A STATISTICAL MANUAL FOR CHEMISTS
CONTENTS CHAPTER 1 GUIDELINES TO INSTALL THE SOFTWARE 1 OBJECTIVES AND SCOPE 1 1 1 SOME GENERAL COMMENTS ABOUT CAT 1 CHAPTER 2 A SUMMARY OF OPERATIONS AND COMMON MATRIX DECOMPOSITIONS IN CHEMOMETRICS 7 OBJECTIVES AND SCOPE 7 2 1 MATRICES 8 2 1 1 MATRICES AND VECTORS 8 2 1 2 MATRIX OPERATIONS 9 2 1 3 MATRIX RANK 16 2 2 INVERSION OF MATRICES 17 2 2 1 SQUARE MATRICES AND INVERSES 18 2 2 2 SYMMETRIC MATRICES AND DISTANCES 20 2 2 3 GENERAL MATRICES AND PSEUDOINVERSES 21 2 3 MATRIX DECOMPOSITION AND DIAGONALISATION 24 2 3 1 EIGENANALYSIS EIGENVECTOR EIGENVALUE PROBLEM 24 2 3 2 SINGULAR VALUE DECOMPOSITION 26 2 4 HIGHER DIMENSIONAL ARRAYS 28 REFERENCES 29 CHAPTER 3 THE BASICS OF UNIVARIATE AND MULTIVARIATE CALIBRATION 31 OBJECTIVES AND SCOPE 31 3 1 INTRODUCTION EPISTEMOLOGICAL PRINCIPLES 32 3 2 UNDERSTANDING THE JARGON CHEMICAL CALIBRATION OR STANDARDISATION 34 3 3 THE LINEAR MODEL IN THE UNIVARIATE CASE 39 3 4 MULTIVARIATE LINEAR REGRESSION BY THE ORDINARY MULTIPLE LINEAR REGRESSION METHOD 48 3 5 INVERSE MULTIVARIATE LINEAR REGRESSION FOR SOLVING SOME ORDINARY MULTIVARIATE LINEAR REGRESSION PROBLEMS 55 3 6 VALIDATING THE MODEL 61 3 7 A PRACTICAL EXAMPLE 65 3 7 1 MODEL DEVELOPMENT STANDARDISATION CALIBRATION 66 3 7 2 MODEL VALIDATION 71 REFERENCES 74 CHAPTER 4 STATISTICAL EXPERIMENTAL DESIGN 77 OBJECTIVES AND SCOPE 77 4 1 INTRODUCTION 77 4 2 THE CHEMICAL PROBLEM 78 4 2 1 COMPLEXITY 79 4 2 2 EXPERIMENTAL ERROR 80 4 3 WHY MULTIVARIATE 81 4 4 MATHEMATICS BEHIND THE EXPERIMENTAL DESIGNS 87 4 4 1 MODEL VALIDATION 93 4 4 2 QUESTIONS BEFORE STATISTICS 98 4 5 SCREENING IMPROVEMENT ROBUSTNESS DESIGNS 99 4 5 1 PLACKETT BURMAN DESIGNS 100 4 5 2 FULL FACTORIAL DESIGNS 105 4 6 OPTIMISATION DESIGNS 112 4 6 1 COMPOSITE DESIGNS 112 4 6 2 MULTICRITERIA DECISION MAKING PARETO OPTIMALITY 119 4 7 DESIGNS FOR QUALITATIVE VARIABLES WITH MORE THAN TWO LEVELS 123 4 8 MIXTURE DESIGNS 131 4 8 1 MIXTURE PROCESS DESIGNS 140 4 9 OPTIMAL DESIGNS 142 4 10 BRIDGE BETWEEN EXPERIMENTAL DESIGN AND MULTIVARIATE ANALYSIS 146 REFERENCES 148 CHAPTER 5 DATA PRE PROCESSING THE IMPORTANCE OF PROPERLY TRANSFORMING YOUR DATA BEFORE