Estimation of multivariate long-memory models parameters: memory parameters and long-run covariance matrix (also called fractal connectivity).

Description

This package computes an estimation of the long-memory parameters and the long-run covariance matrix using a multivariate model (Lobato, 1999; Shimotsu 2007). Two semi-parametric methods are implemented: a Fourier based approach (Shimotsu 2007) and a wavelet based approach (Achard and Gannaz 2014; Achard and Gannaz (2024) <doi:10.1111/jtsa.12719>). Real and complex wavelets are implemented.

Details

Author(s)

Sophie Achard and Irene Gannaz

Maintainer: Sophie Achard <sophie.achard@gipsa-lab.fr>, Irene Gannaz <irene.gannaz@insa-lyon.fr>

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

Examples

rho<-0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d<-c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)

x <- resp$x
long_run_cov <- resp$long_run_cov

#### Compute wavelets this is also included in the functions without _wav
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(1,11)

if(is.matrix(x)){
    N <- dim(x)[1]
    k <- dim(x)[2]
}else{
    N <- length(x)
    k <- 1
}
mat_x <- as.matrix(x,dim=c(N,k))

## Wavelet decomposition
xwav <- matrix(0,N,k)
    for(j in 1:k){
        xx <- mat_x[,j]   
        resw <- DWTexact(xx,filter)
        xwav_temp <- resw$dwt
        index <- resw$indmaxband
        Jmax <- resw$Jmax
        xwav[1:index[Jmax],j] <- xwav_temp;
    }
## we free some memory
new_xwav <- matrix(0,min(index[Jmax],N),k)
    if(index[Jmax]<N){
        new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
    }
    xwav <- new_xwav
    index <- c(0,index)


##### Compute the wavelet functions 
res_psi <- psi_hat_exact(filter,J)
psih<-res_psi$psih
grid<-res_psi$grid


##### Estimate using Fourier #############

m <- floor(N^{0.65}) ## default value of Shimotsu
res_mfw <- mfw(x,m)
res_d_mfw<-res_mfw$d
res_rho_mfw<-res_mfw$cov[1,2]

### Eval MFW

res_mfw_eval <- mfw_eval(d,x,m)
res_mfw_cov_eval <- mfw_cov_eval(d,x,m)

###### Estimate using Wavelets #############

## Using xwav 

if(dim(xwav)[2]==1) xwav<-as.vector(xwav)
res_mww_wav <- mww_wav(xwav,index,psih,grid,LU)

### Eval MWW_wav

res_mww_wav_eval <- mww_wav_eval(d,xwav,index,LU)
res_mww_wav_cov_eval <- mww_wav_cov_eval(d,xwav,index,psih,grid,LU)

## Using directly the time series

res_mww <- mww(x,filter,LU)
res_d_mww<-res_mww$d
res_rho_mww<-res_mww$cov[1,2]

### Eval MWW_wav

res_mww_eval <- mww_eval(d,x,filter,LU)
res_mww_cov_eval <- mww_cov_eval(d,x,filter,LU)

Exact discrete wavelet decomposition with common-factor wavelets

Description

Computes the discrete wavelet transform of the data using the pyramidal algorithm.

Usage

DWTcomplex(x, filter, real)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

S. Achard, M. Clausel, I. Gannaz, F. Roueff (2020). New results on approximate Hilbert pairs of wavelet filters with common factor structure. Applied and Computational Harmonic Analysis, Vol 49, N.3, pp 1025-1045.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

hwlet

Examples

filter_complex <- hwlet(M=4,L=4,type='const')
u <- rnorm(2^10,0,1)
x <- vfracdiff(u,d=0.2)

	resw <- DWTcomplex(x,filter_complex)
		xwav <- resw$dwt
		index <- resw$indmaxband
		Jmax <- resw$Jmax

## Wavelet scale 1
ws_1 <- xwav[1:index[1]]
## Wavelet scale 2
ws_2 <- xwav[(index[1]+1):index[2]]
## Wavelet scale 3
ws_3 <- xwav[(index[2]+1):index[3]]
### upto Jmax

Exact discrete wavelet decomposition

Description

Computes the discrete wavelet transform of the data using the pyramidal algorithm.

