Estimates the spectral density and cross spectral density of vector time series.
spectral.density(
X,
Y = X,
freq = (-1000:1000/1000) * pi,
q = max(1, floor(dim(X)[1]^(1/3))),
weights = c("Bartlett", "trunc", "Tukey", "Parzen", "Bohman", "Daniell",
"ParzenCogburnDavis")
)Returns an object of class freqdom. The list is containing the following components:
operators \(\quad\) an array. The \(k\)-th matrix in this array corresponds to the spectral density matrix evaluated at the \(k\)-th frequency listed in freq.
freq \(\quad\) returns argument vector freq.
a vector or a vector time series given in matrix form. Each row corresponds to a timepoint.
a vector or vector time series given in matrix form. Each row corresponds to a timepoint.
a vector containing frequencies in \([-\pi, \pi]\) on which the spectral density should be evaluated.
window size for the kernel estimator, i.e. a positive integer.
kernel used in the spectral smoothing. By default the Bartlett kernel is chosen.
Let \([X_1,\ldots, X_T]^\prime\) be a \(T\times d_1\) matrix and \([Y_1,\ldots, Y_T]^\prime\) be a \(T\times d_2\) matrix. We stack the vectors and assume that \((X_t^\prime,Y_t^\prime)^\prime\) is a stationary multivariate time series of dimension \(d_1+d_2\). The cross-spectral density between the two time series \((X_t)\) and \((Y_t)\) is defined as
$$
\sum_{h\in\mathbf{Z}} \mathrm{Cov}(X_h,Y_0) e^{-ih\omega}.
$$
The function spectral.density determines the empirical cross-spectral density between the two time series \((X_t)\) and \((Y_t)\). The estimator is of form
$$
\widehat{\mathcal{F}}^{XY}(\omega)=\sum_{|h|\leq q} w(|k|/q)\widehat{C}^{XY}(h)e^{-ih\omega},
$$
with \(\widehat{C}^{XY}(h)\) defined in cov.structure Here \(w\) is a kernel of the specified type and \(q\) is the window size. By default the Bartlett kernel \(w(x)=1-|x|\) is used.
See, e.g., Chapter 10 and 11 in Brockwell and Davis (1991) for details.
Peter J. Brockwell and Richard A. Davis Time Series: Theory and Methods Springer Series in Statistics, 2009