In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced. By doing so, one makes an assumption of the unknown (for example, a driver may extrapolate road conditions beyond what is currently visible and these extrapolations may be correct or incorrect). The extrapolation method can be applied in the interior reconstruction problem.
Extrapolation in the context of Kardashev scale
The Kardashev scale (Russian: шкала Кардашёва, romanized: shkala Kardashyova) is a method of measuring a civilization's level of technological advancement based on the amount of energy it is capable of harnessing and using. The measure was proposed by Soviet astronomer Nikolai Kardashev in 1964, and was named after him.
Kardashev first outlined his scale in a paper presented at the 1964 conference that communicated findings on BS-29-76, Byurakan Conference in the Armenian SSR, which he initiated, a scientific meeting that reviewed the Soviet radio astronomy space listening program. The paper was titled "Передача информации внеземными цивилизациями" ("Transmission of Information by Extraterrestrial Civilizations"). Starting from a functional definition of civilization, based on the immutability of physical laws and using human civilization as a model for extrapolation, Kardashev's initial model was developed. He proposed a classification of civilizations into three types, based on the axiom of exponential growth:
View the full Wikipedia page for Kardashev scaleExtrapolation in the context of Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fitted to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.
For linear-algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical (y-axis) displacement of a point from the curve (e.g., ordinary least squares). However, for graphical and image applications, geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include both axes of displacement of a point from the curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result.
View the full Wikipedia page for Curve fitting