5. Process Improvement
5.5.9. An EDA approach to experimental design
5.5.9.9. Cumulative residual standard deviation plot

Motivation: What is a Model?

Mathematical models: functional form and coefficients A model is a mathematical function that relates the response Y to the factors X1 to Xk. A model has a
1. functional form; and
2. coefficients.
An excellent and easy-to-use functional form that we find particularly useful is a linear combination of the main effects and the interactions (the selected model is a subset of the full model and almost always a proper subset). The coefficients in this linear model are easy to obtain via application of the least squares estimation criterion (regression). A given functional form with estimated coefficients is referred to as a "fitted model" or a "prediction equation".
Predicted values and residuals For given settings of the factors X1 to Xk, a fitted model will yield predicted values. For each (and every) setting of the Xi's, a "perfect-fit" model is one in which the predicted values are identical to the observed responses Y at these Xi's. In other words, a perfect-fit model would yield a vector of predicted values identical to the observed vector of response values. For these same Xi's, a "good-fitting" model is one that yields predicted values "acceptably near", but not necessarily identical to, the observed responses Y.

The residuals (= deviations = error) of a model are the vector of differences (Y - $$\small \hat{Y}$$) between the responses and the predicted values from the model. For a perfect-fit model, the vector of residuals would be all zeros. For a good-fitting model, the vector of residuals will be acceptably (from an engineering point of view) close to zero.