In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

subfield of

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, 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 ''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_^j (t_-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 In computing, data transformation is the process of converting data from one format or structure into another format or structure. It is a fundamental aspect of most data integrationCIO.com. Agile Comes to Data Integration. Retrieved from: https ...

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

Splines (mathematics) {{mathapplied-stub