Least Squares Problem (compute and plot fit using matlab)

Question

I have the following data for a least squares problem: $$x: 0 , 1 , 3, 5 , 7, 9 , 12$$ $$y: 10 ,12, 18 ,15, 20, 25, 36$$

I have already found the normal equations and the pseudoinverse for the problem. my question is, how can I compute and plot the resulting fit for this problem using matlab?

Answer 1

This is a forum about mathematics, not Matlab. I will try to give you a mathematical answer and the relative code to solve this problem with Matlab.

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First of all, we need to define your problem in a matricia fashion.

The equation $y = mx+ q$ contains two parameters: $m$ and $q$. Then, I can write the previous equation in a matricial form:

$$y = [x ~~~ 1]\left[\begin{array}{c}m\\q\end{array}\right].$$

Since you have a lot of $x$s and $ys$, then:

$$\left[\begin{array}{c}10\\12\\18\\\ldots\end{array}\right] = \left[\begin{array}{cc}0 ~ 1\\1 ~ 1\\ 3 ~ 1 \\ \ldots\end{array}\right]\left[\begin{array}{c}m\\q\end{array}\right].$$

Let's pose

$$U = \left[\begin{array}{cc}0 ~ 1\\1 ~ 1\\ 3 ~ 1 \\ \ldots\end{array}\right] ~\text{and}~ \theta = \left[\begin{array}{c}m\\q\end{array}\right].$$

Then, the problem is: $$y = U \theta$$ and the solution is given by calculating the pseudoinverse of $U$, that is:

$$\hat{\theta} = U^{\#}y = (U^\top U)^{-1} U^\top y.$$

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Matlab code

1) If $x$ and $y$ are column vector, go to step 3.

2) If $x$ and $y$ are row vector, then transform them to column vector as follows:

x = x(:);
y = y(:);

3) Build up the matrix $U$:

U = [x ones(7,1)];

where $7$ is the number of elements of $x$.

4) Evaluate the pseudoinverse $U^{\#}$ of U:

Usharp = pinv(U);

5) Finally, evaluate your results:

theta = Usharp*y;
m = theta(1);
q = theta(2);

The first element of theta is $m$, while the second is $q$.

6) To plot the data and the estimated line, use the following:

scatter(x, y);
hold on;
plot(x, m*x+q, 'r');

7) The fitting error can be obtained as follows:

fitErr = norm(y-(m*x+q))^2/7`