Minimum Variance Spectrum

The Minimum Variance Spectrum option in the Spectral menu or the Spectral toolbar offers a low variance spectral estimator where it is possible to graphically compare component powers and test for component significance. The frequency resolution lies somewhere between the FFT and the AR methods.

The reciprocal of the minimum variance (MV) spectral estimator of order n is equal to the average of the reciprocals of the AR estimators from order 0 through n. This averaging effect results in a reduced variance estimator, but the inclusion of lower AR orders in the average significantly degrades the spectral resolution. The area under the MV spectrum is not equal to the input power but the MV peak heights are linearly proportional to the power of the spectral components.

Algorithm

The minimum variance spectrum can be computed using a variety of MV spectral algorithms. It can be computed directly (Capon-Kay method), from its relationship with AR coefficients of all orders through to the order of the MV spectral esimator, or the spectrum can be computed using a very fast procedure (Musicus method).

The Capon-Kay MV procedure is the most accurate of the methods for a given model order in reflecting the power of sinusoidal components, but the algorithm is quite slow. The SumAR Burg, SumARDataF, SumARDataB, SumARDataFB MV algorithms construct the individual AR spectra for each order through to the MV order and average the inverses. The SumARData procedures offer an intermediate accuracy in reflecting the power of harmonic components, but are considerably faster than the direct method. The Musicus MV algorithm computes coefficients that require only the correlations between the AR parameters of the MV model order. By far, this is the fastest of the MV algorithms although it also offers the least accuracy in estimating component powers for a given model order.

The Musicus and SumARBurg procedures produce equivalent results. The SumARBurg algorithm is offered in case you want to create a partial range spectrum. The Musicus procedure exclusively uses the FFT and generates a full frequency range spectrum.

Order

Due to the averaging effect across all AR orders, a minimum variance spectrum may require a higher order than an AR model fit. Averaging the very lowest AR orders does significantly degrade the resolution of the MV estimator. Some of this can be remedied, though, by using very high orders (data size permitting). Performance begins to approach that of a high resolution estimator. The SumAR procedures are supported through order 200, and the Musicus algorithm through order 500.

Spectrum

A minimum variance spectrum can be generated directly, 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 unless the Capon-Kay algorithm is used. 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, 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 drop down box for n are used, since these produce power of 2 FFTs. The MV 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. For the Musicus algorithm, the Full Range option is disabled since only a full range spectrum is available.

Plot

You can choose the plot the MV spectrum directly (Spec), or as decibels (dB(Spec)), decibels normalized to 0 at the principal peak(dB0(Spec)), or as a variance-normalized power(Spec/var). As with the FFT, a dB normalized plot would have a half power peak at -3dB and a quarter power peak at -6dB.

AR(1) Background

AutoSignal offers peak-type critical limits to determine the statistical significance of the largest peak present in the spectrum. These limits are computed for all algorithms and model orders. The default background used for this null hypothesis is white (Gaussian or normally distributed) noise, AR(1)=0.0. A lag-1 autoregressive spectrum can also be specified. An AR(1) coefficient greater than 0.0 can often model red noise (where the noise power decreases with increasing frequency).

The AI Expert option will set the AR(1) value to the lag-1 normalized autocorrelation. This should only be used as a preliminary estimate. For a better estimate of the background, use the one of the Fourier Filtering and Reconstruction, Eigendecomposition Filtering and Reconstruction, or Wavelet Filtering and Reconstruction options to remove the spectral components from the signal, leaving only the background. Then use the AR (AutoRegressive) Spectrum option with an order of 1 to fit the AR model. The desired AR(1) coefficient is listed in AR procedure's numeric summary.

The Show Significance Levels option is in the graph's toolbar. This button is used to toggle the significance levels on and off.

Peaks

The spectral peaks are identified by a local maxima detection algorithm. The sig item sets the target number of peaks (signal components) to detect. Up to 50 peaks can be detected. Peaks are ranked by power. Note that this target signal component count may not be realized as fewer peaks than this target may be detected. Note also that the frequency analysis and linear sinusoidal fits reported in the Numeric Summary use the component count and frequencies from this peak identification.

The wid item sets the tolerance for defining a peak. A peak must exist across this number of spectral frequencies to be counted. The default is one.

The Display Maxima option is used to step through the options for displaying spectral peak labels: frequencies, spectral powers, both frequencies and spectral powers, or none.

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%).

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 a 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 MV spectral analysis 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 will always use a landscape orientation. Beyond a certain size, the graph will utilize a full landscape page.

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.

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.