I-spline
Encyclopedia
In the mathematical
subfield of numerical analysis
, an I-spline is a monotone spline
function.
s 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.
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
subfield of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, an I-spline is a monotone spline
Spline (mathematics)
In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...
function.
Definition
A family of I-spline functions of order k with n free parameters is defined in terms of the M-splineM-spline
In the mathematical subfield of numerical analysis, an M-spline is a non-negative spline function.-Definition:A family of M-spline functions of order k with n free parameters is defined by a set of knots t1 ≤ t2 ≤ .....
s 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 transformationData transformation (statistics)
In statistics, data transformation refers to the application of a deterministic mathematical function to each point in a data set — that is, each data point zi is replaced with the transformed value yi = f, where f is a function...
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).