Eigendecomposition Filtering and Reconstruction

The Eigendecomposition Filtering and Reconstruction option in the Process menu or the Process toolbar offers full eigenmode filtering and reconstruction. Eigendecomposition partitions by signal strength rather than by frequency. Using this option, it is possible to isolate individual oscillatory components in signals. It is also possible to isolate the signal and noise portions of a data series.

This option presents a dual AutoSignal graph with the eigenvalues in the upper graph and a variety of visualization options in the lower graph. In addition to the reconstructed data, the lower graph can optionally consist of the eigenvectors, the principal components, the data components, FFTs of the data components, an FFT of the data, AR spectra of the components, or an AR spectrum of the data.

The graph's toolbar has a button that can toggle on and off and change the proportions of the two graphs in the dialog. The default has the eigenvalue graph using the upper third of the region, and the output graph using the lower two-thirds.

Graphical Selection of Eigenmodes

The eigenmode selection in the upper graph is fully graphical. To select eigenmodes, the Selection Mode must be active. All eigenvalues boxed using the left mouse button are included in the reconstruction while those outside the enclosed region are excluded. If the right mouse button is used to enclose a set of eigenvalues, these are excluded from the reconstruction and all others are included. Individual eigenmodes can be turned on and off by left or right clicking on the point. To select a single eigenmode for reconstruction, it is necessary that you box just that eigenmode using the left mouse button. To exclude just a single eigenmode, you must similarly box the desired mode using the right mouse button.

Two eigenmodes are required to capture an oscillation. For this reason eigenmodes often appear in pairs and should be selected and processed in pairs. Note also that an eigendecomposition is non-parametric. A non-sinusoidal anharmonic oscillation is captured as easily as a sinusoidal harmonic.

Eigenvalue Data Labels

The Toggle Data Labels option displays an rē correlation coefficient for each of the eigenmodes used in the reconstruction. These partial rē values will generally follow the ordering of the eigenvalues. These rē values measure the extent to which each eigenmode captures the variance in the overall data. A perfect rē is 1.0. This will not be observed with oscillatory data since two eigenmodes are needed to capture a single oscillatory component. An rē of 0 indicates no correlation at all between the data set and the data from the reconstructed eigenmode. Eigenmodes containing noise will usually have partial rē values very near zero. Note that the partial rē values derive from the reconstruction of individual components.

Algorithm

The eigendecomposition algorithms vary only by how a lagged covariance matrix is constructed:

· CovM Fwd - the covariance matrix equivalent of processing a forward prediction data matrix

· CovM FB - the covariance matrix equivalent of processing a forward-backward prediction data matrix

· CovM Tpltz - a Toeplitz symmetric covariance matrix constructed using the Burg procedure

In general, the differences between the methods will be small. Because it imposes a matrix structure, the CovM Tpltz algorithm is typically the least effective in mapping data trends. If the AR algorithm properties apply, the CovM FB algorithm should be the most effective of the procedures.

Full data matrix procedures are not needed since no parametric model is computed. The decomposition of a forward or forward-backward data matrix produces the same eigenvalues and eigenvectors as the equivalent covariance matrix, and requires more processing time with large data sets. Similarly, no backward prediction data/covariance matrix algorithm is needed since it would produce the same decomposition as the forward data/covariance matrix.

In this procedure, the main task is to isolate and reconstruct signal components of interest by selecting one or more eigenmodes. The principal components of the signal will be captured in the initial eigenmodes. The noise is broken into low power elements that generally form a sloping floor beyond the signal eigenmodes in the eigenvalue plot.

Order

Establishing the eigendecomposition order for harmonic signals in the absence of noise is a simple matter. A model order of two is needed to fully describe one oscillation. Similarly an order of four is needed to fully model two oscillatory components. For noise-free data, the minimum order needed will be twice the number of signal components introducing oscillations in the data. These oscillatory components can be harmonic, such as undamped or damped sinusoids, sawtooth, and others. Or they can be anharmonic, oscillations where the model is not readily apparent.

Since there is usually some level of noise present in the data, a higher order model is needed to also account for the oscillations introduced by noise. To achieve a reasonable signal-noise separation within an eigendecomposition, it is necessary to use a high enough order so that the primary eigenvectors span only signal space. Further, when isolating components, the partitioning of component signals into different eigenmodes is usually enhanced by higher orders.

When this procedure is invoked, all of the eigenmodes in the decomposition are automatically selected. This produces an exact reconstruction of the input data.

Normalize

The Normalize (%) item plots the eigenvalues on a percent scale. When this normalization is used, all of the eigenvalues sum to 100%. This option makes it easy to see the relative contribution, by eigenvalue, of each component.

Eigenvectors

When the Eigenvectors option is selected, each of the eigenvectors used in the reconstruction is plotted. Each eigenvector plotted will be of the length of the embedding dimension (model order). Sinusoidal harmonics will generally appear in pairs, one offset in index from the other.

Principal Components

The Princ. Comp. option plots the temporal principal components used in the reconstruction. The temporal components consist of the projection of the eigenvectors onto the data series. Each temporal component plotted will be of the length of the data series minus the model order plus one. Again, sinusoidal harmonics will generally appear in pairs, one offset in index from the other. Unlike the eigenvectors, the magnitude of the principal components will be related to signal strength.

