# Data fitting in between following variables x and b for quadratic equation

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MOHD UWAIS el 20 de En. de 2023
Editada: Mathieu NOE el 23 de En. de 2023
clear all
clc
a=[0.450 0.486 0.546 0.589 0.633 0.656];
b=[1.5435 1.536 1.5273 1.523 1.5197 1.5179];
x=1./a.^2;
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MOHD UWAIS el 20 de En. de 2023
Here we can treat x as input and b as output, and coefficient may be p1, p2, ...... (say).
Torsten el 20 de En. de 2023
So you want to determine p0,p1,p2 such that (in the least-squares sense)
p0 + p1*(1/0.45)^2 + p2*(1/0.45)^4 = 1.5435
p0 + p1*(1/0.486)^2 + p2*(1/0.486)^4 = 1.536
p0 + p1*(1/0.546)^2 + p2*(1/0.546)^4 = 1.5273
...
p0 + p1*(1/0.656)^2 + p2*(1/0.656)^4 = 1.5179
?

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Mathieu NOE el 23 de En. de 2023
Editada: Mathieu NOE el 23 de En. de 2023
hello again
seems to me b can be expressed as a first order polynomial of x
b = 1.4952 + 0.0097108*x
you can of course increse the order if you need to.
simply change this value in the code provided below :
degree = 1;
code :
a=[0.450 0.486 0.546 0.589 0.633 0.656];
x=1./a.^2;
b=[1.5435 1.536 1.5273 1.523 1.5197 1.5179];
% Fit a polynomial p of degree "degree" to the (x,y) data:
degree = 1;
p = polyfit(x,b,degree);
% Evaluate the fitted polynomial p and plot:
f = polyval(p,x);
eqn = poly_equation(flip(p)) % polynomial equation (string)
eqn = " y = 1.4952 + 0.0097108*x "
Rsquared = my_Rsquared_coeff(b,f); % correlation coefficient
figure(1);plot(x,b,'.',x,f,'-','markersize',20)
title('b vs x')
xlabel('x')
ylabel('b')
legend('data',eqn)
title(['Data fit , R² = ' num2str(Rsquared)]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function eqn = poly_equation(a_hat)
eqn = " y = "+a_hat(1);
for i = 2:(length(a_hat))
if sign(a_hat(i))>0
str = " + ";
else
str = " ";
end
if i == 2
% eqn = eqn+" + "+a_hat(i)+"*x";
eqn = eqn+str+a_hat(i)+"*x";
else
% eqn = eqn+" + "+a_hat(i)+"*x^"+(i-1)+" ";
eqn = eqn+str+a_hat(i)+"*x^"+(i-1)+" ";
end
end
eqn = eqn+" ";
end
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