NP Algorithm Recommendations¶
The Non-Parametric Menu contains the following items:
Fill Sparse Grid - The data set is assumed to be grid-based with incomplete elements, as arise from incomplete sampling, from removing outliers in a grid of data, and from design of experiments matrices. The output dimensions of the x and y grid are determined by the number of unique x and y data values. The interpolation is performed only at those positions in the grid where data are missing. The data table is augmented by these interpolated grid elements. A complete regular grid is required for the Estimate Gridded Data option. Although that option automatically performs a sparse grid interpolation as a preprocessing step for scattered data, this Fill Sparse Grid option offers the flexibility of algorithm choice. Ten state of the art scattered data algorithms are available.
Interpolate Uniform Grid - The data table is replaced by interpolated data on a uniform grid. The output dimensions of the x and y grid can be independently set. Data reduction (decreasing the data count) and data enhancement (increasing the count) are supported. A regular grid is required for the Estimate Gridded Data option and may possibly be of merit in constraining the fit of certain parametric models if there are data-barren regions. Ten state of the art scattered data algorithms are available. All furnish an exact interpolation of the input nodes and scale based upon the input data.
Estimate Gridded Data - This option requires grid-structured data, although the x and y grids need not be uniformly spaced. If this option is invoked without a regular grid being present, the current data set is assumed to be an incomplete or sparse grid, and the missing elements are interpolated using the Renka I procedure before the processing by one of five gridded interpolation algorithms. These include bivariate B-splines orders 2 through 5 with the x and y order separately adjustable, least-squares reduced knot B-splines, a regular grid procedure using an Akima-type interpolant, a bicubic B-spline that uses the Akima procedure for extrapolations, and non-uniform rational B-splines (NURBS). This procedure produces non-parametric output rather than an XYZ object.
Estimate Scattered Data - These procedures are designed for interpolating and extrapolating scattered data, although gridded data can also be processed. Eight different algorithms are available. All algorithms offer extrapolations. All algorithms also offer once continuously differentiable interpolants, with several offering a twice continuously differentiable interpolant. The Renka I algorithm additionally interpolates gradients, and as such offers smooth first and second order partial derivative surfaces. The algorithms include triangulation, nearest neighbor, and natural neighbor based interpolations. The Loess procedure is available for smoothed estimations. All other algorithms reproduce the input data. As with the gridded data estimation, this procedure produces non-parametric output rather than an XYZ object.
The two non-parametric estimation options are particularly useful when a parametric model is not essential and when the surface contains too many features, ripples, or other trends for a parametric model to yield good results. The local nature of the non-parametric interpolation procedures makes it possible for even the most difficult of surfaces to be successfully mapped.
Algorithm Suggestions¶
Each of the interpolation algorithms within the estimation options are thought to have specific value with certain data features. In general, the most accurate algorithm is likely to be a B-spline. If you have gridded data, a B-spline is encouraged. If you have a sparse grid (an x-y matrix of data with empty positions), the B-spline may also be the best choice. The Estimate Gridded Data option will automatically complete the grid, or for greater control, the Fill Sparse Grid option can be used. If you also need extrapolations, a special option adds the Akima extrapolant for bicubics on rectangles to a bicubic B-spline. This Bicubic+Akima option is especially recommended.
For scattered data, the interpolation options are extensive. Strongly recommended are the Akima II, Preusser, and Renka I, II, and III procedures. These procedures are robust, they tend to be quite accurate. The Renka I procedure offers smooth higher order partial derivatives, even across interpolation-extrapolation boundaries.
For smoothed estimations, the Loess procedure in the Estimate Scattered Data option is recommended over the Least-squares B-spline or NURBS in the Estimate Gridded Data option. The smoothing at any given node in a least-squares B-spline will vary with knot proximity. The smoothing that occurs with a non-uniform rational B-spline is adjustable only via spline order.
Recommendations:¶
Aim is to Make Select Interpolations or Extrapolations and no Parametric Model is Needed - Use the Estimate Gridded Data if the current data table consists of a regular grid of x and y values. Use the Estimate Scattered Data option if a regular grid is absent. The Evaluation procedure is used for individual estimations of the surface, partial derivatives, finding roots, computing volumes, and generating XYZ tables using a uniform grid or input XY data. A Quick Evaluation option allows a single interpolation or extrapolation, surface minimum, and surface maximum to be displayed automatically as algorithms and settings are varied.
Aim is to Smooth Data for Subsequent Fitting - Use the Table menu's Smooth Loess option. It offers a Loess-type nearest-neighbor procedure using planar as well as Taylor second and third order bivariate polynomials. The Loess procedure is an excellent time-domain smoother, and this option automatically updates the data table. The Loess procedure is recommended for both scattered and gridded data.
Aim is to Process Scattered Data for Subsequent Spline Interpolation - Bicubic spline procedures offer excellent accuracy, but require a regular grid. When working with scattered data, you can simply invoke the Estimate Gridded Data option. Incoming scattered data sets are automatically processed as an incomplete grid using the Renka I procedure and the main TableCurve 3D data table is not altered in any way. For full control over the algorithm used to complete the grid, however, you will need to first use the Fill Sparse Grid option.
Aim is to Complete a Design of Experiments Matrix by Interpolation - Use the Fill Sparse Grid option prior to non-parametric estimation or parametric fitting.
Aim is to Generate Data on a Grid to Constrain Parametric Fits in Regions Lacking Data - Use either the Interpolate Uniform Grid or Fill Sparse Grid option. Both options generate output data on grid. Most of the capability of the Estimate Scattered Data option is automatically included. The Fill Sparse Grid option will preserve the accuracy of existing data points, whereas the Interpolate Uniform Grid offers the ability to control the grid counts in each dimension.
Aim is Simply to Visualize the Surface or Partial Derivatives - Use the Estimate Gridded Data if the current data table consists of a regular grid of x and y values. Use the Estimate Scattered Data option if a regular grid is absent. Both options furnish the means to plot the first order partial derivative with respect to x or y, the partial with respect to both x and y, and the second order partials with respect to x or y. For scattered data and smooth partial derivatives, the Renka I algorithm is recommended. For data on a grid, a B-spline of at least order 3 in each dimension should be used for smooth first order partials. A B-spline of at least order 4 in each dimension is needed for smooth continuous second order partial derivatives, although the lack of smoothness in the higher order derivatives with a bicubic spline may be minor.
Aim is to Generate a Partial Derivative Surface for Subsequent Parametric Fitting: Use the Generate Table capability in the Eval option of either the Estimate Gridded Data or Estimate Scatted Data options. The X,Y for the table can be generated or imported from any of TableCurve 3D's supported file formats. The table must then be saved to disk and added to the TableCurve 3D explorer using the Import Data Source option. Alternately, the data can be copied to the clipboard and processed by the Import Clipboard option.
Aim is to Compute an Accurate Volume: Use the Estimate Gridded Data if the current data table consists of a regular grid of x and y values. The computation will be exact for all algorithms. Use the Estimate Scattered Data option if a regular grid is absent. In both cases, the Evaluation procedure contains a very fast and accurate means for computing volumes. A repeat volume evaluation at the same limits allows verification by an alternate integration method. The Watson C0 algorithm may be of interest with topographical data.
Aim is to Digitally Filter Data Table: Use the Interpolate Uniform Grid option to reduce the size of a data table for faster fitting. Be sure to preserve in both the x and y dimensions the degrees of freedom needed for the models desired. For example, in order to have a tenth order bivariate polynomial available for fitting you must generate an 11 x 11 grid. Note that the Import Digital Filter simply samples every nth data point in the input stream and is normally of value only for extremely large data sets.