Gaussian function

Gaussian curves with different parameters for medium and variance. The corresponding values are a=1σ σ 2π π {displaystyle a={tfrac {1}{sigma {sqrt {2pi }}}}}}}, b = μand c = σ.

In statistics, the Gaussian function (also, Gaussian bell or Gaussian curve), named after Carl Friedrich Gauss, is a function defined by the expression:

f(x)=ae− − (x− − b)22c2{displaystyle f(x)=ae^{-{frac {(x-b)^{2}{2c^{2}}}}}}}}}

where a, b, and c are real constants (c > –1). The parameter a is the value of the highest point of the bell, b is the position of the center of the bell and c (the standard deviation, sometimes called the root mean square or root mean square value) controls the width of the bell.

Gaussian functions are frequently used in statistics. In the case of a equal to 1c2π π {displaystyle textstyle {frac {1}{c{sqrt {2pi}}}}}}}, the density function of a random variable corresponds to the normal distribution of average μ =b and variance σ²=c².

Properties

Gaussians are among the elementary functions, although they do not have elementary primitives. However, the exact value of the improper integral over the entire real range can be derived from the value of the Gauss integral, obtaining that:

∫ ∫ − − ∞ ∞ ∞ ∞ ae− − (x− − b)22c2dx=a日本語c日本語2π π .{displaystyle int _{-infty }^{infty }ae^{-{frac {(x-b)^{2}}{2c^{2}}}}}}}}}}{,dx=a info. !

The value of the integral is 1 yes and only if a=1c2π π {displaystyle textstyle a={frac {1}{c{sqrt {2pi }}}}}}}, in which case the gaussian function is the density function of a random variable with normal average distribution μ=b and variance σ²=c². Various graphics of gaussian functions are shown in the attached image.

Gaussian functions with c² = 2 are the eigenfunctions of the Fourier transform. This means that the Fourier transform of a Gaussian function is not just another Gaussian, but also a scalar multiple of the original function. The graph of the function is symmetric with the shape of a bell, known as the Gaussian bell. The parameter a is the height of the bell centered at point b, with c determining the width of the bell.

Integral of a Gaussian function

The integral of any Gaussian function is

∫ ∫ − − ∞ ∞ ∞ ∞ ae− − (x− − b)2/2c2dx=2a日本語c日本語π π {displaystyle int _{-infty }^{infty }a,e^{-left(x-bright)^{2}/2c^{2}}{2}}}{,dx={sqrt {2}a,leftvert crightvert ,{sqrt {pi }}}}}}}}}}}}}

whose alternate form is

∫ ∫ − − ∞ ∞ ∞ ∞ ke− − fx2+gx+hdx=∫ ∫ − − ∞ ∞ ∞ ∞ ke− − f(x− − g/(2f))2+g2/(4f)+hdx=kπ π fExp (g24f+h){infty}{infty}{infty }k,e^{-fx^{2} +gx+h},dx=int _{-infty}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{c}{

where f must be positive for the integral to converge.

Relation to the standard Gaussian integral

The integral

∫ ∫ − − ∞ ∞ ∞ ∞ ae− − (x− − b)2/2c2dx{displaystyle int _{-infty }^{infty }ae^{-(x-b)^{2}/2c^{2}{2}}{,dx}

for some real values a, b, c > 0 can be calculated by representing it in the form of a Gaussian integral. To do this, the constant a can be operated outside the integral, then the variable with respect to which it is integrated(differential) is changed from x to y = x-b.

a∫ ∫ − − ∞ ∞ ∞ ∞ e− − and2/2c2dand{displaystyle aint _{-infty }^{infty }e^{-y^{2}/2c^{2},dy}

and then to z=and/2c2{displaystyle z=y/{sqrt {2c^{2}}}}}}

a2c2∫ ∫ − − ∞ ∞ ∞ ∞ e− − z2dz{displaystyle a{sqrt {2c^{2}}}int _{-infty }^{infty }e^{-z^{2}{2}{,dz}

So, using the Gauss Integral

∫ ∫ − − ∞ ∞ ∞ ∞ e− − z2dz=π π {displaystyle int _{-infty }^{infty }e^{-z^{2}}{,dz={sqrt {pi }}}}}}

we have

∫ ∫ − − ∞ ∞ ∞ ∞ ae− − (x− − b)2/2c2dx=a2π π c2{displaystyle int _{-infty }^{infty }ae^{-(x-b)^{2}/2c^{2}}},dx=a{sqrt {2pi c^{2}}}}}}}

Two-dimensional Gaussian function

Gauss curve in a two-dimensional domain

In two dimensions, the exponent of the power of e within the Gaussian function is any negative value and defined in quadratic form. As a consequence, the levels of the function will always be ellipses.

An example of a two-dimensional function is

f(x,and)=AExp (− − ((x− − x0)22σ σ X2+(and− − and0)22σ σ And2)).{displaystyle f(x,y)=Aexp left(-left({frac {(x-x_{0})}{2}{2sigma _{X}^{2}}}}}}}} +{frac {(y-y_{0})^{2}{2sigma _{Y^{2}{2}}{2}}}{2}}}}{right). !

In the function, the coefficient A is the amplitude, x_o,y_o is the center and σ_x, σ_y are x and y extended to the graph.

The volume under the Gaussian function is given by this integral

V=∫ ∫ − − ∞ ∞ ∞ ∞ ∫ ∫ − − ∞ ∞ ∞ ∞ f(x,and)dxdand=2π π Aσ σ Xσ σ And.{displaystyle V=int _{-infty }^{infty }int _{-infty }^{infty }f(x,y),dx,dy=2pi Asigma _{X}sigma _{Y}. !

Applications

The primitive of a Gaussian function is the error function. These functions appear in numerous contexts in the natural sciences, social sciences, mathematics, and engineering. Some examples:

• In probabilities statistics and theory, Gaussian functions appear as the density function of the normal distribution, which is a distribution of probability limit of complicated sums, according to the theorem of the central limit.
• A Gaussian function is the wave function of the fundamental state of the quantum harmonic oscillator.
• Molecular orbitals used in computational chemistry are linear combinations of gaussian functions called gaussian orbitals.
• Mathematically, the Gaussian function plays an important role in defining Hermite polynomials.
• Consequently, they are also associated with the state of emptiness in quantum field theory.
• Gaussian rays are used in optical and microwave systems.
• Gaussian functions are used as a softening filter in digital image processing.