AR Spectrum with Order Exploration

The AR Spectrum with Order Exploration option in the Spectral menu or the Spectral toolbar extends AR spectral fitting to where multiple orders can be fitted and simultaneously plotted. This procedure is especially useful for the Nrml algorithms and all of the SVD procedures since they do not compute all lesser orders in the process of computing the target order. Even for the AutoCorr, Burg, and Data least-squares methods, which have order selection criteria available in the AR (AutoRegressive) Spectrum option, this procedure offers both 2D and 3D visualization for assessing the impact of model order.

When multiple orders are fitted, an average is computed and full error bars are available. The average spectrum from some number of viable AR orders should result in a lower variance estimator. Spurious peaks that arise from noise at one order may be absent or diminished at other orders.

Only the linear sinusoid fit is available in the Numeric Summary, but it will be based upon the frequencies extracted from the averaged spectra. Some caution is advised. In some cases, this averaging may result in multiple closely-spaced peaks where only a single peak would otherwise occur.

The individual spectra for the various orders and their labels can be toggled on and off with the added reference buttons in the graph's toolbar.

The graph’s toolbar also has a button that offers the full selection of error bars, as well as an error-bar toggle.

Viewing the various spectra in a 3D format is one way to perceive the optimum order. At low orders, narrowband components are blended and fuzzy. As the order progresses, the components become distinct and the peaks sharpen. As the orders become too high with the non-SVD procedures, these components diminish and noise peaks begin to appear.

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. Although AutoSignal offers the traditional non-SVD AR algorithms, an SVD (singular value decomposition) least-squares method is recommended. As an alternative to the FB (forward-backward) prediction, there are also forward prediction (Nrml Fwd, Nrml SVD Fwd, Data Fwd, Data SVD Fwd) and backward prediction (Nrml Bwd, Nrml SVD Bwd, Data Bwd, Data SVD Bwd) least-squares algorithms available.

When more than one order is fitted, peaks are determined from the local maxima in the spectrum formed by the arithmetic averaging of the individual spectra. For a single order, this algorithm extracts the frequencies from the roots in the AR polynomial. For non-SVD algorithms, all local maxima or 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. For multiple orders with an SVD procedure, the component count will be set to half the signal space value.

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.

Model Order Selection

For this procedure, the order minimum (min), maximum (max), and increment (inc) are specified. The AutoCorr, Burg, and Data least-squares methods compute all lesser orders when computing a given order, so these algorithms will be very fast in this procedure, regardless of the count of individual spectra. The Nrml algorithms and all of the SVD procedures compute only the target order, so the number of individual spectra specified can significantly impact processing time.

When multiple orders are fitted, an average is computed that may result in multiple closely-spaced peaks where only a single peak would otherwise occur. This is particularly a problem when the span of orders is too broad. To prevent this "split-peak" effect within the average, you may need to restrict the range of orders used within the average.

The AutoCorr, Burg, and Data least-squares methods compute all lesser orders in the process of computing the target order. The Plot Selection Criteria option will include all orders up the maximum order specified. The Nrml algorithms and all of the SVD procedures compute only the target order. In this case the Plot Selection Criteria option will include only those orders actually specified by the minimum, maximum, and increment values.

Fitting AR models to harmonic signals in the absence of noise is a simple matter. For noise-free data, the minimum order needed will be twice the number of sinusoids comprising the spectrum. In practice, there is usually some level of noise present in the data and a higher order model is needed. It is often the case that the minimum order needed to resolve all components is significantly higher. You may want to be generous when setting the maximum order value when it is your aim to determine this minimum order needed.

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. Provided the order is sufficiently high to produce an effective partitioning of the signal and noise, the actual order of the fit is not critical. 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 an accurate specification of the signal space.

Signal Subspace Specification

The Graphically Select Signal and Noise Sub-Spaces option is available only when an SVD algorithm is being used. Even when the spectral component count is known, you should use this option to insure that a high enough order is being used to achieve the desired signal-noise separation. The eigendecomposition for only the first of the orders fitted, the minimum order, is available for graphical signal space selection. Again, to accommodate both positive and negative frequencies, the signal space value must be twice the number of components.

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 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.

Unlike the single order AR (AutoRegressive) Spectrum procedure, the Adaptive option can be prohibitively costly both in time and memory if a large number of orders are being processed. The spectrum for each order is adaptively integrated, and a different set of frequencies are generated for each. In order to produce a meaningful average spectrum, the frequencies from all orders are combined and sorted, producing what may be a very large frequency set. Then, for each frequency in this combined adaptive set, the AR spectrum is evaluated for each of the orders being processed. The average curve is generated by the arithmetic mean of these values. When large numbers of orders are being processed and plotted, it may be faster and less memory intensive to disable the Adaptive option and set a fixed spectral length, such as 8193 or 4097.

Also, the peak splitting that can be observed with the averaging process is much less likely to occur when a fixed frequency spacing is used. If the Adaptive option is used, it is possible to Normalize the spectrum for each order so that its integrated power matches that of the input data.

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.

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 if only a single order is fitted. Otherwise, the peak frequencies are determined from the local maxima in the averaged spectrum. For the non-SVD procedures, each valid frequency derived from a root (single order) or evidencing a local maximum (multiple orders) 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 can be up to half the signal subspace value.

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, the spectral quantity currently plotted, and for multiple orders the standard deviation (SD) computed in producing the average across orders at this frequency. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies to the clipboard the frequency, the spectral quantity currently plotted, and for multiple orders the standard deviation (SD) computed in producing the average across orders at this frequency. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors). In the ASCII format, the SD values are coded as weights.

Save

The Save Data to Disk option writes to a supported file format the frequency, the spectral quantity currently plotted, and for multiple orders the standard deviation (SD) computed in producing the average across orders at this frequency. 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 spectral analysis report. The report optionally includes 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.

Plot Selection Criteria

The Plot Selection Criteria option will include all orders up the maximum order specified for the AutoCorr, Burg, and Data least-squares methods. These algorithms compute all lesser orders in the process of computing the target order. The Nrml algorithms and all of the SVD procedures compute only the target order. In this case the Plot Selection Criteria option will include only those orders actually specified by the minimum, maximum, and increment values. The MDL (minimum description length) is the most widely accepted AR order selection criterion.

Display as 3D Plot

The Display as 3D Plot option will generate an AutoSignal 3D surface graph using all of the individual spectra from the various orders. The 3D display option is particularly useful in discerning the optimum order. At low orders, narrowband components are blended and fuzzy. As the order progresses, the components become distinct and the peaks sharpen. As the orders become too high with the non-SVD procedures, these components will diminish and noise peaks may begin to appear. In the contour plot that follows, data containing 12 sinusoids plus noise (generated from sample7.sig in the Generate Signal option) is evaluated for AR orders 30 through 60 using the Data FB procedure. It is clear that order 47 is the minimum order that resolves the individual components.

For this option to be available, there must be at least 3 orders processed. The greater the density of orders computed, the greater will be the accuracy of the 3D plot. To properly capture the sharp AR peaks, the surface is rendered on a scattered basis, rather than a rectangular grid. The nodes for each spectrum will consist of the bounds and extrema, and nodes will be added if necessary to create a 10 point minimum. Once the nodes have been determined, the surface is then rendered using the Renka I 3D scattered data interpolation algorithm. Since the surface is constructed from a scattered rather than a gridded basis, large differences in the extrema positions across adjacent orders can result in an anomalous appearance in the surface. For the best rendering accuracy, an order increment of 1 is recommended and if needed, a high mesh count should be used in the 3D rendering.

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.