This book introduces a new approach, denoted RMM, for an empirical modeling of a response variation, relating to both systematic variation and random variation. In the book, the developer of RMM discusses the required properties of empirical modeling and evaluates how current approaches conform to these requirements. In addition, he explains the motivation for the development of the new methodology, introduces in detail the new approach and its estimation procedures, and shows how it may provide an excellent alternative to current approaches for empirical modeling (like Generalized Linear Modeling, GLM). The book also demonstrates that a myriad of current relational models, developed independently in various engineering and scientific disciplines, are in fact special cases of the RMM model, and so are many current statistical distributions, transformations and approximations.
Author(s): Haim Shore
Year: 2005
Language: English
Pages: 460
Contents......Page 14
1 Introduction......Page 24
References......Page 35
2.1. Introduction......Page 38
2.2. Chemistry and Chemical Engineering......Page 40
2.3. Physics......Page 42
2.5. Hardware Reliability Engineering......Page 45
2.6. Software Reliability-Growth Modeling......Page 46
References......Page 48
3.1. Introduction......Page 52
Example 1 (Chemistry)......Page 53
Example 3 (Electrical Engineering)......Page 54
OBSERVATION B......Page 55
3.3. "The Ladder of Fundamental Uniformly Convex/Concave Function"......Page 57
4.1. Introduction......Page 60
4.2. Linear Regression Analysis......Page 62
4.3. Box-Cox Power Transformations......Page 64
4.4. Generalized Linear Models......Page 67
4.5. Conclusions......Page 72
References......Page 74
5.1. Introduction......Page 76
(2) Modeling via a parameter-rich family of distributions......Page 78
(4) Heuristic methods......Page 79
5.2. Parameter-Rich Families of Distributions, Transformations and Expansions......Page 80
5.2.1. The Pearson family of distributions......Page 81
5.2.2. Other families of distributions (Burr, Tukey's g- and h-systems, generalized Lambda, Shore, the exponential family)......Page 82
5.2.3. Transformations (Johnson, Box-Cox) and expansions......Page 86
5.3.1. Why moment matching?......Page 88
5.3.2. How many moments to match......Page 90
5.4. Heuristic Methods in Empirical Modeling of Random Variation......Page 93
5.5. An Alternative Approach to Four-Moment Matching......Page 96
References......Page 97
6.1. Introduction......Page 102
6.2. Desirable Requirements of a General Methodology for Empirical Modeling......Page 104
Requirement 1: Provide monotone convex/concave relationship in modeling systematic variation......Page 105
Requirement 2: The effects to include in the LP and the structure of the model are part and parcel of the empirical modeling process......Page 106
Requirement 4: A dual-error structure......Page 107
Requirement 5: Modeling systematic variation that spans several orders of magnitude should allow the allied error-distribution to change in a major way. In particular, the modeling methodology should allow for asymptotic normality and the resulting decoupling of the variance from the mean......Page 109
Requirement 6: The model's error distribution needs to maintain a degree of flexibility, which would allow it to preserve some of the actual moments (preferably the first three or four) of the modeled distribution. The allied estimation procedures should also ensure preservation of moments......Page 110
Requirement 7: Provide good coverage of the ( 1, 2) plane......Page 111
Requirement 9: Compatibility with current proven-effective methodologies for empirical modeling......Page 113
6.3.1. Modeling systematic variation......Page 114
Linear Regression Analysis......Page 115
Data Transformation......Page 116
Generalized Linear Models (GLM)......Page 117
6.3.2. Modeling random variation......Page 118
References......Page 120
7.1. Introduction......Page 124
7.2.1. The model assumptions......Page 126
7.2.2. The general model......Page 127
7.2.3. Deriving f2......Page 128
