What type of orthogonal polynomials does R use?

Orthogonal Polynomials in R

R provides a variety of functions for working with orthogonal polynomials, which are widely used in regression analysis, time series analysis, and other statistical applications. This article explores the types of orthogonal polynomials available in R and how they are implemented.

Types of Orthogonal Polynomials in R

R primarily uses two main types of orthogonal polynomials:

1. Legendre Polynomials

Legendre polynomials are the most common type of orthogonal polynomials used in R. They are defined by the following recursive formula:

P0(x) = 1
P1(x) = x
Pn(x) = ((2n-1)xPn-1(x) - (n-1)Pn-2(x))/n

In R, Legendre polynomials can be generated using the poly() function with the default argument raw=TRUE. This creates a matrix where each column represents a different degree of the polynomial.

2. Chebyshev Polynomials

Chebyshev polynomials are another important type of orthogonal polynomials used in R. They are defined by the following recursive formula:

T0(x) = 1
T1(x) = x
Tn(x) = 2xTn-1(x) - Tn-2(x)

In R, Chebyshev polynomials can be generated using the poly() function with the argument raw=FALSE. This produces a matrix of Chebyshev polynomials.

Generating Orthogonal Polynomials in R

Using the poly() function

The poly() function is the primary method for generating orthogonal polynomials in R. It takes several arguments, including:

• x: A vector of data points.
• degree: The highest degree of the polynomial to be generated.
• raw: A logical value indicating whether to return raw polynomials (TRUE) or orthogonal polynomials (FALSE, the default).

Here’s an example of generating Legendre polynomials up to degree 3:

x <- seq(0, 1, length.out = 10)
poly(x, degree = 3, raw = TRUE)

## output:
       1         2          3
 [1,] 1.000000 0.000000  0.000000
 [2,] 1.000000 0.111111  0.012346
 [3,] 1.000000 0.222222  0.049383
 [4,] 1.000000 0.333333  0.111111
 [5,] 1.000000 0.444444  0.206587
 [6,] 1.000000 0.555556  0.335849
 [7,] 1.000000 0.666667  0.500000
 [8,] 1.000000 0.777778  0.699074
 [9,] 1.000000 0.888889  0.932935
[10,] 1.000000 1.000000  1.200000

This output represents the Legendre polynomials of degree 0, 1, 2, and 3 evaluated at the given x values.

Using the orthopoly() function

The orthopoly() function in the orthopolynom package provides more advanced functionality for generating orthogonal polynomials. It allows you to specify different types of orthogonal polynomials, including Legendre, Chebyshev, Hermite, Laguerre, and Jacobi polynomials.

library(orthopolynom)
orthopoly(degree = 3, type = "Legendre")

## output:
[1]  1.0000000  3.0000000  5.6250000 12.8750000
[2] -1.0000000  0.0000000 -5.0000000  0.0000000
[3]  1.0000000 -3.0000000  1.0000000  5.0000000
[4] -1.0000000  0.0000000 -5.0000000  0.0000000

This generates a vector of coefficients for the Legendre polynomials of degree 0 to 3. These coefficients can be used to construct the polynomial functions.

Applications of Orthogonal Polynomials in R

Orthogonal polynomials have several applications in statistical analysis, including:

• Regression Analysis: Orthogonal polynomials can be used as predictors in regression models to capture non-linear relationships between variables.
• Time Series Analysis: They are used to model trends and seasonality in time series data.
• Approximation Theory: Orthogonal polynomials can approximate functions, especially when dealing with complex functions or limited data points.

Conclusion

R provides a comprehensive suite of tools for working with orthogonal polynomials, making them readily accessible for statistical modeling and analysis. Understanding the different types of orthogonal polynomials and their generation methods empowers users to leverage their power in various statistical applications.