Question

Question 4) Suppose that the (univariate) variable y is known to be a quadratic function of the variable x; that is, y = a x2 +bx+c, where the coefficients a, b, c are obtained by conducting an experiment in which values y1, .. , Yn of the variable y are measured for corresponding values 21,.. , Un of the variable x. Find the best least-squares fit of the quadratic polynomial using the data: {(-2,-5),(-1, -1),(0,4), (1,7), (2,6), (3,5), (4, -1)}. Plot these points and the computed quadratic curve in the (x, y)-plane. Hint: You need to solve the following optimization problem: min (ax? + bxi +c- yi)?, a,b,c i=1

Answer 1

MATLAB Code:

close all
clear
clc

% To get the coefficients of the quadratic function,
% y = a*x^2 + b*x + c
% We need to solve the system AX = y,
% where:
% A = [x^2 x 1] and X = [a b c]

% Data points
x = (-2:1:4)';
y = [-5 -1 4 7 6 5 -1]';
A = [x.^2 x ones(size(x))];
X = A\y; % Solve the system
fprintf('Quadratic Function, y = (%.4f)*x^2 + (%.4f)*x + (%.4f)\n', X)

plot(x, y, 'o') % Plotting data points
hold on
xx = linspace(min(x), max(x), 100); % Bunch of new samples for a smoother plot
yy = X(1)*xx.^2 + X(2)*xx + X(3);
plot(xx, yy) % Plotting the quadratic curve
xlabel('x'), ylabel('y'), title('Least Squares Quadratic Fit')
legend('Data Points', 'Quadratic Fit', 'Location', 'northwest')

Output:

Quadratic Function, y = (-1.0476)*x^2 + (3.0238)*x + (4.3571)

Plot:

Least Squares Quadratic Fit Data Points Quadratic Fit Y -1 0 1 2