Polynomial Curve Fitting using MATLAB

AIM: To perform a curve fit on given data and estimate the fitness characteristics using MATLAB.

OBJECTIVE:

1. To plot the linear and cubic fit curves along with the raw data points. Write a code to show split wise method.

2. To validate the fitness characteristics using the curve fitting toolbox.

3. Analyze the following questions:

- How to make a curve fit perfectly?

- How to get the best fit?

- What could be done to improve the cubic fit?

THEORY:

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. Curve fitting 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.

Consider N points in the curve. Assume y = f(x) is the curve fitting equation.

The error for any point i can be given by:

`Err(i) = |[y(i) - f(x(i))]|`

To remove the modulus sign, we square the error at each point on the curve and add them to get:

`SSE = sum_(i=1)^n Err(i)^2`

Assume the proposed equation is a `N^(th)` degree polynomial function as:

`y(x) = A + Bx + Cx^2 + Dx^3 + ... + Nx^(n-1)`

Differentiate each SSE with individual coefficient and equate to zero to find the minima and all the coefficients of the above equation:

`[(frac[delSSE][delA]) , (frac[delSSE][delB]) , (frac[delSSE][delC]) , (frac[delSSE][delD]) , (.) , (.) , (.) , (frac[delSSE][delN])] = 0`

This method is called as Least Square Method and commonly used for fitting a polynomial function.

R-squared: It is the correlation between the given values and the predicted values. It determines to what extent the given data is a successful fit. It is also called the square of multiple correlation and coefficient of multiple determination.

The value of R-squared ranges from 0 (worst possible fit) to 1 (best possible fit) and can be calculated as:

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

`SSR = sum_(i=1)^l [f(x(i)) - Mean]^2`

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

`SST = SSR + SSE`

where;

`SSR` = Sum of squares of regression

`SST` = Sum of squared total

Adjusted R-squared: It is a normalizing variation of R-squared that neglects the variation of R-squared due to insignificant independent variables.

As you keep on adding new variables even if they seem completely unrelated, they will still be related in an infinitesimally small manner.

Adjusted R-squared can be formulated as:

`R_(adj)^2 = 1 - frac[(1-R^2)(n-1)][n-k-1]`

where;

`n` = number of data points

`k` = number of independent variables

Root Mean Squared Error: It is the estimate of the standard deviation of a random component in the data. It is also called the fit standard error and standard error of the regression.

Root Mean Squared Error can be formulated as:

`RMSE = sqrtfrac[SSE][n]`

Best Fit: Refers to the curve that has the least error (maximum R-squared value) with respect to the original curve.

Perfect Fit: Refers to the curve that has the zero error (R-squared value = 1) with respect to the original curve.

SOLUTION:

Programming Language: MATLAB

Procedure/Steps:

1. Load the given data file into the working directory. Assign suitable variables to store the given input values.

2. Create a for loop for generation of polynomial degrees as per the requirement.

3. Compute the coefficients and predicted Cp value using polyfit and polyval commands respectively.

polyfit is a function that computes a least squares polynomial for a give set of data. Polyfit generates the coefficients of the polynomial, which can be used to model a curve to fit the data.

polyval evaluates a polynomial for a given set of x values.

4. The parameters for the goodness of fit (SSE, R-sqr, Adjusted R-sqr, RMSE) are defined with proper relations.

5. The curves are plotted with the help of plot function. 

Code #1: The MATLAB program to fit the curve is shown below.

% Program to fit a linear and cubic polynomial for given data.

% Inputs:

file = load('data');
temp = file(:,1);
Cp = file(:,2);


% Solution:

for i=1:3
coeff = polyfit(temp,Cp,i);
Cp_pr(:,i) = polyval(coeff,temp);

% Goodness of Fit
SSE(i) = sum((Cp - Cp_pr(:,i)).^2)   % Sum of squared errors

SSR(i) = sum((Cp_pr(:,i) - mean(Cp)).^2);   % Sum of squares of regression
SST(i) = SSE(i) + SSR(i);   % Sum of squared total
R_sqr(i) = SSR(i)/SST(i)   % R-squared
 
n = length(Cp_pr(:,i));   % number of data points
k = 1;   % number of independent variables
R_sqr_adj(i) = 1 - ((1-R_sqr(i))*(n-1)/(n-k-1))   % Adjsuted R-squared
 
RMSE(i) = sqrt(SSE(i)/n)   % Root Mean Squared Error


% Plot:

plot(temp,Cp,'linewidth',4,'color','k')
hold on
plot(temp,Cp_pr,'linewidth',4,'linestyle','-.')
hold off
xlabel('Temperature (K)')
ylabel('Entropy (kJ/kmol-K)')
title('Curve Fit')
grid on

end

% Labeling
legend('Dataset','Curve Fit Linear','Curve Fit Quadratic','Curve Fit Cubic')

Output #1: The program is executed and goodness of fit parameters are computed as shown below.

