AR (AutoRegressive) Spectrum

The AR (AutoRegressive) Spectrum option in the Spectral menu or the Spectral toolbar is the primary AR spectral procedure. In this option, the spectrum is produced by autoregressive modeling rather than a transform. An AR spectrum offers accurate frequency estimation with short data records.

Algorithm

The AR algorithm list offers fourteen procedures. Unlike the various FFT algorithms, each of the AR methods is likely to produce at least slightly different results. 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. Although AutoSignal offers the traditional non-SVD AR algorithms, an SVD (singular value decomposition) least-squares method is recommended.

In an AR spectral procedure, peaks occur exactly at frequencies corresponding to roots in the AR polynomial. For non-SVD algorithms, all roots are treated as relevant peaks and analyzed accordingly in the linear least-squares component fitting in the Numeric Summary. This full count is also the default in the Non-Linear Optimization procedure. To pare the spectrum to only the signal components of interest, an SVD algorithm must be used. Apart from longer processing times, there are no disadvantages to using an SVD procedure, and the advantages are numerous when extracting harmonics is the primary aim of the modeling.

There is generally no need to fit an AR model non-linearly. Typically, the Data algorithms, both with and without SVD, produce stable estimates where all roots lie on or within the unit circle. Since this is not assured, however, there is a way to insure the least-squares minimum where all roots are constrained to lie within the unit circle. You can use the ARMA (AutoRegressive Moving Average) Spectrum option with the NL SF or NL Svd SF procedures and set the moving average (MA) order to 0. This type of iterative fit will be considerably slower.

Note that the statistical results reported for all AR fits are based on a forward prediction filter with partial backward prediction. This will not exactly match any of the linear AR procedures. The AR filter is defined using backward prediction from the model order down to the initial data element, and forward prediction from the model order up to the final data element. This simplifies the goodness of fit statistics since a single estimate is made for each of the input data elements.

Model Order Selection

The AutoCorr, Burg, and Data least-squares methods compute all lesser orders in the process of computing the target order. For this reason, these algorithms will have order selection criteria available. The Nrml algorithms and all of the SVD procedures will not. For these procedures, multiple orders can be evaluated in the AR Spectrum with Order Exploration option.

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.

An AR model can have both real and complex roots. The real roots, usually at -0.5, 0, or 0.5 normalized frequencies, are not processed since they represent singularities at the bounds. The complex roots produce finite spectral power, and for real data, the positive and negative frequency roots mirror one another. AutoSignal reports only the positive roots, but both sides of the spectrum must be taken into account. This is why the minimum order needed must be twice the number of component sinusoids.

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 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

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 uses 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.

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, and as such, this adaptive integration uses partitions formed by midpoints of the root-determined peak positions. The Numeric Summary offers this numeric integration.

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.

This is not to say that AR spectra should be regarded only as frequency estimators, but it should be kept in mind that power values from peak integration may be quite limited in accuracy. This is true even when the adaptive integration achieves the target fractional error since the envelope of the peak is highly sensitive to AR modeling errors. For true harmonics, the linear least-squares sinusoidal fit in the Numeric Summary will almost certainly be more accurate, and the Non-Linear Optimization better still.

It is for this reason that the AR peak labels consist of frequencies only. They are toggled on and off with the Display Maxima button. The frequencies are determined directly from the AR roots of the model, and are usually computed to at least 1E-12 precision. Unlike the FFT, there is no need for a local maxima detection procedure and there is no specification of peak count. 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 SVD procedures, the spectral peak count should be half the signal subspace value.

Add Noise

It may be instructive to see where a given procedure starts to break down as a consequence of temporarily adding white observation noise to the input data. The zero noise level is S/N=300dB (fractional noise=1E-15, the IEEE double precision threshold for addition). At this value, no noise is added to the data. A value of 280 would add noise in the 14th significant figure, 260 in the 13th, 240 in the 12th and so on. This option assumes that the current data set is entirely signal, and adds noise accordingly. Typical test values are 40dB(1% noise), 20(10%), 10(31.6%), 6(50.1%), 3(70.8%), and 0(100%).

