I-spline
From HandWiki
In the mathematical subfield of numerical analysis, an I-spline[1][2] is a monotone spline function.
Definition
A family of I-spline functions of degree k with n free parameters is defined in terms of the M-splines Mi(x|k, t)
- [math]\displaystyle{ I_i(x|k,t) = \int_L^x M_i(u|k,t)du, }[/math]
where L is the lower limit of the domain of the splines.
Since M-splines are non-negative, I-splines are monotonically non-decreasing.
Computation
Let j be the index such that tj ≤ x < tj+1. Then Ii(x|k, t) is zero if i > j, and equals one if j − k + 1 > i. Otherwise,
- [math]\displaystyle{ I_i(x|k,t) = \sum_{m=i}^j (t_{m+k+1}-t_m)M_m(x|k+1,t)/(k+1). }[/math]
Applications
I-splines can be used as basis splines for regression analysis and data transformation when monotonicity is desired (constraining the regression coefficients to be non-negative for a non-decreasing fit, and non-positive for a non-increasing fit).
References
- ↑ Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". Journal d'Analyse Mathématique 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.
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