FPC_basis_expansion: 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 functional_variable,integer
FPC_basis_expansion(object, npc)

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 $(\zeta_{jk})_{m\times p}$

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

$$\zeta_{jk} = \xi_k(t_j)$$

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.

Author

Heyang Ji

Examples

Run this code

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)

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