Interpolation in the context of "Geostatistics"

Spatial statistics is a field of applied statistics dealing with spatial data.It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional (areal unit) and lattice (gridded) data, point patterns, as well as image analysis and stereology.

In statistics, originally in geostatistics, kriging or Kriging (/ˈkriːɡɪŋ/), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov.

The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. The English verb is to krige, and the most common noun is kriging. The word is sometimes capitalized as Kriging in the literature.

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.

A blind spot, scotoma, is an obscuration of the visual field. A particular blind spot known as the physiological blind spot, "blind point", or punctum caecum in medical literature, is the place in the visual field that corresponds to the lack of light-detecting photoreceptor cells on the optic disc of the retina where the optic nerve passes through the optic disc. Because there are no cells to detect light on the optic disc, the corresponding part of the field of vision is invisible. Via processes in the brain, the blind spot is interpolated based on surrounding detail and information from the other eye, so it is not normally perceived.

Although all vertebrates have this blind spot, cephalopod eyes, which are only superficially similar because they evolved independently, do not. In them, the optic nerve approaches the receptors from behind, so it does not create a break in the retina.

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid-20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.

In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices.

Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount.