Spectral Estimator Analysis Object and Template – ARMA (AutoRegressive Moving Average) Spectral Estimator (Spectral Analysis Option)
The ARMA (AutoRegressive Moving Average) procedure produces a combined pole-zero model that is capable of effectively describing both peaks and valleys. An ARMA model is generally regarded as superior for fitting both signal and noise. Unfortunately, an ARMA model is non-linear in nature, and requires an iterative procedure to resolve the parameters accurately.
Dealing with both AR and MA parameters, whose orders can be independently set, adds a considerably complexity to the modeling and to the spectral interpretation. To embrace this additional complexity and yet accept a suboptimal ARMA parameter set seems incongruous. For this reason we have chosen not to support the various suboptimal linear sequential procedures.
Algorithm
The four Non-Linear ARMA procedures consist of full non-linear Levenburg-Marquardt minimizations.
Unlike many ARMA implementations, the FlexPro ARMA filter in the NL algorithms first proceeds toward the initial data element with backward prediction/averaging and then forward across the full data sequence. Both the ARMA model and a partial derivative for each parameter must be computed point by point at each iteration. The fitting process can be very slow with large data sets and high AR, MA model orders.
The algorithm imposes no constraints as parameters are allowed to vary freely. The algorithm adds full spectral factorization so that both the AR and MA roots will lie within the unit circle. Despite the overhead of the spectral factorization, the algorithm can sometimes be faster since a good measure of a non-linear ARMA fit involves parameters wandering about in regions of instability.
FlexPro also offers the and versions of the two NL procedures. Just as in FlexPro’s AR SVD options, a signal space is selected that should contain the principal singular values of the least-squares problem. While one of the uses of ARMA models is to also characterize observation noise, there are still benefits to truncating eigenmodes with SVD. If considerable fitting time is expended wandering about in n-dimensional space fitting weak noise components, faster fits can be achieved by discarding these eigenmodes. Also, deep nulls and sharp peaks are treated equally in the least-squares problem. A principal eigenmode may be associated with a null if this MA component significantly impacts the least-squares fit merit function. Thus SVD retains nulls that significantly impact the model.
Spectrum Type
For ARMA spectra, there are only four spectral formats. The PSD can reflect the three different power normalizations, Integral=TISA, Time-Integral Squared Amplitude, Integral=MSA, Mean Squared Amplitude, Integral=SSA, Sum Squared Amplitude, or it can be expressed in dB. There is no normalized dB scale where the highest peak is set to 0 dB; sharp peaks are likely to be poorly characterized for height and they will not linearly reflect the power of spectral components. In general, ARMA spectra should be regarded primarily as frequency estimators.
Parameters
One of the greatest obstacles to ARMA fitting is determining the AR and MA orders. Selecting optimum AR and MA orders is difficult. There is no reason that these should be the same; that is, that there should be one spectral null (from an MA root) for each spectral peak (from an AR root). However, to simplify order selection, it is common practice to set the AR and MA orders equal to one another.
To see only an AR fit, the MA coefficient count can be set to zero.
Since the non-linear spectral factorization algorithms fit stable ARMA models with all roots within the unit circle, AR-only fits are likewise constrained. Bear in mind, however, that the linear Data Matrix algorithms in the AR (AutoRegressive) Spectral Estimator option often achieve this stability.
The Signal Subspace selection is enabled only when the Non-Linear SVD and Non-Linear Spectral Factorization SVD algorithm is selected. Just as in FlexPro’s AR SVD options, a signal space is selected that should contain the principal singular values of the least-squares problem. Here though, the principal eigenmodes may be associated with both AR and MA components.
Since nulls are being modeled in addition to peaks, the optimum signal space will not automatically be twice the number of narrowband components present in the data.
A full signal space SVD fit, one where the signal space equals the sum of the AR and MA model orders, produces the same results as the algorithms.
Spectrum
An ARMA spectrum can be generated directly from the AR and MA coefficients, or with some performance benefits using an FFT. The Full range option locks the 0-0.5 Nyquist range. It also causes the spectrum to be generated via an FFT if the Adaptive spacing option is disabled. When the option Full range is on, only the total spectral count (Number of Frequencies) can be specified. Unlike the FFT options, which specify the length of the transform, this option specifies the total frequency count in the output spectrum. An FFT of 16384 points produces 8193 spectral frequencies from 0 to 0.5 normalized frequency. For the Full range option, it will be fastest if the values in the Number of Frequencies drop down list are used, since these produce fast FFTs. The ARMA procedures use the Best Exact N FFT algorithm.
When the option Full range is off, you can select the desired Start and End Frequency as well as the count of spectral frequencies (Number of Frequencies) in this band. It is thus possible to generate a detailed spectrum only in the region of specific interest. This option uses a direct computation for the spectrum and any size can be used.
The Adaptive spacing option always uses a direct computation for the spectrum. An ARMA spectral estimator can consist of astonishingly sharp peaks and nulls, especially in comparison with traditional FFT spectra. For uniform sampling, a size of 8193 uniformly spaced points is not unreasonable in order to get good representation of the peaks and nulls. Even with a large , it is possible to miss some fraction of the power of a peak. As an alternative, FlexPro can use a Runge-Kutta procedure to integrate the spectrum adaptively, saving the points used in the computation of the integral. This results in an adaptive frequency set containing frequencies concentrated near the peaks.
Options - Toggle Labels (Analysis Wizard Only)
You can view the Y and/or X values of the peaks in the spectrum by pressing Toggle Labels. Initial frequency estimates are based upon the local maxima in an 8193 count full-range spectrum. These peaks are then further refined using a one-dimensional minimization procedure with the continuous ARMA spectral model. For the non-SVD algorithms, each local maximum in the 8193 frequency count spectrum is treated as a valid spectral peak. The spectral peak count can therefore be as high as half the model order. For the SVD procedures, the spectral peak count should be half the signal subspace value.
Unlike the FFT, it is not possible to compare power by looking at the magnitude of the AR spectral peaks. The areas under the peaks, however, are indicative of estimated power.
Options - Set/Clear Reference (Analysis Wizard Only)
This function lets you compare various spectral procedures and settings. You can view a copy of the currently displayed spectrum in the lower pane by pressing Set Reference. Next, you can adjust additional settings that affect the display in the upper pane. With Clear Reference you can remove the copy and the time signal will appear again.