Nonlinear Regression Modeling for Engineering Applications

Since mathematical models express our understanding of how nature behaves, we use them to validate our understanding of the fundamentals about systems (which could be processes, equipment, procedures, devices, or products). Also, when validated, the model is useful for engineering applications related to diagnosis, design, and optimization. First, we postulate a mechanism, then derive a model grounded in that mechanistic understanding. If the model does not fit the data, our understanding of the mechanism was wrong or incomplete. Patterns in the residuals can guide model improvement. Alternately, when the model fits the data, our understanding is sufficient and confidently functional for engineering applications. This book details methods of nonlinear regression, computational algorithms,model validation, interpretation of residuals, and useful experimental design. The focus is on practical applications, with relevant methods supported by fundamental analysis. This book will assist either the academic or industrial practitioner to properly classify the system, choose between the various available modeling options and regression objectives, design experiments to obtain data capturing critical system behaviors, fit the model parameters based on that data, and statistically characterize the resulting model. The author has used the material in the undergraduate unit operations lab course and in advanced control applications.

Auteur

R. Russell Rhinehart, Oklahoma State University, USA.
Professor Rhinehart obtained his Ph.D. in Chemical Engineering in 1985 from North Carolina State University, USA. His research interests include process improvement (modeling, optimization, and control), and product improvement (modeling and design). In 2004 he was named as one of InTECHs 50 most influential industry innovators of the past 50 years, and was inducted into the Automation Hall of Fame for the Process Industries in 2005. He has written extensively for numerous journals and refereed articles.

Contenu

Series Preface xiii

Preface xv

Acknowledgments xxiii

Nomenclature xxv

Symbols xxxvii

Part I INTRODUCTION

1 Introductory Concepts 3

1.1 Illustrative Example Traditional Linear Least-Squares Regression 3

1.2 How Models Are Used 7

1.3 Nonlinear Regression 7

1.4 Variable Types 8

1.5 Simulation 12

1.6 Issues 13

1.7 Takeaway 15

Exercises 15

2 Model Types 16

2.1 Model Terminology 16

2.2 A Classification of Mathematical Model Types 17

2.3 Steady-State and Dynamic Models 21

2.3.1 Steady-State Models 22

2.3.2 Dynamic Models (Time-Dependent, Transient) 24

2.4 Pseudo-First Principles Appropriated First Principles 26

2.5 Pseudo-First Principles Pseudo-Components 28

2.6 Empirical Models with Theoretical Grounding 28

2.6.1 Empirical Steady State 28

2.6.2 Empirical Time-Dependent 30

2.7 Empirical Models with No Theoretical Grounding 31

2.8 Partitioned Models 31

2.9 Empirical or Phenomenological? 32

2.10 Ensemble Models 32

2.11 Simulators 33

2.12 Stochastic and Probabilistic Models 33

2.13 Linearity 34

2.14 Discrete or Continuous 36

2.15 Constraints 36

2.16 Model Design (Architecture, Functionality, Structure) 37

2.17 Takeaway 37

Exercises 37

Part II PREPARATION FOR UNDERLYING SKILLS

3 Propagation of Uncertainty 43

3.1 Introduction 43

3.2 Sources of Error and Uncertainty 44

3.2.1 Estimation 45

3.2.2 Discrimination 45

3.2.3 Calibration Drift 45

3.2.4 Accuracy 45

3.2.5 Technique 46

3.2.6 Constants and Data 46

3.2.7 Noise 46

3.2.8 Model and Equations 46

3.2.9 Humans 47

3.3 Significant Digits 47

3.4 Rounding Off 48

3.5 Estimating Uncertainty on Values 49

3.5.1 Caution 50

3.6 Propagation of Uncertainty Overview Two Types, Two Ways Each 51

3.6.1 Maximum Uncertainty 51

3.6.2 Probable Uncertainty 56

3.6.3 Generality 58

3.7 Which to Report? Maximum or Probable Uncertainty 59

3.8 Bootstrapping 59

3.9 Bias and Precision 61

3.10 Takeaway 65

Exercises 66

4 Essential Probability and Statistics 67

4.1 Variation and Its Role in Topics 67

4.2 Histogram and Its PDF and CDF Views 67

4.3 Constructing a Data-Based View of PDF and CDF 70

4.4 Parameters that Characterize the Distribution 71

4.5 Some Representative Distributions 72

4.5.1 Gaussian Distribution 72

4.5.2 Log-Normal Distribution 72

4.5.3 Logistic Distribution 74

4.5.4 Exponential Distribution 74

4.5.5 Binomial Distribution 75

4.6 Confidence Interval 76

4.7 Central Limit Theorem 77

4.8 Hypothesis and Testing 78

4.9 Type I and Type II Errors, Alpha and Beta 80

4.10 Essential Statistics for This Text 82

4.10.1 t-Test for Bias 83

4.10.2 Wilcoxon Signed Rank Test for Bias 83

4.10.3 r-lag-1 Autocorrelation Test 84

4.10.4 Runs Test 87

4.10.5 Test for Steady State in a Noisy Signal 87

4.10.6 Chi-Square Contingency Test 89

4.10.7 KolmogorovSmirnov Distribution Test 89

4.10.8 Test for Proportion 90

4.10.9 F-Test for Equal Variance 90

4.11 Takeaway 91

Exercises 91

5 Simulation 93

5.1 Introduction 93

5.2 Three Sources of Deviation: Measurement, Inputs, Coefficients 93

5.3 Two Types of Perturbations: Noise (Independent) and Drifts (Persistence) 95 5.4 Two Types of Influence:...