Question

Suppose I choose to use a quadratic spline (k=2) with two knots. This is an ordinary spline, not a natural spline. How many effective number of parameters (degrees of freedom) does this spline have?

Answer 1

The quadratic spline with two knots has five effective degrees of freedom, calculated by initially having nine parameters (three per interval across three intervals) and then subtracting the four constraints imposed by the two knots for continuity.

Step-by-step explanation:

When using a quadratic spline with two knots, the total number of parameters or degrees of freedom is determined by the polynomial pieces and the continuity conditions imposed by the knots. For a quadratic polynomial, there are three parameters (coefficients) per interval. With two knots, we divide the data into three intervals, thus having initially 9 parameters. However, the conditions of continuity at the knots lead to constraints. Each knot enforces two conditions: one for function continuity and one for the first derivative continuity. Therefore, 2 knots give us 4 constraints. Subtracting those from the initial 9 parameters, we end up with 5 parameters, which represent the effective number of parameters or degrees of freedom for the spline.