M-spline

In the mathematical subfield of numerical analysis, an M-spline[1][2] is a non-negative spline function.

An M-spline family of order three with four interior knots.

Definition

A family of M-spline functions of order k with n free parameters is defined by a set of knots t1   t2    ...    tn+k such that

  • t1 = ... = tk
  • tn+1 = ... = tn+k
  • ti < ti+k for all i

The family includes n members indexed by i = 1,...,n.

Properties

An M-spline Mi(x|k, t) has the following mathematical properties

  • Mi(x|k, t) is non-negative
  • Mi(x|k, t) is zero unless ti  x < ti+k
  • Mi(x|k, t) has k  2 continuous derivatives at interior knots tk+1, ..., tn1
  • Mi(x|k, t) integrates to 1

Computation

M-splines can be efficiently and stably computed using the following recursions:

For k = 1,

if ti  x < ti+1, and Mi(x|1,t) = 0 otherwise.

For k > 1,

Applications

M-splines can be integrated to produce a family of monotone splines called I-splines. M-splines can also be used directly as basis splines for regression analysis involving positive response data (constraining the regression coefficients to be non-negative).

References


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