slp: Shifted Legendre Polynomials

In Mqrcm: M-Quantile Regression Coefficients Modeling

Description

Computes shifted Legendre polynomials.

Usage

slp(p, k = 3, intercept = FALSE)

Arguments

p	the variable for which to compute the polynomials. Must be 0 <= p <= 1.
k	the degree of the polynomial.
intercept	logical. If TRUE, the polynomials include the constant term.

Details

Shifted Legendre polynomials (SLP) are orthogonal polynomial functions in (0,1) that can be used to build a spline basis, typically within a call to iMqr. The constant term is omitted unless intercept = TRUE: for example, the first two SLP are (2*p - 1, 6*p^2 - 6*p + 1), but slp(p, k = 2) will only return (2*p, 6*p^2 - 6*p).

Value

An object of class “slp”, i.e., a matrix with the same number of rows as p, and with k columns named slp1, slp2, ... containing the SLP of the corresponding orders. The value of k is reported as attribute.

Note

The default for iMqr is formula.p = ~ slp(p, k = 3).

Author(s)

Paolo Frumento [e-mail]

References

Refaat El Attar (2009), Legendre Polynomials and Functions, CreateSpace, ISBN 978-1-4414-9012-4.

See Also

plf, for piecewise linear functions in the unit interval.

Examples

  p <- seq(0,1,0.1)
  slp(p, k = 1) # = 2*p
  slp(p, k = 1, intercept = TRUE) # = 2*p - 1 (this is the true SLP of order 1)
  slp(p, k = 3) # a linear combination of (p, p^2, p^3), with slp(0,k) = 0

Example output

Loading required package: pch
Loading required package: survival
      slp1
 [1,]  0.0
 [2,]  0.2
 [3,]  0.4
 [4,]  0.6
 [5,]  0.8
 [6,]  1.0
 [7,]  1.2
 [8,]  1.4
 [9,]  1.6
[10,]  1.8
[11,]  2.0
attr(,"k")
[1] 1
attr(,"class")
[1] "slp"
      slp1
 [1,] -1.0
 [2,] -0.8
 [3,] -0.6
 [4,] -0.4
 [5,] -0.2
 [6,]  0.0
 [7,]  0.2
 [8,]  0.4
 [9,]  0.6
[10,]  0.8
[11,]  1.0
attr(,"k")
[1] 1
attr(,"class")
[1] "slp"
      slp1  slp2 slp3
 [1,]  0.0  0.00 0.00
 [2,]  0.2 -0.54 0.92
 [3,]  0.4 -0.96 1.36
 [4,]  0.6 -1.26 1.44
 [5,]  0.8 -1.44 1.28
 [6,]  1.0 -1.50 1.00
 [7,]  1.2 -1.44 0.72
 [8,]  1.4 -1.26 0.56
 [9,]  1.6 -0.96 0.64
[10,]  1.8 -0.54 1.08
[11,]  2.0  0.00 2.00
attr(,"k")
[1] 3
attr(,"class")
[1] "slp"

Mqrcm documentation built on Feb. 2, 2021, 9:06 a.m.