.N.1
( n=0 r.(n)2)2
J = (6.25)
.N.1 .∞
r.(n).r(n . k)
n=0 k=.∞
As d is a low frequency process, any contribution from d in r.will be con-.ned to the .rst few DFT coe.cients, and will thus unbalance the otherwise white spectral content of r.. The function J determines the whiteness of r.. This function has an expected value of 0.5 for white noise, and less than
0.5 for non-white signals. If r.is not su.ciently white, the model order is incremented and the model parameters reestimated. This process is repeated until an acceptable model is found.
The whiteness acceptance criterion is more suitable for this application than traditional goodness-of-.t criteria, like r2 or adjusted r2, as these are energy-based metrics. In this context, one can however not make any as-sumption about the energy distribution between d and r. The only identify-ing mark remains the power spectrum of r.
This approach can be implemented as a wrapping around the two previous methods, thus identifying both the optimal model order and the optimal value for each parameter. The method was tested using Trust Region and a spectral
acceptance
criterion
of
0.45,
with
.nal
results
in
.gure
6.19.
Spectral
validation produces repeatedly good results when used with Trust Region. Execution time is low, but the method seems inherently robust.
Iterative Evolutionary Estimation (IEO)
Although evolutionary optimization is a robust method in the presence of local optimums, the algorithm struggles when the parameter space grows to waste. In order to maintain robustness, the algorithm should maintain a number of individuals su.cient to span the space containing the optimal solution with a certain population along each dimension. If for instance a one dimensional problem is solved using a population of k individuals, then a two dimensional problem will require k2 individuals to maintain the same population density. This becomes a problem when the order of the sigmoid model increases, as this causes the number of dimension in parameter space to increase three times as fast, thus causing the number of individuals necessary to maintain population density to increase exponentially.
The problem of approximating an indicator series containing several tran-sitions is one where a problem consisting of several sub problems is more complex than the sum complexity of the sub problems. This because an Nth order model can have N! di.erent orderings of its sigmoids which for the optimization algorithms are seen as N! di.erent solutions, even though they indeed are equivalent.
A simpli.cation will thus be to solve one sub-problem at a time, instead of trying to solve all the problems at once. This can be attempted by splitting the input series into uniform segments, and approximating one segment at a time. The algorithm starts by estimating a .rst order model to the .rst K observations of the input series. It then extends the estimation window by K observations, and estimates a model consisting of two sigmoids, treating the previously estimated dc component as static. I.e. the estimate for the dc component is kept while the sigmoid is re-estimated. The estimation window is extended by another K points for the third iteration, and the dc component
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