Definitions

linear-extrapolation

Extrapolation

[ik-strap-uh-leyt]

In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty.

Extrapolation methods

A sound choice of which extrapolation method to apply relies on a prior knowledge of the process that created the existing data points. Crucial questions are for example if the data can be assumed to be continuous, smooth, possibly periodic etc.

Linear extrapolation

Linear extrapolation means creating a tangent line at the end of the known data and extending it beyond that limit. Linear extrapolation will only provide good results when used to extend the graph of an approximately linear function or not too far beyond the known data.

If the two data points nearest the point x_* to be extrapolated are (x_{k-1},y_{k-1}) and (x_k, y_k), linear extrapolation gives the function (identical to linear interpolation if x_{k-1} < x_* < x_k),

y(x_*) = y_{k-1} + frac{x_* - x_{k-1}}{x_{k}-x_{k-1}}(y_{k} - y_{k-1}).

It is possible to include more than two points, and averaging the slope of the linear interpolant, by regression-like techniques, on the data points chosen to be included. This is similar to linear prediction.

Polynomial extrapolation

A polynomial curve can be created through the entire known data or just near the end. The resulting curve can then be extended beyond the end of the known data. Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences to create a Newton series that fits the data. The resulting polynomial may be used to extrapolate the data.

High order polynomial extrapolation must be used with due care. For the example data set and problem in the figure above, anything above order 1 (linear extrapolation) will possibly yield unusable values, an error estimate of the extrapolated value will grow with the degree of the polynomial extrapolation. This is related to Runge's phenomenon.

Conic extrapolation

A conic section can be created using five points near the end of the known data. If the conic section created is an ellipse or circle, it will loop back and rejoin itself. A parabolic or hyperbolic curve will not rejoin itself, but may curve back relative to the X-axis. This type of extrapolation could be done with a conic sections template (on paper) or with a computer.

French curve extrapolation

A method of extrapolation suitable for any distribution that has a tendency to be exponential but with accelerating or decelerating factors is French curve extrapolation. This method has been used successfully in providing forecast projections of the growth of HIV/AIDS in the UK since 1987 and variant CJD in the UK for a number of years

Quality of extrapolation

Typically, the quality of a particular method of extrapolation is limited by the assumptions about the function made by the method. If the method assumes the data are smooth, then a non-smooth function will be poorly extrapolated.

Even for proper assumptions about the function, the extrapolation can diverge strongly from the function. The classic example is truncated power series representations of sin(x) and related trigonometric functions. For instance, taking only data from near the x = 0, we may estimate that the function behaves as sin(x) ~ x. In the neighborhood of x = 0, this is an excellent estimate. Away from x = 0 however, the extrapolation moves arbitrarily away from the x-axis while sin(x) remains in the interval [−1,1]. I.e., the error increases without bound.

Taking more terms in the power series of sin(x) around x = 0 will produce better agreement over a larger interval near x = 0, but will produce extrapolations that eventually diverge away from the x-axis even faster than the linear approximation.

This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated. For particular problems, this additional information may be available, but in the general case, it is impossible to satisfy all possible function behaviors with a workably small set of potential behaviors.

Extrapolation in the complex plane

In complex analysis, a problem of extrapolation may be converted into an interpolation problem by the change of variable hat{z} = 1/z. This transform exchanges the part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular, the compactification point at infinity is mapped to the origin and vice versa. Care must be taken with this transform however, since the original function may have had "features", for example poles and other singularities, at infinity that were not evident from the sampled data.

Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function onto a larger region.

Again, analytic continuation can be thwarted by function features that were not evident from the initial data.

Also, one may use sequence transformations like Padé approximants and Levin-type sequence transformations as extrapolation methods that lead to a summation of power series that are divergent outside the original radius of convergence. In this case, one often obtains rational approximants.

See also

Notes

References

  • Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.

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