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 ≤ ... ≤ 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,

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

if t_i ≤ x < t_i+1, and M_i(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).

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References

Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". J. Analyse Math. 17: 71–107. doi:10.1007/BF02788653.
Ramsay, J.O. (1988). "Monotone Regression Splines in Action". Statistical Science. 3 (4): 425–441. doi:10.1214/ss/1177012761. JSTOR 2245395.

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[1] Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". J. Analyse Math. 17: 71–107. doi:10.1007/BF02788653.

[2] Ramsay, J.O. (1988). "Monotone Regression Splines in Action". Statistical Science. 3 (4): 425–441. doi:10.1214/ss/1177012761. JSTOR 2245395.