naginterfaces.library.fit.dim2_​spline_​ts_​sctr

naginterfaces.library.fit.dim2_spline_ts_sctr(x, y, f, lsminp, lsmaxp, nxcels, nycels, comm)

dim2_spline_ts_sctr computes a spline approximation to a set of scattered data using a two-stage approximation method.

The computational complexity of the method grows linearly with the number of data points; hence large datasets are easily accommodated.

Note: this function uses optional algorithmic parameters, see also: opt_set(), opt_get().

For full information please refer to the NAG Library document for e02jd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e02/e02jdf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The $(x_i,y_i,f_i)$ data values to be fitted.

yfloat, array-like, shape $\left(n\right)$

The $(x_i,y_i,f_i)$ data values to be fitted.

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

The $(x_i,y_i,f_i)$ data values to be fitted.

lsminpint

$lsminp$ and $lsmaxp$ are control parameters for the local approximations.

Each local approximation is computed on a local domain containing one of the triangles in the discretization of the bounding box.

The size of each local domain will be adaptively chosen such that if it contains fewer than $lsminp$ sample points it is expanded, else if it contains greater than $lsmaxp$ sample points a thinning method is applied. $lsmaxp$ mainly controls computational cost (in that working with a thinned set of points is cheaper and may be appropriate if the input data is densely distributed), while $lsminp$ allows handling of different types of scattered data.

Setting $lsmaxp<lsminp$, and, therefore, forcing either expansion or thinning, may be useful for computing initial coarse approximations.

In general smaller values for these arguments reduces cost.

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate values for $lsminp$ and $lsmaxp$.

lsmaxpint

$lsminp$ and $lsmaxp$ are control parameters for the local approximations.

Each local approximation is computed on a local domain containing one of the triangles in the discretization of the bounding box.

The size of each local domain will be adaptively chosen such that if it contains fewer than $lsminp$ sample points it is expanded, else if it contains greater than $lsmaxp$ sample points a thinning method is applied. $lsmaxp$ mainly controls computational cost (in that working with a thinned set of points is cheaper and may be appropriate if the input data is densely distributed), while $lsminp$ allows handling of different types of scattered data.

Setting $lsmaxp<lsminp$, and, therefore, forcing either expansion or thinning, may be useful for computing initial coarse approximations.

In general smaller values for these arguments reduces cost.

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate values for $lsminp$ and $lsmaxp$.

nxcelsint

$nxcels$ (respectively $nycels$) is the number of cells in the $x$ (respectively $y$) direction that will be used to create the triangulation of the bounding box of the domain of the function to be fitted.

Greater efficiency generally comes when $nxcels$ and $nycels$ are chosen to be of the same order of magnitude and are such that $\text{n}$ is $O(nxcels\times nycels)$.

Thus for a ‘square’ triangulation — when $nxcels=nycels$ — the quantities $\sqrt{n}$ and $nxcels$ should be of the same order of magnitude.

See also Further Comments.

nycelsint

$nxcels$ (respectively $nycels$) is the number of cells in the $x$ (respectively $y$) direction that will be used to create the triangulation of the bounding box of the domain of the function to be fitted.

Greater efficiency generally comes when $nxcels$ and $nycels$ are chosen to be of the same order of magnitude and are such that $\text{n}$ is $O(nxcels\times nycels)$.

Thus for a ‘square’ triangulation — when $nxcels=nycels$ — the quantities $\sqrt{n}$ and $nxcels$ should be of the same order of magnitude.

See also Further Comments.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to opt_set().

Other Parameters

‘Averaged Spline’str

Default $=\text{'NO'}$

When the bounding box is triangulated there are $8$ equivalent configurations of the mesh. Setting $\text{‘Averaged Spline'}=\text{'YES'}$ will use the averaged value of the $8$ possible local polynomial approximations over each triangle in the mesh. This usually gives better results but at (about 8 times) higher computational cost.

Constraint: $\text{‘Averaged Spline'}=\text{'YES'}$ or $\text{'NO'}$.

