regularizeNd

Create a gridded lookup table of scattered data in n dimensions.

regularizeNd Fits a nD lookup table with smoothness to scattered data. Constraints are possible.
regularizeNd answers the question what is the best possible lookup table that the scattered data input x and output y in the least squares sense with smoothing? regularizeNd is meant to calculate a smooth lookup table given n-D scattered data. regularizeNd supports extrapolation from a scattered data set.
The calculated lookup table, yGrid, is meant to be used with griddedInterpolant class with the conservative memory form. Call griddedInterpolant like

xGrid = cell array of grid vectors
smoothness = smoothness value or vector
yGrid = regularizeNd(xData, yData, xGrid, smoothness);
F = griddedInterpolant(xGrid, yGrid).

Desirable properties of regularizeNd:
-Calculates a relationship between the input x and the output y without definition of the functional form of x to y.
-Often the fit is superior to polynomial type fitting without the wiggles.
-Extrapolation is possible from a scattered data set.
-After creating the lookup table yGrid and using it with griddedInterpolant, as the query point moves away from the scattered data, the relationship between the input x and output y becomes more linear because of the smoothness equations and no nearby fidelity equations. The linear relationship is a good choice when the relationship between x and y is unknown in extrapolation.
-regularizeNd can handle 1D, 2D, nD input data to 1D output data. RegularizeData3D and gridfit can only handle 2D input and 1D out (total 3D).
-regularizeNd can handle setting the smoothness to 0 in any/some axis/dimension. This means no smoothing is applied in a particular axis/dimension and the data is just a least squares fit of a lookup table in that axis/dimension. Note this is not recommended and often can lead to an ill-conditioned fitting problem. However, I have found it useful so I left this as an option.
- Constraints are possible with the function regularizeNdMatrices. See the example.

The source code is locate here:
https://github.com/jasonnicholson/regularizeNd

For an introduction on how regularization of a lookup table works, start here: https://mathformeremortals.wordpress.com/2013/01/29/introduction-to-regularizing-with-2d-data-part-1-of-3/

Acknowledgement
Special thanks to Peter Goldstein, author of RegularizeData3D, for his coaching and help through writing regularizeNd.