Statistics for Environmental Engineers

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which has residuals:


ei = Уi — Po — Pi xi


Similarly, if one proposed the nonlinear model n = 01exp(-02x), the observed response is:


Уi = 0i exp(-02xi) + ei


with residuals:


ei = уi — 0i exp(-02xi)


The relation of the residuals to the data and the fitted model is shown in Figure 33.1. The lines represent the model functions evaluated at particular numerical values of the parameters. The residual (e{ = yt — п,) is the vertical distance from the observation to the value on the line that is calculated from the model. The residuals can be positive or negative.


The position of the line obviously will depend upon the particular values that are used for po and P1 in the linear model and for 01 and 02 in the nonlinear model. The regression problem is to select the values for these parameters that best fit the available observations. “Best” is measured in terms of making the residuals small according to a least squares criterion that will be explained in a moment.


If the model is correct, the residual et = уi — n will be nothing more than random measurement error. If the model is incorrect, et will reflect lack-of-fit due to all terms that are needed but missing from the model specification. This means that, after we have fitted a model, the residuals contain diagnostic information.

FIGURE 33.1 Definition of residual error for a linear model and a nonlinear model.


Residuals that are normally and independently distributed with constant variance over the range of values studied are persuasive evidence that the proposed model adequately fits the data. If the residuals show some pattern, the pattern will suggest how the model should be modified to improve the fit. One way to check the adequacy of the model is to check the properties of the residuals of the fitted model by plotting them against the predicted values and against the independent variables.

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