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''₁  ≤ ''t''₂  ≤  ...  ≤  ''t''_''n''+''k'' such that

* ''t''₁ = ... = ''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,

:$M_i(x, 1,t) = \frac$

if ''t''_''i'' ≤ ''x'' < ''t''_''i''+1, and ''M''_''i''(''x'', 1,''t'') = 0 otherwise.

For ''k'' > 1,

:$M_i(x, k,t) = \frac.$

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

Splines (mathematics)

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