One method of judging the quality of a particular model is by *
residuals*. That means the model is fit using all the data points and
the prediction for each data point is compared with its actual output.
The absolute value of each error is taken and the mean of those values
is computed to arrive at the mean absolute residual error. Models
with lower values of this measure are deemed to be better.

**Figure 24:** Approximating a one-dimensional data set with A90:9, L90:9, L10:9
metacodes. The residual error for each data point is the distance along a
vertical line between it and the fitted line. The result is very large, large,
and zero residual error, respectively.

**Figure 25:** Approximating a one-dimensional data set, with A90:9, L90:9, L10:9
metacodes. The residual error for each data point is the distance along a
vertical line between it and the fitted line. The result is very large, small,
and near zero residual error, respectively.

Fig. 24 shows an example where choosing the model
with the lowest residual error is a good idea (the data comes from
*b1.mbl* if you want to load it into Vizier). The fit on the
right is clearly the best fit of the three for the data and its mean
absolute residual error is near zero. However, choosing models by
residual error is a risky thing to do.

Fig. 25 shows an example (from *a1.mbl*) where
residuals can lead us astray. Again, the residual error in the middle
plot is moderate, and the residual error on the rightmost plot is near
zero. The middle plot is a much better fit though. The reason is
that the rightmost plot is fitting the noise in the data. This
phenomenon is referred to as ``overfitting'' and is a common problem
that must be avoided in learning systems. Overfitting in this example
means that errors in predicting future data points from this curve
will actually be higher than if we use the middle plot's fit instead.
In general, it is preferable to use something more trustworthy than
residual error to choose a good model and avoid overfitting.

Fri Feb 7 18:00:08 EST 1997