I-spline

A family of I-spline functions of degree k with n free parameters is defined in terms of the M-splines M_i(x|k, t)

${\displaystyle I_{i}(x|k,t)=\int _{L}^{x}M_{i}(u|k,t)du,}$

where L is the lower limit of the domain of the splines.

Since M-splines are non-negative, I-splines are monotonically non-decreasing.

Let j be the index such that t_j ≤ x < t_j+1. Then I_i(x|k, t) is zero if i > j, and equals one if j − k + 1 > i. Otherwise,

${\displaystyle I_{i}(x|k,t)=\sum _{m=i}^{j}(t_{m+k+1}-t_{m})M_{m}(x|k+1,t)/(k+1).}$

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).