Parametric Interpolation and Prediction

The Parametric Interpolation and Prediction option in the Process menu or the Process toolbar is a powerful composite algorithm that generates a parametric (sinusoids or damped sinusoids) model of the signal. The algorithm has three stages. In the first stage, an AR, Prony, Eigenanalysis, or Fourier procedure is used to estimate the frequencies and component count. In the second stage a linear fit is made to determine the amplitudes and phases. These are the starting estimates for the third stage, the non-linear optimization.

While it is possible to accomplish these same steps from any of the spectral procedures, this option combines all of the steps into a single integrated procedure designed specifically for interpolation and prediction.

This option presents a dual AutoSignal graph with a frequency domain bar plot in the upper graph. The non-linear optimization is shown in the lower graph.

Frequency-Amplitude Bar Plot

The upper graph is a parametric amplitude spectrum that uses a bar plot. There is a single bar for each component fitted.

The Toggle Data Labels option in the upper graph toggles the display of frequency labels. These are the frequencies from the first stage procedure, not the final frequencies from the non-linear optimization.

Non-Linear Optimization Plot

The following is the non-linear optimization graph for data consisting of three sinusoids and noise. The three component functions are shown in the Y-axis plot. The Y2 plot contains the fitted curve and the data that were fitted. By default, the parametric model (the fitted curve) is white and the data points are colored by relation to the fit standard error. When the magnitude of the residual is less than 1 std error, the point will be cyan. When less than 2 std errors, the point will be green. When less than 3 std errors, the point will be yellow. A point beyond three std errors will be red by default.

In this example, only the data up to 0.012 in time were processed for estimating the model, even though data are present through time 0.051. The increased prediction error with time is apparent as many of the subsequent points (those not used in the estimation) start to fall outside 2 and 3 fit standard errors. Also, the prediction interval widens appreciably, indicating reduced confidence in the prediction.

The non-linear optimization graph offers the option of displaying Confidence and Prediction Intervals about the fitted curve. The Set Confidence/Prediction Intervals, % Confidence button is used to set confidence interval type and percentage. The option opens a simple dialog where you can individually check Confidence Intervals and Prediction Intervals, and where you can select from 90% Confidence, 95% Confidence, and 99% Confidence.

The Show Confidence/Prediction Intervals button toggles the currently set intervals on and off.

The Select Function Labels button offers the means to display amplitude, frequency, or TISA power labels for each component in the plot.

The Toggle Display of Reference Data toggles the display of the generated data. The generated data curve will be cyan by default. It will normally be hidden by the parametric model curve upon which it based, although it will be apparent when the plotted generated data density is sufficiently different from the single pixel per point rendering used to plot the model function.

Algorithm

The estimation of component count and frequencies can be done by one of eight different procedures.

· AR,Data FB - Autoregressive modeling. The Data Svd FB algorithm in the AR (AutoRegressive) Spectrum option.

· AR,NmlEq FB - Autoregressive modeling. The Nrml Svd FB algorithm in the AR (AutoRegressive) Spectrum option.

· Prony - Parametric modeling. The Undmp Svd (sinusoids) or Dmp Svd (damped sinusoids) algorithm in the Prony Spectrum option.

· MUSIC Eigenanalysis. The MUSIC FB algorithm in the Eigenanalysis Spectrum option.

· FFT - Fourier analysis. The Exact N non-windowed FFT in the Fourier Spectrum option.

· Lomb 1x - Fourier Analysis. The Fast procedure in the Fourier Spectrum of Unevenly Sampled Data option, spectrum to 1x avg Nyquist limit.

· Lomb 2x - Fourier Analysis. The Fast procedure in the Fourier Spectrum of Unevenly Sampled Data option, spectrum to 2x avg Nyquist limit.

· Lomb 4x - Fourier Analysis. The Fast procedure in the Fourier Spectrum of Unevenly Sampled Data option, spectrum to 4x avg Nyquist limit.

There are five options for equally spaced data, and three ranges of the Lomb algorithm for data that are not uniformly sampled. The Prony algorithm is recommended for fitting damped sines since it provides accurate starting estimates for the damping factors.

