The Gaussian function (named after Carl Friedrich Gauss) is a classic bell-shaped curve generally defined as:
- exp means “exponential” (i.e. ex),
- a, b, and c (non-zero) are adjustable constants:
- a (height of peak),
- b (position of peak),
- c (standard deviation or “spread”).
The above formula is used when the constant a is time. The Gaussian can also be specified with a standard deviation (σ or S), where 2 * S * S appears in the denominator of the exponent (Hahn, 1995).
Global Max and Inflection Points
The graph has two inflection points, at x = b ± √(c/2).
Practical Uses of a Gaussian Function
The Gaussian function has a myriad of uses in mathematics and sciences, including machine learning, physics and biomedical sciences. For example, it is the usual choice for radial basis function (RBF) neural network because it generalizes a global mapping and refines local features without altering the learned mapping (Sundararajan & Lu, 1999). In physics, a Gaussian beam is described by a Gaussian function. The Gaussian is the only function that provides the minimum possible time-bandwidth product along all smooth (analytic) functions (Smith,2020).
Hahn, L. (1995). The Gaussian Function. Retrieved July 22, 2020 from: http://retina.anatomy.upenn.edu/~rob/lance/gaussian.html
Kurzweg, U. (2020). Properties of the Gaussian Function. Retrieved July 22, 2020 from: https://mae.ufl.edu/~uhk/GAUSSIAN-NEW.pdf
Smith, J. (2011). Spectral Audio Signal Processing. W3K Publishing.
Sundararajan, N. & Lu, Y. (1999). Radial Basis Function Neural Networks with Sequential Learning. MRAN and Its Applications. World Scientific.
Stephanie Glen. "Gaussian Function: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/gaussian-function-simple-definition-examples/
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