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M-spline

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 t<sub>1</sub> &nbsp;≤&nbsp;t<sub>2</sub> &nbsp;≤&nbsp; ... &nbsp;≤&nbsp; t<sub>n+k</sub> such that

  • t<sub>1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;t<sub>k</sub>
  • t<sub>n+1</sub>&nbsp;=&nbsp;...&nbsp;=&nbsp;t<sub>n+k</sub>
  • t<sub>i</sub>&nbsp;<&nbsp;t<sub>i+k</sub> for all i

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

Properties

An M-spline M<sub>i</sub>(x|k,&nbsp;t) has the following mathematical properties

  • M<sub>i</sub>(x|k,&nbsp;t) is non-negative
  • M<sub>i</sub>(x|k,&nbsp;t) is zero unless t<sub>i</sub>&nbsp;≤&nbsp;x&nbsp;<&nbsp;t<sub>i+k</sub>
  • M<sub>i</sub>(x|k,&nbsp;t) has k&nbsp;&minus;&nbsp;2 continuous derivatives at interior knots t<sub>k+1</sub>, ..., t<sub>n&minus;1</sub>
  • M<sub>i</sub>(x|k,&nbsp;t) integrates to 1

Computation

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

For k&nbsp;=&nbsp;1,

if t<sub>i</sub>&nbsp;≤&nbsp;x&nbsp;<&nbsp;t<sub>i+1</sub>, and M<sub>i</sub>(x|1,t)&nbsp;=&nbsp;0 otherwise.

For k&nbsp;>&nbsp;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