This project addresses quantification of the measurement noise induced error in the estimation of transfer function type model structures of the form

where is an oberved output, is an observed input with power spectral density and is a zero mean white noise corruption with variance . The above transfer functions are assumed of the form

with

The parameter estimate is assumed to be found by least squares estimation:

It has been widely accepted that the variability of the ensuing estimate of the system frequency response can be approximated by the formula

Existing Variance Quantification

In the Ouput Error or Box–Jenkins model structure cases, if then .

This approximation depends on theoretical analysis that calculates the variance when , and then uses that quantity for the finite in question. The quality of the approximation therefore depends on the speed of convergence with increasing .

This project examines new analysis methods based on the principles of reproducing kernels and orthonormal parameterizations of model structures to derive a new variance approximation

New Variance Quantification

where the are the zeros of the true underlying denominator

In some cases, such as white input excitation , this quantification is exact for finite . In the non-white case it has been observed to offer quicker convergence with increasing , and hence better approximation for finite .

For example, consider the system

At 1 second sample period, this has zero order hold equivalent representation with poles as

We conduct a simulation experiment where a 'th order output error model structure is estimated on the basis of observing a length sample input-output record from this system, and for which the output is corrupted by white Gaussian noise of variance , and with input which is a realisation of a stationary Gaussian process with spectral density

The sample mean square error over estimation experiments with different input and noise realisations is used as an estimate of .

This is plotted as the green solid line in the figure below. The pre-exiting approximation is shown as the magenta dash-dot line. Our new approximation is shown as the blue dashed line.

Clearly, in this case it is a much better representation of the true variability (green line) than the pre-existing approximation.

Of course, this is just one example. If you are interesting exploring further, you may wish to download the following Matlab script which profiles both variance expressions together with the true variance (estimated via Monte-Carlo analysis) for a range of models.

For more details on the theoretical analysis and other aspects underlying this approximation, please refer to the following publications. As you will see from the author list, this work is the results of a collaboration with Hakan Hjalmarsson and Fredrik Gustafsson.