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Since log(1/x)=-log(x) your equation model has redundant terms. It is equivalent to
y = a0+(a1-a2)*log(x) = A+B*log(x)
where a0 has been relabeled as A and B has replaced a1-a2.
You could fit A and B, I suppose, by doing
Any fixed degree that you use will result in a polynomial that tends to be infinitely wrong as x tends to infinity.
sisay, try this:
clc; % Clear the command window. close all; % Close all figures (except those of imtool.) clear; % Erase all existing variables. Or clearvars if you want. workspace; % Make sure the workspace panel is showing. format long g; format compact; fontSize = 24;
% Construct x. x = linspace(.01, 40, 50); a0 = 1; a1 = 2; a2 = 3;
% Create the perfect equation. y = a0 + a1 * log(x)+ a2 * log(1./x); subplot(3,1,1); plot(x, y, 'b.-'); title('Noise-free signal', 'FontSize', fontSize);
% Enlarge figure to full screen. set(gcf, 'units','normalized','outerposition',[0 0 1 1]);
% Add some noise to make a noisy signal that we will fit. yNoisy = y + 1.5 * rand(1, length(y)); subplot(3,1,2); plot(x, yNoisy, 'b.-'); title('Noisy signal', 'FontSize', fontSize);
% Now get the fit % y = a0 + a1 * log(x) - a2 * log(x) % y = a0 + (a1 - a2) * log(x) % y = (a1 - a2) * log(x) + a0 % Let newX = log(x), and (a1-a2) = coeffs(1), then % y = coefficients(1) * newX + coefficients(2) % so now we can use polyfit to fit a line. newX = log(x); coefficients = polyfit(newX, yNoisy, 1);
% Now get the fitted values a0 = coefficients(2); a1 = coefficients(1); a2 = 0; % Might as well be 0 as any other value. yFitted = a0 + a1 * log(x)+ a2 * log(1./x); % and plot them subplot(3,1,3); plot(x, yNoisy, 'b.'); hold on; plot(x, yFitted, 'r-', 'LineWidth', 3); title('Fitted signal', 'FontSize', fontSize); legend('Noisy data', 'Fitted signal');