I've been working on a fascinating problem involving measurements that reveal a relationship between two variables, $t$ and $y$. I have a set of four data points: $(-1, 0)$, $(0, 1)$, $(1, 2)$, and $(2, 4)$. The first component corresponds to the $t$-values, while the second component corresponds to the $y$-values. My goal is to approximate a linear equation that describes the relationship $y(t)$.
I'd really appreciate some guidance on the next steps in my analysis. Here are the specific questions I have:
(a) By using the equation of a straight line, $kt + m = y$, along with the four data points, how can I set up an equation system $Ax = b$?
(b) How can I show that there is no exact solution to the equation system $Ax = b$?
(c) What's the process to formulate a least squares problem and find the least squares solution to $Ax = b$?
(d) Could someone explain how to calculate the differences in $y$ values between the measured data and the values predicted by the least squares fit?
Absolutely, linear regression is indeed a suitable approach for this scenario. Given the data points $\{(-1, 0), (0, 1), (1, 2), (2, 4)\}$, you can use linear regression to find the best-fit line in the form $y = mx + b$. The goal is to minimize the sum of squared vertical distances between the observed $y$ values and the values predicted by the line.
To formulate the linear regression problem, we can represent the data points as a matrix equation:
$$\begin{bmatrix}-1&1\\ 0&1\\ 1&1\\ 2&1\end{bmatrix} \begin{bmatrix}m\\ b\end{bmatrix} =\begin{bmatrix}0\\ 1\\ 2\\ 4\end{bmatrix}$$
Here, the matrix on the left represents the design matrix $X$, the vector on the right is the response vector $\mathbf{y}$, and $\mathbf{m} = [m, b]^T$ contains the coefficients we want to find.
To solve for $\mathbf{m}$ using the least squares solution, we can compute:
$$\mathbf{m} = (X^T X)^{-1} X^T \mathbf{y}$$
Once we have the values of $m$ and $b$, the best-fit line equation is $y = mx + b$, which can be used for approximating the relationship between $y$ and $t$.
I think it's important here to perform appropriate calculations to find the numerical values of $m$ and $b$ using the provided data.
Your insights and advice would be immensely helpful in guiding me through these steps. Thank you so much in advance!
You probably will want to use linear regression to find a best-fit line, given these data points. There are standard textbooks that explain all of these aspects of linear regression; there is little point in us repeating that here. I suggest you spend some time studying this topic in textbooks on statistics.