Meixner curve (hyperbolic secant distribution)

In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.

Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution.

y = 2.*p(1)./(exp(p(2).*(x10-p(4))) + exp(-p(3).*(x10-p(4)))) + p(5);

SCRIPT : (www.octave.org)

clc;clear all;

global x10 y10

function delta = effe(p); % MINIMIZING FUNCTION

global x10 y10

y = meixner(x10,p);

delta = sumsq(y - y10);

endfunction

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

y = 2.*p(1)./(exp(p(2).*(x10-p(4))) + exp(-p(3).*(x10-p(4)))) + p(5);

endfunction

% Start Guess is defined (4 parameters)

p0(1) = 195; %200 Maximum intensity

p0(2) = 0.6; %0.2 right skewness

p0(3) = 1.5; %1 left skewness

p0(4) = 18; %15; Max position

p0(5) = 200; %200; baseline

% Some noisy data are generated

x10 = linspace (0,40,100)';

y10 = meixner(x10,[200 0.2 1 15 200]) + 10*randn(100,1);

% fMinSearch on the sum of squares of delta

fer = @(p) effe(p);

[p1,fer1] = fminsearch(fer,p0);

% final plot

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

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

axis([0,40,140,500]);grid on;grid minor on;hold on;

y11 = meixner(x10,p1);

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

title(["Max : ",num2str(p1(1))," MaxPos : ",num2str(p1(4))," right skew : ",num2str(p1(2)),...

" left skew : ",num2str(p1(3));" BaseLine : ",num2str(p1(5))," Correlation : ",num2str(corr(y10,y11)) ]);