Usage

DWTexact(x, filter)

Arguments

Value

Note

This function was rewritten from an original matlab version by Fay et al. (2009)

Author(s)

S. Achard and I. Gannaz

References

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

scaling_filter

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
u <- rnorm(2^10,0,1)
x <- vfracdiff(u,d=0.2)

	resw <- DWTexact(x,filter)
		xwav <- resw$dwt
		index <- resw$indmaxband
		Jmax <- resw$Jmax

## Wavelet scale 1
ws_1 <- xwav[1:index[1]]
## Wavelet scale 2
ws_2 <- xwav[(index[1]+1):index[2]]
## Wavelet scale 3
ws_3 <- xwav[(index[2]+1):index[3]]
### upto Jmax

Evaluation of function K

Description

Computes the function K as defined in (Achard and Gannaz 2014) and in (Achard and Gannaz 2024).

Usage

K_eval(psi_hat,u,d)

Arguments

Details

K_eval computes the matrix K with elements

K(d_l,d_m)=\int u^{(d_l+d_m)} |\code{psi}\_\code{hat}(u)|^2 du

Value

value of function K as a matrix.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

psi_hat_exact

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha
res_psi <- psi_hat_exact(filter,J=10)
K_eval(res_psi$psih,res_psi$grid,d=c(0.2,0.2))

Time series obtained by an fMRI experiment on the brain

Description

Time series for each region of interest in the brain. These series are obtained by SPM preprocessing.

Usage

data(brainHCP)

Format

A data frame with 1200 observations on the following 89 variables.

Source

contact S. Achard (sophie.achard@gipsa-lab.fr)

References

M. Termenon, A. Jaillard, C. Delon-Martin, S. Achard (2016) Reliability of graph analysis of resting state fMRI using test-retest dataset from the Human Connectome Project, Neuroimage, Vol 142, pages 172-187.

Examples

data(brainHCP)
## maybe str(brainHCP) ; plot(brainHCP) ...

Wavelets coefficients utilities

Description

Computes the number of wavelet coefficients at each scale.

Usage

compute_nj(n, N)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

DWTexact, scaling_filter

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
n <- 5^10
N <- length(filter)
compute_nj(n,N)

Convolution matrix

Description

Returns the convolution matrix, A, associated to the filter v such that the product of A and an n-element vector, x, is the convolution of v and x.

Usage

convmtx(v,n)

Arguments

Value

The convolution matrix, A, associated to the filter v such that the product of A and an n-element vector, x, is the convolution of v and x.

Author(s)

Achard, Clausel, Gannaz, Roueff (2017)

References

S. Achard, M. Clausel, I. Gannaz, F. Roueff (2020). New results on approximate Hilbert pairs of wavelet filters with common factor structure. Applied and Computational Harmonic Analysis, Vol 49, N.3, pp 1025-1045.

simulation of FIVARMA process

Description

Generates N observations of a realisation of a multivariate FIVARMA process X.

Usage

fivarma(N, d = 0, cov_matrix = diag(length(d)), VAR = NULL,
            VMA = NULL,skip = 2000)

Arguments

Details

Let (e(t))_t be a multivariate gaussian process with a covariance matrix cov_matrix. The values of the process X are given by the equations:

VAR(L)U(t) = VMA(L)e(t),

and

diag((1-L)^d)X(t) = U(t)

where L is the lag-operator.

