Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form with

 f(x) =  exp(-x^2)
and with parametric extension
f(x) = aexp(-(x-b)^2)/2c^2)

for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".


Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define the 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. They are also abundantly used in quantum chemistry to from basic sets.
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The most common method for estimating the Gaussian parameters is to take the logarithm of data and fit a parabola to the resulting data set. While this provides a simple curve fitting procedure, the resulting algorithm may be biased by excessively weighting small data values, which can produce large errors in the profile estimate. One can partially compensate for this problem through weighted least squares estimation, reducing the weight of small data values, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iteratively reweighted least squares procedure, in which the weights are updated at each iteration. It is also possible to perfrom non-linear regression directly on the data without involving the logarithmic data transformation.


Parameter Precision

Once one has an algorithm for estimating the Gaussian parameters, it is also important to know how precise those estimates are. Any least squares estimation algorithm can provide numerical estimates for the variance of each parameter ( i.e, the variance of the estimated height, position, and width of the function) . Once can also use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given certain assumptions about the data.

1. The noise in the measured profile is either independent and identically distributed  Gaussian, or the noise is Poissin-distributed.
2. The spacing between each sampling ( i.e the distance between pixels measuring the data) is uniform.
3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak(area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region.
4. The width of the peak is much larger than the distance between sample locations( i.e. the detector pixels must be al least 5 time smaller than the Gaussian FWHM).


Applications

Gaussian functions appear in many context in natural sciences, the social sciences, mathematics and engineering. Some examples include: