fdapace (version 0.6.0)

FCCor: Calculation of functional correlation between two simultaneously observed processes.

Description

Calculation of functional correlation between two simultaneously observed processes.

Usage

FCCor(
  x,
  y,
  Lt,
  bw = stop("bw missing"),
  kern = "epan",
  Tout = sort(unique(unlist(Lt)))
)

Value

A list with the following components:

corr

A vector of the correlation corr(X(t), Y(t)) evaluated at Tout.

Tout

Same as the input Tout.

bw

The bandwidths used for E(X), E(Y), var(X), var(Y), and cov(X, Y).

Arguments

x

A list of function values corresponding to the first process.

y

A list of function values corresponding to the second process.

Lt

A list of time points for both x and y.

bw

A numeric vector for bandwidth of length either 5 or 1, specifying the bandwidths for E(X), E(Y), var(X), var(Y), and cov(X, Y). If bw is a scalar then all five bandwidths are chosen to be the same.

kern

Smoothing kernel for mu and covariance; "rect", "gauss", "epan", "gausvar", "quar" (default: "gauss")

Tout

Output time points. Default to the sorted unique time points.

Details

FCCor calculate only the concurrent correlation corr(X(t), Y(t)) (note that the time points t are the same). It assumes no measurement error in the observed values.

Examples

set.seed(1)
n <- 200
nGridIn <- 50
sparsity <- 1:5 # must have length > 1
bw <- 0.2
kern <- 'epan'
T <- matrix(seq(0.5, 1, length.out=nGridIn))

## Corr(X(t), Y(t)) = 1/2
A <- Wiener(n, T)
B <- Wiener(n, T) 
C <- Wiener(n, T) + matrix((1:nGridIn) , n, nGridIn, byrow=TRUE)
X <- A + B
Y <- A + C
indEach <- lapply(1:n, function(x) sort(sample(nGridIn, sample(sparsity, 1))))
tAll <- lapply(1:n, function(i) T[indEach[[i]]])
Xsp <- lapply(1:n, function(i) X[i, indEach[[i]]])
Ysp <- lapply(1:n, function(i) Y[i, indEach[[i]]])

plot(T, FCCor(Xsp, Ysp, tAll, bw)[['corr']], ylim=c(-1, 1))
abline(h=0.5)