2.2.1.9 xyz_shep_nag
Brief Information
Convert XYZ data to matrix using Modified Shepard gridding
Command Line Usage
1. xyz_shep_nag iz:=Col(3);
2. xyz_shep_nag iz:=Col(3) rows:=10 cols:=10;
4. xyz_shep_nag iz:=Col(3) q:=18 w:=9;
5. xyz_shep_nag iz:=Col(3) om:=[MBook]MSheet!Mat(1);
Variables
Display Name

Variable Name

I/O and Type

Default Value

Description

Input

iz

Input
XYZRange

<active>

Specifies the input XYZ range.

Rows

rows

Input
int

20

Rows in the output matrix.

Columns

cols

Input
int

20

Columns in the output matrix.

Quadratic

q

Input
int

18

The quadratic interplant locality factor, which is used to calculate the influence radius of local approximate quadratic fitted function for each node. By default, q equals to 18. It is better to make q ≈ 2w and it should be satisfy that 0<w≤q. Modifying these factors could increase gridding accuracy, though note that the computation time can be greatly increased for large values (i.e. values that decrease the locality of the method.)

Weight

w

Input
int

9

The weight function locality factor, which is used to calculate the weighting radius for each node. By default, w equals to 9. It is better to make q≈2w and it should be satisfy that 0<w<q. Modifying these factors could increase gridding accuracy, though note that the computation time can be greatly increased for large values (i.e. values that decrease the locality of the method.)

Output Matrix

om

Output
MatrixObject

<new>

Specify the output matrix object.
See the syntax here.

Description
This function calls NAG library to perform modified Shepard gridding method which described by Franke and Nielson^{[}^{1]}. This is a distancebased method and improves the Shepard's method by some local strategies. During gridding, only the data points that lying within certain ranges, and , to the grid nodes are considered. To make it easier for setting, two integers, and are used to calculate and (parameters q and w of the function, and called Quadratic Interplant Locality Factor and Weight Function Locality Factor, respectively). Increase the value of and will make the calculation more global, vice versa. Generally speaking, setting ≈ works fine and by default, Nq=18, Nw=9. However, the following constraints: 0< ≤ should be satisfied.
The value of and in this function is fixed and there is another similar XFunction xyz_shep which described by Renka^{[2]} uses a vary and strategy.
Examples
1. Import XYZ Random Gaussian.dat on the \Samples\Matrix Conversion and Gridding folder.
2. Type xyz_shep_nag 3 in the command window. Or type xyz_shep_nag d to bring up the dialog.
Algorithm
This is a distancebased weighted gridding method which interpolate data by:
,
where is the underlying function at nodes (, ), and is the weights. To make the function more local, and are calculated only by the data points lying in the circle with center (, ) and some radius R..
Firstly, the weights are defined as:
.
Given a radius , the relative weight is:
for
,
and is the Euclidean distance between (x, y) and (, ):
.
For any >0, we have:
.
Secondly, the nodal function is replaced by a local approximation function :
is the weighted leastsquare quadratic fitted function to the data located within of nodal points. So the coefficients minimize:
for
.
It can be seen above that the interpolate function is a local approximation function and depends on the radius of influence about nodal points, and . In this method, two integers and are used to calculate and :
and ,
where n is the number of data points and D is the maximum distance between any pair of data points. So and can be considered to be the average numbers of data points lying within distance and respectively for each node.
References
[1]. Franke R and Nielson G. smooth Interpolation of Large Sets of Scattered Data. Internat. J.Num. Methods Engrg. 1980, 15, pp:16911704.
[2]. Renka, R. J., Multivariate Interpolation of Large Sets of Scattered Data. ACM Transactions on Mathematical Software, Vol. 14, No. 2, June 1988, pp:139148.
Related XFunctions
xyz_regular, xyz_renka, xyz_renka_nag, xyz_shep, xyz_sparse, xyz_tps
Keywords:worksheet
