FPC_basis_expansion: Functional principal component basis expansion for functional...

In MECfda: Scalar-on-Function Regression with Measurement Error Correction

Functional principal component basis expansion for functional variable data

Description

For a function f(t), t\in\Omega, and a basis function sequence \{\rho_k\}_{k\in\kappa}, basis expansion is to compute \int_\Omega f(t)\rho_k(t) dt. Here we do basis expansion for all f_i(t), t\in\Omega = [t_0,t_0+T] in functional variable data, i=1,\dots,n. We compute a matrix (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt. The basis we use here is the functional principal component (FPC) basis induced by the covariance function of the functional variable. Suppose K(s,t)\in L^2(\Omega\times \Omega), f(t)\in L^2(\Omega). Then K induces an linear operator \mathcal{K} by

(\mathcal{K}f)(x) = \int_{\Omega} K(t,x)f(t)dt

If \xi(\cdot)\in L^2(\Omega) such that

\mathcal{K}\xi = \lambda \xi

where \lambda\in {C}, we call \xi an eigenfunction/eigenvector of \mathcal{K}, and \lambda an eigenvalue associated with \xi.
For a stochastic process \{X(t),t\in\Omega\} we call the orthogonal basis \{\xi_k\}_{k=1}^\infty corresponding to eigenvalues \{\lambda_k\}_{k=1}^\infty (\lambda_1\geq\lambda_2\geq\dots), induced by

K(s,t)=\text{Cov}(X(t),X(s))

a functional principal component (FPC) basis for L^2(\Omega).

Usage

FPC_basis_expansion(object, npc)

## S4 method for signature 'functional_variable,integer'
FPC_basis_expansion(object, npc)

Arguments

object	a functional_variable class object. The minimum and maximum of the slot t_points should be respectively equal to the slot t_0 and slot t_0 plus slot period.
npc	The number of functional principal components. See npc in fpca.sc.

Value

Returns a numeric matrix, (b_{ik})_{n\times p}, with an extra attribute numeric_basis, which represents the FPC basis. The attribute numeric_basis is a numeric_basis object. See numeric_basis. The slot basis_function is also a numeric matrix, denoted as (\zeta_{jk})_{m\times p}

b_{ik} = \int_\Omega f(t)\xi_k(t) dt

\zeta_{jk} = \xi_k(t_j)

Author(s)

Heyang Ji

Examples

n<-50; ti<-seq(0,1,length.out=101)
X<-t(sin(2*pi*ti)%*%t(rnorm(n,0,1)))
object = functional_variable(X = X, t_0 = 0, period = 1, t_points = ti)
a = FPC_basis_expansion(object,3L)
dim(a)

MECfda documentation built on April 3, 2025, 10:07 p.m.