I-spline explained

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 M_i(x|k, t)

I _{i(x|k,t)} = \int ^{x} _{L} 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 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,

I _{i(x|k,t)} = \sum ^{j} _{m = i} (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).

References

1. On Polya frequency functions. IV. The fundamental spline functions and their limits . Curry . H.B. . Schoenberg, I.J. . 1966 . 17 . 71 - 107 . . 10.1007/BF02788653 .
2. Ramsay . J.O. . Statistical Science . 1988 . 3 . 425 - 441 . Monotone Regression Splines in Action . 2245395 . 10.1214/ss/1177012761 . 4. free .