Statistics - Cubic spline


Cubic splines is a step cubic piecewise polynomial in the left and the right and continuous at the knot.

There's the following constraints:

  • The first derivatives continues at the knot,
  • and the second derivatives continues at the knot.

We couldn't make the third derivative continuous, because then it would just be a global cubic polynomial. It's believed that a cubic spline, the discontinuity which exists in the third derivative is not detectable by the human eye.

The degree of Statistics - Piecewise polynomials is 2 because it's a third degree polynomial.

Definition / Representation

A cubic spline with knots at <math>\xi_k</math> with <math>k = 1, \dots, K</math> is a piecewise cubic polynomial with continuous derivatives up to order 2 at each knot.

As a linear spline, this model can be represented with truncated power basis (functions|expansion|transformation) in the original variables

<MATH> y_i = \beta_0 + \beta_1.b_1(x_i) + \beta_2.b_2(x_i) + \dots + \beta_{K+3}b_{K+3}(x_i) + \epsilon_i </MATH>

where the <math>b_k</math> are basis functions and are:

  • the variable itself. The first basis functions is just the variable itself

<MATH> \begin{array}{rll} b_1(x_i) & = & x_i \\ \end{array} </MATH>

  • the variable squared

<MATH> \begin{array}{rll} b_2(x_i) & = & x_i^2 \\ \end{array} </MATH>

  • the variable cubed

<MATH> \begin{array}{rll} b_3(x_i) & = & x_i^3 \\ \end{array} </MATH>

  • and additional variables that are a collection of truncated basis transformation functions at each of the knots. They are called truncated power functions and this one is to power 3 (cubic).

<MATH> \begin{array}{rll} b_{k+3}(xi) & = & (xi - \xi_k)_+^3 \text{ where } k = 1, \dots, K \\ \text{where the ()+ means positive part. i.e. } (xi - \xi_k)^3_+ & = & \left\{ \begin{matrix} (xi - \xi_k)^3 & \text{ if } x_i > \xi_k & \\ 0 & \text{Otherwise} & \text{It's not allowed to go negative.} \end{matrix} \right. \end{array} </MATH>


At the knot, the basis function:

  • is 0
  • got 0 first derivative and 0 second derivative.

Therefore, when we add it, we don't change the derivatives of the function and so has the required continuity of a cubic spline.

Degree of freedom

A cubic spline with K knots has K plus 4 parameters or degrees of freedom.

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