Frequency domain basde analysis: dynamic PCA

Description

Implementation of dynamic principle component analysis (DPCA), simulation of VAR and VMA processes and frequency domain tools. The package also provides a toolset for developers simplifying construction of new frequency domain based methods for multivariate signals.

Details

freqdom package allows you to manipulate time series objects in both time and frequency domains. We implement dynamic principal component analysis methods, enabling spectral decomposition of a stationary vector time series into uncorrelated components.

Dynamic principal component analysis enables estimation of temporal filters which transform a vector time series into another vector time series with uncorrelated components, maximizing the long run variance explained. There are two key differnces between classical PCA and dynamic PCA:

• Components returned by the dynamic procedure are uncorrelated in time, i.e. for any i \neq j and l \in Z, Y_i(t) and Y_j(t_l) are uncorrelated,

• The mapping maximizes the long run variance, which, in case of stationary vector time series, means that the process reconstructed from and d > 0 first dynamic principal components better approximates your vector time series process than the first d classic principal components.

For details, please refer to literature below and to help pages of functions dpca for estimating the components, dpca.scores for estimating scores and dpca.KLexpansion for retrieving the signal from components.

Apart from frequency domain techniques for stationary vector time series, freqdom provides a toolset of operators such as the vector Fourier Transform (fourier.transform) or a vector spectral density operator (spectral.density) as well as simulation of vector time series models rar, rma generating vector autoregressive and moving average respectively. These functions enable developing new techniques based on the Frequency domain analysis.

References

Hormann Siegfried, Kidzinski Lukasz and Hallin Marc. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Hormann Siegfried, Kidzinski Lukasz and Kokoszka Piotr. Estimation in functional lagged regression. Journal of Time Series Analysis 36.4 (2015): 541-561.

Hormann Siegfried and Kidzinski Lukasz. A note on estimation in Hilbertian linear models. Scandinavian journal of statistics 42.1 (2015): 43-62.

Frequency-wise product of freqdom objects

Description

Frequency-wise product of freqdom objects

Usage

A %*% B

Frequency-wise sum of freqdom objects

Description

Frequency-wise sum of freqdom objects

Usage

## S3 method for class 'freqdom'
e1 + e2

Time-wise sum of freqdom objects

Description

Time-wise sum of freqdom objects

Usage

## S3 method for class 'timedom'
e1 + e2

Frequency-wise difference of freqdom objects

Description

Frequency-wise difference of freqdom objects

Usage

## S3 method for class 'freqdom'
e1 - e2

Time-wise difference of freqdom objects

Description

Time-wise sum of freqdom objects

Usage

## S3 method for class 'timedom'
e1 - e2

Estimate cross-covariances of two stationary multivariate time series

Description

This function computes the empirical cross-covariance of two stationary multivariate time series. If only one time series is provided it determines the empirical autocovariance function.

Usage

cov.structure(X, Y = X, lags = 0)

Arguments

Details

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. This function determines empirical lagged covariances between the series (X_t) and (Y_t). More precisely it determines \widehat{C}^{XY}(h) for h\in lags, where \widehat{C}^{XY}(h) is the empirical version of \mathrm{Cov}(X_h,Y_0). For a sample of size T we set \hat\mu^X=\frac{1}{T}\sum_{t=1}^T X_t and \hat\mu^Y=\frac{1}{T}\sum_{t=1}^T Y_t and

\hat C^{XY}(h) = \frac{1}{T}\sum_{t=1}^{T-h} (X_{t+h} - \hat\mu^X)(Y_{t} - \hat\mu^Y)'

and for h < 0

\hat C^{XY}(h) = \frac{1}{T}\sum_{t=|h|+1}^{T} (X_{t+h} - \hat\mu^X)(Y_{t} - \hat\mu^Y)'.

Value

An object of class timedom. The list contains

• operators \quad an array. Element [,,k] contains the covariance matrix related to lag \ell_k.

• lags \quad returns the lags vector from the arguments.

