The goal for this chapter is to present the background and specific analysis techniques needed to construct a statistical model that describes a particular scientific or engineering process. The types of models discussed in this chapter are limited to those based on an explicit mathematical function. These types of models can be used for prediction of process outputs, for calibration, or for process optimization.
1. Introduction to Process Modeling  [4.1.]
1. What is process modeling?  [4.1.1.]
2. What terminology do statisticians use to describe process models?  [4.1.2.]
3. What are process models used for?  [4.1.3.]
1. Estimation  [4.1.3.1.]
2. Prediction  [4.1.3.2.]
3. Calibration  [4.1.3.3.]
4. Optimization  [4.1.3.4.]
4. What are some of the different statistical methods for model building?  [4.1.4.]
1. Linear Least Squares Regression  [4.1.4.1.]
2. Nonlinear Least Squares Regression  [4.1.4.2.]
3. Weighted Least Squares Regression  [4.1.4.3.]
4. LOESS (aka LOWESS)  [4.1.4.4.]

2. Underlying Assumptions for Process Modeling  [4.2.]
1. What are the typical underlying assumptions in process modeling?  [4.2.1.]
1. The process is a statistical process.  [4.2.1.1.]
2. The means of the random errors are zero.  [4.2.1.2.]
3. The random errors have a constant standard deviation.  [4.2.1.3.]
4. The random errors follow a normal distribution.  [4.2.1.4.]
5. The data are randomly sampled from the process.  [4.2.1.5.]
6. The explanatory variables are observed without error.  [4.2.1.6.]

3. Data Collection for Process Modeling  [4.3.]
4. Data Analysis for Process Modeling  [4.4.]
5. Use and Interpretation of Process Models  [4.5.]
6. Case Studies in Process Modeling  [4.6.]
1. Background & Data  [4.6.1.1.]
2. Selection of Initial Model  [4.6.1.2.]
3. Model Fitting - Initial Model  [4.6.1.3.]
4. Graphical Residual Analysis - Initial Model  [4.6.1.4.]
5. Interpretation of Numerical Output - Initial Model  [4.6.1.5.]
6. Model Refinement  [4.6.1.6.]
7. Model Fitting - Model #2  [4.6.1.7.]
8. Graphical Residual Analysis - Model #2  [4.6.1.8.]
9. Interpretation of Numerical Output - Model #2  [4.6.1.9.]
10. Use of the Model for Calibration  [4.6.1.10.]
11. Work This Example Yourself  [4.6.1.11.]
1. Background and Data  [4.6.2.1.]
2. Check for Batch Effect  [4.6.2.2.]
3. Initial Linear Fit  [4.6.2.3.]
4. Transformations to Improve Fit and Equalize Variances  [4.6.2.4.]
5. Weighting to Improve Fit  [4.6.2.5.]
6. Compare the Fits  [4.6.2.6.]
7. Work This Example Yourself  [4.6.2.7.]
3. Ultrasonic Reference Block Study  [4.6.3.]
1. Background and Data  [4.6.3.1.]
2. Initial Non-Linear Fit  [4.6.3.2.]
3. Transformations to Improve Fit  [4.6.3.3.]
4. Weighting to Improve Fit  [4.6.3.4.]
5. Compare the Fits  [4.6.3.5.]
6. Work This Example Yourself  [4.6.3.6.]
4. Thermal Expansion of Copper Case Study  [4.6.4.]
1. Background and Data  [4.6.4.1.]
2. Rational Function Models  [4.6.4.2.]
3. Initial Plot of Data  [4.6.4.3.]
5. Cubic/Cubic Rational Function Model  [4.6.4.5.]
6. Work This Example Yourself  [4.6.4.6.]

7. References For Chapter 4: Process Modeling  [4.7.]

8. Some Useful Functions for Process Modeling  [4.8.]
1. Univariate Functions  [4.8.1.]
1. Polynomial Functions  [4.8.1.1.]
1. Straight Line  [4.8.1.1.1.]