View source: R/fts.spectral.density.R
fts.spectral.density | R Documentation |
Estimates the spectral density operator and cross spectral density operator of functional time series.
fts.spectral.density( X, Y = X, freq = (-1000:1000/1000) * pi, q = ceiling((dim(X$coefs)[2])^{ 0.33 }), weights = "Bartlett" )
X |
an object of class |
Y |
an object of class |
freq |
a vector containing frequencies in [-π, π] on which the spectral density should be evaluated.
By default |
q |
window size for the kernel estimator, i.e. a positive integer. By default we choose |
weights |
kernel used in the spectral smoothing. For possible choices see
|
Let X_1(u),…, X_T(u) and Y_1(u),…, Y_T(u) be two samples of functional data. The cross-spectral density kernel between the two time series (X_t(u)) and (Y_t(u)) is defined as
f^{XY}_ω(u,v)=∑_{h\in\mathbf{Z}} \mathrm{Cov}(X_h(u),Y_0(v)) e^{-ihω}.
The function fts.spectral.density
determines the empirical
cross-spectral density kernel between the two time series. The estimator is of the
form
\widehat{f}^{XY}_ω(u,v)=∑_{|h|≤q q} w(|k|/q)\widehat{c}^{XY}_h(u,v)e^{-ihω},
with \widehat{c}^{XY}_h(u,v) defined in fts.cov.structure
.
The other paremeters are as in cov.structure
.
Since X_t(u)=\boldsymbol{b}_1^\prime(u)\mathbf{x}_t and Y_t(u)=\mathbf{y}_t^\prime \boldsymbol{b}_2(u) we can write
\widehat{f}^{XY}_ω(u,v)=\boldsymbol{b}_1^\prime(u)\widehat{\mathcal{F}}^{\mathbf{xy}}(ω)\boldsymbol{b}_2(v),
where \widehat{\mathcal{F}}^{\mathbf{xy}}(ω) is defined as for the function spectral.density
for series of coefficient vectors
(\mathbf{x}_t\colon 1≤q t≤q T) and (\mathbf{y}_t\colon 1≤q t≤q T).
Returns an object of class fts.timedom
. The list is containing the following components:
operators
\quad an array. Element [,,k]
in the coefficient matrix of the spectral density matrix evaluated at the k-th frequency listed in freq
.
lags
\quad returns the lags vector from the arguments.
basisX
\quad returns X$basis
, an object of class basis.fd
(see create.basis
).
basisY
\quad returns Y$basis
, an object of class basis.fd
(see create.basis
)
The multivariate equivalent in the freqdom
package: spectral.density
data(pm10) X = center.fd(pm10) # Compute the spectral density operator with Bartlett weights SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 2, weight="Bartlett") fts.plot.operators(SD, freq = -2:2) # Compute the spectral density operator with Tukey weights SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 2, weight="Tukey") fts.plot.operators(SD, freq = -2:2) # Note relatively small difference between the two plots # Now, compute the spectral density operator with Tukey weights and larger q SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 5, weight="Tukey") fts.plot.operators(SD, freq = -2:2)
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