Compute Dynamic Principal Components and dynamic Karhunen Loeve extepansion

Description

Dynamic principal component analysis (DPCA) decomposes multivariate time series into uncorrelated components. Compared to classical principal components, DPCA decomposition outputs components which are uncorrelated in time, allowing simpler modeling of the processes and maximizing long run variance of the projection.

Usage

dpca(X, q = 30, freq = (-1000:1000/1000) * pi, Ndpc = dim(X)[2])

Arguments

Details

This convenience function applies the DPCA methodology and returns filters (dpca.filters), scores (dpca.scores), the spectral density (spectral.density), variances (dpca.var) and Karhunen-Leove expansion (dpca.KLexpansion).

See the example for understanding usage, and help pages for details on individual functions.

Value

A list containing

• scores \quad DPCA scores (dpca.scores)

• filters \quad DPCA filters (dpca.filters)

• spec.density \quad spectral density of X (spectral.density)

• var \quad amount of variance explained by dynamic principal components (dpca.var)

• Xhat \quad Karhunen-Loeve expansion using Ndpc dynamic principal components (dpca.KLexpansion)

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R., and Stoffer, D. Time series analysis and its applications: with R examples (2010), Springer Science & Business Media

Examples

X = rar(100,3)

# Compute DPCA with only one component
res.dpca = dpca(X, q = 5, Ndpc = 1)

# Compute PCA with only one component
res.pca = prcomp(X, center = TRUE)
res.pca$x[,-1] = 0

# Reconstruct the data
var.dpca = (1 - sum( (res.dpca$Xhat - X)**2 ) / sum(X**2))*100
var.pca = (1 - sum( (res.pca$x %*% t(res.pca$rotation) - X)**2 ) / sum(X**2))*100

cat("Variance explained by DPCA:\t",var.dpca,"%\n")
cat("Variance explained by PCA:\t",var.pca,"%\n")

Dynamic KL expansion

Description

Computes the dynamic Karhunen-Loeve expansion of a vector time series up to a given order.

Usage

dpca.KLexpansion(X, dpcs)

Arguments

Details

We obtain the dynamic Karhnunen-Loeve expansion of order L, 1\leq L\leq d. It is defined as

\sum_{\ell=1}^L\sum_{k\in\mathbf{Z}} Y_{\ell, t+k} \phi_{\ell k},

where \phi_{\ell k} are the dynamic PC filters as explained in dpca.filters and Y_{\ell k} are dynamic scores as explained in dpca.scores. For the sample version the sum in k extends over the range of lags for which the \phi_{\ell k} are defined.

For more details we refer to Chapter 9 in Brillinger (2001), Chapter 7.8 in Shumway and Stoffer (2006) and to Hormann et al. (2015).

Value

A (T\times d)-matix. The \ell-th column contains the \ell-th data point.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.

See Also

dpca.filters, filter.process, dpca.scores

Compute DPCA filter coefficients

Description

For a given spectral density matrix dynamic principal component filter sequences are computed.

Usage

dpca.filters(F, Ndpc = dim(F$operators)[1], q = 30)

Arguments

Details

Dynamic principal components are linear filters (\phi_{\ell k}\colon k\in \mathbf{Z}), 1 \leq \ell \leq d. They are defined as the Fourier coefficients of the dynamic eigenvector \varphi_\ell(\omega) of a spectral density matrix \mathcal{F}_\omega:

\phi_{\ell k}:=\frac{1}{2\pi}\int_{-\pi}^\pi \varphi_\ell(\omega) \exp(-ik\omega) d\omega.

The index \ell is referring to the \ell-th #'largest dynamic eigenvalue. Since the \phi_{\ell k} are real, we have

\phi_{\ell k}^\prime=\phi_{\ell k}^*=\frac{1}{2\pi}\int_{-\pi}^\pi \varphi_\ell^* \exp(ik\omega)d\omega.