The SVD procedures have the greatest noise resistance. This noise option is also helpful in ascertaining at what level the SVD procedures can no longer evidence the eigenmode signal to noise transition.

Considerations

An AR model, of sufficient order and especially using one of the Data algorithms, is an excellent frequency estimator since the frequencies depend only on the roots of the fitted polynomial. It is the power within an AR spectrum that will be far less certain.

An AR model emphasizes the narrowband spectral content and de-emphasizes wideband trends and noise. AR models can achieve far greater separation of nearly adjacent spectral components than is possible with FFT methods, again supporting this strength in isolating sinusoidal frequencies. On the other hand, this weakness in power characterization may make it impossible for the AR procedures to find low amplitude components that FFT methods clearly reveal.

In general, an AR method is not a good candidate for high dynamic range signal detection. The prominent signals may be captured by most of the estimation capability of the algorithm, and signals just above the noise threshold can easily be lost. Used in conjunction with Eigendecomposition Filtering and Reconstruction, however, AR models can produce smooth spectra when low threshold components are extracted from the eigenmodes. In fact, this eigenspace procedure offers the Nrml SVD FB algorithm for displaying the AR frequency content of selected eigenmodes. This eigenspace filtering is available as a local option in all spectral procedures.

List

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

Copy

The Copy Data to Clipboard option copies the frequency and the spectral quantity currently plotted 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 and the spectral quantity currently plotted 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.

Numeric Summary

The Numeric Summary offers a full AR report. The report optionally includes a listing of the coefficients, component powers by numeric integration, AR fit estimation details, and a linear sinusoidal least-squares fit summary.

Non-Linear Optimization

The Non-Linear Optimization offers the means to refine the parameter estimates given in the linear sinusoidal fit that is reported in the Numeric Summary. Constrained least-squares and robust (maximum likelihood) non-linear fitting is available with either sinusoid or damped sinusoid models.

Rich-Text Format Export

The Export Numeric Summary and Graph to RTF File option writes the numeric summary and spectral plot to an RTF file. The numeric portion of the file is based upon the current settings in the Numeric Summary option. The text data will be written to portrait orientation pages. The graph uses the current settings and size of the spectral plot, and is inserted as a Windows Metafile. The graph always uses a landscape orientation. Beyond a certain size, the graph utilizes a full landscape page.

View Residuals

Because a fitting occurs, the residuals can be inspected to see if they are normally distributed. The SNP plot is particularly useful.

Plot Selection Criteria

When the coefficients for all orders are computed en route to the target order, this order selection criteria option will be available. The MDL (minimum description length) is the most widely accepted AR order selection criterion.

Plot Roots

The poles of the AR model can be inspected with the Plot Roots option. Roots consisting of signal will rest on or close to the unit circle while those corresponding with noise tend to be found in the interior of the unit circle.

Toggle Popup Information Window

Because the AR spectrum is a fitting procedure, a host of statistics are available to describe the AR model fit. The Toggle Popup Information Window is used to show or hide this information. The rē goodness of fit index may be particularly useful, since spectra that visually appear to be well fitted may be the result of a poor deterministic fit. A smooth AR spectrum is not an indicator of an accurate model fit. A high rē (0.95+) is not needed for good frequency estimation, but it is necessary for accuracy in the numerical integrations.

The statistical results reported for all AR fits are based on a forward prediction filter with partial backward prediction. This will not exactly match the model fitted in any of the linear AR procedures. This AR filter is defined using backward prediction from the model order down to the initial data element, and forward prediction from the model order up to the final data element. This simplifies the goodness of fit statistics since a single estimate is made for each of the input data elements.

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.

Red Noise AR Order 1 Parameter Estimation

For the non-SVD procedures, it is sometimes useful to remove the signal elements and fit the reconstructed noise to an AR model. If the signals can be isolated by frequency or signal strength, the Fourier Filtration or Eigendecomposition Filtration local options offer the means to reconstruct a good snapshot of the noise that is present in the signal. This is one way to estimate the AR(1) red noise background parameter used in those spectral procedures offering critical significance limits. Model orders of 1 are supported for all algorithms in this procedure.

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.