‘Global Smoothing Level’int

Default $=1$

The smoothness level for the global spline approximation.

$\text{‘Global Smoothing Level'}=1$

Will use $C^1$ piecewise cubics.

$\text{‘Global Smoothing Level'}=2$

Will use $C^2$ piecewise sextics.

‘Interpolation Only RBF’str

Default $=\text{'YES'}$

If $\text{‘Interpolation Only RBF'}=\text{'YES'}$, each local RBF approximation is computed by interpolation.

If $\text{‘Interpolation Only RBF'}=\text{'NO'}$, each local RBF approximation is computed by a discrete least squares approach. This is likely to be more accurate and more expensive than interpolation.

If $\text{‘Local Method'}=\text{'HYBRID'}$ or $\text{'POLYNOMIAL'}$, this option setting is ignored.

‘Local Method’str

Default $=\text{'POLYNOMIAL'}$

The local approximation scheme to use.

$\text{‘Local Method'}=\text{'POLYNOMIAL'}$

Uses least squares polynomial approximations.

$\text{‘Local Method'}=\text{'HYBRID'}$

Uses hybrid polynomial and RBF approximations.

$\text{‘Local Method'}=\text{'RBF'}$

Uses pure RBF approximations.

In general $\text{'POLYNOMIAL'}$ is less computationally expensive than $\text{'HYBRID'}$ is less computationally expensive than $\text{'RBF'}$ with the reverse ordering holding for accuracy of results.

‘Minimum Singular Value LHA’float

Default $=1.0$

A tolerance measure for accepting or rejecting a local hybrid approximation (LHA) as reliable.

The solution of a local least squares problem solved on each triangle subdomain is accepted as reliable if the minimum singular value $\sigma$ of the collocation matrix (of polynomial and radial basis function terms) associated with the least squares problem satisfies $\text{‘Minimum Singular Value LHA'}\le \sigma$.

In general the approximation power will be reduced as ‘Minimum Singular Value LHA’ is reduced. (A small $\sigma$ indicates that the local data has hidden redundancies which prevent it from carrying enough information for a good approximation to be made.) Setting ‘Minimum Singular Value LHA’ very large may have the detrimental effect that only approximations of low degree are deemed reliable.

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate value for this parameter.

If $\text{‘Local Method'}=\text{'POLYNOMIAL'}$ or $\text{'RBF'}$, this option setting is ignored.

‘Minimum Singular Value LPA’float

Default $=1.0$

A tolerance measure for accepting or rejecting a local polynomial approximation (LPA) as reliable. Clearly this setting is relevant when $\text{‘Local Method'}=\text{'POLYNOMIAL'}$, but it also may be used when $\text{‘Local Method'}=\text{'HYBRID'}$ (see Further Comments.)

The solution of a local least squares problem solved on each triangle subdomain is accepted as reliable if the minimum singular value $\sigma$ of the matrix (of Bernstein polynomial values) associated with the least squares problem satisfies $\text{‘Minimum Singular Value LPA'}\le \sigma$.

In general the approximation power will be reduced as ‘Minimum Singular Value LPA’ is reduced. (A small $\sigma$ indicates that the local data has hidden redundancies which prevent it from carrying enough information for a good approximation to be made.) Setting ‘Minimum Singular Value LPA’ very large may have the detrimental effect that only approximations of low degree are deemed reliable.

‘Minimum Singular Value LPA’ will have no effect if $\text{‘Polynomial Starting Degree'}=0$, and it will have little effect if the input data is ‘smooth’ (e.g., from a known function).

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate value for this parameter.

If $\text{‘Local Method'}=\text{'RBF'}$, this option setting is ignored.

‘Polynomial Starting Degree’int

Default $=5$ if $\text{‘Local Method'}=\text{'HYBRID'}$, Default $=1$ otherwise

The degree to be used for the polynomial part in the initial step of each local approximation.

