How to fit an integral with data points in Matlab, where the integration limit can vary
By varying the integration limit how do I fit some data points in Matlab?
MATLAB: How to fit an integral with data points in Matlab, where the integration limit can vary
curve fittingintegrationMATLAB
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You can do something like this:
y_fcn = @(CT,x) CT(1) * 74.826 * (x./CT(4)).^3 * integral(t.^4.*exp(t)./(exp(t)-1).^2), 0, CT(4)./x) + ... CT(2) * 24.942 * (CT(5)./x).^2 .* exp(CT(5)./x)./(exp(CT(5)./x)-1).^2 + CT(3)*24.942*(CT(6)./x).^2.*exp(CT(6)./x)./(exp(CT(6)./x)-1).^2;err_fcn = @(CT,x,y) sum((y-y_fcn(CT,x)).^2);CT0 = [1 2 3 4 5 7]; % Some sensible initial guess for C and T, that we concatenate together as CT = [C,T]
CT_best = fminsearch(@(CT) err_fcn(CT,x_obs,y_obs),CT0)You most likely have to use element-wise operations in the equation for y (I might have missed one or two multiplictions or power-operations). Then it should be fine to use ordinary least-square fitting of the C and T parameters (provided you have enough data-points, and they have the same standard deviation - but I expect you to know this already).
HTH
It looks like a parabols. If you have optimization toolbox, you can use lsqcurvefit to fit this equation (y=a*x^2+b*x+c) to the dataset.
angle = [45,50,55,60,65,70,75,80,85]';P = [55.51, 69.5, 78.07, 82.06, 81.81, 77.3, 68.12, 53.42, 31.59]';fun = @(a,b,c,angles) a*angles.^2 + b.*angles + c;param_sol = lsqcurvefit(@(param, angles) fun(param(1),param(2),param(3),angles), rand(1,3), angle, P);a_sol = param_sol(1);b_sol = param_sol(2);c_sol = param_sol(3);plot(angle, P, 'bx', angle, fun(a_sol, b_sol, c_sol, angle), 'r-');
You can also do it without any toolbox. Following also fit a parabolic equation of form (y=a*x^2+b*x+c)
angle = [45,50,55,60,65,70,75,80,85]';P = [55.51, 69.5, 78.07, 82.06, 81.81, 77.3, 68.12, 53.42, 31.59]';X = [angle(:).^2 angle(:) ones(size(angle(:)))];params = X\P(:);P_estimated = X*params;plot(angle, P, 'bx', angle, P_estimated, 'r-')
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