7.2.4. Deriving f1......Page 129
7.2.5. The RMM Model......Page 130
7.3. The Response Moments......Page 133
7.4. Exploring the Relationship between the CV and......Page 136
References......Page 138
8.1. Introduction......Page 140
8.2.1. Introduction and motivation......Page 143
8.2.2. Stage I - Approximating a transformed response via a Taylor series expansion and estimating the parameters via CCA......Page 146
(A) "How many terms to keep in the Taylor approximation for the transformed response?"......Page 148
(C) "What are the assumptions and are they valid"?......Page 151
(D) "Is there a single correct LP"?......Page 153
(F) "Handling observational data vs. data from designed experiments"......Page 154
8.4.1. Introduction......Page 156
8.4.2. Stage I - Estimating the RMM parameters { , , 2, }......Page 157
8.4.3. Stage II - Estimating the RMM "Error Parameters" { , 1, 2}......Page 160
8.4.4. Summary of the estimation procedure (Phase 2)......Page 161
8.5.1. Example 1 - The Wave-Soldering Process......Page 162
8.5.2. Example 2 - The Resistivity Data......Page 167
Appendix A - Canonical Correlation Analysis - Background......Page 170
(4) Ill-Conditioned Correlation Matrix......Page 172
9.1. Introduction......Page 174
9.2. Derivation of the RMM Error Distribution......Page 175
9.3. Properties of the Error Distribution......Page 177
9.4. Variations of the RMM Error Distribution......Page 182
References......Page 183
10.1. Introduction......Page 184
10.2. Brief Review of Current Methodologies......Page 186
An Example......Page 188
10.3. Fitting via "Moment Matching"......Page 189
10.4. Fitting via "Quantile Matching"......Page 194
10.5.1. A moment-matching example......Page 196
10.5.2. A quantile-matching example......Page 197
References......Page 199
11.1. Introduction......Page 200
11.2.1. The estimation procedure......Page 203
Calculating a percentile by "The weighted average at Y(n+1)p"......Page 204
Method B. Estimating the CDF values associated with the sample order statistics......Page 205
A detailed numerical example (to prepare a sample for NL-LS)......Page 206
Example 1. Birth weights of twins - The Indiana Twin Study......Page 207
Example 2. Distribution of Intra-Galactic velocities......Page 211
11.3.1. Introduction......Page 212
11.3.2. Procedure I......Page 217
11.3.3. Procedure II......Page 219
Example 1. Birth weights of twins - The Indiana Twin Study......Page 221
Procedure I......Page 222
Procedure II......Page 223
References......Page 224
12.1. Current Relational Models as Special Cases of RMM......Page 226
12.1.1. Chemistry and Chemical Engineering......Page 227
12.1.2. Physics......Page 230
12.1.3. Electrical engineering......Page 231
12.2.1. The Johnson families of distributions......Page 232
12.2.2. Tukey g- and h-Systems of distributions......Page 233
12.2.3. Fisher's transformation of the sample correlation......Page 234
12.2.4. Haldane power-transformation and Wilson-Hilferty approximation to 2......Page 235
12.2.5. Box-Cox normalizing transformation......Page 236
12.2.7. Generalized Inverse Gaussian distribution and the Levy distribution......Page 237
12.2.8. Generalized Gamma distributions......Page 239
References......Page 240
13.2. Compliance for Modeling Systematic Variation......Page 242
13.3. Compliance in Modeling Random Variation......Page 246
References......Page 249
14.1. Introduction......Page 252
14.2.1. Example 1 - The Windshield Experiment......Page 253
14.2.2. Example 2 - The Economist Big Mac Parity Index......Page 258
14.3.1. Example 3 - The Wave-Soldering Process......Page 264
14.3.2. Example 4 - The Resistivity data......Page 267
14.4.1. Mallow's Cp......Page 269
14.4.2. Akaike's Information Criterion (AIC)......Page 270
References......Page 271
15.1. Introduction......Page 274
15.2. RMM Estimating with Censored Data......Page 275
Comments......Page 279
15.3. A Numerical Example - The RFL model......Page 280
References......Page 286
16.1. Introduction......Page 288
16.2. Example 1 - Musa's M1 Data-Set......Page 289
