legendre.polynomials: Orthogonal Legendre Polynomials Basis System

In cSFM: Covariate-adjusted Skewed Functional Model (cSFM)

Description

Generate Legendre polynomials as an orthogonal basis system in the interval [0, 1]. The set of polynomial functions are used to be the eigenfunctions to generate the covariance matrix of a latent process.

Usage

legendre.polynomials(t, degree, normalized = TRUE)

Arguments

t	the set of values to be evaluated, taking values from [0, 1]
degree	the polynomial degree of the function, taking values as 0 (constant), 1 (linear), 2 (quadratic) and 3 (cubic)
normalized	logical value; If TRUE (default) then the values are normalized such that the L2 norm of the function values is 1

Details

Legendre polynomials can take any integers as degree, however, it is sufficent for most applications up to degree = 3. It is also messy to obtain the exploit expression for a degree greater than 3.

The logical value normalized take TRUE as the default, which means the generated eigenfunction basis matrix will be an orthogonal matrix.

Value

A vector which are the evaluations of the Legendre polynomial function at t.

See Also

DFT.basis

Examples

# time points 
t <- seq(from = 0, to = 1, length = 100)
# basis matrix evaluated at the time points t
# an intercept column is included
Phi <- cbind(legendre.polynomials(t, degree = 0), legendre.polynomials(t, degree = 1),
             legendre.polynomials(t, degree = 2), legendre.polynomials(t, degree = 3))
# check the orthogonality
crossprod(Phi) # is equal to I_4 up to rounding errors
# plot the basis system excluding the constant column
matplot(t, Phi[,2:4], type = "l", lwd = 2, lty = 1:3, 
        ylab = "basis function", main = "Legendre Polynomials (normalized)")
legend("top", c("linear", "quadratic", "cubic"), col = 1:3, lwd = 2, lty = 1:3)

Example output

Loading required package: sn
Loading required package: stats4

Attaching package: 'sn'

The following object is masked from 'package:stats':

    sd

              [,1]         [,2]         [,3]          [,4]
[1,]  1.000000e+00 1.769418e-16 2.213718e-02 -4.510281e-17
[2,]  1.769418e-16 1.000000e+00 2.567391e-16  4.504553e-02
[3,]  2.213718e-02 2.567391e-16 1.000000e+00  3.191891e-16
[4,] -4.510281e-17 4.504553e-02 3.191891e-16  1.000000e+00

cSFM documentation built on May 29, 2017, 6:10 p.m.