Non-Parametric Estimation

The Non-Parametric Estimation option in the Time menu or the Time toolbar offers an adjustable order Loess-type (locally-weighted least-squares) procedure with two different weighting functions. Although this procedure lacks both the continuity and smoothness of a spline of sufficient order, it can sometimes extract an underlying low frequency data pattern in extremely noisy data. This procedure is also useful for both upsampling and downsampling since the range and number of output points are specified. Uniform data are not required.

The input data and local regression curve are shown in an AutoSignal Graph for rapid visualization of the effectiveness of the fit.

Algorithm

This procedure is a fitting and smoothing algorithm that performs a series of weighted least squares fits in a moving window across the data. The AutoSignal non-parametric local regression estimation algorithm is similar to the Loess non-parametric estimation procedure. The AutoSignal algorithm also offers a Gaussian weighting function in addition to the tricube weighting function used in Loess. It also offers variable order fitting, from linear through quartic. For more information on the Loess algorithm, you may wish to refer to William S. Cleveland, "Visualizing Data", 1993, Hobart Press, Summit, NJ, ISBN 0-9634884-0-6.

Weighting Function

The tricube weighting function gives a much greater weight to nearby points as compared to a Gaussian. The following graph depicts both the tricube function and the particular form of Gaussian AutoSignal implements for this algorithm:

Model

You can choose to locally fit a linear (order 1), quadratic (order 2), cubic (order 3), or quartic (order 4) model to the data.

Data Window

The window minimum value incurs the minimum of smoothing. Higher window counts can be used to increasingly smooth the data.

Sectioning Data

You may enable or disable regions of data or individual points just as within autosignal's main Section option. When the state of one or more points is changed, the local regression is recomputed and the graph is updated.

Goodness of Fit

AutoSignal computes a Standard Error and rē value based upon the estimation at the X data values. It is recommended that you use these values only to assess the different models and data window sizes. Note that the best goodness of fit values occur at the smallest window count, one that tends to produce significantly discontinuous derivatives, especially with noisy data and higher order models. Local regression does not produce continuous derivatives. You should use the Spline Estimation option if continuous derivatives are required.

Output

For the output, the number of elements n is generated between x start and x end. Up to 65536 values can be generated. The starting and ending X values should be at or within the range of the input data. The default values will perform a 2x upsampling within the x range of the data.

List

The List Data option lists the index, time, and output 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 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 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.

Evaluation

The Evaluation option offers the means to more extensively evaluate the spline 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.

Residuals

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

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 interpolated/smoothed data.