**Textile Processes: Quality Control and Design of Experiments**

By Georgi Borisov Damyanov and Diana Stoyanova

**CONTENTS**

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

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