Contents -- 1 Introduction -- References -- 2 Relational Models in Engineering and the Sciences (Monotone Convex/Concave Relationships) -- 2.1. Introduction -- 2.2. Chemistry and Chemical Engineering -- 2.3. Physics -- 2.4. Electrical Engineering -- 2.5. Hardware Reliability Engineering -- 2.6. Software Reliability-Growth Modeling -- 2.7. Growth Models -- References -- 3 Shared Features and "The Ladder" -- 3.1. Introduction -- 3.2. Shared Features -- Example 1 (Chemistry) -- Example 2 (Chemical Engineering) -- Example 3 (Electrical Engineering) -- OBSERVATION A -- OBSERVATION B -- 3.3. "The Ladder of Fundamental Uniformly Convex/Concave Function" -- 4 Approaches to Model Systematic Variation -- 4.1. Introduction -- 4.2. Linear Regression Analysis -- 4.3. Box-Cox Power Transformations -- 4.4. Generalized Linear Models -- 4.5. Conclusions -- References -- 5 Approaches to Model Random Variation -- 5.1. Introduction -- (1) "The true distribution is from a specified family of distributions" -- (2) Modeling via a parameter-rich family of distributions -- (3) Modeling via a general transformation that transforms data into a recognized and well-studied distribution (like the normal) -- (4) Heuristic methods -- 5.2. Parameter-Rich Families of Distributions, Transformations and Expansions -- 5.2.1. The Pearson family of distributions -- 5.2.2. Other families of distributions (Burr, Tukey's g- and h-systems, generalized Lambda, Shore, the exponential family) -- 5.2.3. Transformations (Johnson, Box-Cox) and expansions -- 5.3. Moments and Their Role in Empirical Modeling of a Distribution -- 5.3.1. Why moment matching? -- 5.3.2. How many moments to match -- 5.4. Heuristic Methods in Empirical Modeling of Random Variation -- 5.5. An Alternative Approach to Four-Moment Matching -- References -- 6 The Requirements and Evaluation of Compliance.

6.1. Introduction -- 6.2. Desirable Requirements of a General Methodology for Empirical Modeling -- Requirement 1: Provide monotone convex/concave relationship in modeling systematic variation -- Requirement 2: The effects to include in the LP and the structure of the model are part and parcel of the empirical modeling process -- Requirement 3: Provide linear relationship as a special case -- Requirement 4: A dual-error structure -- 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 -- 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 -- Requirement 7: Provide good coverage of the ( 1, 2) plane -- Requirement 8: Parsimony in modeling -- Requirement 9: Compatibility with current proven-effective methodologies for empirical modeling -- Requirement 10: Ease of application -- 6.3. An Evaluation of Compliance of Current Methodologies -- 6.3.1. Modeling systematic variation -- Linear Regression Analysis -- Data Transformation -- Generalized Linear Models (GLM) -- 6.3.2. Modeling random variation -- References -- 7 The RMM Model -- 7.1. Introduction -- 7.2. An Axiomatic Derivation of the RMM Model -- 7.2.1. The model assumptions -- 7.2.2. The general model -- 7.2.3. Deriving f2 -- 7.2.4. Deriving f1 -- 7.2.5. The RMM Model -- 7.3. The Response Moments -- 7.4. Exploring the Relationship between the CV and -- References -- 8 Estimating the Relational Model -- 8.1. Introduction.

8.2. Phase 1 - Estimating the Linear Predictor (LP) -- 8.2.1. Introduction and motivation -- 8.2.2. Stage I - Approximating a transformed response via a Taylor series expansion and estimating the parameters via CCA -- 8.2.3. Stage II - Stepwise linear regression analysis with canonical scores as response values -- 8.3. Issues Related to Implementation of Phase 1 -- (A) "How many terms to keep in the Taylor approximation for the transformed response?" -- (B) "Which effects to include in the initial LP (Stage I)?" -- (C) "What are the assumptions and are they valid"? -- (D) "Is there a single correct LP"? -- (E) "Integration with existing methodologies" -- (F) "Handling observational data vs. data from designed experiments" -- 8.4. Phase 2 - Estimating the RMM Model -- 8.4.1. Introduction -- 8.4.2. Stage I - Estimating the RMM parameters { , , 2, } -- 8.4.3. Stage II - Estimating the RMM "Error Parameters" { , 1, 2} -- 8.4.4. Summary of the estimation procedure (Phase 2) -- 8.5. Two Numerical Examples -- 8.5.1. Example 1 - The Wave-Soldering Process -- 8.5.2. Example 2 - The Resistivity Data -- References -- Appendix A - Canonical Correlation Analysis - Background -- Appendix B - The Assumptions of CCA and Major Threats to the Reliability and Validity of Results -- (1) Multivariate Normality -- (2) Sample Size Sufficient for Reliable Estimated of Factor Loadings -- (3) Sensitivity to Outliers -- (4) Ill-Conditioned Correlation Matrix -- 9 The RMM Error Distribution -- 9.1. Introduction -- 9.2. Derivation of the RMM Error Distribution -- 9.3. Properties of the Error Distribution -- 9.4. Variations of the RMM Error Distribution -- References -- 10 Fitting Procedures (for the Error Distribution) -- 10.1. Introduction -- 10.2. Brief Review of Current Methodologies -- An Example -- 10.3. Fitting via "Moment Matching".

