Extrapolation

From Academic Kids

In mathematics, extrapolation is a type of interpolation. When a tabulated function is interpolated not between given values, but outside of the given range, this is called extrapolation. Extrapolation often looks sensible at first glance, but its results may sometimes be invalid or subject to substantial uncertainty.

Extrapolation also means to generalize, making use of a specific case and adjusting it to fit several cases in general.

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 is smooth, then a non-smooth function will be poorly extrapolated.

Even for proper assumptions about the function, the extrapolation can diverge exponentially 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 still produce extrapolations that diverge away from the x-axis.

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 complex analysis

In complex analysis, a problem of extrapolation may be converted into an interpolation problem by the change of variable 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.

See also

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