**2.A.1** Further Results for Line Fitting

This appendix presents a closer analysis of the line-fitting example introduced in this chapter. Some various situations are considered as examples.

If the data are not Gaussian, use of higher-order moments can be used to improve the identifiability properties. This is illustrated with an example.

Next evaluate the asymptotic expressions for the estimates \( \hat{a} _\mathrm{LS}\), \( \hat{a} _\mathrm{DLS}\), and \( \hat{a} _\mathrm{TLS}\) when *N*, the number of data points, tends to infinity.

It is important to note that the line-fitting examples above, even if very simple, still are a bit special. A specific property is that the line is constrained to go through the origin. Another way to express this constraint is to add the origin as a further given data point with no measurement error. Now generalize to an arbitrary straight line.

Can the situation improve in the multivariable case? Unfortunately, the answer is negative, as shown in the following example.

In the above examples, the parameters of the model (the parameters *a* and *b*) were considered to be estimated from second-order moments of the data (the time series \(\{x_i\}\) and \(\{y_i\}\). The unknown *x* coordinates were characterized by their mean (*m*) and variance (\(\sigma \)). It is also possible, however, to change the setting and formulate another identification problem. This is done next.

Return to the situation treated in Example 2.9. One might believe that the root of the identifiability problem is that all the *N* unknowns \(\{x_{oi}\}\) need to be determined as auxiliary unknowns. This is not so, as shown next.

The estimate of the noise variance \(\lambda _x\) is treated next.

So far, linear models have been considered. When even a simple nonlinearity is introduced, the complexity is increased. This is illustrated in the next example.

Needless to say, a parabola as treated in Example 2.12 is an extremely simple case of a nonlinear static model. To base an estimate for a nonlinear EIV model in general on higher-order moments of the data will soon become very complex when more advanced parameterizations are considered, if at all possible.