fts.cov.structure {freqdom.fda}

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

X

an object of class fd containing T functional observations.

Y

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.

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).

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)

See Also

The multivariate equivalent in the freqdom package: cov.structure

Examples

# 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)