Gaussian Function Instance

A Gaussian function instance is an exponential function from a Gaussian function family (of the form [math]\displaystyle{ f(x,a,b,c) = ae^{-k/m} }[/math], such that [math]\displaystyle{ k=(x-b)^2 }[/math] and [math]\displaystyle{ m=2c^2 }[/math]).

• Context:
  • It can arise by applying the Exponential Function to a general quadratic function.
  • It can be used to define Gaussian Statistical Models (based on Gaussian probability distributions).
• Example(s):
  • [math]\displaystyle{ f }[/math](1,2,3,4) ⇒ ...
  • …
• Counter-Example(s):
  • a Gabor Function.
  • a Logistic Function.
• See: Gaussian Error Function.

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References

2009

• (Wikipedia, 2011) http://en.wikipedia.org/wiki/Gaussian_function
  • In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form: [math]\displaystyle{ f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } } }[/math] for some real constants [math]\displaystyle{ a }[/math], b, [math]\displaystyle{ c }[/math] > 0, and [math]\displaystyle{ e }[/math] ≈ 2.718281828 (Euler's number). The graph of a Gaussian is a characteristic symmetric "bell curve" shape that quickly falls off towards plus/minus infinity. The parameter [math]\displaystyle{ a }[/math] is the height of the curve's peak, [math]\displaystyle{ b }[/math] is the position of the centre of the peak, and [math]\displaystyle{ c }[/math] controls the width of the "bell". Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform. Gaussian functions arise by applying the exponential function to a general quadratic function. The Gaussian functions are thus those functions whose logarithm is a quadratic function.

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