Example relevant for the exercises

In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\boldsymbol{y}$, we condiser a simple polynomial fit. We assume our data can represented by a fourth-order polynomial. For the $i$th component we have

$$\tilde{y}_i = \beta_0+\beta_1x_i+\beta_2x_i^2+\beta_3x_i^3+\beta_4x_i^4.$$

we have five predictors/features. The first is the intercept $\beta_0$. The other terms are $\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an $n\times p$ matrix $\boldsymbol{X}$.

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