naginterfaces.library.interp.dimn_​scat_​shep

naginterfaces.library.interp.dimn_scat_shep(x, f, nw=- 1, nq=- 1)

dimn_scat_shep generates a multidimensional interpolant to a set of scattered data points, using a modified Shepard method. When the number of dimensions is no more than five, there are corresponding functions in submodule interp which are specific to the given dimensionality. dim2_scat_shep() generates the two-dimensional interpolant, while dim3_scat_shep(), dim4_scat_shep() and dim5_scat_shep() generate the three-, four - and five-dimensional interpolants respectively.

For full information please refer to the NAG Library document for e01zm

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e01/e01zmf.html

Parameters

xfloat, array-like, shape $(d,m)$

The $\text{d}$ components of the first data point must be stored in elements $0,1,\dots ,d-1$ of $x$. The second data point must be stored in elements $d,d+1,\dots ,2\times d-1$ of $x$, and so on. In general, the $\text{m}$ data points must be stored in $x[\text{j},\text{i}]$, for $\text{j}=0,1,\dots ,d-1$, for $\text{i}=0,1,\dots ,m-1$.

ffloat, array-like, shape $\left(m\right)$

$f[r-1]$ must be set to the data value $f_{\text{r}}$, for $\text{r}=1,2,\dots ,m$.

nwint, optional

The number $N_w$ of data points that determines each radius of influence $R_w$, appearing in the definition of each of the weights $w_{\text{r}}$, for $\text{r}=1,2,\dots ,m$ (see Notes). Note that $R_w$ is different for each weight. If $nw\le 0$ the default value $nw=min(2\times (d+1)\times (d+2),m-1)$ is used instead.

nqint, optional

The number $N_q$ of data points to be used in the least squares fit for coefficients defining the quadratic functions $q_r\left(x\right)$ (see Notes). If $nq\le 0$ the default value $nq=min((d+1)\times (d+2)\times 6/5,m-1)$ is used instead.

Returns

iqint, ndarray, shape $(2\times m+1)$

Integer data defining the interpolant $Q\left(x\right)$.

rqfloat, ndarray, shape $\left(\text{lrq}\right)$

Real data defining the interpolant $Q\left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $nw=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $nw\le m-1$.

(errno $1$)

On entry, $nq=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $nq\le m-1$.

(errno $1$)

On entry, $nq=⟨value⟩$ and $d=⟨value⟩$.

Constraint: $nq\le 0$ or $nq\ge (d+1)\times (d+2)/2-1$.

(errno $1$)

On entry, $m=⟨value⟩$ and $d=⟨value⟩$.

Constraint: $m\ge (d+1)\times (d+2)/2+2$.

(errno $1$)

On entry, $((d+1)\times (d+2)/2)\times m+2\times d+1$ exceeds the largest machine integer.

$d=⟨value⟩$ and $m=⟨value⟩$.

(errno $1$)

On entry, $d=⟨value⟩$.

Constraint: $d\ge 2$.

(errno $2$)

There are duplicate nodes in the dataset. $x[k-1,i-1]=x[k-1,j-1]$, for $i=⟨value⟩$, $j=⟨value⟩$ and $k=1,2,\dots ,d$. The interpolant cannot be derived.

(errno $3$)

On entry, all the data points lie on the same hypersurface. No unique solution exists.

Notes

dimn_scat_shep constructs a smooth function $Q\left(x\right)$, $x\in R^d$ which interpolates a set of $m$ scattered data points $(x_r,f_r)$, for $r=1,2,\dots ,m$, using a modification of Shepard’s method. The surface is continuous and has continuous first partial derivatives.

The basic Shepard method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean

$$Q\left(x\right)=\frac{{\sum }_{r=1}^m{w}_r\left(x\right)q_r}{{\sum }_{r=1}^m{w}_r\left(x\right)}\text{,}$$

where $q_r=f_r$, $w_r\left(x\right)=\frac{1}{{\parallel x-x_r\parallel }_2^2}$.

The basic method is global in that the interpolated value at any point depends on all the data, but dimn_scat_shep uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each $w_r\left(x\right)$ to be zero outside a hypersphere with centre $x_r$ and some radius $R_w$. Also, to improve the performance of the basic method, each $q_r$ above is replaced by a function $q_r\left(x\right)$, which is a quadratic fitted by weighted least squares to data local to $x_r$ and forced to interpolate $(x_r,f_r)$. In this context, a point $x$ is defined to be local to another point if it lies within some distance $R_q$ of it.

The efficiency of dimn_scat_shep is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of $3$.

The radii $R_w$ and $R_q$ are chosen to be just large enough to include $N_w$ and $N_q$ data points, respectively, for user-supplied constants $N_w$ and $N_q$. Default values of these parameters are provided, and advice on alternatives is given in Further Comments.

dimn_scat_shep is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).

Values of the interpolant $Q\left(x\right)$ generated by dimn_scat_shep, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to dimn_scat_shep_eval().

References

Bentley, J L and Friedman, J H, 1979, Data structures for range searching, ACM Comput. Surv. (11), 397–409

Berry, M W, Minser, K S, 1999, Algorithm 798: high-dimensional interpolation using the modified Shepard method, ACM Trans. Math. Software (25), 353–366

Franke, R and Nielson, G, 1980, Smooth interpolation of large sets of scattered data, Internat. J. Num. Methods Engrg. (15), 1691–1704

Renka, R J, 1988, Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software (14), 139–148

Renka, R J, 1988, Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data, ACM Trans. Math. Software (14), 151–152

Shepard, D, 1968, A two-dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, 517–523, Brandon/Systems Press Inc., Princeton