At this initial step the method will attempt to fit with a local approximation having polynomial part of degree ‘Polynomial Starting Degree’. If $\text{‘Local Method'}=\text{'POLYNOMIAL'}$ and the approximation is deemed unreliable (according to ‘Minimum Singular Value LPA’), the degree will be decremented by one and a new local approximation computed, ending with a constant approximation if no other is reliable. If $\text{‘Local Method'}=\text{'HYBRID'}$ and the approximation is deemed unreliable (according to ‘Minimum Singular Value LHA’), a pure polynomial approximation of this degree will be tried instead. The method then proceeds as in the ‘POLYNOMIAL’ case.

‘Polynomial Starting Degree’ is bounded from above by the maximum possible spline degree, which is $6$ (when performing $C^2$ global super-smoothing). Note that the best-case approximation error (see Accuracy) for $C^2$ smoothing with $\text{‘Supersmooth C2'}=\text{'NO'}$ is achieved for local polynomials of degree $5$; that is, for this level of global smoothing no further benefit is gained by setting $\text{‘Polynomial Starting Degree'}=6$.

The default value gives a good compromise between efficiency and accuracy. In general the best approximation can be obtained by setting:

If $\text{‘Local Method'}=\text{'POLYNOMIAL'}$

if $\text{‘Global Smoothing Level'}=1$, $\text{‘Polynomial Starting Degree'}=3$;

if $\text{‘Global Smoothing Level'}=2$;

if $\text{‘Supersmooth C2'}=\text{'NO'}$, $\text{‘Polynomial Starting Degree'}=5$;

otherwise $\text{‘Polynomial Starting Degree'}=6$.

If $\text{‘Local Method'}=\text{'HYBRID'}$, ‘Polynomial Starting Degree’ as small as possible.

If $\text{‘Local Method'}=\text{'RBF'}$, this option setting is ignored.

‘Radial Basis Function’str

Default $=\text{'MQ'}$

‘Radial Basis Function’ selects the RBF to use in each local RBF approximation, while ‘Scaling Coefficient RBF’ selects the scale factor to use in its evaluation, as described below.

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate scale factor and RBF.

If $\text{‘Local Method'}=\text{'POLYNOMIAL'}$, these option settings are ignored.

If $\text{‘Local Method'}=\text{'HYBRID'}$ or $\text{'RBF'}$, the following (conditionally) positive definite functions may be chosen.

Define $R=\sqrt{x^2+y^2}$ and $\rho =R/r$.

$\text{'GAUSS'}$	Gaussian $exp(-{\rho }^2)$
$\text{'IMQ'}$	inverse multiquadric $1/\sqrt{r^2+R^2}$
$\text{'IMQ2'}$	inverse multiquadric $1/(r^2+R^2)$
$\text{'IMQ3'}$	inverse multiquadric $1/{(r^2+R^2)}^{\left(3/2\right)}$
$\text{'IMQ0\_5'}$	inverse multiquadric $1/{(r^2+R^2)}^{\left(1/4\right)}$
$\text{'WENDLAND31'}$	Wendland’s $C^2$ function $max{(0,1-\rho )}^4(4\rho +1)$
$\text{'WENDLAND32'}$	Wendland’s $C^4$ function $max{(0,1-\rho )}^6(35{\rho }^2+18\rho +3)$
$\text{'WENDLAND33'}$	Wendland’s $C^6$ function $max{(0,1-\rho )}^8(32{\rho }^3+25{\rho }^2+8\rho +1)$
$\text{'BUHMANNC3'}$	Buhmann’s $C^3$ function $112/45{\rho }^{\left(9/2\right)}+16/3{\rho }^{\left(7/2\right)}-7{\rho }^4-14/15{\rho }^2+1/9$ if $\rho \le 1$, $0$ otherwise
$\text{'MQ'}$	multiquadric $\sqrt{r^2+R^2}$
$\text{'MQ1\_5'}$	multiquadric ${(r^2+R^2)}^{\left(1.5/2\right)}$
$\text{'POLYHARMONIC1\_5'}$	polyharmonic spline ${\rho }^{1.5}$
$\text{'POLYHARMONIC1\_75'}$	polyharmonic spline ${\rho }^{1.75}$

If $\text{‘Local Method'}=\text{'HYBRID'}$ the following conditionally positive definite functions may also be chosen.

