# MATLAB: Write program to fit a straight line, 2nd & 3rd degree polynomial, exponential curves & plot the functions for the given values & the experimental data in the same graph. [Add comments beside every coding line] & comment which one is the best fitted curve.

 x 1 2 3 4 5 y 0.5 1.7 3.4 5.7 8.4

Solution:
Fit with a Straight Line:
clc
clear all
close all

x = [1 2 3 4 5];                     % Provided x
y = [0.5 1.7 3.4 5.7 8.4];           % provided y

n = length(x);                       % determine the length of x

X = [n sum(x); sum(x) sum(x.^2)];
Y = [sum(y); sum(x.*y)];

A = inv(X) * Y                      % value of coefficient a, b in matrix form

y_bar = sum(y)/n;

s_t = 0;                            % set initial value of s_t as 0
for i = 1:n
s_t = s_t + sum(y(i)-y_bar)^2;  % sum of squares of errors/residue between
the actual data and their arithmetic mean
end
s_t

s_r = 0;                            % set initial value of s_r as 0
for i = 1:n
s_r = s_r + sum(y(i)-A(1)-A(2)*x(i))^2; % sum of the squares of
errors/residue between the actual data and our predicted model
end
s_r

r = sqrt(1-(s_r/s_t));               % correlation coefficient
r

scatter (x, y, '*');                 % plot scatter of x and y
hold on
syms xi                              % define xi as a symbolic number
yi = A(1)+A(2)*xi;                   % define yi function
xi = linspace (x(1), x(n));          % generate linearly spaced vector of xi

yi = subs(yi);                       % symbolic substitution of yi
plot (xi, yi);                       % plot curve of xi and yi
grid on                              % turn on grid view

Output:
A   =     - 2.0000
1.9800
s_t =     40.1320
s_r =     0.9280
r     =    0.9884
Curve:

Fit with a 2nd degree Polynomial:
clc
clear all
close all

x = [1 2 3 4 5];                         % Provided x
y = [0.5 1.7 3.4 5.7 8.4];               % provided y

n = length(x);                           % determine the length of x

X = [n sum(x) sum(x.^2); sum(x) sum(x.^2) sum(x.^3); sum(x.^2) sum(x.^3) sum(x.^4)];
Y = [sum(y); sum(x.*y); sum((x.^2).*y)];

A = inv(X) * Y                           % value of coefficient a, b, c in
matrix form
y_bar = sum(y)/n;

s_t = 0;                                 % set initial value of s_t as 0
for i = 1:n
s_t = s_t + sum(y(i)-y_bar)^2;       % sum of squares of errors/residue
between the actual data and their
arithmetic mean
end
s_t

s_r = 0;                                % set initial value of s_r as 0
for i = 1:n
s_r = s_r + sum(y(i)-A(1)-A(2)*x(i) - A(3)*x(i)^2)^2;   % sum of the squares
of errors/residue between the actual
data and our predicted model
end
s_r

r = sqrt(1-(s_r/s_t));                 % correlation coefficient
r

scatter (x, y, '*');                   % plot scatter of x and y
hold on
syms xi                                % define xi as a symbolic number
yi = A(1)+A(2)*xi+A(3)*xi*xi;          % define yi function
xi = linspace (x(1), x(n));            % generate linearly spaced vector of xi
yi = subs(yi);                         % symbolic substitution of yi
plot (xi, yi);                         % plot curve of xi and yi
grid on                                % turn on grid view
Output:
A =
-0.2000
0.4371
0.2571
s_t = 40.1320
s_r = 0.0023
r = 1.0000
Curve:

Fit With a 3rd degree Polynomial:
clc
clear all
close all

x = [1 2 3 4 5];               % Provided x
y = [0.5 1.7 3.4 5.7 8.4];     % provided y

n = length(x);                 % determine the length of x

X = [n sum(x) sum(x.^2) sum(x.^3); sum(x) sum(x.^2) sum(x.^3) sum(x.^4); sum(x.^2) sum(x.^3) sum(x.^4) sum(x.^5); sum(x.^3) sum(x.^4) sum(x.^5) sum(x.^6)];
Y = [sum(y); sum(x.*y); sum((x.^2).*y); sum((x.^3).*y)];

A = inv(X) * Y                 % value of coefficient a, b, c, d in matrix form

y_bar = sum(y)/n;

s_t = 0;                       % set initial value of s_t as 0
for i = 1:n
s_t = s_t + sum(y(i)-y_bar)^2;  % sum of squares of errors/residue between
the actual data and their arithmetic mean
end
s_t

s_r = 0;                       % set initial value of s_r as 0
for i = 1:n
s_r = s_r + sum(y(i)-A(1)-A(2)*x(i) - A(3)*x(i)^2 - A(4)*x(i)^3)^2;
% sum of the squares of errors/residue between the actual data and our predicted model
end
s_r
r = sqrt(1-(s_r/s_t));        % correlation coefficient
r

scatter (x, y, '*');          % plot scatter of x and y
hold on
syms xi                       % define xi as a symbolic number
yi = A(1) + A(2)*xi + A(3)*xi*xi + A(4)*xi*xi*xi;   % define yi function
xi = linspace (x(1), x(n));   % generate linearly spaced vector of xi
yi = subs(yi);                % symbolic substitution of yi
plot (xi, yi);                % plot curve of xi and yi
grid on
Output:
A =
-0.0600
0.2405
0.3321
-0.0083
s_t =   40.1320
s_r = 0.0013
r =   1.0000

Curve:

Fit with a Exponential Curve:
clc
clear all
close all

x = [1 2 3 4 5];                        % Provided x
w = [0.5 1.7 3.4 5.7 8.4];              % set provided y as w

y = log(w);                             % turn the value of w in terms of log

n = length(x);                          % determine length of x

X = [n sum(x); sum(x) sum(x.^2)];
Y = [sum(y); sum(x.*y)];

B = inv(X) * Y            % value of coefficient log(a), b in matrix form
A = [exp(B(1)); B(2)]     % revert back to coefficien of a and b in matrix form

y_bar = sum(w)/n;

s_t = 0;                             % set initial value of s_t as 0
for i = 1:n
s_t = s_t + sum(w(i)-y_bar)^2;    % sum of squares of errors/residue between
the actual data and their arithmetic mean
end
s_t

s_r = 0;                             % set initial value of s_r as 0
for i = 1:n
s_r = s_r + sum(y(i)-log(A(1))-A(2)*x(i))^2;    % sum of the squares of
errors/residue between the actual data and our predicted model
end
s_r

r = sqrt(1-(s_r/s_t));                 % correlation coefficient
r

scatter (x, w, 'p');                   % plot scatter of x and y
hold on
syms xi                                % define xi as a symbolic number
yi = A(1)*exp(A(2)*xi);                % define yi function
xi = linspace (x(1), x(n), 100);        % generate linearly spaced vector of xi
yi = subs(yi);                         % symbolic substitution of yi
plot (xi, yi);                         % plot curve of xi and yi
grid on                                % turn grid view on
Output:
B =
-1.0698
0.6853
A =
0.3431
0.6853
s_t = 40.1320
s_r = 0.2615
r =     0.9967
Curve:

Comparison:
The correlation coefficient ‘r’ determines the best fitted curve. In best fitted curve the value of ‘r’ is close to 1. And the worst fitting curve ‘r’ is close to 0. Here by the analysis of the above curve fittings we can see the value of correlation coefficient ‘r’ is 1 in the 2nd order and 3rd order polynomial. Hence these two are best fitted curve.