Interpolation

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In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. [1] [2]

Contents

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.

An interpolation of a finite set of points on an epitrochoid. The points in red are connected by blue interpolated spline curves deduced only from the red points. The interpolated curves have polynomial formulas much simpler than that of the original epitrochoid curve. Splined epitrochoid.svg
An interpolation of a finite set of points on an epitrochoid. The points in red are connected by blue interpolated spline curves deduced only from the red points. The interpolated curves have polynomial formulas much simpler than that of the original epitrochoid curve.

Example

This table gives some values of an unknown function .

Plot of the data points as given in the table Interpolation Data.svg
Plot of the data points as given in the table
00
10.8415
20.9093
30.1411
40.7568
50.9589
60.2794

Interpolation provides a means of estimating the function at intermediate points, such as

We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function.

Piecewise constant interpolation

Piecewise constant interpolation, or nearest-neighbor interpolation Piecewise constant.svg
Piecewise constant interpolation, or nearest-neighbor interpolation

The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.

Linear interpolation

Plot of the data with linear interpolation superimposed Interpolation example linear.svg
Plot of the data with linear interpolation superimposed

One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.

Generally, linear interpolation takes two data points, say (xa,ya) and (xb,yb), and the interpolant is given by:

This previous equation states that the slope of the new line between and is the same as the slope of the line between and

Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point xk.

The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, and suppose that x lies between xa and xb and that g is twice continuously differentiable. Then the linear interpolation error is

In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomial interpolation and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.

Polynomial interpolation

Plot of the data with polynomial interpolation applied Interpolation example polynomial.svg
Plot of the data with polynomial interpolation applied

Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function. We now replace this interpolant with a polynomial of higher degree.

Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:

Substituting x = 2.5, we find that f(2.5) = ~0.59678.

Generally, if we have n data points, there is exactly one polynomial of degree at most n1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation.

However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon).

Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈ 1.566, f(x) ≈ 1.003 and a local minimum at x ≈ 4.708, f(x) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false vertical asymptotes.

More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials.

Spline interpolation

Plot of the data with spline interpolation applied Interpolation example spline.svg
Plot of the data with spline interpolation applied

Linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.

For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by

In this case we get f(2.5) = 0.5972.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress. [3]

Mimetic interpolation

Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).

A key feature of mimetic interpolation is that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem. As a result, mimetic interpolation conserves line, area and volume integrals. [4] Conservation of line integrals might be desirable when interpolating the electric field, for instance, since the line integral gives the electric potential difference at the endpoints of the integration path. [5] Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path.

Linear, bilinear and trilinear interpolation are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed. [6]

Function approximation

Interpolation is a common way to approximate functions. Given a function with a set of points one can form a function such that for (that is, that interpolates at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if (four times continuously differentiable) then cubic spline interpolation has an error bound given by where and is a constant. [7]

Via Gaussian processes

Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as Kriging.

Other forms

Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Padé approximant, and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series. Another possibility is to use wavelets.

The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite or if the function to be interpolated has compact support.

Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems.

When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to the displacement interpolation problem used in transportation theory.

In higher dimensions

Comparison of some 1- and 2-dimensional interpolations.
Black and red
/yellow
/green
/blue dots correspond to the interpolated point and neighbouring samples, respectively.
Their heights above the ground correspond to their values. Comparison of 1D and 2D interpolation.svg
Comparison of some 1- and 2-dimensional interpolations.
Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively.
Their heights above the ground correspond to their values.

Multivariate interpolation is the interpolation of functions of more than one variable. Methods include bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to dimensional spaces where . [8] [9]

In digital signal processing

In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (Upsampling) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of the original signal above the original Nyquist limit of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing. [10]

The term extrapolation is used to find data points outside the range of known data points.

In curve fitting problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to least squares approximation.

Approximation theory studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function.

Generalization

If we consider as a variable in a topological space, and the function mapping to a Banach space, then the problem is treated as "interpolation of operators". [11] The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.

See also

Related Research Articles

<span class="mw-page-title-main">B-spline</span> Spline function

In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data.

<span class="mw-page-title-main">Linear interpolation</span> Method of curve fitting to construct new data points within the range of known data points

In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

<span class="mw-page-title-main">Runge's phenomenon</span> Failure of convergence in interpolation

In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by Carl David Tolmé Runge (1901) when exploring the behavior of errors when using polynomial interpolation to approximate certain functions. The discovery shows that going to higher degrees does not always improve accuracy. The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations.

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset.

In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments.

<span class="mw-page-title-main">Spline (mathematics)</span> Mathematical function defined piecewise by polynomials

In mathematics, a spline is a function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees.

<span class="mw-page-title-main">Extrapolation</span> Method for estimating new data outside known data points

In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a knowledge of the unknown. The extrapolation method can be applied in the interior reconstruction problem.

In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-nine polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline. Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials.

<span class="mw-page-title-main">Curve fitting</span> Process of constructing a curve that has the best fit to a series of data points

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval.

<span class="mw-page-title-main">Bilinear interpolation</span> Method of interpolating functions on a 2D grid

In mathematics, bilinear interpolation is a method for interpolating functions of two variables using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of arbitrary convex quadrilaterals.

In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ). These constants are named after Henri Lebesgue.

The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm L sense. It is sometimes referred to as Remes algorithm or Reme algorithm.

In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension.

In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.

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

Smoothing splines are function estimates, , obtained from a set of noisy observations of the target , in order to balance a measure of goodness of fit of to with a derivative based measure of the smoothness of . They provide a means for smoothing noisy data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where is a vector quantity.

In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous.

Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate and stable for large numbers of nodes even in high dimensions.

References

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  11. Colin Bennett, Robert C. Sharpley, Interpolation of Operators, Academic Press 1988