FURTHER INSIGHTS 151 OBJECTIVES AND SCOPE 151 5 1 INTRODUCTION 151 5 2 COLUMN SCALING 157 5 2 1 COLUMN CENTRING 157 5 2 2 AUTOSCALING 159 5 2 3 RANGE SCALING 160 5 3 ROW SCALING 160 5 3 1 ROW AUTOSCALING 161 5 3 2 MULTIPLICATIVE SCATTER CORRECTION 163 5 3 3 DETRENDING 164 5 3 4 SMOOTHING 166 5 3 5 DERIVATIVES 169 5 4 THE IMPACT OF SIGNAL PREPROCESSING 172 5 5 PREPROCESSING FUSED DATA 175 5 6 WORKED EXAMPLE 176 APPENDIX 179 REFERENCES 182 FURTHER READING 183 CHAPTER 6 CLASSI CATION 185 OBJECTIVES AND SCOPE 185 6 1 INTRODUCTION 186 6 1 1 MOTIVATIONS 186 6 1 2 CASE STUDY 189 6 1 3 PREPROCESSING AND EXPLORATORY DATA ANALYSIS 189 6 2 TWO CLASS CLASSI ERS 197 6 2 1 HARD BOUNDARIES 197 6 2 2 LINEAR DISCRIMINANT ANALYSIS 198 6 3 ONE CLASS CLASSI ERS 205 6 3 1 SOFT BOUNDARIES 205 6 3 2 QUADRATIC DISCRIMINANT ANALYSIS 207 6 4 VALIDATION 216 6 5 DISCRIMINATORY VARIABLES 224 6 6 A REAL CASE STUDY 228 6 7 CONCLUSION 233 BIBLIOGRAPHY 234 CHAPTER 7 EXPLORING MULTIVARIATE DATA MODELS BASED ON LATENT VARIABLES 235 OBJECTIVES AND SCOPE 235 7 1 TWO WAY MODELS PCA 236 7 1 1 FUNDAMENTALS OF PRINCIPAL COMPONENT ANALYSIS 236 7 1 2 DECOMPOSITION OF THE SUM OF SQUARES 243 7 1 3 BILINEAR MODEL 245 7 1 4 ALGEBRAIC SOLUTION 246 7 1 5 ALGORITHMIC IMPLEMENTATION 246 7 1 6 SOME COMMENTS ABOUT AUTOSCALING IN PCA 247 7 1 7 HOW MANY PRINCIPAL COMPONENTS 250 7 1 8 INDETERMINACY AND ROTATION IN PCA 255 7 1 9 PROJECTION SPACE AND ITS BOUNDARIES Q AND T 2 STATISTICS 257 7 1 10 WORKED EXAMPLES AND CASE STUDIES OF PCA 260 7 1 10 1 A MIXTURE OF TWO FOOD COLOURANTS 260 7 1 10 2 FRAUD DETECTION 266 7 1 10 3 VINEGARS 270 7 2 THREE WAY MODELS PARAFAC 274 7 2 1 FUNDAMENTALS OF PARAFAC 275 7 2 2 ANALYSIS OF A PARAFAC MODEL 278 7 2 3 COMPATIBILITY OF A PARAFAC MODEL TRILINEARITY 280 7 2 4 WORKED EXAMPLES 281 7 2 4 1 ENRO OXACIN IN WATER FROM POULTRY FARMS 281 7 2 4 2 SOLVING OVERESTIMATION IN MIGRATION OF MELAMINE FROM FOOD CONTACT MATERIALS 285 7 3 SOME OTHER METHODS FOR ANALYSING DATA MATRICES OR CUBES 289 7 3 1 ROBUST PCA AND PARAFAC 290 7 3 2 METHODS FOR TWO WAY DATA MATRICES 290 7 3 3 METHODS FOR THREE WAY DATA CUBES 294 7 4 LESSONS LEARNT 295 REFERENCES 297