A Matrix Representation for Spline Basis Functions

Description

This package provides functions for manipulation of spline basis functions. A matrix representation for the basis functions is at the center of the functions. The matrix representation simplifies the process of orthogonalization as well as differentiation and integration.

Author(s)

Andrew Redd Maintainer: Andrew Redd andrew.redd@hsc.utah.edu

Computing the Gram Matrix for a set of Spline Basis

Description

Function for computing the Gram matrix of a spline basis.

Usage

		GramMatrix(object)

Arguments

Details

Compute the Gram Matrix. If object denotes the basis functions b(t) = \{b_1(t),\ldots,b_J(t)\} then the Gram Matrix is,

G = \int b^T(t)b(t) \mathrm{d}t

Value

a matrix as defined above.

Generating a Hankel Matrix

Description

Functions to generate a Hankel matrix.

Usage

Hankel(x, nrow = length(x)%/%2, ncol = length(x)%/%2)

Arguments

Details

Computes a Hankel matrix. If we denote the vector x=(x_1,\dots,x_n) the Hankel matrix is defined and formed as

H=\left( \begin{array}{ccccc} x_1&x_2&x_3&\cdots&x_{1/2}\\ x_2&x_3&&\vdots&\vdots\\ x_3&&\vdots&&\vdots\\ \vdots&\vdots&&&\vdots\\ x_{1/2}&\cdots&\cdots&\cdots&x_n \end{array}\right).

Value

a matrix as defined above.

Examples

Hankel(1:6)

Matrix Power

Description

Performs the matrix power operation.

Usage

MatrixPower(A, n)

Arguments

Details

Only well defined for integers the matrix power operation is a convenience function to multiply a matrix, A, with itself n times.

Value

A matrix of the same dimension as A.

Examples

A<-rbind(0,cbind(diag(1:5),0))  #a nilpotent matrix
A
MatrixPower(A,3)
MatrixPower(A,5)
MatrixPower(A,6)    #Gets to a zero matrix

Orthogonalize a Spline Basis

Description

Specific function for orthogonalizing the functions in a SplineBasis object.

Usage

OrthogonalizeBasis(object, ...)

Arguments

Value

An OrthogonalSplineBasis object.

See Also

OrthogonalSplineBasis, SplineBasis,orthogonalize

Outer Product of Second Derivatives of Spline Bases

Description

Provides the functional outer product of second derivatives of a set of basis functions in a SplineBasis object. It a convenient form for forming a penalty on curve smoothness when fitting a spline curve.

Usage

OuterProdSecondDerivative(basis)

Arguments

Value

A square matrix of order nrow(basis).

See Also

SplineBasis,fitLS

Creating SplineBasis Objects.

Description

The function to create SplineBasis and OrthogonalSplineBasis Objects

Usage

	SplineBasis(knots, order=4, keep.duplicates=FALSE)
	OrthogonalSplineBasis(knots, ...)
	OBasis(...)

Arguments

Details

SplineBasis produces an object representing the basis functions used in spline fitting. Provides a compact easily evaluated representation of the functions. Produces a class of object SplineBasis. OrthogonalSplineBasis is a shortcut to obtain a set of orthogonalized basis functions from the knots. OBasis is an alias for OrthogonalSplineBasis. Both provide an object of class OrthogonalSplineBasis. The class OrthogonalSplineBasis inherits directly from SplineBasis meaning all functions that apply to SplineBasis functions also apply to the orthogonalized version.

Value

Object of class SplineBasis or OrthogonalSplineBasis

References

General matrix representations for B-splines Kaihuai, Qin, The Visual Computer 2000 16:177–186

See Also

SplineBasis, spline, orthogonalsplinebasis-package

Examples

knots<-c(0,0,0,0:10,10,10,10)
plot(SplineBasis(knots))
obase<-OBasis(knots)
plot(obase)
dim(obase)[2] #number of functions
evaluate(obase, 1:10-.5)

Classes SplineBasis and OrthogonalSplineBasis

Description

Contains the matrix representation for spline basis functions. The OrthogonalSplineBasis class has the basis functions orthogonalized.

