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 Mi(x|k, t)

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,

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).

gollark: I mean, I can't really provide a more useful answer than "it is a genre which encompasses a lot of music I like listening to".
gollark: I figure that, having had some time to think, I'll answer the bot pretty late, then: Erra, Motionless in White, Brothers of Metal, Fit For A King, Rising Insane, Thornhill.
gollark: I just have songs picked at random from the list of ones I quite like.
gollark: Probably. I'm just terrible at answering "favourite X" questions.
gollark: Probably not!

References

  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.


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