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

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 k2 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