
Chapter 60. Comparison of Goodness of Fit Statistics for Linear Regression: Part 3 – Computing Confidence Limits for the Correlation Coefficient 408
TESTING CORRELATION FOR DIFFERENT SIZE POPULATIONS 411
REFERENCES 412
Chapter 61. Comparison of Goodness of Fit Statistics for Linear Regression: Part 4 – Confidence Limits for Slope and Intercept 414
REFERENCES 416
Supplement 417
MathCad Worksheets for Correlation, Slope and Intercept 417
REFERENCES 427
Chapter 62. Correction and Discussion Regarding Derivatives 428
REFERENCES 434
Chapter 63. Linearity in Calibration: Act III Scene I – Importance of Nonlinearity 436
WHY IS NONLINEARITY IMPORTANT? 436
REFERENCES 441
Chapter 64. Linearity in Calibration: Act III Scene II – A Discussion of the Durbin-Watson Statistic, a Step in the Right Direction 442
REFERENCES 449
Chapter 65. Linearity in Calibration: Act III Scene III – Other Tests for Nonlinearity 450
F -TEST 450
NORMALITY OF RESIDUALS 452
Chapter 66. Linearity in Calibration: Act III Scene IV – How to Test for Nonlinearity 454
CONCLUSION 460
APPENDIX A: DERIVATION AND DISCUSSION OF THE FORMULA IN EQUATION 66–11 462
REFERENCES 464
Chapter 67. Linearity in Calibration: Act III Scene V – Quantifying Nonlinearity 466
REFERENCES 473
Chapter 68. Linearity in Calibration: Act III Scene VI – Quantifying Nonlinearity, Part II, and a News Flash 474
NEWS FLASH!! 478
REFERENCES 483
Chapter 69. Connecting Chemometrics to Statistics: Part 1 – The Chemometrics Side 486
REFERENCES 490
Chapter 70. Connecting Chemometrics to Statistics: Part 2 – The Statistics Side 492
MULTIVARIATE ANOVA 492
REFERENCES 495
Chapter 71. Limitations in Analytical Accuracy: Part 1 – Horwitz’s Trumpet 496
REFERENCES 500
Chapter 72. Limitations in Analytical Accuracy: Part 2 – Theories to Describe the Limits in Analytical Accuracy 502
DETECTION LIMIT FOR CONCENTRATIONS NEAR ZERO 503
REFERENCES 504
Chapter 73. Limitations in Analytical Accuracy: Part 3 – Comparing Test Results for Analytical Uncertainty 506
UNCERTAINTY IN AN ANALYTICAL MEASUREMENT 506
COMPARISON TEST FOR A SINGLE SET OF MEASUREMENTS VERSUS A TRUE ANALYTICAL RESULT 506
COMPARISON TEST FOR A TWO SETS OF MEASUREMENTS 507
CALCULATING THE NUMBER OF MEASUREMENTS REQUIRED TO ESTABLISH A MEAN VALUE (OR ANALYTICAL RESULT) WITH A PRESCRIBED UNCERTAINTY (ACCURACY) 508
THE Q-TEST FOR OUTLIERS [1–3] 509
SUMMATION OF VARIANCE FROM SEVERAL DATA SETS 509
REFERENCES 510
Chapter 74. The Statistics of Spectral Searches 512
COMMON SPECTRAL MATCHING APPROACHES 512
MAHALANOBIS DISTANCE MEASUREMENTS 512
EUCLIDEAN DISTANCE 514
COMMON SPECTRAL MATCHING (CORRELATION OR DOT PRODUCT) 514
REFERENCES 515
Chapter 75. The Chemometrics of Imaging Spectroscopy 518
IMAGE PROJECTION OF SPECTROSCOPIC DATA 518
REFERENCES 522
Glossary of Terms 524
Index 528
COLOUR PLATE SECTION 542