M-spline

In the mathematical subfield of numerical analysis, an M-spline is a non-negative spline function.
[Image:Mspline order3.svg|thumb|325px|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 ≤ ... ≤ t_n+''k such thatt''₁ = ... = t_kt_n+1 = ... = t_n+''kt''_i < t_i+''k for all i''
The family includes n members indexed by i = 1,...,n.

Properties

An M-spline ''Mi'' has the following mathematical propertiesM_i is non-negativeM_i is zero unless t_i ≤ x < t_i+''kM''_i has k − 2 continuous derivatives at interior knots t_k+1,..., t_n−1M_i integrates to 1

Computation

M-splines can be efficiently and stably computed using the following recursions:
For k = 1,
if t_i ≤ x < t_i+1, and M_i = 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.