spectral.density: Compute empirical spectral density

Description

Estimates the spectral density and cross spectral density of vector time series.

Usage

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")
)

Value

Returns an object of class freqdom. The list is containing the following components:

operators 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 returns argument vector freq.

Arguments

X: a vector or a vector time series given in matrix form. Each row corresponds to a timepoint.
Y: a vector or vector time series given in matrix form. Each row corresponds to a timepoint.
freq: a vector containing frequencies in $[- π, π]$ on which the spectral density should be evaluated.
q: window size for the kernel estimator, i.e. a positive integer.
weights: kernel used in the spectral smoothing. By default the Bartlett kernel is chosen.

Details

Let $[X_{1}, \dots, X_{T}]^{'}$ be a $T \times d_{1}$ matrix and $[Y_{1}, \dots, Y_{T}]^{'}$ be a $T \times d_{2}$ matrix. We stack the vectors and assume that $(X_{t}^{'}, Y_{t}^{'})^{'}$ 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 $\sum_{h \in Z} Cov (X_{h}, Y_{0}) e^{- i h ω} .$ 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 ${\hat{F}}^{X Y} (ω) = \sum_{| h | \leq q} w (| k | / q) {\hat{C}}^{X Y} (h) e^{- i h ω},$ with ${\hat{C}}^{X Y} (h)$ 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.

References