Plot #1: Entropy varies with respect to Temperature for the original and predicted dataset as shown below.

Curve Fitting Toolbox #1:

Linear:

Quadratic:

Cubic:

Curve Fit Validation #1:

`SSE_(Cubic) < SSE_(Quadratic) < SSE_(L i n ear)`

`R_(Cubic)^2 > R_(Quadratic)^2 > R_(L i n ear)^2`

`RMSE_(Cubic) < RMSE_(Quadratic) < RMSE_(L i n ear)`

We can observe that the Cubic form is extrapolated with highest accuracy amongst all.

Code #2: The MATLAB program to split wise fit the curve is shown below.

% Program to split wise curve fit a linear and cubic polynomial for given data.

% Inputs:

file = load('data');
temp = file(:,1);
Cp = file(:,2);


% Solution:

% Splitting the data
temp_1 = file(1:800,1);
temp_2 = file(801:1600,1);
temp_3 = file(1601:2400,1);
temp_4 = file(2401:3200,1);

Cp1 = file(1:800,2);
Cp2 = file(801:1600,2);
Cp3 = file(1601:2400,2);
Cp4 = file(2401:3200,2);

coeff = polyfit(temp,Cp,3);
Cp_pr = polyval(coeff,temp);


% Coefficients in split-wise order
coeff_1 = polyfit(temp_1,Cp1,3);
Cp_pr_1 = polyval(coeff_1,temp_1);

coeff_2 = polyfit(temp_2,Cp2,3);
Cp_pr_2 = polyval(coeff_2,temp_2);

coeff_3 = polyfit(temp_3,Cp3,3);
Cp_pr_3 = polyval(coeff_3,temp_3);

coeff_4 = polyfit(temp_4,Cp4,3);
Cp_pr_4 = polyval(coeff_4,temp_4);

for i = 1:length(Cp)

% Goodness of Fit
SSE = sum((Cp(i) - Cp_pr(i)).^2);   % Sum of squared errors

SSR = sum((Cp_pr(i) - mean(Cp(i))).^2);   % Sum of squares of regression
SST = SSE + SSR;   % Sum of squared total
R_sqr = SSR/SST;   % R-squared
 
n = length(Cp);   % number of data points
k = 1;   % number of independent variables
R_sqr_adj = 1 - ((1-R_sqr)*(n-1)/(n-k-1));   % Adjsuted R-squared
 
RMSE = sqrt(SSE/n);   % Root Mean Squared Error
end


% Plot:

plot(temp,Cp,'linewidth',5,'color','c')
hold on
plot(temp_1,Cp_pr_1,'linewidth',3,'linestyle','-.')
hold on
plot(temp_2,Cp_pr_2,'linewidth',3,'linestyle','-.')
hold on
plot(temp_3,Cp_pr_3,'linewidth',3,'linestyle','-.')
hold on
plot(temp_4,Cp_pr_4,'linewidth',3,'linestyle','-.')
hold off
xlabel('Temperature (K)')
ylabel('Entropy (kJ/kmol-K)')
title('Curve Fit (Split Wise)')
legend('Dataset','Curve Fit 1','Curve Fit 2','Curve Fit 3','Curve Fit 4')
grid on

Output #2: The program is executed as per given parameters as shown below.

Plot #2: Entropy varies with respect to Temperature for the original and predicted dataset (in split fashion) as shown below.

Curve Fitting Toolbox #2:

Cp1 vs temp_1

Cp2 vs temp_2

Cp3 vs temp_3

Cp4 vs temp_4

Curve Fit Validation #2:

We can observe that the sections after the first split are the same without any variation and the curve fit is extrapolated with the highest accuracy thereafter.

CONCLUSION: From the above results, we can infer that:

- A perfect curve fit can be achieved by incorporating a split wise method by dividing the given data and polynomials with a different degree at each range.

- A best fit can be achieved by mimicking original dataset and High R2 value with less error.

- Improvement of the cubic fit can be achieved by 'Center and Scale' command in curve fit toolbox or by increasing the degree of the polynomial.

REFERENCES:

1. Curve Fitting and criterion to choose the best fit

2. Curve Fitting Toolbox