Model Order

The model order is set in accord with the AR (AutoRegressive) Spectrum option for the AR,Data FB and AR,NmlEq FB algorithms, in accord with the Prony Spectrum option for the Prony algorithm, and as fitting the Eigenanalysis Spectrum option for the MUSIC algorithm. There is no model order in a Fourier procedure.

In general, all four algorithms will use approximately the same order for an optimized frequency analysis.

Signal Subspace

The Graphically Select Signal and Noise Sub-Spaces signal selection is enabled for the AR,Data FB, AR,NmlEq FB, Prony, and MUSIC algorithms. The number of spectral components to be modeled in the parametric fit will be half the signal subspace. For an odd signal subspace, the component count will be (signal space+1)/2.

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.

Sinusoidal Model

The sinusoidal model can be Undamped or Damped. These models are implemented in AutoSignal as follows:

Sine : Y=Ampl*sin(2*Pi*Freq*X+Phase)

Sine, Exp Damped : Y=Ampl*exp(-k*X)*sin(2*Pi*Freq*X+Phase)

NL Optimization

The Prony algorithm directly furnishes estimates for the amplitudes, frequencies, phases, and damping factors. For the other algorithms, a separate linear fit is automatically made to determine the amplitudes, phases. Damping factors are initially set to zero. These are the starting estimates for the third stage, the non-linear optimization. If Enable is checked, all of the parameters are refined non-linearly. If Enable is not checked, the original estimates are used and the non-linear optimization is not made.

The Non-Linear Optimization Preferences option can be used to adjust the controls used in the non-linear fitting. The default preferences should suffice for most optimizations.

The non-linear optimization should produce the true least-squares or robust minimization, provided the first stage algorithm generated good frequency estimates. In many cases, the final parametric model will be the same, regardless of the first stage algorithm. Of the five uniformly spaced procedures, the FFT should be used cautiously since it may not produce accurate frequency estimates.

Data Processed

To facilitate prediction tests, it is possible to specify that only a portion of the data set be processed. The x start and x end values specify that starting and ending x values within the data stream that will be used to create the parametric model. Subsequent (or earlier) data can then be compared with the predicted values. The default values will specify the actual range of the input data.

Note that the order will be reduced automatically when it is required by the size of the subset being processed.

Output

For the output, the number of elements n are generated between x start and x end. Up to 65536 values can be generated. The starting and ending X values can precede or extend beyond the data to whatever degree is desired. The default limits will add an additional 25% to the upper end of the x range, and the default n will maintain the x spacing in the input data.

List

The List Data option lists the index, time, and output signal in a three column table. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the time and output signal values to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors).

Save

The Save Data to Disk option writes the time and output values 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 processed data can be exported to an Excel 95 or Excel 97 file.

Numeric

The Numeric button is used to display a full Numeric Summary. The report optionally includes fitted parameters, parameter statistics, an analysis of variance, and the fit details.

Data

The Data button opens a Data Summary which displays a point by point data summary for the fit with the X and Y value of each data point, the predicted value, the residual and %residual, the confidence limits, and the prediction limits. Only those points used in the fit are displayed.

Evaluation

The Evaluation option offers the means to more extensively evaluate the fit model. The evaluation can be be direct, or the computation can consist of first or second derivatives, roots, or integrated areas. This option can be used to generate a table or file of any size using a generated X grid or by importing X data from supported file formats. While the Evaluation procedure is active, all other processes are suspended.

Residuals

The Residuals button opens an AutoSignal Graph containing the residuals from the fit. The residuals are the difference between the data and fitted curve.

Export

All AutoSignal Graphs have the means to export the graph image to the clipboard or file as various forms of bitmaps and metafiles. Also included is an option that copies to the clipboard in spreadsheet format all of the numeric information used to produce the graph.

The Export button is used specifically to create ASCII, Excel, Lotus 123, Quattro Pro, or SigmaPlot disk files. These files use a generated x data stream and contain columns for the individual components as well as the overall composite curve.

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 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.

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