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

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Faisal on 19 Jan 2023
Commented: Matt J on 19 Jan 2023
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?

Matt J on 19 Jan 2023
Edited: Matt J on 19 Jan 2023
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
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Matt J on 19 Jan 2023
Probably similar, but with 3 unknowns fminsearch is not guaranteed to converge, so no rigorous predictions are possible.

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 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)