# Statistics for Environmental Engineers

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Figure 33.3 shows the smooth sum of squares curves obtained by following this approach. The minimum sum of squares — the minimum point on the curve — is called the residual sum of squares and the corresponding parameter values are called the least squares estimates. The least squares estimate of в is b = 0.1. The least squares estimate of 0 is k = 0.2. The fitted models are y = 0.1x and y = exp(-0.2x). y is the predicted value of the model using the least squares parameter estimate.

The sum of squares function of a linear model is always symmetric. For a univariate model it will be a parabola. The curve in Figure 33.3a is a parabola. The sum of squares function for nonlinear models is not symmetric, as can be seen in Figure 33.3b.

When a model has two parameters, the sum of squares function can be drawn as a surface in three dimensions, or as a contour map in two dimensions. For a two-parameter linear model, the surface will be a parabaloid and the contour map of S will be concentric ellipses. For nonlinear models, the sum of squares surface is not defined by any regular geometric function and it may have very interesting contours.

### The Precision of Estimates of a Linear Model

Calculating the “best” values of the parameters is only part of the job. The precision of the parameter estimates needs to be understood. Figure 33.3 is the basis for showing the confidence interval of the example one-parameter models.

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