For a given spectral density (provided as on object of class freqdom) the function dpca.filters() computes (\phi_{\ell k}) for |k| \leq q and 1 \leq \ell \leq Ndpc.

For more details we refer to Chapter 9 in Brillinger (2001), Chapter 7.8 in Shumway and Stoffer (2006) and to Hormann et al. (2015).

Value

An object of class timedom. The list has the following components:

• operators \quad an array. Each matrix in this array has dimension Ndpc \times d and is assigned to a certain lag. For a given lag k, the rows of the matrix correpsond to \phi_{\ell k}.

• lags \quad a vector with the lags of the filter coefficients.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.

See Also

dpca.var, dpca.scores, dpca.KLexpansion

Obtain dynamic principal components scores

Description

Computes dynamic principal component score vectors of a vector time series.

Usage

dpca.scores(X, dpcs = dpca.filters(spectral.density(X)))

Arguments

Details

The \ell-th dynamic principal components score sequence is defined by

Y_{\ell t}:=\sum_{k\in\mathbf{Z}} \phi_{\ell k}^\prime X_{t-k},\quad 1\leq \ell\leq d,

where \phi_{\ell k} are the dynamic PC filters as explained in dpca.filters. For the sample version the sum extends over the range of lags for which the \phi_{\ell k} are defined. The actual operation carried out is filter.process(X, A = dpcs).

We for more details we refer to Chapter 9 in Brillinger (2001), Chapter 7.8 in Shumway and Stoffer (2006) and to Hormann et al. (2015).

Value

A T\times Ndpc-matix with Ndpc = dim(dpcs$operators)[1]. The \ell-th column contains the \ell-th dynamic principal component score sequence.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.

See Also

dpca.filters, dpca.KLexpansion, dpca.var

Proportion of variance explained

Description

Computes the proportion of variance explained by a given dynamic principal component.

Usage

dpca.var(F)

Arguments

Details

Consider a spectral density matrix \mathcal{F}_\omega and let \lambda_\ell(\omega) by the \ell-th dynamic eigenvalue. The proportion of variance described by the \ell-th dynamic principal component is given as

v_\ell:=\int_{-\pi}^\pi \lambda_\ell(\omega)d\omega/\int_{-\pi}^\pi \mathrm{tr}(\mathcal{F}_\omega)d\omega.

This function numerically computes the vectors (v_\ell\colon 1\leq \ell\leq d).

For more details we refer to Chapter 9 in Brillinger (2001), Chapter 7.8 in Shumway and Stoffer (2006) and to Hormann et al. (2015).

Value

A d-dimensional vector containing the v_\ell.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.

See Also

dpca.filters, dpca.KLexpansion, dpca.scores

Convolute (filter) a multivariate time series using a time-domain filter

Description

This function applies a linear filter to some vector time series.

Usage

filter.process(X, A)

X %c% A

Arguments

Details

Let [X_1,\ldots, X_T]^\prime be a T\times d matrix corresponding to a vector series X_1,\ldots,X_T. This time series is transformed to the series Y_1,\ldots, Y_T, where

Y_t=\sum_{k=-q}^p A_k X_{t-k},\quad t\in\{p+1,\ldots, T-q\}.

The index k of A_k is determined by the lags defined for the time domain object. When index t-k falls outside the domain \{1,\ldots, T\} we set X_t=\frac{1}{T}\sum_{k=1}^T X_k.

Value

A matrix. Row t corresponds to Y_t.

Functions

• filter.process(): Multivariate convolution (filter) in the time domain

• X %c% A: Convenience operator for filter.process function

See Also

timedom

Coefficients of a discrete Fourier transform

Description

Computes Fourier coefficients of some functional represented by an object of class freqdom.

Usage

fourier.inverse(F, lags = 0)

Arguments

Details

Consider a function F \colon [-\pi,\pi]\to\mathbf{C}^{d_1\times d_2}. Its k-th Fourier coefficient is given as

\frac{1}{2\pi}\int_{-\pi}^\pi F(\omega) \exp(ik\omega)d\omega.

