Curve Fitting

Introduction

Curve fitting is the process of constructing a curve or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.

Best Fit is the best possible way to represent a set of values by curve fitting. It is not necessary the polynomial represents all values.

Perfect fit is a way to represent a set of values such that all values could be found by this polynomial.

Formulas Used

`SSE = Σ"Error"(i)^2`

`SSR = Σ(f(x(i))-Mean)^2`

`SST = SSR + SSE`

`R^2 = "SSR"/"SST"`

`R^2adj = 1-[((1-R^2)*(n-1))/(n-k-1)]`

`RSME = ("SSE"/n)^0.5`

Code

1. Firstly the given data is uploaded in a matrix.
2. Using that matrix, cp and temperature are stored in column vectors.
3. A polynomial of order `1` and `3` is created using the polyfit function which takes input some parameters (cp, temperature matrix, and order of polynomial).
4. Then the predicted values of cp using polyval function are found out.
5. The original and predicted curve is plotted.
6. `SSR, SSE, and SST` are calculated.
7. Using these values `R^2`, `R^2adj` and `RSME` are calculated.
8. Using these values we can determine the goodness of fit criteria.

% A program to Curve fit to given set of parameters
close all
clear all
clc

% Loading the data to which curve fit is to be done
cp_data = load('data');
temprature = cp_data(:,1);
cp = cp_data(:,2);

% Performing curve fit using polyfit and polyval
co_effs = polyfit(temprature,cp,1);
predicted_cp = polyval(co_effs,temprature);
%constant = 

% Plotting original and predicted curve
plot(temprature, cp,'linewidth', 3);
hold on;
plot(temprature, predicted_cp,'linewidth', 3);
xlabel('Temperature[K]');
ylabel('Specific Heat [KJ/Kmol-K]');
legend('Original Curve','Predicted Curve');
grid on;
title('Curve Fit (1st degree polynomial)');
hold on;
Error = cp-predicted_cp;

% Calculating SSE
SSE = 0;
sum = 0;
for i=1:length(cp)
  SSE = SSE+Error(i)^2;
  sum = sum+cp(i);
endfor
SSE

n = length(cp)
k = 2

mean = sum/n

%Calculating SSR
SSR = 0;
for j=1:length(predicted_cp)
  SSR = SSR+(predicted_cp(i)-mean)^2;
endfor
SSR
SST = SSR+SSE

% Calculating R^2 
R_sq = SSR/SST

% Calculating Radj^2
R_sq_adj = 1-((1-R_sq)*(n-1)/(n-k-1))
RMSE = (SSE/n)^0.5


% Saving the plot
file_text=sprintf('plot1.png',n);        
saveas (gca, file_text);

Output

For 1st degree polynomial :

For 2nd degree polynomial :