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Full Description
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.
Contents
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 Kolmogorov-Smirnov 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: Additive and Scaled with Level 98
5.5 Using the Inverse CDF to Generate n and u from UID(0, 1) 99
5.6 Takeaway 100
Exercises 100
6 Steady and Transient State Detection 101
6.1 Introduction 101
6.1.1 General Applications 101
6.1.2 Concepts and Issues in Detecting Steady State 104
6.1.3 Approaches and Issues to SSID and TSID 104
6.2 Method 106
6.2.1 Conceptual Model 106
6.2.2 Equations 107
6.2.3 Coefficient, Threshold, and Sample Frequency Values 108
6.2.4 Noiseless Data 111
6.3 Applications 112
6.3.1 Applications of the R-Statistic Approach for Process Monitoring 112
6.3.2 Applications of the R-Statistic Approach for Determining Regression Convergence 112
6.4 Takeaway 114
Exercises 114
Part III REGRESSION, VALIDATION, DESIGN
7 Regression Target - Objective Function 119
7.1 Introduction 119
7.2 Experimental and Measurement Uncertainty - Static and Continuous Valued 119
7.3 Likelihood 122
7.4 Maximum Likelihood 124
7.5 Estimating σx and σy Values 127
7.6 Vertical SSD - A Limiting Consideration of Variability Only in the Response Measurement 127
7.7 r-Square as a Measure of Fit 128
7.8 Normal, Total, or Perpendicular SSD 130
7.9 Akaho's Method 132
7.10 Using a Model Inverse for Regression 134
7.11 Choosing the Dependent Variable 135
7.12 Model Prediction with Dynamic Models 136
7.13 Model Prediction with Classification Models 137
7.14 Model Prediction with Rank Models 138
7.15 Probabilistic Models 139
7.16 Stochastic Models 139
7.17 Takeaway 139
Exercises 140
8 Constraints 141
8.1 Introduction 141
8.2 Constraint Types 141
8.3 Expressing Hard Constraints in the Optimization Statement 142
8.4 Expressing Soft Constraints in the Optimization Statement 143
8.5 Equality Constraints 147
8.6 Takeaway 148
Exercises 148
9 The Distortion of Linearizing Transforms 149
9.1 Linearizing Coefficient Expression in Nonlinear Functions 149
9.2 The Associated Distortion 151
9.3 Sequential Coefficient Evaluation 154
9.4 Takeaway 155
Exercises 155
10 Optimization Algorithms 157
10.1 Introduction 157
10.2 Optimization Concepts 157
10.3 Gradient-Based Optimization 159
10.3.1 Numerical Derivative Evaluation 159
10.3.2 Steepest Descent - The Gradient 161
10.3.3 Cauchy's Method 162
10.3.4 Incremental Steepest Descent (ISD) 163
10.3.5 Newton-Raphson (NR) 163
10.3.6 Levenberg-Marquardt (LM) 165
10.3.7 Modified LM 166
10.3.8 Generalized Reduced Gradient (GRG) 167
10.3.9 Work Assessment 167
10.3.10 Successive Quadratic (SQ) 167
10.3.11 Perspective 168
10.4 Direct Search Optimizers 168
10.4.1 Cyclic Heuristic Direct Search 169
10.4.2 Multiplayer Direct Search Algorithms 170
10.4.3 Leapfrogging 171
10.5 Takeaway 173
11 Multiple Optima 176
11.1 Introduction 176
11.2 Quantifying the Probability of Finding the Global Best 178
11.3 Approaches to Find the Global Optimum 179
11.4 Best-of-N Rule for Regression Starts 180
11.5 Interpreting the CDF 182
11.6 Takeaway 184
12 Regression Convergence Criteria 185
12.1 Introduction 185
12.2 Convergence versus Stopping 185
12.3 Traditional Criteria for Claiming Convergence 186
12.4 Combining DV Influence on OF 188
12.5 Use Relative Impact as Convergence Criterion 189
12.6 Steady-State Convergence Criterion 190
12.7 Neural Network Validation 197
12.8 Takeaway 198
Exercises 198
13 Model Design - Desired and Undesired Model Characteristics and Effects 199
13.1 Introduction 199
13.2 Redundant Coefficients 199
13.3 Coefficient Correlation 201
13.4 Asymptotic and Uncertainty Effects When Model is Inverted 203
13.5 Irrelevant Coefficients 205
13.6 Poles and Sign Flips w.r.t. the DV 206
13.7 Too Many Adjustable Coefficients or Too Many Regressors 206
13.8 Irrelevant Model Coefficients 215
