2.2.1.9 xyz_shep_nag

1 Brief Information
2 Command Line Usage
3 X-Function Execution Options
4 Variables
5 Description
6 Examples
7 Algorithm
8 References
9 Related X-Functions

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);

X-Function Execution Options

Please refer to the page for additional option switches when accessing the x-function from script

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 distance-based method and improves the Shepard's method by some local strategies. During gridding, only the data points that lying within certain ranges, $R_q$ and $R_w$ , to the grid nodes are considered. To make it easier for setting, two integers, $N_q$ and $N_w$ are used to calculate $R_q$ and $R_w$ (parameters q and w of the function, and called Quadratic Interplant Locality Factor and Weight Function Locality Factor, respectively). Increase the value of $N_q$ and $N_w$ will make the calculation more global, vice versa. Generally speaking, setting $N_q$ ≈ $2*N_w$ works fine and by default, Nq=18, Nw=9. However, the following constraints: 0< $N_w$ ≤ $N_q$ should be satisfied.

The value of $R_q$ and $R_w$ in this function is fixed and there is another similar X-Function xyz_shep which described by Renka^[2] uses a vary $R_q$ and $R_w$ 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 distance-based weighted gridding method which interpolate data by:

where $F_i$ is the underlying function at nodes ( $x_i$ , $y_i$ ), and $W_i(x, y)$ is the weights. To make the function more local, $F_i$ and $W_i$ are calculated only by the data points lying in the circle with center ( $x_i$ , $y_i$ ) and some radius R..

Firstly, the weights are defined as:

Given a radius $R_w$ , the relative weight $w_k$ is:

for

and $d_k$ is the Euclidean distance between (x, y) and ( $x_k$ , $y_k$ ):

For any $R_w$ >0, we have:

Secondly, the nodal function $F_i$ is replaced by a local approximation function $Q_k$ :

$Q_k$ is the weighted least-square quadratic fitted function to the data located within $R_q$ 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, $R_q$ and $R_w$ . In this method, two integers $N_q$ and $N_w$ are used to calculate $R_q$ and $R_w$ :

and ,

where n is the number of data points and D is the maximum distance between any pair of data points. So $N_q$ and $N_w$ can be considered to be the average numbers of data points lying within distance $R_q$ and $R_w$ 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:1691-1704.

[2]. Renka, R. J., Multivariate Interpolation of Large Sets of Scattered Data. ACM Transactions on Mathematical Software, Vol. 14, No. 2, June 1988, pp:139-148.