Program to curve fit a linear and cubic polynomial using MATLAB

Curve fitting - Curve fiiting is a process of constructing a curve to co-relate the real series of data and mathematical function representing the curve fit.Curve fit can involve either , interpolation,where an exact fit to the data is required,or smoothing,in which a 'smooth' function is constructed that approximately fits the data.

Here in this program a linear and cubic polynomial is used:

Linear polynomial equation is `a*x+b`

Cubic polynomial equation is `a*x^3+a*x^2+a*x+b`

Functions used:

Polyfit function polyfit(x,y,n) is used to return the coefficient for a polynomial p(x) of degree n that is a best fit for the data in y.

Polyval function polyval(p,x) evaluates the polynomial p at each point in x.

Data set used has a set of 3200 diffrent values of temperature and their respective specific heat(as specific heat changes with change in temperature)

Code for curve fitting:

clear all
close all
clc

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

%curve fit for linear polynomial
%cp = a*t+b
coefficient1 = polyfit(temperature,cp,1);
predicted_cp1 = polyval(coefficient1,temperature);

%curve fit for cubic polynomial
%cp = a*t^3+a*t^2+a*t+b
coefficient2 = polyfit(temperature,cp,3);
predicted_cp2 = polyval(coefficient2,temperature);

%comparison of curve fit with the original data for linear  polynomial
figure(1)
plot(temperature,cp,'linewidth',3)
hold on
plot(temperature,predicted_cp1,'linewidth',3,'color','r')
xlabel('temperature[k]')
ylabel('specific heat[kj/kmol-k]')
legend('original dataset','curve fit')
title('Linear polynomial')

%comparison of curve fit with the original data for cubic polynomial 
figure(2)
plot(temperature,cp,'linewidth',3)
hold on
plot(temperature,predicted_cp2,'linewidth',3,'color','r')
xlabel('temperature[k]')
ylabel('specific heat[kj/kmol-k]')
legend('original data set','curve fit')
title('Cubic polynomial')

Curve fit for linear polynomial:

Curve fit for cubic polynomial:

Alternate method:

We can do the curve fitting using an alternate method called splitwise method.In this method the data set is taken in splits.However,in this program we have taken the data set in two splits(0-1600)(1600-3200)

Code for curve fitting using splitwise method :

%splitwise curve fitting for linear polynomial
coefficient3 = polyfit(temperature(1:1600),cp(1:1600),1);
predicted_cp3 = polyval(coefficient3,temperature(1:1600));

figure(3)
plot(temperature,cp,'linewidth',3)
hold on
plot(temperature(1:1600),predicted_cp3,'linewidth',3,'color','r')
xlabel('temperature[k]')
ylabel('specific heat[kj/kmol-k]')
legend('original dataset','curve fit')
title('Linear polynomial')

%splitwise curve fitting for cubic polynomial
coefficient4 = polyfit(temperature(1600:3200),cp(1600:3200),3);
predicted_cp4 = polyval(coefficient4,temperature(1600:3200));

figure(4)
plot(temperature,cp,'linewidth',3)
hold on
plot(temperature(1600:3200),predicted_cp4,'linewidth',3,'color','r')
xlabel('temperature[k]')
ylabel('specific heat[kj/kmol-k]')
legend('original dataset','curve fit')
title('Cubic polynomial')

Linear curve fit using splitwise method:

Cubic curve fit using splitwise method:

Fitness character parameters:

Four parameters are used to measure the fitness characters of a curve

1. The sum of squares due to error(SSE)

2. R-square

3. Adjusted R-square

4. Root mean squared error(RMSE)

Sum of squatres due to error:

This statistic measures the total deviation of the response values from the fit to the response values. It is also called the summed square of residuals and is usually labeled as SSE.