# Dictionary Definition

extrapolation

### Noun

1 (mathematics) calculation of the value of a
function outside the range of known values

2 an inference about the future (or about some
hypothetical situation) based on known facts and observations

# User Contributed Dictionary

## English

### Pronunciation

- Rhymes with: -eɪʃǝn

### Noun

#### Translations

calculation of an estimate

- Dutch: extrapolatie
- Finnish: ekstrapolaatio
- Swedish: extrapolering

inference

- Dutch: extrapolatie
- Finnish: ekstrapolaatio
- Swedish: extrapolation

### Related terms

## Swedish

### Noun

extrapolation- Extrapolation; inference of a hypothetical situation.

#### See also

# Extensive Definition

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 its results 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_,y_) and (x_k, y_k), linear extrapolation
gives the function (identical to linear
interpolation if x_ ),

- y(x_*) = y_ + \frac(y_ - y_).

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 http://www.AIDSCJDUK.info.

## 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 = 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.

## References

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

## See also

extrapolation in Afrikaans: Ekstrapolasie

extrapolation in Arabic: استكمال خارجي

extrapolation in Bulgarian: Екстраполация

extrapolation in Danish: Ekstrapolation

extrapolation in German: Extrapolation

extrapolation in Persian: برونیابی

extrapolation in Italian: Estrapolazione

extrapolation in Hebrew: אקסטרפולציה

extrapolation in Dutch: Extrapolatie

extrapolation in Japanese: 外挿

extrapolation in Polish: Ekstrapolacja

extrapolation in Russian: Экстраполяция

extrapolation in Sundanese: Ékstrapolasi

extrapolation in Swedish: Extrapolering

extrapolation in Ukrainian: Екстраполяція

extrapolation in Chinese: 外推

# Synonyms, Antonyms and Related Words

accession, accessory, accompaniment, actualization, addenda, addendum, additament, addition, additive, additory, additum, adjunct, adjuvant, annex, annexation, appanage, appendage, appendant, approximation, appurtenance, appurtenant, attachment, augment, augmentation, coda, complement, concomitant, continuation, corollary, differentiation,
division, equation, evolution, extension, exteriorization,
externalization,
fixture, increase, increment, integration, interpolation, inversion, involution, multiplication, notation, objectification,
offshoot, pendant, practice, projection, proportion, reduction, reinforcement, side
effect, side issue, subtraction, supplement, tailpiece, transformation, undergirding