Curve Fitting Analysis for a given data in Matlab

In this challenge, we are constructing a curve using the given set of data. The curve constructed needs to be as close in value as to the given set as possible.

Here, we construct the curve using a linear and cubic polynomial and then check the results by plotting it. The code is given below.

clear all
close all
clc

% Preparaing the data
cp_data = load(\'data\')
temperature = cp_data(:,1);
cp = cp_data(:,2);

% Curve fit

% cp = a*T^2 + b*T + c
co_effs = polyfit(temperature,cp,1);

predicted_cp = polyval(co_effs, temperature);

% Compare my cureve fit with original data
figure(1)
plot(temperature, cp, \'linewidth\' , 3)
hold on
plot(temperature, predicted_cp, \'linewidth\', 3 , \'color\' , \'m\')
xlabel(\'Temperature[K]\')
ylabel(\'Specific Heat [KJ/Kmol-L]\')
legend(\'Orginal dataset\',\'Curve fit\')
title(\'Linear\')

% Calculation of sum of squared error
error = predicted_cp(:,1) - cp(:,1);
SSE = sum(error.^2);

% Calculation of sum of squares of regression
SSR = sum((cp(:,1) - mean(cp(:,1))).^2);

% Calculation of R_square
SST = SSR + SSE;
R1_square = SSR/SST

if R1_square>0.95
    display(\'Fit is good\')
else
    display(\'Fit is not good\')
end

% Cubic
co_effs = polyfit(temperature,cp,3);
predicted_cp = polyval(co_effs, temperature);

figure(2)
plot(temperature, cp, \'linewidth\' , 3)
hold on
plot(temperature, predicted_cp, \'linewidth\', 3 , \'color\' , \'m\')
xlabel(\'Temperature[K]\')
ylabel(\'Specific Heat [KJ/Kmol-L]\')
legend(\'Orginal dataset\',\'Curve fit\')
title(\'Cubic\')

% Calculation of sum of squared error
error = predicted_cp(:,1) - cp(:,1);
SSE = sum(error.^2);

% Calculation of sum of squares of regression
SSR = sum((cp(:,1) - mean(cp(:,1))).^2);

% Calculation of R_square
SST = SSR + SSE;
R2_square = SSR/SST

if R2_square>0.95
    display(\'Fit is good\')
else
    display(\'Fit is not good\')
end

The figure for the linear curve is -

And the figure for cubic polynomial is -

Just by visualization, we can see that the cubic polynomial is a better approximation than the linear one. This is because the cubic curve is a higher degree polynomial. Mathematically, we need to check the goodness of fit for each of the figures. This is done by using the equations below-

The value of SST,SSR,SSE and R^2 is calculated for each of the curves. Ideally, the curse is called a good fit if R^2 is greater than 0.95.

The output of the code for linear polynomial is -

And for the cubic polynomial it is -

Hence, it can be seen that using a cubic polynomial is better than the linear polynomial.

The curve can be made more perfect by increasing the order of the polynomial used.