AR Spectrum with Algorithm Comparison

This AR Spectrum with Algorithm Comparison option in the Spectral menu or the Spectral toolbar is a specialty AR (autoregressive) spectral procedure that offers the means to explore up to three different algorithms simultaneously.

An algorithm field, model order field, signal subspace field, and signal subspace graphical selection button is furnished for up to three separate algorithms. The second algorithm is toggled on and off using the 2 checkbox. Similarly, the third algorithm is toggled on and off with the 3 checkbox.

In this option, each spectrum is independently generated and plotted. There is no average spectrum.

Spectral peaks are labeled only for the first of the algorithms.

Algorithm

The AR algorithm list offers fourteen procedures. The Data SVD FB algorithm is the most robust and accurate of the methods, although it is also the slowest. If performance is an issue with large data set sizes, the Nrml SVD FB algorithm may be a viable alternative.

It may be instructive to compare the FB (forward-backward), Fwd (forward), and Bwd (backward) prediction variants of the Data (or Nrml) algorithms. Comparing the SVD and non-SVD form of one of the Data or Nrml algorithms is also quite useful. It may be helpful to see the difference between the Data and Nrml algorithms for the FB (or Fwd or Bwd) prediction type. Similarly, you may want to compare the autocorrelation-based AutoCorr algorithm or the reflection coefficient-based Burg procedure with one of the least-squares algorithms.

For a given SVD algorithm, it may be useful to compare different signal spaces.

Model Order Selection

Fitting AR models to harmonic signals in the absence of noise is a simple matter. A model order of two is needed to fully describe one sinusoid. Similarly an order of four is needed to fully model two components. For noise-free data, the minimum order needed will be twice the number of sinusoids comprising the spectrum. The Data procedures will achieve a perfect fit in this instance, exactly resolving the frequencies; the other algorithms will not. In practice, there is usually some level of noise present in the data and a higher order model is needed. The additional coefficients go primarily into modeling at least some of the noise. To achieve a reasonable signal-noise separation with SVD, it is necessary to fit a high enough order so that the primary singular vectors (eigenvectors) span only signal space.

With the SVD routines, the order of the fit ceases to be critical. A tolerably high order is needed, one that is sufficient to produce an effective partitioning of the signal and noise. The quality of the fit for the noise components is not a consideration, since these eigenvectors are discarded in the SVD processing. All that is needed is to accurately determine the signal space threshold. For most data sets, this is far easier than determining an optimum AR order.

Signal Subspace Selection

The Graphically Select Signal and Noise Sub-Spaces signal selection is enabled only when an SVD procedure is being used. You can enter the signal space value numerically if you know with a certainty the number of spectral components present in the data. To accommodate both positive and negative frequencies, you must enter a value that is twice the number of components. If three spectral components are known to exist, the signal subspace must be set to 6. Even when the spectral component count is known, you should use this Graphically Select Signal and Noise Sub-Spaces option to insure that a high enough order is being used to achieve the desired signal-noise separation.

When there is sufficient signal-noise separation in the eigenmodes, the singular value plot reveals one or more sharp transitions between the signal subspace and the noise subspace floor. The last eigenmode before the long sloping noise floor represents the last element of signal space. Assuming a high-enough AR model order is used, this signal-noise space separation does not become difficult until the noise level approaches that of the signal. At this point, the sharp characteristic transition disappears. An earlier diminishing of this transition occurs when the noise is red.

A full signal space SVD fit, one where the signal space equals the model order, produces the same results as the non-SVD algorithms.

Spectrum

These options apply to all of the spectra being plotted.

An AR spectrum can be generated directly from the AR 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 option is disabled. When the Full Range option is on, only the total spectral count (n) can be specified. Unlike the FFT options, which specify the length of the transform, the AR options specify 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 drop down box for n are used, since these produce power of 2 FFTs. The AR procedures use the Best Exact n FFT procedure.

If the Full Range option is off, you can select the desired start and end frequencies as well as the count of spectral frequencies (n) 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 option always uses a direct computation for the spectrum. An AR spectrum can consist of astonishingly sharp peaks, 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. Even with a large n, it is possible to miss some fraction of the power of a peak. As an alternative, AutoSignal can use a Runge-Kutta procedure to integrate the spectrum adaptively, saving the points used in the computation of the integral. This results not only in an adaptive frequency set containing frequencies concentrated near the peaks, but also in an accurate area under the spectrum.

If the Adaptive option is used, it is possible to Normalize the spectrum so that its integrated power matches that of the input data. With an FFT, this is intrinsic, but it is not so for an AR algorithm since the magnitude of AR spectral elements is directly proportional to the estimated white noise variance. The Adaptive integration seeks 1E-5 fractional convergence, and is usually successful with most real world data sets.

Plot

For AR spectra, there are only four formats. The PSD can reflect the three different power normalizations, 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.

The AR peak labels consist of frequencies only and are toggled on and off with the Display Maxima button. These frequencies are determined directly from the AR roots of the model produced by the first of the algorithms. For the non-SVD procedures, each valid frequency derived from a root is treated as a valid spectral peak. Thus the spectral peak count can be as high as half the model order for the first algorithm. For the SVD procedures, the spectral peak count should be half the signal subspace value for this initial algorithm.

List

The List Data option lists the index and the frequency and spectral quantity currently plotted for each of the algorithms. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the frequency and spectral quantity currently plotted for each of the algorithms to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors). You can generally find a Paste As option in most applications if you want specific control over the format imported.

Save

The Save Data to Disk option writes the frequency and spectral quantity currently plotted for each of the algorithms to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat.

Production Facility

The AutoSignal Automation facility allows unattended processing of large numbers of data sets. The data sets can be consolidated in an Excel file or acquired using a DLL. The numeric summaries and graphs can be exported to an MS Word RTF file, while the extended data summaries or the current spectra can be exported to an Excel 95 or Excel 97 file.

Local Options

A local option changes the data set for the duration of the current procedure only. The main data table is not altered. AutoSignal offers four local options in most of the spectral procedures.

Section the data to isolate specific regions for processing.

Detrend for removing mean or subtracting a least-squares trend model.

Fourier Filtration for isolating spectral components by frequency.

Eigendecomposition Filtration for isolating spectral components by signal strength. Note that the use of this option strictly for noise removal is redundant when an SVD procedure is used. For the SVD algorithms, this option should be used to isolate specific oscillatory components for analysis.

The Reset button restores the data to its state when first entering the procedure. Note that if you implement sequential local procedures, all of the revisions are discarded upon reset. If an Automation Session is in progress, the Reset button can be used to terminate the automated processing.