I am currently trying to understand the linear regression fit by least squares for my machine learning homework, where I implement it and have to plot the result:
I have given two data sets, containing each a matrix $X$ and a vector $y$.
The goal is to predict the output $\hat Y$ for arbitrary data, isn't it? For this
In order to compute optimal parameters $\beta$ which will be used as coefficients:
$\hat Y = \beta_0 + \sum_{j = 1}^p (X_j \cdot \beta_j)$
For an easier computation I add the constant variable $1$ into each row of the matrix $X$, this way the prediction vector $\hat Y$can be calculated by:
$\hat Y = X^T \beta$
This way I compute $\beta = (X^T X)^{-1} X^T y$
Have I understand this correctly, $y$ is the training data?
For this exercise I have to generate a test data along a grid and collect it in a matrix $Z$. Then I should compute the prediction vector by $\hat y = Z \beta$. This works fine, but now I should plot it.
Here is my question, what, respectively how is this data plotted?
(I have 1-dim test data $\rightarrow X^{100 \times 1}$ and 2-dim test data $\rightarrow X^{100 \times 2}$)
My guesses so far:
The function I received through this method $f(x) = \beta_0 + x \cdot \beta_1$
The generated test data $Z$ and the prediction vector, but how is this plotted? How do you plot 1-dim data? What are exactly the $(x,y)$ points I should plot?
I tried for the coordinates, $x$ from the test data matrix $Z$ and $y$ from the prediction vector. This way all points will be put along the function $f(x)$ I don't know whether this is right, I would have thought that the points are scattered around the line.