Statistics

Preface xiii About the Author xix Supplementary Material xxi Part 1: The Basics 1 1. Introduction 3 2. Application to Process Control 5 2.1 Benefit Estimation 5 2.2 Inferential Properties 7 2.3 Controller Performance Monitoring 7 2.4 Event Analysis 8 2.5 Time Series Analysis 9 3. Process Examples 11 3.1 Debutaniser 11 3.2 De-ethaniser 11 3.3 LPG Splitter 12 3.4 Propane Cargoes 17 3.5 Diesel Quality 17 3.6 Fuel Gas Heating Value 18 3.7 Stock Level 19 3.8 Batch Blending 22 4. Characteristics of Data 23 4.1 Data Types 23 4.2 Memory 24 4.3 Use of Historical Data 24 4.4 Central Value 25 4.5 Dispersion 32 4.6 Mode 33 4.7 Standard Deviation 35 4.8 Skewness and Kurtosis 37 4.9 Correlation 46 4.10 Data Conditioning 47 5. Probability Density Function 51 5.1 Uniform Distribution 55 5.2 Triangular Distribution 57 5.3 Normal Distribution 59 5.4 Bivariate Normal Distribution 62 5.5 Central Limit Theorem 65 5.6 Generating a Normal Distribution 69 5.7 Quantile Function 70 5.8 Location and Scale 71 5.9 Mixture Distribution 73 5.10 Combined Distribution 73 5.11 Compound Distribution 75 5.12 Generalised Distribution 75 5.13 Inverse Distribution 76 5.14 Transformed Distribution 76 5.15 Truncated Distribution 77 5.16 Rectified Distribution 78 5.17 Noncentral Distribution 78 5.18 Odds 79 5.19 Entropy 80 6. Presenting the Data 83 6.1 Box and Whisker Diagram 83 6.2 Histogram 84 6.3 Kernel Density Estimation 90 6.4 Circular Plots 95 6.5 Parallel Coordinates 97 6.6 Pie Chart 98 6.7 Quantile Plot 98 7. Sample Size 105 7.1 Mean 105 7.2 Standard Deviation 106 7.3 Skewness and Kurtosis 107 7.4 Dichotomous Data 108 7.5 Bootstrapping 110 8. Significance Testing 113 8.1 Null Hypothesis 113 8.2 Confidence Interval 116 8.3 Six-Sigma 118 8.4 Outliers 119 8.5 Repeatability 120 8.6 Reproducibility 121 8.7 Accuracy 122 8.8 Instrumentation Error 123 9. Fitting a Distribution 127 9.1 Accuracy of Mean and Standard Deviation 130 9.2 Fitting a CDF 131 9.3 Fitting a QF 134 9.4 Fitting a PDF 135 9.5 Fitting to a Histogram 138 9.6 Choice of Penalty Function 141 10. Distribution of Dependent Variables 147 10.1 Addition and Subtraction 147 10.2 Division and Multiplication 148 10.3 Reciprocal 153 10.4 Logarithmic and Exponential Functions 153 10.5 Root Mean Square 162 10.6 Trigonometric Functions 164 11. Commonly Used Functions 165 11.1 Euler's Number 165 11.2 Euler-Mascheroni Constant 166 11.3 Logit Function 166 11.4 Logistic Function 167 11.5 Gamma Function 168 11.6 Beta Function 174 11.7 Pochhammer Symbol 174 11.8 Bessel Function 176 11.9 Marcum Q-Function 178 11.10 Riemann Zeta Function 180 11.11 Harmonic Number 180 11.12 Stirling Approximation 182 11.13 Derivatives 183 12. Selected Distributions 185 12.1 Lognormal 186 12.2 Burr 189 12.3 Beta 191 12.4 Hosking 195 12.5 Student t 204 12.6 Fisher 208 12.7 Exponential 210 12.8 Weibull 213 12.9 Chi-Squared 216 12.10 Gamma 221 12.11 Binomial 225 12.12 Poisson 231 13. Extreme Value Analysis 235 14. Hazard Function 245 15. CUSUM 253 16. Regression Analysis 259 16.1 F Test 275 16.2 Adjusted R2 278 16.3 Akaike Information Criterion 279 16.4 Artificial Neural Networks 281 16.5 Performance Index 286 17. Autocorrelation 291 18. Data Reconciliation 299 19. Fourier Transform 305 Part 2: Catalogue of Distributions 315 20. Normal Distribution 317 20.1 Skew-Normal 317 20.2 Gibrat 320 20.3 Power Lognormal 320 20.4 Logit-Normal 321 20.5 Folded Normal 321 20.6 Lévy 323 20.7 Inverse Gaussian 325 20.8 Generalised Inverse Gaussian 329 20.9 Normal Inverse Gaussian 330 20.10 Reciprocal Inverse Gaussian 332 20.11 Q-Gaussian 334 20.12 Generalised Normal 338 20.13 Exponentially Modified Gaussian 345 20.14 Moyal 347 21. Burr Distribution 349 21.1 Type I 349 21.2 Type II 349 21.3 Type III 349 