M-spline

In the mathematical subfield of numerical analysis, an M-spline[1][2] is a non-negative spline function.

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    ...    tn+k such that

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

Properties

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

Computation

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

For k = 1,


M_i(x|1,t) = \frac{1}{t_{i+1}-t_i}

if ti  x < ti+1, and Mi(x|1,t) = 0 otherwise.

For k > 1,


M_i(x|k,t) = \frac{k\left[(x-t_i)M_i(x|k-1,t) + (t_{i+k}-x)M_{i+1}(x|k-1,t)\right]}{(k-1)(t_{i+k}-t_i)}.

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

  1. Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". J. Analyse Math. 17: 71107. doi:10.1007/BF02788653.
  2. Ramsay, J.O. (1988). "Monotone Regression Splines in Action". Statistical Science 3 (4): 425441. doi:10.1214/ss/1177012761. JSTOR 2245395.


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