# Deriving the basis functions for natural cubic spline

Im looking through Section 5.2.1 of Elements of statistical learning, and am confused by this section

I understand that, from this answer , the definition of $$d_k(X)$$ is important in order to derive the following:

My questions are:

1. What does this $$d_k(X)$$ term represent? I can't figure out what is is.

2. Why is $$N_{k+2}$$ then given as it is above, using these $$d(X)$$ terms I am uncertain about?

3. Why are there only 3 basis functions shown here, when above it says there are K basis functions. Does this mean there will be a $$N_{3}, N_{4},...$$? If so, would they take the same format as $$N_{k+2}$$?

4. If we are representing a natural cubic spline, then where are the $$X^2$$ and $$X^3$$ basis functions?

1. It sounds like you want an intuitive way to interpret $$d_k(x)$$. Nice, intuitive ways of understanding things don't always exist, and I doubt such an interpretation exists for the $$d_k$$. Hastie, Tibshirani, and Friedman certainly don't mention one. Nonetheless, we can visualize the functions $$d_k(x)$$, however, which might help. Let's take the knots to be the set $$\{ 0, 1/3, 2/3, 1 \}$$. Then we can plot the functions $$d_k$$ and $$N_k$$, along with an example of a natural cubic spline given by the linear combination of the $$N_k$$ with weights $$(\beta_1, \beta_2, \beta_3, \beta_4) = (1, 10, -50, 100)$$.
4. (I know this is out of order) The natural cubic splines differ from cubic splines in that they are linear outside the interval $$(\xi_1, \xi_K)$$. The functions $$x^2$$ and $$x^3$$ can't be part of the basis because they are not linear outside the interval $$(\xi_1, \xi_K)$$. Any function in the basis has to itself be an element of the function space, and $$x^2$$ and $$x^3$$ are not natural cubic splines because they don't satisfy the linearity condition outside $$(\xi_1, \xi_K)$$.
2. Now, the reason we need to the basis functions $$N_k$$ is the same reason that $$x^2$$ and $$x^3$$ can't be basis functions: every basis function must be linear outside the interval $$(\xi_1, \xi_K)$$. The $$N_k$$ are constructed in the post you linked in order to satisfy this constraint. By taking the differences $$d_k(x) - d_{K - 1}$$, the quadratic and cubic parts of these functions cancel out on the interval $$(\xi_K, \infty)$$, which makes the resulting functions linear on that interval.
3. Perhaps the authors could have mentioned that $$k$$ is an index/variable that ranges over the values $$k = 1, \dots, K-2$$. This means that the expression $$N_{k + 2} (x) = d_k (x) - d_{K - 1} (x)$$ for $$k = 1, \dots, K-2$$ represents the $$3^{rd}, 4^{th}, \dots, K^{th}$$ basis functions.