In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of It is similar to the process of interpolation, which constructs new points between known points, but its results are often less meaningful, and are subject to greater uncertainty. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of Uncertainty is a term used in subtly different ways in a number of fields including Philosophy, Statistics, Economics, Finance, Insurance

Example illustration of the extrapolation problem, consisting of assigning a meaningful value at the blue box, at x = 7, given the red data points.

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),

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. Linear interpolation is a method of Curve fitting using linear polynomials In statistics regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a Dependent variable (response This is similar to linear prediction. Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples

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. In Numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation Polynomial for a given set of data points A finite difference is a mathematical expression of the form f ( x + b) &minus f ( x + a) In Mathematics, a difference operator maps a function, f ( x) to another function f ( x + a) &minus f ( x 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. In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of

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^[1]. 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  [1].

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. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

Even for proper assumptions about the function, the extrapolation can diverge strongly from the function. In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or The classic example is truncated power series representations of sin(x) and related trigonometric functions. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + 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]. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set 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 . Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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 Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Mathematics, a unit circle is In particular, the compactification point at infinity is mapped to the origin and vice versa. The point at infinity, also called ideal point, is a point which when added to the real Number line yields a Closed curve called the Real 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. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be

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 Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a 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. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

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 Mathematics, a sequence transformation is an Operator acting on a given space of Sequences Sequence transformations include linear mappings such as Padé approximant is the "best" approximation of a function by a Rational function of given order In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a In this case, one often obtains rational approximants.

References

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

See also

• Forecasting
• Minimum polynomial extrapolation
• Multigrid method
• Prediction interval
• Regression analysis
• Richardson extrapolation
• Trend estimation
• Interpolation