21.4 Type IV 350 21.5 Type V 351 21.6 Type VI 351 21.7 Type VII 353 21.8 Type VIII 354 21.9 Type IX 354 21.10 Type X 355 21.11 Type XI 356 21.12 Type XII 356 21.13 Inverse 357 22. Logistic Distribution 361 22.1 Logistic 361 22.2 Half-Logistic 364 22.3 Skew-Logistic 365 22.4 Log-Logistic 367 22.5 Paralogistic 369 22.6 Inverse Paralogistic 370 22.7 Generalised Logistic 371 22.8 Generalised Log-Logistic 375 22.9 Exponentiated Kumaraswamy-Dagum 376 23. Pareto Distribution 377 23.1 Pareto Type I 377 23.2 Bounded Pareto Type I 378 23.3 Pareto Type II 379 23.4 Lomax 381 23.5 Inverse Pareto 381 23.6 Pareto Type III 382 23.7 Pareto Type IV 383 23.8 Generalised Pareto 383 23.9 Pareto Principle 385 24. Stoppa Distribution 389 24.1 Type I 389 24.2 Type II 389 24.3 Type III 391 24.4 Type IV 391 24.5 Type V 392 25. Beta Distribution 393 25.1 Arcsine 393 25.2 Wigner Semicircle 394 25.3 Balding-Nichols 395 25.4 Generalised Beta 396 25.5 Beta Type II 396 25.6 Generalised Beta Prime 399 25.7 Beta Type IV 400 25.8 PERT 401 25.9 Beta Rectangular 403 25.10 Kumaraswamy 404 25.11 Noncentral Beta 407 26. Johnson Distribution 409 26.1 SN 409 26.2 SU 410 26.3 SL 412 26.4 SB 412 26.5 Summary 413 27. Pearson Distribution 415 27.1 Type I 416 27.2 Type II 416 27.3 Type III 417 27.4 Type IV 418 27.5 Type V 424 27.6 Type VI 425 27.7 Type VII 429 27.8 Type VIII 433 27.9 Type IX 433 27.10 Type X 433 27.11 Type XI 434 27.12 Type XII 434 28. Exponential Distribution 435 28.1 Generalised Exponential 435 28.2 Gompertz-Verhulst 435 28.3 Hyperexponential 436 28.4 Hypoexponential 437 28.5 Double Exponential 438 28.6 Inverse Exponential 439 28.7 Maxwell-Jüttner 439 28.8 Stretched Exponential 440 28.9 Exponential Logarithmic 441 28.10 Logistic Exponential 442 28.11 Q-Exponential 442 28.12 Benktander 445 29. Weibull Distribution 447 29.1 Nukiyama-Tanasawa 447 29.2 Q-Weibull 447 30. Chi Distribution 451 30.1 Half-Normal 451 30.2 Rayleigh 452 30.3 Inverse Rayleigh 454 30.4 Maxwell 454 30.5 Inverse Chi 458 30.6 Inverse Chi-Squared 459 30.7 Noncentral Chi-Squared 460 31. Gamma Distribution 463 31.1 Inverse Gamma 463 31.2 Log-Gamma 463 31.3 Generalised Gamma 467 31.4 Q-Gamma 468 32. Symmetrical Distributions 471 32.1 Anglit 471 32.2 Bates 472 32.3 Irwin-Hall 473 32.4 Hyperbolic Secant 475 32.5 Arctangent 476 32.6 Kappa 477 32.7 Laplace 478 32.8 Raised Cosine 479 32.9 Cardioid 481 32.10 Slash 481 32.11 Tukey Lambda 483 32.12 Von Mises 486 33. Asymmetrical Distributions 487 33.1 Benini 487 33.2 Birnbaum-Saunders 488 33.3 Bradford 490 33.4 Champernowne 491 33.5 Davis 492 33.6 Fréchet 494 33.7 Gompertz 496 33.8 Shifted Gompertz 497 33.9 Gompertz-Makeham 498 33.10 Gamma-Gompertz 499 33.11 Hyperbolic 499 33.12 Asymmetric Laplace 502 33.13 Log-Laplace 504 33.14 Lindley 506 33.15 Lindley-Geometric 507 33.16 Generalised Lindley 509 33.17 Mielke 509 33.18 Muth 510 33.19 Nakagami 512 33.20 Power 513 33.21 Two-Sided Power 514 33.22 Exponential Power 516 33.23 Rician 517 33.24 Topp-Leone 517 33.25 Generalised Tukey Lambda 519 33.26 Wakeby 521 34. Amoroso Distribution 525 35. Binomial Distribution 529 35.1 Negative-Binomial 529 35.2 Pulya 531 35.3 Geometric 531 35.4 Beta-Geometric 535 35.5 Yule-Simon 536 35.6 Beta-Binomial 538 35.7 Beta-Negative Binomial 540 35.8 Beta-Pascal 541 35.9 Gamma-Poisson 542 35.10 Conway-Maxwell-Poisson 543 35.11 Skellam 546 36. Other Discrete Distributions 549 36.1 Benford 549 36.2 Borel-Tanner 552 36.3 Consul 555 36.4 Delaporte 556 36.5 Flory-Schulz 558 36.6 Hypergeometric 559 36.7 Negative Hypergeometric 561 36.8 Logarithmic 561 36.9 Discrete Weibull 563 36.10 Zeta 564 36.11 Zipf 565 36.12 Parabolic Fractal 567 Appendix 1 Data Used in Examples 569 Appendix 2 Summary of Distributions 577 References 591 Index 593