We represent the function F by an object of class freqdom and approximate the integral via

\frac{1}{|F\$freq|}\sum_{\omega\in {F\$freq}} F(\omega) \exp(i k\omega),

for k\in lags.

Value

An object of class timedom. The list has the following components:

• operators \quad an array. The k-th matrix in this array corresponds to the k-th Fourier coefficient.

• lags \quad the lags of the corresponding Fourier coefficients.

See Also

fourier.transform, freqdom

Examples

Y = rar(100)
grid = c(pi*(1:2000) / 1000 - pi) #a dense grid on -pi, pi
fourier.inverse(spectral.density(Y, q=2, freq=grid))

# compare this to
cov.structure(Y)

Computes the Fourier transformation of a filter given as timedom object

Description

Computes the frequency response function of a linear filter and returns it as a freqdom object.

Usage

fourier.transform(A, freq = pi * -100:100/100)

Arguments

Details

Consider a filter (a sequence of vectors or matrices) (A_k)_{k\in A\$lags}. Then this function computes

\sum_{k\in A\$lags} A_k e^{-ik\omega}

for all frequencies \omega listed in the vector freq.

Value

An object of class freqdom.

See Also

fourier.inverse

Examples

# We compute the discrete Fourier transform (DFT) of a time series X_1,..., X_T.

X = rar(100)
T=dim(X)[1]
tdX = timedom(X/sqrt(T),lags=1:T)
DFT = fourier.transform(tdX, freq= pi*-1000:1000/1000)

Create an object corresponding to a frequency domain functional

Description

Creates an object of class freqdom. This object corresponds to a functional with domain [-\pi,\pi] and some complex vector space as codomain.

Usage

freqdom(F, freq)

Arguments

Details

This class is used to describe a frequency domain functional (like a spectral density matrix, a discrete Fourier transform, an impulse response function, etc.) on selected frequencies. Formally we consider a collection [F_1,\ldots,F_K] of complex-valued matrices F_k, all of which have the same dimension d_1\times d_2. Moreover, we consider frequencies \{\omega_1,\ldots, \omega_K\}\subset[-\pi,\pi]. The object this function creates corresponds to the mapping f: \mathrm{freq}\to \mathbf{C}^{d_1\times d_2}, where \omega_k\mapsto F_k.

Consider, for example, the discrete Fourier transform of a vector time series X_1,\ldots, X_T:. It is defined as

D_T(\omega)=\frac{1}{\sqrt{T}}\sum_{t=1}^T X_t e^{-it\omega},\quad \omega\in[-\pi,\pi].

We may choose \omega_k=2\pi k/K-\pi and F_k=D_T(\omega_k). Then, the object freqdom creates, is corresponding to the function which associates \omega_k and D_T(\omega_k).

Value

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

• operators \quad the array F as given in the argument.

• freq \quad the vector freq as given in the argument.

See Also

fourier.transform

Examples

i = complex(imaginary=1)
OP = array(0, c(2, 2, 3))
OP[,,1] = diag(2) * exp(i)/2
OP[,,2] = diag(2)
OP[,,3] = diag(2) * exp(-i)/2
freq = c(-pi/3, 0, pi/3)
A = freqdom(OP, freq)

Eigendecompose a frequency domain operator at each frequency

Description

Gives the eigendecomposition of objects of class freqdom.

Usage

freqdom.eigen(F)

Arguments

Details

This function makes an eigendecomposition for each of the matrices F\$operator[,,k].

Value

Returns a list. The list is containing the following components:

• vectors \quad an array containing d matrices. The i-th matrix contains in its k-th row the conjugate transpose eigenvector belonging to the k-th largest eigenvalue of F\$operator[,,i].

• values \quad matrix containing in k-th column the eigenvalues of F\$operator[,,k].

• freq \quad vector of frequencies defining the object F.

