In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

subfield of

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

, an I-spline is a

monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...

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: htt ...

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