Curve fitting of Specific Heat data using MATLAB

AIM

Write a program to fit a linear and cubic polynomial for the Cp data and measure the fitness characteristics for both the curves.

OBJECTIVES

• Write code to fit a linear and cubic polynomial for the Cp data.
• Plot the linear and cubic fit curves.
• Measure the fitness characteristics of the curves.

EQUATIONS

Curve Fitting

Polynomial curve fitting is when we fit our data to the graph of a polynomial function.

• Polynomial p(x) of degree n

`p(x)=p_1*x^n+p_2*x^(n−1)+...+p_n*x+p(n+1)`

Measure the fitness characteristics of the curves.

`"Error = "|y(i)- f(x(i))|`

Here,

`y(i)= "Data at point " i`

`f(x(i))= "Data at polynomial function point " i`

• Goodness-of-Fit Statistics

  • The sum of squares due to error (SSE)

  • R-square

  • Adjusted R-square

  • Root mean squared error (RMSE)

• Residual Analysis
• Confidence and Prediction Bounds

The sum of squares due to error (SSE)

`SSE ("sum of square errors")=sum_(i=1)^n "Error"(i)^2`

R-square

`SSR = sum_(i=1)^n (f(x(i))-"Mean")^2`

`SSE = sum_(i=1)^n (y(i)-f(x(i)))^2`

`SST = SSR + SSE`

`R^2 = (SSR) / (SST)`

Adjusted R-square

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

where,

`n = "Number of datapoints"`

`k = "Number of independent variables"`

Root mean squared error (RMSE)

`RMSE = sqrt((sum_(i=1)^n (y(i)-f(x(i)))^2)/n)=sqrt("SSE"/2)`

MATLAB PROGRAM

% Curve fitting by a linear and cubic polynomial for the Cp data. 

clear all
close all
clc

% load data

cp_data = load('data');
temperature = cp_data(:,1);
cp = cp_data(:,2);
legendInfo{1} = 'Original dataset'; 

% Range of Polynomial Degree
maxPolyDegree = 5;

for polyDegree = 1:maxPolyDegree
    
    co_effs = polyfit(temperature,cp,polyDegree);   % polyfit to fit curve
    predicted_cp(:,polyDegree) = polyval(co_effs, temperature);   % polyval to get predicted data

    SSE = sum((cp - predicted_cp(:,polyDegree)).^2); % Sum of squares due to error (SSE)

    SSR = sum((predicted_cp(:,polyDegree) - mean(cp)).^2);
    SST = SSR + SSE;

    R_square = SSR/SST;   % R-square

    k=1;
    n=length(cp);
    R_squareAdj = ((1 - R_square)*((n-1)/(n-k-1)));   % Adjusted R-square

    RMS = sqrt(SSE/n);    % Root mean squared error (RMSE)
    
    fitError(:,polyDegree) = [SSE;R_square;R_squareAdj;RMS];   % fiError

    legendInfo{polyDegree+1} = ['FitDegree ' num2str(polyDegree)];   % legendInfo Strings
end

% plot curve fit 
figure(1)
set(gcf,'position',[10,10,900,600])
plot(temperature, cp, 'linewidth',1);
hold on
plot(temperature, predicted_cp, 'linewidth',1);
xlabel('Temperture in [K]');
ylabel('Specific Heat in [KJ/Kmol-K]');
title('Curve fitting of Cp data');
legend(legendInfo,'location','southeast');

% plot error of curve fit 
for polyDegree = 1:maxPolyDegree

    % 'Sum of squares due to error (SSE)'
    figure(2)
    set(gcf,'position',[10,10,900,600])
    hold on
    plot(polyDegree, fitError(1,polyDegree),'b*');
    text(polyDegree, fitError(1,polyDegree),legendInfo{1,polyDegree+1},'VerticalAlignment','bottom','HorizontalAlignment','center');
    xlabel('Degree of Polynomial');
    ylabel('Sum of squares due to error (SSE)');
    title('Sum of squares due to error (SSE)');

    % 'R-square '
    figure(3)
    set(gcf,'position',[10,10,900,600])
    hold on
    plot(polyDegree, fitError(2,polyDegree),'c*');
    text(polyDegree, fitError(2,polyDegree),legendInfo{1,polyDegree+1},'VerticalAlignment','top','HorizontalAlignment','center');
    xlabel('Degree of Polynomial');
    ylabel('R-square ');
    title('R-square ');

    % 'Adjusted R-square'
    figure(4)
    set(gcf,'position',[10,10,900,600])
    hold on
    plot(polyDegree, fitError(3,polyDegree),'b*');
    text(polyDegree, fitError(3,polyDegree),legendInfo{1,polyDegree+1},'VerticalAlignment','bottom','HorizontalAlignment','center');
    xlabel('Degree of Polynomial');
    ylabel('Adjusted R-square');
    title('Adjusted R-square');

    % 'Root mean squared error (RMSE)'
    figure(5)
    set(gcf,'position',[10,10,900,600])
    hold on
    plot(polyDegree, fitError(4,polyDegree),'b*');
    text(polyDegree, fitError(4,polyDegree),legendInfo{1,polyDegree+1},'VerticalAlignment','bottom','HorizontalAlignment','center');
    xlabel('Degree of Polynomial');
    ylabel('Root mean squared error (RMSE)');
    title('Root mean squared error (RMSE)');

end

% Display plot by order and save as .jpj
figure(5)
saveas(gcf,strcat('Root mean squared error (RMSE)','.jpg'))
figure(4)
saveas(gcf,strcat('Adjusted R-square','.jpg'))
figure(3)
saveas(gcf,strcat('R-square','.jpg'))
figure(2)
saveas(gcf,strcat('Sum of squares due to error (SSE)','.jpg'))
figure(1)
saveas(gcf,strcat('Curve fitting of Cp data','.jpg'))

EXPLANATION

Step 1: Load Cp data from file 'data' into the MATLAB workspace.

Step 2: Solve the fit data into polynomials using `"MATLAB polyfit function"`.

Step 3: Get the predicted data ployfit coefficients using `" MATLAB polyval function"`.

Step 3: To evaluate the Goodness-of-Fit use Statistics and create fitness characteristics of the curve fits.

RESULTS

CONCLUSION

Cp data was fitted as a linear and cubic polynomial by using MATLAB code and the fitness characteristics of the curves measured using Goodness-of-Fit Statistics methods.

ERRORS

Correction:

plot(polyDegree, fitError(1,polyDegree),'b*');

Correction:

figure(1)
saveas(gcf,strcat('Curve fitting of Cp data','.jpg'))

REFERENCES

1. Evaluating Goodness of Fit
2. Residual Analysis
3. Confidence and Prediction Bounds
4. Polynomial curve fitting
5. Polynomial evaluation