Suppose we fit a curve with basis functions \(b_1(X) = X\), \(B_2(X) = (X - 1)^2 I(X \geq 1)\). Note that \(I(X \geq 1)\) equals 1 for \(X \geq 1\) and 0 otherwise. We fit the linear regression model
\[Y = \beta_0 + \beta_1b_1(X) + \beta_2b_2(X) + \varepsilon\]
and obtain the coefficient estimates \(\hat \beta_0 = 1\), \(\hat \beta_1 = 1\), \(\hat \beta_2 = -2\). Sketch the estimated curve between \(X = -2\) and \(X = 2\). Note the intercepts, slopes and other relevant information.
Suppose we fit a curve with basis functions \(b_1(X) = I(0 \leq X \leq 2) - (X-1)I(1 \leq X \leq 2)\), \(B_2(X) = (X - 3) I(3 \leq X \leq 4) + I(4 < X \leq 5)\). We fit the linear regression model
\[Y = \beta_0 + \beta_1b_1(X) + \beta_2b_2(X) + \varepsilon\]
and obtain the coefficient estimates \(\hat \beta_0 = 1\), \(\hat \beta_1 = 1\), \(\hat \beta_2 = 3\). Sketch the estimated curve between \(X = -2\) and \(X = 2\). Note the intercepts, slopes and other relevant information.
Explain what happens to the bias/variance trade-off of our model estimates use regression splines.
In this exercise you will analyze the Wage data set. It is found in the ISLR package and can be loaded like so
wage using age. Polynomial regression can be performed in tidymodels by using a linear regression model (linear_reg()) with a recipe that performs polynomial expansion (step_poly()). Use cross-validation to select the optimal degree.Optional: Make a plot of the resulting polynomial fit to he data.
step_discretize()) and perform cross-validation to choose the optimal number of cuts.Optional: Make a plot of the fit.