将数据拟合到已知函数MATLAB（无曲线拟合工具箱）

Question

I have a function with three parameters and some data that I want to fit. How can I do this optimally? I am not even sure of the range of the three parameters in the equation.

The function has free parameters alpha , beta and gamma and is given by

 y = (1 - alpha + alpha./sqrt(1 + 2*beta*(gamma*x).^2./alpha)).^(-1) - 1;

I have arrays of x and y data points (around 50 points in each set) to which I want to find the best fit (defined as minimizing least squares) using any alpha , beta and gamma .

The solutions online recommend the curve fitting toolbox, which I do not have on my machine and am unable to install. I only have the barebones MATLAB 2015b version.

Answer 1

You need an optimization algorithm for smooth, R^n -> R functions. Since you have only access to barebone Matlab, a good idea is to take an algorithm from File Exchange. For illustration I picked LMFnlsq , which should suffice since you have a small problem, although it seems to be more general and a little bit overkill here.

Download LMFnlsq and add to your Matlab path.

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Example

For convenience make a function called regr_fun :

function y = regr_fun(par, x)
    alpha   = par(1);
    beta    = par(2);
    gamma   = par(3); 
    y = (1 - alpha + alpha./sqrt(1 + 2*beta*(gamma*x).^2./alpha)).^(-1) - 1;
end

Curve fitting (in the same folder as regr_fun ):

%---------------------------------------------------------------------
% DUMMY DATA
%---------------------------------------------------------------------
% Generate data from known model contaminated with random noise
rng(333) % for reproducibility

alpha   = 2;
beta    = 0.1;
gamma   = 0.1;
par     = [alpha, beta, gamma];

xx      = 1:50;
y_true  = regr_fun(par, xx);
yy      = y_true + normrnd(0,1,1,50);

%---------------------------------------------------------------------
% FIT MODEL
%---------------------------------------------------------------------
% intial point of solver
p0      = [1,1,1];

obj_fun = @(p) sum((regr_fun(p, xx) - yy).^2);

% optimization
p_fit   = LMFnlsq(obj_fun, p0);

y_fit   = regr_fun(p_fit, xx);

%---------------------------------------------------------------------
% PLOT
%---------------------------------------------------------------------
plot(xx, yy, 'o')
hold on
plot(xx, y_true)
plot(xx, y_fit, '--')

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Note

Although matlab.codetools.requiredFilesAndProducts lists the symbolic toolbox as well, for this problem it is not needed and the function should run withouth that as well.