10.4. Fitting via "Quantile Matching" -- 10.5. Two Numerical Examples -- 10.5.1. A moment-matching example -- 10.5.2. A quantile-matching example -- References -- 11 Estimating the Error Distribution -- 11.1. Introduction -- 11.2. Percentile-Based Estimation -- 11.2.1. The estimation procedure -- Method A. Selecting a sample according to a pre-determined set of CDF values -- Calculating a percentile by "The weighted average at Y(n+1)p" -- Method B. Estimating the CDF values associated with the sample order statistics -- Preparing the Sample for NL-LS Estimating -- A detailed numerical example (to prepare a sample for NL-LS) -- 11.2.2. Two numerical examples (percentile-based estimation) -- Example 1. Birth weights of twins - The Indiana Twin Study -- Example 2. Distribution of Intra-Galactic velocities -- 11.3. Moment-Based Estimation -- 11.3.1. Introduction -- 11.3.2. Procedure I -- 11.3.3. Procedure II -- 11.3.4. Two numerical examples (moment-based estimation) -- Example 1. Birth weights of twins - The Indiana Twin Study -- Example 2. Distribution of Intra-Galactic velocities -- Procedure I -- Procedure II -- References -- 12 Special Cases of the RMM Model -- 12.1. Current Relational Models as Special Cases of RMM -- 12.1.1. Chemistry and Chemical Engineering -- 12.1.2. Physics -- 12.1.3. Electrical engineering -- 12.1.4. Growth models -- 12.2. Current Models of Random Variation as RMM Models -- 12.2.1. The Johnson families of distributions -- 12.2.2. Tukey g- and h-Systems of distributions -- 12.2.3. Fisher's transformation of the sample correlation -- 12.2.4. Haldane power-transformation and Wilson-Hilferty approximation to 2 -- 12.2.5. Box-Cox normalizing transformation -- 12.2.6. Cauchy distribution -- 12.2.7. Generalized Inverse Gaussian distribution and the Levy distribution -- 12.2.8. Generalized Gamma distributions -- References.

13 Evaluating RMM for Compliance -- 13.1. Introduction -- 13.2. Compliance for Modeling Systematic Variation -- 13.3. Compliance in Modeling Random Variation -- References -- 14 Comparative Solutions for Relational Models -- 14.1. Introduction -- 14.2. Two New Problems -- 14.2.1. Example 1 - The Windshield Experiment -- 14.2.2. Example 2 - The Economist Big Mac Parity Index -- 14.3. Two Familiar Problems (Cont'd from Chapter 8) -- 14.3.1. Example 3 - The Wave-Soldering Process -- 14.3.2. Example 4 - The Resistivity data -- 14.4. Comparison of Models -- 14.4.1. Mallow's Cp -- 14.4.2. Akaike's Information Criterion (AIC) -- References -- 15 Reliability Engineering (with Censoring) -- 15.1. Introduction -- 15.2. RMM Estimating with Censored Data -- Comments -- 15.3. A Numerical Example - The RFL model -- References -- 16 Software Reliability-Growth Models -- 16.1. Introduction -- 16.2. Example 1 - Musa's M1 Data-Set -- 16.3. Example 2 - Musa's M3 Data-Set -- References -- 17 Modeling a Chemo-Response -- 17.1. Introduction -- 17.2. Applying RMM to a Chemo-Response- First Variation -- 17.2.1. Example 1 - Temperature dependence of vapor pressure -- 17.2.2. Example 2 - Temperature dependence of solid heat capacity -- 17.3. Applying RMM to a Chemo-Response- Second Variation -- 17.3.1. Example 1 - Temperature dependence of vapor pressure -- 17.3.2. Example 2 - Heat capacity of solids and liquids -- 17.3.3. Other temperature-dependent properties -- References -- 18 Forecasting S-Shaped Diffusion Processes -- 18.1. Introduction -- 18.2. Theoretical Background for S-shaped Diffusion Processes -- 18.3. Modeling and Forecasting S-shaped Processes -- Procedure I: For a Given Time Series {Pt}- Model and Forecase TP in Terms of a specified P -- (A) Fitting the model -- (B) Forecast TP, given P.

Procedure II: For a Given Time Series {Pt}- Model and Forecast PT in Terms of a specified T.