$\text{'MQ2'}$	multiquadric $(r^2+R^2)\log (r^2+R^2)$
$\text{'MQ3'}$	multiquadric ${(r^2+R^2)}^{\left(3/2\right)}$
$\text{'TPS'}$	thin plate spline ${\rho }^2\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC3'}$	polyharmonic spline ${\rho }^3$
$\text{'TPS4'}$	thin plate spline ${\rho }^4\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC5'}$	polyharmonic spline ${\rho }^5$
$\text{'TPS6'}$	thin plate spline ${\rho }^6\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC7'}$	polyharmonic spline ${\rho }^7$
$\text{'POLYHARMONIC9'}$	polyharmonic spline ${\rho }^9$

‘Scaling Coefficient RBF’float

Default $=1.0$

‘Radial Basis Function’ selects the RBF to use in each local RBF approximation, while ‘Scaling Coefficient RBF’ selects the scale factor to use in its evaluation, as described below.

A calibration procedure (experimenting with a small subset of the data to be fitted and validating the results) may be needed to choose the most appropriate scale factor and RBF.

If $\text{‘Local Method'}=\text{'POLYNOMIAL'}$, these option settings are ignored.

If $\text{‘Local Method'}=\text{'HYBRID'}$ or $\text{'RBF'}$, the following (conditionally) positive definite functions may be chosen.

Define $R=\sqrt{x^2+y^2}$ and $\rho =R/r$.

$\text{'GAUSS'}$	Gaussian $exp(-{\rho }^2)$
$\text{'IMQ'}$	inverse multiquadric $1/\sqrt{r^2+R^2}$
$\text{'IMQ2'}$	inverse multiquadric $1/(r^2+R^2)$
$\text{'IMQ3'}$	inverse multiquadric $1/{(r^2+R^2)}^{\left(3/2\right)}$
$\text{'IMQ0\_5'}$	inverse multiquadric $1/{(r^2+R^2)}^{\left(1/4\right)}$
$\text{'WENDLAND31'}$	Wendland’s $C^2$ function $max{(0,1-\rho )}^4(4\rho +1)$
$\text{'WENDLAND32'}$	Wendland’s $C^4$ function $max{(0,1-\rho )}^6(35{\rho }^2+18\rho +3)$
$\text{'WENDLAND33'}$	Wendland’s $C^6$ function $max{(0,1-\rho )}^8(32{\rho }^3+25{\rho }^2+8\rho +1)$
$\text{'BUHMANNC3'}$	Buhmann’s $C^3$ function $112/45{\rho }^{\left(9/2\right)}+16/3{\rho }^{\left(7/2\right)}-7{\rho }^4-14/15{\rho }^2+1/9$ if $\rho \le 1$, $0$ otherwise
$\text{'MQ'}$	multiquadric $\sqrt{r^2+R^2}$
$\text{'MQ1\_5'}$	multiquadric ${(r^2+R^2)}^{\left(1.5/2\right)}$
$\text{'POLYHARMONIC1\_5'}$	polyharmonic spline ${\rho }^{1.5}$
$\text{'POLYHARMONIC1\_75'}$	polyharmonic spline ${\rho }^{1.75}$

If $\text{‘Local Method'}=\text{'HYBRID'}$ the following conditionally positive definite functions may also be chosen.

