polyfit and polyval funtions [duplicate] - matlab

Question

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MATLAB curve fitting - least squares method - wrong "fit" using high degrees
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Hello my question is about curve tiffing of P3(x) I write my code with using polyfit and polyval functions but I need to make it without using these functions.
this is my code that I wrote
n=input('Enter a value: ');
if(n<3)
fprintf('Enter a different value larger than 3')
end
if(n>=3)
x = 1:n;
y = -0.3*x + 2*randn(1,n);
[p,S] = polyfit(x,y,3);
[y_fit,delta] = polyval(p,x,S);
plot(x,y,'bo')
hold on
plot(x,y_fit,'r-')
title('Linear-Fit Output')
legend('Data','Linear Fit')
end
this is code that I write it is working but I am suppose to write it without using polfite and polyval functions

Answer 1

Without Using syms
y = a0*x^0 + a1*x^1 + a2*x^2 + a3*x^3
For n data point --> y = X*a
where
X = [x1^0 , a1*x1^1, a2*x1^2 , a3*x1^3; x2^0 , a1*x2^1, a2*x2^2 , a3*x2^3;...;xn^0 , a1*xn^1 , a2*xn^2 , a3*xn^3 ]
and a = [a0, a1, a2, a3]; y = [y1, y2, ..., yn]
a is computed as follow
y = X*a ---> a = X\y
The code is as follows
n is given
x = 1:n;
y = -0.3*x + 2*randn(1,n);
x0 = ones(n, 1);
x1 = x';
x2 = (x.^2)';
x3 = (x.^3)';
X = [x0 x1 x2 x3];
a = X\(y');
f =#(t)a(1) + a(2).*t + a(3).*(t.^2)+ a(4).*(t.^3);
Use least square method to find best fit cubic polynomial
n=input('Enter a value: ');
if(n<3)
fprintf('Enter a different value larger than 3')
else
x = 1:n;
y = -0.3*x + 2*randn(1,n);
% Cubic regression
syms a0 a1 a2 a3
yq = a0 + a1.*x + a2.*(x.^2) + a3.*(x.^3) ;
rq = yq - y;
f = sum(rq.^2);
fa0 = diff(f,a0);
fa1 = diff(f,a1);
fa2 = diff(f,a2);
fa3 = diff(f,a3);
sol = solve(fa0 == 0, fa1 == 0, fa2 == 0, a0, a1, a2, a3);
a0 = sol.a0;
a1 = sol.a1;
a2 = sol.a2;
a3 = sol.a3;
% Cubic Regression Curve Function
f =#(t)a0 + a1.*t + a2.*(t.^2)+ a3.*(t.^3);
% Plot Data and Cubic Regression Curve
h = figure(1);
% Data
plot3 = scatter(x, y, 100, '+', 'MarkerEdgeColor', 'red', 'linewidth', 5);
hold on
% cubic Regression Curve
xx = linspace(0,n,100);
plot4 = plot(xx, f(xx), 'linewidth', 5);
[~,b] = legend([plot3 plot4],{'Real Data','Cubic Regression'}, 'FontSize',30);
set(findobj(b,'-property','MarkerSize'),'MarkerSize',30);
xlabel('x-axis','color', 'k', 'fontSize', 25)
ylabel('y-axis', 'color','k', 'fontSize', 25)
hYLabel = get(gca,'YLabel');
set(hYLabel,'rotation',0,'VerticalAlignment','middle', 'HorizontalAlignment','right')
grid on
grid minor
set(gca,'FontSize',20)
set(get(h,'CurrentAxes'),'GridAlpha',0.8,'MinorGridAlpha',0.5);
xticks(x);
title('Cubic Regression', 'color', 'r');
whitebg('w');
end
n = 5
n = 20