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 t1 ≤ t2 ≤ ... ≤ tn+k such that
t1 = ... = tk tn+1 = ... = tn+k ti < ti+k for all i The family includes n members indexed by i = 1,...,n.
Properties An M-spline Mi(x|k, t) has the following mathematical properties
Mi(x|k, t) is non-negative Mi(x|k, t) is zero unless ti ≤ x < ti+k Mi(x|k, t) has k − 2 continuous derivatives at interior knots tk+1, ..., tn−1 Mi(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 ) = 1 t i + 1 − t i {\displaystyle 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 ) = k [ ( x − t i ) M i ( x | k − 1 , t ) + ( t i + k − x ) M i + 1 ( x | k − 1 , t ) ] ( k − 1 ) ( t i + k − t i ) . {\displaystyle 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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