To find exponent in power law equation of the form y = ax^m + b

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I have X and Y points for a curve to be of the form Y = ax^m + b.
I want to find the exponent m, lets just say that m could be inbetween 1.2 - 2.5.
How can I find exact value for m?

Accepted Answer

Matt J
Matt J on 19 Jan 2023
Edited: Matt J on 19 Jan 2023
fminspleas downloadable from
is especially appropriate for power law fits.
a = 0.55;
m = 1.3;
b = -0.78;
% dummy data
x = (1:25)';
y = a*x.^m + b + randn(size(x));
m=fminspleas( {@(m,x)x.^m , 1}, 2,x,y, 1.2,2.5 )
m = 1.2933
  2 Comments
Faisal
Faisal on 19 Jan 2023
thank you for your simple code.
I have found the value of m.
Could you please explain that the following code will give the similar answer or it would be different?
file='03_L.xlsx';
[Fs] = xlsread(file,1,'B2:B1216');
[dep] = xlsread(file,1,'A2:A1216');
fcn = @(b,dep) b(1).*dep.^b(2) + b(3); % Objective Function
B = fminsearch(@(b) norm(Fs - fcn(b,dep)), rand(3,1)*2) % Estimate Parameters
figure
plot(dep, Fs, '.')
hold on
plot(dep, fcn(B,dep), '-r')
hold off
grid
Matt J
Matt J on 19 Jan 2023
Probably similar, but with 3 unknowns fminsearch is not guaranteed to converge, so no rigorous predictions are possible.

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More Answers (2)

Mathieu NOE
Mathieu NOE on 19 Jan 2023
hello
try this
may need some refinement for the initial guess for the parameters depending of your data
a = 0.55;
m = 1.3;
b = -0.78;
% dummy data
x = (1:25);
y = a*x.^m + b + randn(size(x));
% equation model y = a*x^m + b
f = @(a,m,b,x) (a*x.^m + b);
obj_fun = @(params) norm(f(params(1), params(2), params(3),x)-y);
% IC guessed
sol = fminsearch(obj_fun, rand(3,1));
a_sol = sol(1)
a_sol = 0.4997
m_sol = sol(2)
m_sol = 1.3288
b_sol = sol(3)
b_sol = -0.4661
y_fit = f(a_sol, m_sol, b_sol, x);
Rsquared = my_Rsquared_coeff(y,y_fit); % correlation coefficient
figure(1)
plot(x,y,'rd',x,y_fit,'b-');
title(['Power Fit / R² = ' num2str(Rsquared) ], 'FontSize', 15)
ylabel('Intensity (arb. unit)', 'FontSize', 14)
xlabel('x(nm)', 'FontSize', 14)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Rsquared = my_Rsquared_coeff(data,data_fit)
% R² correlation coefficient computation
% The total sum of squares
sum_of_squares = sum((data-mean(data)).^2);
% The sum of squares of residuals, also called the residual sum of squares:
sum_of_squares_of_residuals = sum((data-data_fit).^2);
% definition of the coefficient of correlation is
Rsquared = 1 - sum_of_squares_of_residuals/sum_of_squares;
end

Matt J
Matt J on 19 Jan 2023
If you have the Curve Fitting Toolbox,
a = 0.55;
m = 1.3;
b = -0.78;
% dummy data
x = (1:25)';
y = a*x.^m + b + randn(size(x));
fobj=fit(x,y,'power2','Lower',[-inf,1.2,-inf],'Upper',[+inf,2.5,+inf])
fobj =
General model Power2: fobj(x) = a*x^b+c Coefficients (with 95% confidence bounds): a = 0.5425 (0.2922, 0.7927) b = 1.295 (1.157, 1.434) c = -0.3202 (-1.66, 1.02)
plot(fobj,x,y)

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