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

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;

[Mathieu NOE]

it's unclear what the input and output are

usually a and b would be parameters / coefficients, x and y the input / output arrays

[MOHD UWAIS]

Here we can treat x as input and b as output, and coefficient may be p1, p2, ...... (say).

[Torsten]

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

?

Answer 1

[Mathieu NOE]

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