Multivariate interpolation: Difference between revisions

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{{short description|Interpolation on functions of more than one variable}}

In [[numerical analysis]], '''multivariate interpolation''' is [[interpolation]] on functions of more than one variable; when the variates are [[spatial coordinate]]s, it is also known as '''spatial interpolation'''.

In [[numerical analysis]], '''multivariate interpolation''' or '''multidimensional interpolation''' is [[interpolation]] on ''[[multivariate function]]s'', having more than one variable or defined over a [[multi-dimensional]] ___domain.<ref>Jetter, Kurt; Buhmann, Martin D.; Haussmann, Werner; Schaback, Robert; and Stöckler, Joachim: ''Topics in Multivariate Approximation and Interpolation'', Elsevier, ISBN 0-444-51844-4 (2006)</ref> A common special case is '''bivariate interpolation''' or '''two-dimensional interpolation''', based on [[bivariate function|two variables]] or [[two-dimensional space|two dimensions]]. When the variates are [[spatial coordinate]]s, it is also known as '''spatial interpolation'''.

The function to be interpolated is known at given points <math>(x_i, y_i, z_i, \dots)</math> and the interpolation problem consist of yielding values at arbitrary points <math>(x,y,z,\dots)</math>.

The function to be interpolated is known at given points <math>(x_i, y_i, z_i, \dots)</math> and the interpolation problem consistconsists of yielding values at arbitrary points <math>(x,y,z,\dots)</math>.

Multivariate interpolation is particularly important in [[geostatistics]], where it is used to create a [[digital elevation model]] from a set of points on the Earth's surface (for example, spot heights in a [[topographic survey]] or depths in a [[hydrographic survey]]).

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===Any dimension===

* [[Nearest-neighbor interpolation]]

* n-linear interpolation (see [[bilinear interpolation|bi-]] and [[trilinear interpolation]] and [[multilinear polynomial]])

* n-cubic interpolation (see [[bicubic interpolation|bi-]] and [[tricubic interpolation]])

* [[Kriging]]

* [[Inverse distance weighting]]

* [[Natural -neighbor interpolation]]

* [[Spline interpolation]]

* [[Radial basis function interpolation]]

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* [[Barnes interpolation]]

* [[Bilinear interpolation]]

* [[Bicubic interpolation]]

* [[Bézier surface]]

* [[Lanczos resampling]]

* [[Delaunay triangulation]]

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===Tensor product splines for ''N'' dimensions===

Catmull–Rom splines can be easily generalized to any number of dimensions. The [[cubic Hermite spline]] article will remind you that <math>\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \mathbf{b}(x) \cdot \left( f_{-1} f_0 f_1 f_2 \right)</math> for some 4-vector <math>\mathbf{b}(x)</math> which is a function of ''x'' alone, where <math>f_j</math> is the value at <math>j</math> of the function to be interpolated.

Catmull-Rom splines can be easily generalized to any number of dimensions.

The [[cubic Hermite spline]] article will remind you that <math>\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \mathbf{b}(x) \cdot \left( f_{-1} f_0 f_1 f_2 \right)</math> for some 4-vector <math>\mathbf{b}(x)</math> which is a function of ''x'' alone, where <math>f_j</math> is the value at <math>j</math> of the function to be interpolated.

Rewrite this approximation as

:<math>

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\mathrm{CR}(x_1,\dots,x_N) = \sum_{i_1,\dots,i_N=-1}^2 f_{i_1\dots i_N} \prod_{j=1}^N b_{i_j}(x_j)

</math>

Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines. In regards to efficiency, the general formula can in fact be computed as a composition of successive <math>\mathrm{CINT}</math>-type operations for any type of tensor product splines, as explained in the [[tricubic interpolation]] article. However, the fact remains that if there are <math>n</math> terms in the 1-dimensional <math>\mathrm{CR}</math>-like summation, then there will be <math>n^N</math> terms in the <math>N</math>-dimensional summation.

In regards to efficiency, the general formula can in fact be computed as a composition of successive <math>\mathrm{CINT}</math>-type operations for any type of tensor product splines, as explained in the [[tricubic interpolation]] article.

However, the fact remains that if there are <math>n</math> terms in the 1-dimensional <math>\mathrm{CR}</math>-like summation, then there will be <math>n^N</math> terms in the <math>N</math>-dimensional summation.

== Irregular grid (scattered data) ==

Schemes defined for scattered data on an [[irregular grid]] are more general.

They should all work on a regular grid, typically reducing to another known method.

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* [[Triangulated irregular network]]-based [[natural neighbor]]

* [[Triangulated irregular network]]-based [[linear interpolation]] (a type of [[piecewise linear function]])

** n-[[simplex]] (e.g. tetrahedron) interpolation (see [[barycentric coordinate system]])

* [[Inverse distance weighting]]

* [https://surgeweb.mzf.cz/AOSIM/ABOS.htm ABOS - approximation based on smoothing]

* [[Kriging]]

* [[Gradient-enhanced kriging]] (GEK)

* [[Thin plate spline|Thin-plate spline]]

* [[Polyharmonic spline]] (theThe thin-plate- spline is a special case of a polyharmonic spline.)

* [[Radial basis function]] ([[Polyharmonic spline]]s are a special case of radial basis functions with low degree polynomial terms.)

* Least-squares [[spline (mathematics)|spline]]

* [[Natural -neighbour interpolation]]

{{anchor|Gridding}}''Gridding'' is the process of converting irregularly spaced data to a regular grid ([[gridded data]]).

==See also==

* [[Smoothing]]

* [[Surface fitting]]

==Notes==