fts.cov.structure: Estimate autocovariance and cross-covariances operators

Description

This function is used to estimate a collection of cross-covariances operators of two stationary functional series.

Usage

fts.cov.structure(X, Y = X, lags = 0)

Arguments

an object of class fd containing $T$ functional observations.

lags

an integer-valued vector $(\ell_1,\ldots, \ell_K)$ containing the lags for which covariances are calculated.

Value

An object of class fts.timedom. The list contains the following components:

operators $\quad$ an array. Element [,,k] contains the covariance matrix of the coefficient vectors of the two time series related to lag $\ell_k$.
lags $\quad$ the lags vector from the arguments.
basisX $\quad$ X$basis, an object of class basis.fd (see create.basis)
basisY $\quad$ Y$basis, an object of class basis.fd (see create.basis)

Details

Let $X_1(u),\ldots, X_T(u)$ and $Y_1(u),\ldots, Y_T(u)$ be two samples of functional data. This function determines empirical lagged covariances between the series $(X_t(u))$ and $(Y_t(u))$. More precisely it determines $$ (\widehat{c}^{XY}_h(u,v)\colon h\in lags ), $$ where $\widehat{c}^{XY}_h(u,v)$ is the empirical version of the covariance kernel $\mathrm{Cov}(X_h(u),Y_0(v))$. For a sample of size $T$ we set $\hat\mu^X(u)=\frac{1}{T}\sum_{t=1}^T X_t(u)$ and $\hat\mu^Y(v)=\frac{1}{T}\sum_{t=1}^T Y_t(v)$. Now for $h \geq 0$ $$\frac{1}{T}\sum_{t=1}^{T-h} (X_{t+h}(u)-\hat\mu^X(u))(Y_t(v)-\hat\mu^Y(v))$$ and for $h < 0$ $$\frac{1}{T}\sum_{t=|h|+1}^{T} (X_{t+h}(u)-\hat\mu^X(u))(Y_t(v)-\hat\mu^Y(v)).$$ Since $X_t(u)=\boldsymbol{b}_1^\prime(u)\mathbf{x}_t$ and $Y_t(u)=\mathbf{y}_t^\prime \boldsymbol{b}_2(u)$ we can write $$ \widehat{c}^{XY}_h(u,v)=\boldsymbol{b}_1^\prime(u)\widehat{C}^{\mathbf{xy}}\boldsymbol{b}_2(v), $$ where $\widehat{C}^{\mathbf{xy}}$ is defined as for the function ``cov.structure'' for series of coefficient vectors $(\mathbf{x}_t\colon 1\leq t\leq T)$ and $(\mathbf{y}_t\colon 1\leq t\leq T)$.

Examples

Run this code

# NOT RUN {
# Generate an autoregressive process
fts = fts.rar(d=3)

# Get covariance at lag 0
fts.cov.structure(fts, lags = 0)

# Get covariance at lag 10
fts.cov.structure(fts, lags = 10)

# Get entire covariance structure between -20 and 20
fts.cov.structure(fts, lags = -20:20)

# Compute covariance with another process
fts0 = fts + fts.rma(d=3)
fts.cov.structure(fts, fts0, lags = -2:2)
# }