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Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints. [ 4 ] [ 5 ] Curve fitting can involve either interpolation , [ 6 ] [ 7 ] where an exact fit to the data is required, or smoothing , [ 8 ] [ 9 ] in which a "smooth ...
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y -axis which are half the maximum amplitude.
Chopin (2011) proposed an algorithm inspired from the Ziggurat algorithm of Marsaglia and Tsang (1984, 2000), which is usually considered as the fastest Gaussian sampler, and is also very close to Ahrens's algorithm (1995). Implementations can be found in C, C++, Matlab and Python.
"Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials". Mathematics of Computation. 18 (88): 598– 616. doi: 10.1090/S0025-5718-1964-0166397-1. JSTOR 2002946. MR 0166397. Ehrich, S. (2002). "On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas".
Another generalization is to calculate the number of coprime integer solutions , to the inequality m 2 + n 2 ≤ r 2 . {\displaystyle m^{2}+n^{2}\leq r^{2}.\,} This problem is known as the primitive circle problem , as it involves searching for primitive solutions to the original circle problem. [ 9 ]
Fitting of a noisy curve by an asymmetrical peak model () with parameters by mimimizing the sum of squared residuals () = at grid points , using the Gauss–Newton algorithm. Top: Raw data and model. Bottom: Evolution of the normalised sum of the squares of the errors.
The idea of local linear regression is to fit locally a straight line (or a hyperplane for higher dimensions), and not the constant (horizontal line). After fitting the line, the estimation Y ^ ( X 0 ) {\displaystyle {\hat {Y}}(X_{0})} is provided by the value of this line at X 0 point.