Statistics for Environmental Engineers

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The equation may be an empirical model (simply descriptive) or mechanistic model (based on fundamental science). A response variable or dependent variable (y) has been measured at several settings of one or more independent variables (x), also called input variables, regressors, or predictor variables. Regression is the process of fitting an equation to the data. Sometimes, regression is called curve fitting or parameter estimation.

The purpose of this chapter is to explain that certain basic ideas apply to fitting both linear and nonlinear models. Nonlinear regression is neither conceptually different nor more difficult than linear regression. Later chapters will provide specific examples of linear and nonlinear regression. Many books have been written on regression analysis and introductory statistics textbooks explain the method. Because this information is widely known and readily available, some equations are given in this chapter without much explanation or derivation. The reader who wants more details should refer to books listed at the end of the chapter.

Linear and Nonlinear Models

The fitted model may be a simple function with one independent variable, or it may have many independent variables with higher-order and nonlinear terms, as in the examples given below.

Nonlinear models n = ——-—-

1 — exp (-в2x)

To maintain the distinction between linear and nonlinear we use a different symbol to denote the parameters. In the general linear model, n = f (x, в), x is a vector of independent variables and в are parameters that will be estimated by regression analysis. The estimated values of the parameters вь вг, —

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