Objects from the Class

Objects can be created by calls of the form SplineBasis(knots, order) or to generate orthogonal spline basis functions directly OrthogonalSplineBasis(knots, order) or the short version OBasis(knots,order).

Slots

transformation:

Object of class "matrix" Only applicable on OrthogonalSplineBasis class, shows the transformation matrix use to get from regular basis functions to orthogonal basis functions.

knots:

Object of class "numeric"

order:

Object of class "integer"

Matrices:

Object of class "array"

Methods

deriv

signature(expr = "SplineBasis"): Computes the derivative of the basis functions. Returns an object of class SplineBasis.

dim

signature(x = "SplineBasis"): gives the dim as the order and number of basis functions. Returns numeric of length 2.

evaluate

signature(object = "SplineBasis", x = "numeric"): Evaluates the basis functions and the points provided in x. Returns a matrix with length(x) rows and dim(object)[2] columns.

integrate

signature(object = "SplineBasis"): computes the integral of the basis functions defined by \int\limits_{k_0}^x b(t)dt where k_0 is the first knot. Returns an object of class SplineBasis.

orthogonalize

signature(object = "SplineBasis"): Takes in a SplineBasis object, computes the orthogonalization transformation and returns an object of class OrthogonalSplineBasis.

plot

signature(x = "SplineBasis", y = "missing"): Takes an object of class SplineBasis and plots the basis functions for the domain defined by the knots in object.

plot

signature(x = "SplineBasis", y = "vector"): Interprets y as a vector of coefficients and plots the resulting curve.

plot

signature(x = "SplineBasis", y = "matrix"): Interprets y as a matrix of coefficients and plots the resulting curves.

References

General matrix representations for B-splines Kaihuai Qin, The Visual Computer 2000 16:177–186

See Also

SplineBasis

Examples

showClass("SplineBasis")

knots<-c(0,0,0,0:5,5,5,5)
(base <-SplineBasis(knots))
(obase<-OBasis(knots))
plot(base)
plot(obase)

Generic evaluate method

Description

Methods for function evaluate.

Methods

object = "SplineBasis", x = "numeric"

Evaluates a SplineBasis object for the spline basis curves at points x. See SplineBasis

Expands knots for appropriate number of knots in B-splines

Description

This function is for convenience of specifying knots for B-splines. Since the user usually only want to specify the interval that they are interested in the end knots are usually duplicated. This function interprets the first and last knots as the end points and duplicates them.

Usage

expand.knots(interior, order = 4)

Arguments

Value

A vector of knots with the order specified as an attribute

Author(s)

Andrew Redd

See Also

SplineBasis, ~~~

Examples

(knots<-expand.knots(1:10))
plot(OBasis(knots))

Fitting splines with penalized least squares.

Description

Estimates the control vector for a spline fit by penalized least squares. The penalty being the penalty parameter times the functional inner product of the second derivative of the spline curve.

Usage

fitLS(object, x, y, penalty = 0)

Arguments

Details

For numeric vector y, and x, and a set of basis functions, represented in object, defined on the knots (k_0,\ldots,k_m). The likelihood is defined by

\sum\limits_{i=1}^n(y_i-b(x_i)\mu) + \int\limits_{k_0}^{k_m} \mu^Tb^{\prime\prime}(t)^Tb^{\prime\prime}(t)\mu dt

The function estimates \mu.

Value

a vector of the control points.

See Also

SplineBasis

Examples

knots<-c(0,0,0,0:5,5,5,5)
base<-SplineBasis(knots)
x<-seq(0,5,by=.5)
y<-exp(x)+rnorm(length(x),sd=5)
fitLS(base,x,y)

Methods for Function integrate

Description

Methods for function integrate. integrate integrates generic objects for which an integral is defined.

Methods

object = "SplineBasis"

Returns a new SplineBasis object for the integral of the basis functions. See SplineBasis

Methods for Function orthogonalize

Description

A generic function for orthogonalizing an object and returning the orthogonal object

Methods

object = "SplineBasis"

Orthogonalize the spline basis functions. See SplineBasis