Spectral Density — spectrald • RLDM

Compute the spectral density of an ARMA process or a process defined by a state space model.

Usage

spectrald(obj, n.f, ...)

# S3 method for armamod
spectrald(obj, n.f = 128, ...)

# S3 method for stspmod
spectrald(obj, n.f = 128, ...)

# S3 method for autocov
spectrald(obj, n.f = 128, ...)

# S3 method for impresp
spectrald(obj, n.f = 128, ...)

# S3 method for default
spectrald(obj, n.f = NULL, demean = TRUE, ...)

Arguments

obj: armamod(), stspmod(), autocov(), impresp() object or a "time series" object, i.e. an object which may be coerced to a data matrix by y = as.matrix(obj).
n.f: number of frequencies.
...: not used.
demean: (logical) should the data be demeaned before computing the periodogram?

Value

freqresp object, i.e. a list with slots

spd: rationalmatrices::zvalues() object.
names: (m)-dimensional character vector or NULL. This optional slot stores the names for the components of the time series/process.
label: character string or NULL.
n.obs: (optional) integer or NULL.

Details

The spectral density of a stationary process with an absolutely summable autocovariance function $(\gamma_j)$ is given by $$ \Gamma(\lambda) = \frac{1}{2\pi}\sum_{j=-\infty}^{\infty} \gamma_j e^{-i\lambda j}. $$

For an ARMA process, or process defined by a state space model the spectral density is equal to $$ \Gamma(\lambda) = \frac{1}{2\pi} K(\lambda) \Sigma K^*(\lambda) $$ where $\Sigma$ is the noise covariance, $K()$ is the frequency response of the model and $K^*(\lambda)$ is the Hermitean transpose of $K(\lambda)$. See also autocov() and freqresp().

Note that $\Gamma()$ is (up to a factor $2\pi$) the discrete-time Fourier transform (DTFT) of the autocovariance function and therefore the ACF $\gamma_j$ may be reconstructed from the spectral density via the inverse DTFT $$ \gamma_j = \int_{-\pi}^{\pi} \Gamma(\lambda) e^{i\lambda j} d\lambda $$

The S3 methods spectrald.* evaluate the spectral density function on a grid of angular frequencies $\lambda_j = 2\pi j/N$, $j=0,\ldots,N-1$ and store the result in a spectrald object.

There are several possible ways to specify the process. One may provide the ARMA (armamod) respectively the state space model (stspmod), the autocovariance function (autocov) or the impulse response function (impresp) which maps the noise to the outputs. Note however, that if we have only given an autocov or impresp object then the computed spectral density is only an approximation of the true spectral density since only a finite number of covariances respectively impulse response coefficients are given. The type of the autocovariance function ("covariances", "correlations" or "partial correlations") is irrelevenat since the procedure alwayas uses the slot "gamma" which contains the covariances.

The default method spectrald.default assumes that obj is a "time series" object and tries to coerce this object to a data matrix via y = as.matrix(obj). In this case the procedure computes the periodogram which is a simple estimate of the spectral density. The periodgram may also be computed by the call spectrald(autocov(obj, max.lag = n.obs-1)), i.e. by first computing the sample auto covariance function and then computing the corresponding spectral density.

Note that we use a different scaling than the stats::[spectrum][stats::spectrum] routine.

Examples

#' ### generate random 3-dimensional ARMA(1,1) model
# "bpoles = 1.1" implies that the poles have moduli larger than 1.1
# and therefore the impulse response coefficients decay with a rate (1.1)^k
arma_model = test_armamod(dim = c(3,3), degrees = c(1,1), bpoles = 1.1)

# spectral density
spd = spectrald(arma_model)

# compute the spectral density via the impulse response
spd1 = spectrald(impresp(arma_model, lag.max = 100))

# since the impulse response quickly decays
# the "truncated" spectral density should be close to the true one
all.equal(spd, spd1)
#> [1] TRUE

# compute the spectral density via the autocovariance function
spd1 = spectrald(autocov(arma_model, lag.max = 100))

# since the ACF quickly decays
# the "truncated" spectral density should be close to the true one
all.equal(spd, spd1)
#> [1] TRUE

# create an equivalent state space model
stsp_model = as.stspmod(arma_model)

# of course the state space model gives the same spectrum
# as the original ARMA model
spd1 = spectrald(stsp_model)
all.equal(spd, spd1)
#> [1] TRUE