How to calculate basis functions by hand

Question

I am learning about natural splines and basis functions and am struggling with it a lot.

I understand the concepts of knots being the part where first and second derivatives are equal on either side. But when it comes to calculating basis functions and eventually a basis matrix I get confused.

I understand that for a cubic spline $d_i(x) = \frac{(x-x_i)^{3}_+ - (x-x_n)^{3}_+}{x_n - x_i}, i=1,\dots,n-1.$

for the example n = 4 ,

$d_i(x) = \frac{(x-x_i)^{3}_+ - (x-x_4)^{3}_+}{x_4 - x_i}$

From here onwards I get lost. How do I find $d_1(x_1)$? I would assume you just replace i with 1 and x with $x_1$ however this would result in $$d_1(x_1) = \frac{(x_1-x_i)^{3}_+ - (x_1-x_4)^{3}_+}{x_4 - x_1} = \frac{(0)^{3}_+ - (x_1-x_4)^{3}_+}{x_4 - x_1} = \frac{- (x_1-x_4)^{3}_+}{x_4 - x_1}$$

However the text book says the answer is 0. For $d_1(x_2)$ it gives an answer of $\frac{(x_2-x_1)^3}{x_4-x_1}$.

I do not understand why it appears that for every answer it doesn't include the second term of the numerator.

Any clarification as to how $d_1(x_1)$ or $d_3(x_2)$ is 0 would be very appreciated.

Answer 1

Not sure what book or reference you are using, but it is standard to assume you ordered the data so that $x_1 \leq x_2 \leq \dots \leq x_n$. So for $d_1(x_1)$, recalling that $(w)_+ = \max(w,0)$, we have

$$d_1(x_1) = \frac{(x_1 - x_4)_+^3}{x_4-x_1} = \frac{\big(\max(0,x_1-x_4)\big)^3}{x_4-x_1} = 0$$

where the last equality followed because $x_1-x_4 \leq 0$ so that the numerator is $0$.

Also, just in case, this question on the site may be of interest to you: Why are the basis functions for natural cubic splines expressed as they are? (ESL)