Data Components

The Data Comp. option plots the reconstructed data components. This is normally the starting point for component inspection, as compared to the eigenvectors or principal components. Each data component plotted will be of the length of the data series and each element will correspond in time with the original data. Phase is also fully restored. Separated sinusoidal harmonics will still appear in pairs, although the data plots will now closely overlap one another. The offset observed in the eigenvectors or principal components representing an oscillatory pair will not be present in the reconstructed data components.

Data

The Data option plots the overall reconstruction of data using all selected eigenmodes. There is but a single plot.

The Toggle Display of Reference Data button in the AutoSignal graph's toolbar is used to toggle the original data on and off.

FFT Components

The FFT Comp. option is a convenience for inspecting the Fourier spectra of the individual reconstructed data components. For these frequency spectra, a windowed FFT is offered. A Kaiser-Bessel window is used, its variable width automatically determined as a function of data count. The width will be 1.5 for n<=256, and increase to 4.0 for n>=2048. Since the windowing takes care of edge effects, a minimum 1024 point FFT is used to give a good peak resolution with short data records. For data sets beyond 1024 points, an exact n FFT is used. There will thus be 513 points in each spectrum for all data sets 1024 points in size and smaller. Larger sets will result in n/2+1 frequencies. A decibel (dB) scale is used so that individual components can be compared for power.

FFT

The FFT option is used for inspecting the Fourier spectrum of the overall data reconstruction. The same Kaiser-Bessel windowing, 1024 minimum size FFT, and dB scaling is used.

The Toggle Display of Reference Data button in the AutoSignal graph's toolbar is used to toggle the FFT of the original data on and off.

Autoregressive Components

The AR Comp. option is a convenience for inspecting the AR spectra of the individual reconstructed data components. For performance reasons, the AR frequency spectra are generated using the Data FB algorithm. These AR spectra are full range, non-adaptive, and FFT-generated with a 513 minimum size (this produces the same number of frequencies as the FFT spectra). Note that more accurate AR spectra can be generated in the AR (AutoRegressive) Spectrum procedure by using one of the SVD AR algorithms and an adaptive spectrum. As with the FFT, a decibel (dB) scale is used. Comparisons between the spectra should be cautious ones since the spectrum size is insufficient to capture the often sharp AR peaks, and AR spectra are not linearly proportional to signal power.

AR Spectrum

The AR option is used for inspecting the AR spectrum of the overall data reconstruction. The same Data FB algorithm and fixed 513 point spectrum is used.

The Toggle Display of Reference Data button in the AutoSignal graph's toolbar is used to toggle the AR spectrum of the original data on and off.

Component Labels

When components are plotted, eigenmode numbers can be plotted as data labels. These are automatically positioned at the peaks in the data. The Toggle Reference Labels button in the AutoSignal graph's toolbar is used to toggle these labels on and off.

List

The List Data option lists the index, x-values, and the output currently plotted in a table that will include up to 12 components. The listing uses the AutoSignal text viewer facility. Note that all information will not be present if more than 12 eigenmodes are reconstructed.

Copy

The Copy Data to Clipboard option copies the x values and the output currently plotted to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors). All information will be present, regardless of the number of eigenmodes reconstructed.

Save

The Save Data to Disk option writes the x-values and the output currently plotted to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat. All information will be present, regardless of the number of eigenmodes reconstructed.

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 processed data can be exported to an Excel 95 or Excel 97 file.

Numeric Summary

The Numeric Summary offers an eigendecomposition report. The report includes a table of eigenvalues, normalized eigenvalues, and partial rē correlation coefficients.

Rich-Text Format Export

The Export Numeric Summary and Graph to RTF File option writes the numeric summary and both plots to an RTF file. The text data will be written to portrait orientation pages. The graphs use the current settings and sizes of the plots, and are inserted as Windows Metafiles. The graphs always use a landscape orientation. Beyond a certain size, the graphs utilize a full landscape page.

Popup Information

The Toggle Popup Information Window option is used to report estimated noise and power reductions as well as a correlation coefficient between the filtered and unfiltered data.

AutoSignal offers a robust noise estimation procedure that may be of some value for low-frequency signals. A cubic polynomial interpolation is made for each point using the two points to the left and the two to the right (excluding the current point). The difference between the interpolated and signal values is used to generate a measure of the white noise present in the signal. This assumes that the signal can be locally characterized by a smooth cubic interpolant. Also, the signal component(s) should exist only in the lower quarter of the Nyquist range. If a high frequency signal component is present, these estimates of noise will be invalid. The Noise In value reports the estimated white noise in the incoming data, the Noise Out value the estimated white noise for the filtered signal. The Noise % is given as the amount of estimated noise remaining after filtration.

The TISA In value reports the TISA power in the incoming data. The TISA Out value is the TISA power for the filtered signal. The TISA % is given as the amount of power remaining after filtration.

The r-squared correlation coefficient is also reported. An rē of 1 is a perfect correlation while a value of 0 means the filtered and unfiltered signals are completely uncorrelated.

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.

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.

When exiting this procedure with the OK button, an option will be presented to update AutoSignal's main data table with the reconstructed data.