$\text{'MQ2'}$	multiquadric $(r^2+R^2)\log (r^2+R^2)$
$\text{'MQ3'}$	multiquadric ${(r^2+R^2)}^{\left(3/2\right)}$
$\text{'TPS'}$	thin plate spline ${\rho }^2\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC3'}$	polyharmonic spline ${\rho }^3$
$\text{'TPS4'}$	thin plate spline ${\rho }^4\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC5'}$	polyharmonic spline ${\rho }^5$
$\text{'TPS6'}$	thin plate spline ${\rho }^6\log \left({\rho }^2\right)$
$\text{'POLYHARMONIC7'}$	polyharmonic spline ${\rho }^7$
$\text{'POLYHARMONIC9'}$	polyharmonic spline ${\rho }^9$

‘Separation LRBFA’float

Default $=16.0/\text{‘Scaling Coefficient RBF'}$

A knot-separation parameter used to control the condition number of the matrix used in each local RBF approximation (LRBFA). A smaller value may mean greater numerical stability but fewer knots.

If $\text{‘Local Method'}=\text{'HYBRID'}$ or $\text{'POLYNOMIAL'}$, this option setting is ignored.

‘Supersmooth C2’str

Default $=\text{'NO'}$

If $\text{‘Supersmooth C2'}=\text{'YES'}$, the $C^2$ spline is generated using additional smoothness constraints. This usually gives better results but at higher computational cost.

If $\text{‘Global Smoothing Level'}=1$ this option setting is ignored.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $4$)

On entry, $lsminp=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1\le lsminp\le n$.

(errno $5$)

On entry, $lsmaxp=⟨value⟩$.

Constraint: $lsmaxp\ge 1$.

(errno $6$)

On entry, $nxcels=⟨value⟩$.

Constraint: $nxcels\ge 1$.

(errno $7$)

On entry, $nycels=⟨value⟩$.

Constraint: $nycels\ge 1$.

(errno $9$)

Option arrays are not initialized or are corrupted.

(errno $11$)

An unexpected algorithmic failure was encountered. Please contact NAG.

(errno $12$)

On entry, all elements of $x$ or of $y$ are equal.

(errno $20$)

The selected radial basis function cannot be used with the RBF local method.

(errno $21$)

The value of option ‘Polynomial Starting Degree’ was invalid.

Notes

dim2_spline_ts_sctr determines a smooth bivariate spline approximation to a set of data points $(x_{\text{i}},y_{\text{i}},f_{\text{i}})$, for $\text{i}=1,2,\dots ,n$. Here, ‘smooth’ means $C^1$ or $C^2$. (You may select the degree of smoothing using the option ‘Global Smoothing Level’.)

The approximation domain is the bounding box $[x_{min},x_{max}]\times [y_{min},y_{max}]$, where $x_{min}$ (respectively $y_{min}$) and $x_{max}$ (respectively $y_{max}$) denote the lowest and highest data values of the $\left(x_i\right)$ (respectively $\left(y_i\right)$).

The spline is computed by local approximations on a uniform triangulation of the bounding box. These approximations are extended to a smooth spline representation of the surface over the domain. The local approximation scheme is controlled by the option ‘Local Method’. The schemes provided are: by least squares polynomial approximation (Davydov and Zeilfelder (2004)); by hybrid polynomial and radial basis function (RBF) approximation (Davydov et al. (2006)); or by pure RBF approximation (Davydov et al. (2005)).

The two-stage approximation method employed by dim2_spline_ts_sctr is derived from the TSFIT package of O. Davydov and F. Zeilfelder.

Values of the computed spline can subsequently be computed by calling dim2_spline_ts_evalv() or dim2_spline_ts_evalm().

References

Davydov, O, Morandi, R and Sestini, A, 2006, Local hybrid approximation for scattered data fitting with bivariate splines, Comput. Aided Geom. Design (23), 703–721

Davydov, O, Sestini, A and Morandi, R, 2005, Local RBF approximation for scattered data fitting with bivariate splines, Trends and Applications in Constructive Approximation, M. G. de Bruin, D. H. Mache, and J. Szabados, Eds (ISNM Vol. 151), Birkhauser, 91–102

Davydov, O and Zeilfelder, F, 2004, Scattered data fitting by direct extension of local polynomials to bivariate splines, Advances in Comp. Math. (21), 223–271

See also

naginterfaces.library.examples.fit.dim2_spline_ts_sctr_ex.main()