By Georgi Borisov Damyanov and Diana Stoyanova
CONTENTS
PREFACE xi
ABOUT THE AUTHORS xiii
LIST OF FIGURES xv
LIST OF TABLES xvii
PART I: INTRODUCTION TO MATHEMATICAL STATISTICS 1
I.1. GENERAL TERMS AND DEFINITIONS 3
Parameters and numerical characteristics of the random variable 6
Characteristics for location 6
Dispersion characteristics 7
Moments 9
Properties of numerical characteristics 11
I.2. LAWS OF RANDOM VARIABLES DISTRIBUTION 13
Continuous distributions 13
Discrete distributions 16
I.3. STATISTICAL ESTIMATES 21
Conducting the test 21
Point estimates 24
Interval estimates 25
Confidence intervals 28
Confidence intervals of the estimates in cases of normal distribution 28
Confidence interval of the l parameter for Poisson distribution 30
Confidence interval of the parameter p for binomial distribution 31
I.4. STATISTICAL PROCESS CONTROL AND CONTROL CHARTS 33
Statistical process control 33
Control charts 36
Design of statistical control charts 37
Types of control charts 38
I.5. CORRELATION ANALYSIS 65
Coeffi cient of linear correlation 66
Coeffi cient of determination, anticorrelation, and indeterminate coefficients 66
Correlation in case of alternative indicators—the four-fi eld method 68
Multiple and partial correlation 70
I.6. ANALYSIS OF VARIANCE 73
Single-factor analysis of variance 73
Multifactor ANOVA 75
PART II: DESIGN OF AN EXPERIMENT 79
II.1. MAIN CONCEPTS IN MATHEMATICAL MODELING AND OPTIMIZATION 81
II.2. CHOICE OF PARAMETERS OF OPTIMIZATION 85
II.3. CHOICE OF INPUT FACTORS 89
Methods of rank correlation 93
Coeffi cient of rank correlation 94
Coeffi cient of concordance 98
Random balance method 103
Design of the experiment 103
Construction of a diagram of dispersion 104
Separation of essential factors 107
II.4. MAIN STAGES OF EXPERIMENTAL MODELING 109
II.5. REGRESSION ANALYSIS 113
II.6. FULL FACTORIAL EXPERIMENT 123
Properties of the extended matrix of FFE 124
Types of matrices 125
Stages of derivation of the model 125
Calculation of coeffi cients 129
Verifi cation of reproducibility of the process 129
Calculation of the test variance 130
Determination of the variance of regression coeffi cients 130
Verifi cation of the signifi cance of regression coeffi cients 130
Registration of the derived model 130
Calculation of the values of the output variable on the model 130
Verifi cation of the model adequacy 130
II.7. FRACTIONAL FACTORIAL EXPERIMENT 133
Stages of construction of a fractional factorial experiment 134
Determination of the minimum number of tests for deriving a linear model 135
Choice of main and additional factors 135
Composing the design of the experiment 135
Setting the determining contrasts 136
Setting the generalized determining contrast 136
Mixing the coeffi cients 137
II.8. STATISTICAL METHODS FOR MOVEMENT TO AN OPTIMAL AREA 139
Box–Wilson method 139
Condition for application of the method 140
Method principle 140
Method application conditions 141
Disadvantages of the method 141
Application 141
Simplex method 147
Simplex property 147
Criterion for reaching the optimal area 148
Specifying the optimal area 148
Construction of the initial simplex 148
Calculating the values of the coordinate points 150
Calculating the coordinates of the mirrored point apex 150
Determining the coordinates of the starting points 151
Filling in the simplex table 151
Determining the coordinates of an additional point 152
Formation of a new simplex 152
II.9. INVESTIGATION OF THE OPTIMUM AREA: COMPOSITE DESIGNS OF SECOND ORDER 153
Orthogonal central composite experiment 154
Number of tests 154
Design of the experiment 154
Determination of regression coeffi cients 155
Determination of regression coeffi cients’ variances 156
Signifi cance of coeffi cients of regression equation 156
Recording of regression equation 157
Verifi cation of adequacy of the model 157
Determination of the number of tests 158
Determination of the size of the star arm, a, and the value, k 158
Plan of the experiment 158
Determination of the regression coeffi cients 158
Determination of the variances of the regression coeffi cients 160
Signifi cance of the coeffi cients of the regression equation 161
Record of the regression equation 162
Verifi cation of the model adequacy 162
Rotatable central composite experiment 163
Number of tests 163
Design of the experiment 164
Determination of regression coefficients 165
Variance of regression coeffi cients 165
Signifi cance of coeffi cients of regression equation 166
Verifi cation of adequacy of the model 166
Determination of regression coeffi cients 167
Determination of variance of reproducibility 169
Determination of variances of regression coeffi cients 169
Verifi cation of signifi cance of regression coeffi cients 170
Recording of the model 171
Verifi cation of model adequacy 171
Optimal composite experiment 171
Number of tests 171
Design of the experiment 172
Determination of regression coeffi cients 172
Variance of regression coeffi cients 173
Signifi cance of coeffi cients of regression equation 173
Verifi cation of adequacy 173
Determination of the regression coeffi cients 174
Determination of the variances of the regression coeffi cients 176
Verifi cation of the signifi cance of the coeffi cients of the regression equation 176
Record of the model 177
Verifi cation of the model adequacy 177
II.10. OPTIMIZATION OF TARGET FUNCTION 179
Canonical analysis 180
Algorithm for reduction to a canonical form 180
Determination of the surface type 181
Differentiation of target function 184
Solving the system of linear equations 185
Determination of the extreme value of output parameter 185
Determination of the rotation angle 185
Determination of regression coeffi cients in canonical equation 185
Additional verifi cation of calculation correctness 186
Determination of surface type 186
Determination of the optimal parameters of the cylinder drawing device 187
II.11. TAGUCHI METHODS 189
Innovations in the sphere of the designed experiment 189
Off-line methods of control 190
System design 190
Parameter design 195
Design of the parameter tolerances (tolerance plan) 202
Loss function 202
Application of the function 204
APPENDIX 1: STUDENT’S T-DISTRIBUTION 209
APPENDIX 2: CHI-SQUARED ÷2-DISTRIBUTION 211
APPENDIX 3: FISHER’S F-DISTRIBUTION a = 0.05 213
APPENDIX 4: FISHER’S F-DISTRIBUTION a = 0.01 217
APPENDIX 5: COCHRAN’S CRITICAL VALUES 221
APPENDIX 6: DENSITY OF NORMAL DISTRIBUTION N(0, 1) 225
APPENDIX 7: PROBABILITY P FOR Sd ≤ Sdo 227
APPENDIX 8: RANDOM NUMBERS 229
BIBLIOGRAPHY 231
INDEX 235