See Also

freqdom

Compute a matrix product of two frequency-domain operators

Description

For given frequency-domain operators F and G (freqdom) the function freqdom.kronecker computes their matrix product frequency-wise.

Usage

freqdom.product(F, G)

Arguments

Details

Let F = \{ F_\theta : \theta \in S \}, G = \{ G_\theta : \theta \in S \}, where S is a finite grid of frequencies in [-\pi,\pi], F_\theta \in \mathbf{C}^{p \times q} and G_\theta \in \mathbf{C}^{q \times r}.

We define

H_\theta = F_\theta G_\theta

as a matrix product of F_\theta and G_\theta, i.e. H_\theta \in \mathbf{R}^{p\times r}. Function freqdom.product returns H = \{ H_\theta : \theta \in S \}.

Value

Function returns a frequency domain object (freqdom) of dimensions L \times p \times r, where L is the size of the evaluation grid. The elements correspond to F_\theta * G_\theta defined above.

Functions

• freqdom.product(): Frequency-wise matrix product of two frequency-domain operators

Examples

n = 100
X = rar(n)
Y = rar(n)
SX = spectral.density(X)
SY = spectral.density(Y)
R = freqdom.product(SY,SX)

Compute a transpose of a given frequency-domain operator at each frequency

Description

For a given frequency-domain operator S (freqdom) the function freqdom.transpose computes the transpose of S_\theta' at each frequency from the evaluation grid.

Usage

freqdom.transpose(x)

Arguments

Details

Let S = \{ S_\theta : \theta \in G \}, where G is some finite grid of frequencies in [-\pi,\pi] and S_\theta \in \mathbf{C}^{p \times p}. At each frequency \theta \in G function freqdom.transpose transposes

Resulting object is defined as

S' = \{ S_\theta': \theta \in G \}.

Value

Function returns a frequency domain object (freqdom) of dimensions L \times p_2 \times p_1, where L is the size of the grid. The elements of the object correspond to S_\theta' as defined above.

Checks if an object belongs to the class freqdom

Description

Checks if an object belongs to the class freqdom.

Usage

is.freqdom(X)

Arguments

Value

TRUE if X is of type freqdom, FALSE otherwise

See Also

freqdom, timedom, is.timedom

Checks if an object belongs to the class timedom

Description

Checks if an object belongs to the class timedom.

Usage

is.timedom(X)

Arguments

Value

TRUE if X is of type timedom, FALSE otherwise

See Also

freqdom, timedom, is.freqdom

Frequency-wise sum of freqdom objects

Description

Frequency-wise sum of freqdom objects

Usage

plus.freqdom(e1, e2)

Print freqdom object

Description

Print freqdom object

Usage

## S3 method for class 'freqdom'
print(x, ...)

Print timedom object

Description

Print timedom object

Usage

## S3 method for class 'timedom'
print(x, ...)

Simulate a multivariate autoregressive time series

Description

Generates a zero mean vector autoregressive process of a given order.

Usage

rar(
  n,
  d = 2,
  Psi = NULL,
  burnin = 10,
  noise = c("mnormal", "mt"),
  sigma = NULL,
  df = 4
)

Arguments

Details

We simulate a vector autoregressive process

X_t=\sum_{k=1}^p \Psi_k X_{t-k}+\varepsilon_t,\quad 1\leq t\leq n.

The innovation process \varepsilon_t is either multivariate normal or multivariate t with a predefined covariance/scale matrix sigma and zero mean. The noise is generated with the package mvtnorm. For Gaussian noise we use rmvnorm. For Student-t noise we use rmvt. The parameters sigma and df are imported as arguments, otherwise we use default settings. To initialise the process we set [X_{1-p},\ldots,X_{0}]=[\varepsilon_{1-p},\ldots,\varepsilon_{0}]. When burnin is set equal to K then, n+K observations are generated and the first K will be trashed.

Value

A matrix with d columns and n rows. Each row corresponds to one time point.

