In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 subfield of numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, an M-spline is a non-negative spline

Spline (mathematics)

In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...

 function.

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 that

• t₁ = ... = t_k
• t_n+1 = ... = t_n+k
• t_i < t_i+k for all i

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

Properties

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

• M_i(x|k, t) is non-negative

• M_i(x|k, t) is zero unless t_i ≤ x < t_i+k

• M_i(x|k, t) has k − 2 continuous derivatives at interior knots t_k+1, ..., t_n−1

• M_i(x|k, t) 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(x|1,t) = 0 otherwise.

For k > 1,

Applications

M-splines can be integrated to produce a family of monotone splines called I-spline

I-spline

In the mathematical subfield of numerical analysis, an I-spline is a monotone spline function.-Definition:A family of I-spline functions of order k with n free parameters is defined in terms of the M-splines Mi...

s. 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).