Gaussian function

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

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 a is the height of the curve's peak, b is the position of the center of the peak, and c (the Half Width at Half Maximun (HWHM) controls the width of the "bell". ‘d’ is the background constant value. ( y-shift)

Gaussian functions are often used to represent the probability density function of a normally distributed random variable.

The following script is an example of gaussian interpolation. It generates some noisy data and interpolates them with a gaussian curve. It is used in this web-site here for various climatological dataset interpolation.

SCRIPT : (www.octave.org)

clc;clear all;

global x10 y10

function delta = effe(p); % MINIMIZING FUNCTION

global x10 y10

y = gauss(x10,p);

delta = sumsq(y - y10);

endfunction

function y = gauss(x10,p); % INTERPOLATING FUNCTION

w = log(2)/p(3)^2;

y = p(1)*exp(-w*(x10-p(2)).^2) + p(4);

endfunction

% Start Guess is defined (4 parameters)

p0(1) = 95 %100 Maximum intensity

p0(2) = 0.56 %0.5 Maximum position (x axis)

p0(3) = 0.15 %0.1 HWHM

p0(4) = 17 %20; Baseline (constant value)

% Some noisy data are generated

x10 = linspace (0,1,100)';

y10 = gauss(x10,[100 0.5 0.1 20]) + 20*(rand(100,1)-0.5);

% fMinSearch on the sum of squares of delta

[p1,fer1] = fminsearch('effe',p0);

% final plot

figure (1,'position',[200 100 700 500]);

plot(x10,y10,'r','marker','+','linestyle','none'); % points

axis([0,1,0,140]);grid on;grid minor on;hold on;

y11 = gauss(x10,p1);

plot(x10,y11,'b','linewidth',2); % curve

title(["Max : ",num2str(p1(1))," MaxPos : ",num2str(p1(2))," HWHM : ",num2str(p1(3));" BaseLine: ",num2str(p1(4))," Correlation : ",num2str(corr(y10,y11))],'fontname','verdana','fontsize',12);

disp(['correlation = ',num2str(corr(y10,y11))]);

disp(['spearman = ',num2str(spearman(y10,y11))]);

disp(['kendall = ',num2str(kendall(y10,y11))]);