13.8.1 Standard Error of the Estimate 216
13.8.2 Backward Elimination 216
13.8.3 Logical Tests 216
13.8.4 Propagation of Uncertainty 216
13.8.5 Bootstrapping 217
13.9 Scale-Up or Scale-Down Transition to New Phenomena 217
13.10 Takeaway 218
Exercises 218
14 Data Pre- and Post-processing 220
14.1 Introduction 220
14.2 Pre-processing Techniques 221
14.2.1 Steady- and Transient-State Selection 221
14.2.2 Internal Consistency 221
14.2.3 Truncation 222
14.2.4 Averaging and Voting 222
14.2.5 Data Reconciliation 223
14.2.6 Real-Time Noise Filtering for Noise Reduction (MA, FoF, STF) 224
14.2.7 Real-Time Noise filtering for Outlier Removal (Median Filter) 227
14.2.8 Real-Time Noise Filtering, Statistical Process Control 228
14.2.9 Imputation of Input Data 230
14.3 Post-processing 231
14.3.1 Outliers and Rejection Criterion 231
14.3.2 Bimodal Residual Distributions 233
14.3.3 Imputation of Response Data 235
14.4 Takeaway 235
Exercises 235
15 Incremental Model Adjustment 237
15.1 Introduction 237
15.2 Choosing the Adjustable Coefficient in Phenomenological Models 238
15.3 Simple Approach 238
15.4 An Alternate Approach 240
15.5 Other Approaches 241
15.6 Takeaway 241
Exercises 241
16 Model and Experimental Validation 242
16.1 Introduction 242
16.1.1 Concepts 242
16.1.2 Deterministic Models 244
16.1.3 Stochastic Models 246
16.1.4 Reality! 249
16.2 Logic-Based Validation Criteria 250
16.3 Data-Based Validation Criteria and Statistical Tests 251
16.3.1 Continuous-Valued, Deterministic, Steady State, or End-of-Batch 251
16.3.2 Continuous-Valued, Deterministic, Transient 263
16.3.3 Class/Discrete/Rank-Valued, Deterministic, Batch, or Steady State 264
16.3.4 Continuous-Valued, Stochastic, Batch, or Steady State 265
16.3.5 Test for Normally Distributed Residuals 266
16.3.6 Experimental Procedure Validation 266
16.4 Model Discrimination 267
16.4.1 Mechanistic Models 267
16.4.2 Purely Empirical Models 268
16.5 Procedure Summary 268
16.6 Alternate Validation Approaches 269
16.7 Takeaway 270
Exercises 270
17 Model Prediction Uncertainty 272
17.1 Introduction 272
17.2 Bootstrapping 273
17.3 Takeaway 276
18 Design of Experiments for Model Development and Validation 277
18.1 Concept - Plan and Data 277
18.2 Sufficiently Small Experimental Uncertainty - Methodology 277
18.3 Screening Designs - A Good Plan for an Alternate Purpose 281
18.4 Experimental Design - A Plan for Validation and Discrimination 282
18.4.1 Continually Redesign 282
18.4.2 Experimental Plan 283
18.5 EHS&LP 286
18.6 Visual Examples of Undesired Designs 287
18.7 Example for an Experimental Plan 289
18.8 Takeaway 291
Exercises 292
19 Utility versus Perfection 293
19.1 Competing and Conflicting Measures of Excellence 293
19.2 Attributes for Model Utility Evaluation 294
19.3 Takeaway 295
Exercises 296
20 Troubleshooting 297
20.1 Introduction 297
20.2 Bimodal and Multimodal Residuals 297
20.3 Trends in the Residuals 298
20.4 Parameter Correlation 298
20.5 Convergence Criterion - Too Tight, Too Loose 299
20.6 Overfitting (Memorization) 300
20.7 Solution Procedure Encounters Execution Errors 300
20.8 Not a Sharp CDF (OF) 300
20.9 Outliers 301
20.10 Average Residual Not Zero 302
20.11 Irrelevant Model Coefficients 302
20.12 Data Work-Up after the Trials 302
20.13 Too Many rs! 303
20.14 Propagation of Uncertainty Does Not Match Residuals 303
20.15 Multiple Optima 304
20.16 Very Slow Progress 304
20.17 All Residuals are Zero 304
20.18 Takeaway 305
Exercises 305
Part IV CASE STUDIES AND DATA
21 Case Studies 309
21.1 Valve Characterization 309
21.2 CO2 Orifice Calibration 311
21.3 Enrollment Trend 312
21.4 Algae Response to Sunlight Intensity 314
21.5 Batch Reaction Kinetics 316
Appendix A: VBA Primer: Brief on VBA Programming - Excel in Office 2013 319
Appendix B: Leapfrogging Optimizer Code for Steady-State Models 328
Appendix C: Bootstrapping with Static Model 341
References and Further Reading 350
Index 355
