I-spline

In the mathematical subfield of numerical analysis, an I-spline[1][2] is a monotone spline function.

An I-spline family of order three with four interior knots.

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 _{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 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 _{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).

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References

Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". J. Analyse Math. 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. JSTOR 2245395.

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[1] Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits". J. Analyse Math. 17: 71–107. doi:10.1007/BF02788653.

[2] Ramsay, J.O. (1988). "Monotone Regression Splines in Action". Statistical Science. 3 (4): 425–441. doi:10.1214/ss/1177012761. JSTOR 2245395.