slp: Shifted Legendre Polynomials In qrcm: Quantile Regression Coefficients Modeling

Description

Computes shifted Legendre polynomials.

Usage

 `1` ```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 `iqr`. 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 estimation algorithm of `iqr` is optimized for objects of class “slp”, which means that using `formula.p = ~ slp(p, k)` instead of `formula.p = ~ p + I(p^2) + ... + I(p^k)` will result in a quicker computation, even with k = 1, with equivalent results. The default for `iqr` is `formula.p = ~ slp(p, k = 3)`.

Author(s)

Paolo Frumento [email protected]

References

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

`plf`, for piecewise linear functions in the unit interval.
 ```1 2 3 4``` ``` 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 ```