Non-Linear Optimization

Most spectral procedures offer this option for non-linearly fitting a sum of narrowband models (sine, exponentially-damped sine, or first order formation sine) to the time domain data.

In all cases, this optimization proceeds in three steps:

(1) The component count and frequencies are taken from the spectral estimation procedure (this will be specific to the algorithm)

(2) Amplitudes and phases are then estimated using a linear least-squares fit (this will be in the Numeric Summary of the procedure)

(3) These frequencies, amplitudes, and phases are then optimized in a non-linear fit that will produce the the optimum least-squares or robust parametric solution

When the Non-Linear Optimization is invoked, the first two steps are automatically completed and a Non-Linear Optimization Preferences dialog is presented before the fitting in the third step is initiated. Apart from perhaps reducing the component count to be fitted, or selecting a damped sinusoid model, the default preferences should suffice for most optimizations.

A Numeric Fitting dialog will display the status of the fit as the iterations proceed.

The Non-Linear Optimization Review offers a full numeric and graphical review of the non-linear fit. Pay close attention to the overall goodness of fit, as well as the confidence limits on the amplitude, frequency, and phase parameters for each component. Again, bear in mind that the accuracy of the parametric model depends on the three-step procedure. The native spectral algorithm is used only to determine the frequencies and component count. The linear fit estimates amplitudes and phases as reliably as these original frequencies permit. Finally, the non-linear optimization refines all of the parameters, including the frequencies, producing the final parametric analysis. If the starting estimates for the frequencies are poor or if the component count is incorrect, the parametric estimates will not represent the true optimum. In all cases, however, the non-linear optimization will reflect at least some improvement relative to the linear sinusoid fit.

Another element of interest in signal analysis is whether or not the data, minus the spectral components, is a random noise sampling, or whether additional signal components may be lurking in the noise. Inspecting the residuals to see if they are normally distributed is the same as subtracting the isolated components from a raw signal to see if only white noise remains. An excellent test to ascertain if only white noise remains is to view the residuals as a Stabilized Normal Probability Plot. The procedure offers 90, 95, 99, and 99.9% critical limits. The 99% critical limit indicates not that 1% of the data is probable to rest above this limit, but that in only 1 of 100 instances of true Gaussian noise will even a single point in the SNP plot violate this threshold. When the residuals are not normally distributed, this may mean that additional narrowband components remain, or it may mean that it is not possible to describe the signal components using narrowband models.

Filtering and subsequent time-domain reconstruction are important in signal processing. A parametric model can be viewed as a filter since it describes only the isolated signal components (noise is excluded). The predicted values of a parametric model represent a continuous reconstruction capability (the model is continuous; it is not limited to discrete values, nor to the initial sampling). The Review in the Non-Linear Optimization offers the Evaluation option which enables not only data reconstruction using the original time values, but any set of values desired.

The Parametric Interpolation and Prediction procedure in the Process menu combines this non-linear optimization for the key spectral algorithms with this reconstruction functionality.