I-spline

In the mathematical subfield of numerical analysis, an I-spline 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)



I_i(x|k,t) = \int_L^x M_i(u|k,t)du, $$

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 &minus; k + 1 > i. Otherwise,



I_i(x|k,t) = \sum_{m=i}^j (t_{m+k+1}-t_m)M_m(x|k+1,t)/(k+1). $$

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).