See Also

rma

Invert order of lags or grid parameters of a timedom or freqdom object, respectively

Description

• For a given freqdom object x, the function rev reverts the order of the evaluation grid.

Usage

## S3 method for class 'freqdom'
rev(x)

Arguments

Value

Returns object of same class as x.

rev Reverts order of lags in an object of class timedom

Description

• For a given timedom object x, the function rev reverts the order of lags.

Usage

## S3 method for class 'timedom'
rev(x)

Moving average process

Description

Generates a zero mean vector moving average process.

Usage

rma(n, d = 2, Psi = NULL, noise = c("mnormal", "mt"), sigma = NULL, df = 4)

Arguments

Details

This simulates a vector moving average process

X_t=\varepsilon_t+\sum_{k \in lags} \Psi_k \varepsilon_{t-k},\quad 1\leq t\leq n.

The innovation process \varepsilon_t is either multivariate normal or multivarite t with a predefined covariance/scale matrix sigma and zero mean. The noise is generated with the package mvtnorm. For Gaussian noise we use rmvnorm. For Student-t noise we use rmvt. The parameters sigma and df are imported as arguments, otherwise we use default settings.

Value

A matrix with d columns and n rows. Each row corresponds to one time point.

See Also

rar

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")
)

Arguments

Details

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.

Value

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.

References

Peter J. Brockwell and Richard A. Davis Time Series: Theory and Methods Springer Series in Statistics, 2009

Print object summary

Description

Print object summary

Usage

## S3 method for class 'freqdom'
summary(object, ...)

Print object summary

Description

Print object summary

Usage

## S3 method for class 'timedom'
summary(object, ...)

Defines a linear filter

Description

Creates an object of class timedom. This object corresponds to a multivariate linear filter.

Usage

timedom(A, lags)

Arguments

Details

This class is used to describe a linear filter, i.e. a sequence of matrices, each of which correspond to a certain lag. Filters can, for example, be used to transform a sequence (X_t) into a new sequence (Y_t) by defining

Y_t=\sum_k A_kX_{t-k}.

See filter.process(). Formally we consider a collection [A_1,\ldots,A_K] of complex-valued matrices A_k, all of which have the same dimension d_1\times d_2. Moreover, we consider lags \ell_1<\ell_2<\cdots<\ell_K. The object this function creates corresponds to the mapping f: \mathrm{lags}\to \mathbf{R}^{d_1\times d_2}, where \ell_k\mapsto A_k.

Value

Returns an object of class timedom. An object of class timedom is a list containing the following components:

• operators \quad returns the array A as given in the argument.

• lags \quad returns the vector lags as given in the argument.

See Also

freqdom, is.timedom

Examples

# In this example we apply the difference operator: Delta X_t= X_t-X_{t-1} to a time series
X = rar(20)
OP = array(0,c(2,2,2))
OP[,,1] = diag(2)
OP[,,2] = -diag(2)
A = timedom(OP, lags = c(0,1))
filter.process(X, A)

Compute operator norms of elements of a filter

Description

This function determines the norms of the matrices defining some linear filter.

Usage

timedom.norms(A, type = "2")

Arguments

Details

Computes \|A_h\| for h in the set of lags belonging to the object A. When type is 2 then \|A\| is the spectral radius of A. When type is F then \|A\| is the Frobenius norm (or the Hilbert-Schmidt norm, or Schatten 2-norm) of A. Same options as for the function norm as in base package.

Value

A list which contains the following components:

• lags \quad a vector containing the lags of A.

• norms \quad a vector containing the norms of the matrices defining A.

Examples

d = 2

A = array(0,c(d,d,2))
A[1,,] = 2 * diag(d:1)/d
A[2,,] = 1.5 * diag(d:1)/d
OP = timedom(A,c(-2,1))
timedom.norms(OP)

Choose lags of an object of class timedom

Description

This function removes lags from a linear filter.

Usage

timedom.trunc(A, lags)

Arguments

Value

An object of class timedom.