Editing Extrapolation (section)

==Method==

A sound choice of which extrapolation method to apply relies on ''a priori knowledge'' of the process that created the existing data points. Some experts have proposed the use of causal forces in the evaluation of extrapolation methods.<ref>{{cite journal | title = Causal Forces: Structuring Knowledge for Time-series Extrapolation | author1 = J. Scott Armstrong | author2 = Fred Collopy | journal = Journal of Forecasting | volume = 12 | issue = 2 | pages = 103–115 | year = 1993 | doi = 10.1002/for.3980120205 | citeseerx = 10.1.1.42.40 | s2cid = 3233162 | access-date = 2012-01-10 | url = https://repository.upenn.edu/cgi/viewcontent.cgi?article=1072&context=marketing_papers }}</ref> Crucial questions are, for example, if the data can be assumed to be continuous, smooth, possibly periodic, etc.

===Linear===

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 <math>x_*</math> to be extrapolated are <math>(x_{k-1},y_{k-1})</math> and <math>(x_k, y_k)</math>, linear extrapolation gives the function:

:<math>y(x_*) = y_{k-1} + \frac{x_* - x_{k-1}}{x_{k}-x_{k-1}}(y_{k} - y_{k-1}).</math>

(which is identical to [[linear interpolation]] if <math>x_{k-1} < x_* < x_k</math>). It is possible to include more than two points, and averaging the slope of the linear interpolant, by [[Regression analysis|regression]]-like techniques, on the data points chosen to be included. This is similar to [[linear prediction]].

===Polynomial===
[[File:Lagrange polynomials for continuations of sequence 1,2,3.gif|thumb|right|Lagrange extrapolations of the sequence 1,2,3. Extrapolating by 4 leads to a polynomial of minimal degree ({{color|#006060|cyan}} line).]]
A polynomial curve can be created through the entire known data or just near the end (two points for linear extrapolation, three points for quadratic extrapolation, etc.).  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===

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]], when extrapolated it will loop back and rejoin itself.  An extrapolated [[parabola]] or [[hyperbola]] 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===

[[French curve]] extrapolation is a method suitable for any distribution that has a tendency to be exponential, but with accelerating or decelerating factors.<ref>[http://www.AIDSCJDUK.info AIDSCJDUK.info Main Index<!-- Bot generated title -->]</ref> 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. Another study has shown that extrapolation can produce the same quality of forecasting results as more complex forecasting strategies.<ref>{{cite journal | title = Forecasting by Extrapolation: Conclusions from Twenty-Five Years of Research | author = J. Scott Armstrong | journal = Interfaces | volume = 14 | issue = 6 | pages = 52–66 | year = 1984 | doi = 10.1287/inte.14.6.52 | citeseerx = 10.1.1.715.6481 | s2cid = 5805521 | access-date = 2012-01-10 | url = https://repository.upenn.edu/cgi/viewcontent.cgi?article=1083&context=marketing_papers }}</ref>

===Geometric Extrapolation with error prediction===

Can be created with 3 points of a sequence and the "moment" or "index", this type of extrapolation have 100% accuracy in predictions in a big percentage of known series database (OEIS).<ref>{{Cite web |last=V. Nos |year=2021 |title=Probnet: Geometric Extrapolation of Integer Sequences with error prediction |url=https://hackage.haskell.org/package/Probnet |access-date=2023-03-14}}</ref>

Example of extrapolation with error prediction :

: <math>\text{sequence} = [1, 2, 3, 5]</math>
: <math>{f_1(x, y) = \frac{x}{y}}</math>
: <math>d_1 = f_1(3, 2)</math>
: <math>{d_2 = f_1(5, 3)}</math>
: <math>m = \text{sequence}(5)</math>
: <math>n = \text{sequence}(3)</math>
: <math>\begin{align}
\text{f}(m, n, d_1, d_2) &= \text{round}\left( (n \cdot d_1 - m) + (m \cdot d_2) \right) \\
&= \text{round}\left(( 3 \times 1.5 - 5 \right) + (5 \times 1.66) ) = 8
\end{align}</math>