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

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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Faisal 2023-1-19

@Matt J

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 2023-1-19

Probably similar, but with 3 unknowns fminsearch is not guaranteed to converge, so no rigorous predictions are possible.

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Answer 2

Mathieu NOE 2023-1-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1897040-to-find-exponent-in-power-law-equation-of-the-form-y-ax-m-b#answer_1152055

在 MATLAB Online 中打开

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

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Answer 3

Matt J 2023-1-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1897040-to-find-exponent-in-power-law-equation-of-the-form-y-ax-m-b#answer_1152060

在 MATLAB Online 中打开

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)