16.3. Example 2 - Musa's M3 Data-Set......Page 294
References......Page 299
17.1. Introduction......Page 300
17.2.1. Example 1 - Temperature dependence of vapor pressure......Page 301
17.2.2. Example 2 - Temperature dependence of solid heat capacity......Page 307
17.3. Applying RMM to a Chemo-Response- Second Variation......Page 310
17.3.1. Example 1 - Temperature dependence of vapor pressure......Page 311
17.3.2. Example 2 - Heat capacity of solids and liquids......Page 313
17.3.3. Other temperature-dependent properties......Page 316
References......Page 317
18.1. Introduction......Page 320
18.2. Theoretical Background for S-shaped Diffusion Processes......Page 321
18.3. Modeling and Forecasting S-shaped Processes......Page 326
Procedure I: For a Given Time Series {Pt}- Model and Forecase TP in Terms of a specified P......Page 328
(A) Fitting the model......Page 329
18.4.1. Forecasting TP, given P......Page 330
References......Page 333
Appendix A. Current Forecasting Models......Page 334
Appendix B. Description of Data Sets......Page 335
19.1. Introduction......Page 340
19.2. Fitting RMM with Normal or Log-normal Errors......Page 341
19.3. Fitting with a Logistic Error Term......Page 345
19.4. Approximations for the Normal and the Poisson Distributions......Page 346
19.4.1. Approximating the Poisson quantile......Page 347
19.4.2. Approximating the CDF of the Standard Normal......Page 348
References......Page 357
20.1. Introduction......Page 358
20.2. Derivation of the "Origin" INT......Page 359
20.3. Four-Moment Matching - The Problem and a Solution......Page 360
20.4.1. Off-spring INT I......Page 365
20.4.3. Off-spring INT III......Page 366
20.5. Distribution Fitting Procedures......Page 368
20.5.2. A fitting procedure for INT II (Section 20.4.2)......Page 369
20.5.3. A fitting procedure for INT III (Section 20.4.3)......Page 370
(II) Fitting by the median (M), and the means of W and Y......Page 371
20.6. Normalizing Transformations......Page 373
References......Page 374
21.1. Introduction......Page 376
21.2. The Basic Modified (Normal) Approximation......Page 379
21.3. A Variation of the Basic Model with a Fitting Procedure......Page 380
Case 2. Z is Standard Logistic......Page 382
Case 3. Shore's Approximations to the Standard Normal Inverse Distribution Function......Page 383
21.4. A Simplified Fitting Procedure......Page 384
Case 3. Shore's Approximations to the Standard Normal......Page 385
21.5. A Fitting Procedure Using First- and Second-Degree Moments......Page 386
21.6. Review of Related Published References......Page 387
21.7. A Numerical Example......Page 388
References......Page 390
22.1. Introduction......Page 392
22.2.1. Introduction......Page 393
22.2.2. Modified control limits for attributes......Page 396
22.2.3. Simplified limits......Page 398
22.2.4. Probability limits with "inflated" skewness......Page 400
22.2.5. Probability limits for some attribute distributions......Page 402
22.2.6 Numerical assessment......Page 403
22.3.1. Introduction......Page 407
(I) Control limits for the end-points of the process distribution......Page 408
(II) Control limits for the parameters of the process distribution......Page 409
22.3.2. INT-based control schemes for variables......Page 410
Scheme I. Data requirements- Estimates of M, and (W)......Page 411
2. Applying the control charts......Page 413
A Numerical Example......Page 414
Scheme II. Data requirements- Estimates of M, and......Page 417
References......Page 418
23.1. Introduction......Page 420
23.2. First Approach - The Quantile Function and the Loss Function......Page 421
23.3. Second Approach - The Quantile Function and Loss Function......Page 424
23.4. First Approach - Newsboy Problem with Order-up-to Policy......Page 430
23.5.1. The Continuous-Review (Q,R) Model......Page 432
A numerical example......Page 433
23.5.2. Safety lead-times for purchased components......Page 435
A numerical example......Page 437
References......Page 438
Review Questions......Page 440
Author Index......Page 444
Subject Index......Page 450