Value

Author(s)

S. Achard and I. Gannaz

References

R. J. Sela and C. M. Hurvich (2009) Computationaly efficient methods for two multivariate fractionnaly integrated models. Journal of Time Series Analysis, Vol 30, N. 6, pages 631-651.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

varma, vfracdiff

Examples

rho1 <- 0.3
rho2 <- 0.8
cov <- matrix(c(1,rho1,rho2,rho1,1,rho1,rho2,rho1,1),3,3)
d <- c(0.2,0.3,0.4)

J <- 9
N <- 2^J
VMA <- diag(c(0.4,0.1,0))
### or another example VAR <- array(c(0.8,0,0,0,0.6,0,0,0,0.2,0,0,0,0,0.4,0,0,0,0.5),dim=c(3,3,2))
VAR <- diag(c(0.8,0.6,0))
resp <- fivarma(N, d, cov_matrix=cov, VAR=VAR, VMA=VMA)
x <- resp$x
long_run_cov <- resp$long_run_cov
d <- resp$d

Common-Factor wavelet filter coefficients

Description

Provides the Hilbert transform pair of orthogonal wavelet bases given by Common factor construction of Selesnick (2001), with perfect reconstruction condition

Usage

hwlet(M,L,type)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

I.W. Selesnick (2001) Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters, Vol 8, N.6, pp 170-173.

I.W. Selesnick (2002) The design of approximate Hilbert transform pairs of wavelet bases, IEEE Transactions on Signal Processing, Vol 50, N.5, pp 1144-1152.

S. Achard, M. Clausel, I. Gannaz, F. Roueff (2020). New results on approximate Hilbert pairs of wavelet filters with common factor structure. Applied and Computational Harmonic Analysis, Vol 49, N.3, pp 1025-1045.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

DWTcomplex

Examples

filter_complex <- hwlet(M=4,L=4,type='const')

Evaluation of the roots for spectral factorization

Description

This program orders the values x_in (supposed to be the roots of a polynomial) in this way that computing the polynomial coefficients by using the function poly yields numerically accurate results.

Usage

leja(x_in)

Arguments

Value

Reordering of x_in

Author(s)

Matlab codes provided by Markus Lang : <lang@dsp.rice.edu> in 1993, Rice University. R code by Achard, Clausel, Gannaz, Roueff (2017).

References

I.W. Selesnick (2001) Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters, Vol 8, N.6, pp 170-173.

See Also

seprts, sfact,hwlet

Examples

z = exp(1i*(1:100)*2*pi/100)
p1 = signal::poly(z)
p2 = signal::poly(leja(z))

Multivariate complex (or real) wavelet Whittle estimation

Description

Computes the multivariate complex (or real) wavelet Whittle estimation for the long-memory parameter vector d and the long-run covariance matrix, using DWTcomplex for the wavelet decomposition.

Usage

mcw(x, filter_complex, LU = NULL, J=10)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

mcw_wav,mcw_wav_eval,mcw_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

res_mww <- mww(x,filter,LU)

Multivariate complex (or real) wavelet Whittle estimation for data as wavelet coefficients

Description

Computes the multivariate complex (or real) wavelet Whittle estimation of the long-memory parameter vector d and the long-run covariance matrix for the already wavelet decomposed data.

Usage

mcw_wav(xwav, index, psih, grid_K, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

mcw,mcw_wav_eval,mcw_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

##### Compute the wavelet functions 
res_psi <- psi_hat_exact(filter,10)
psih <- res_psi$psih
grid <- res_psi$grid

res_mww <- mww_wav(xwav,index, psih, grid,LU)

Multivariate complex (or real) wavelet Whittle estimation of the long-run covariance matrix

Description

Computes the multivariate complex (or real) wavelet Whittle estimation of the long-run covariance matrix given the long-memory parameter vector d for the already wavelet decomposed data.

Usage

mcw_wav_cov_eval(d, xwav, index, psih, grid_K, LU)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Long-run covariance matrix estimation.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

mcw,mcw_wav,mcw_wav_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho<-0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d<-c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

##### Compute the wavelet functions 
res_psi <- psi_hat_exact(filter,10)
psih<-res_psi$psih
grid<-res_psi$grid

res_mww <- mww_wav_cov_eval(d,xwav,index, psih, grid,LU)

Multivariate real wavelet Whittle estimation for data as wavelet coefficients

Description

Evaluates the multivariate complex (or real) wavelet Whittle objective function at a given long-memory parameter vector d for the already wavelet decomposed data.

Usage

mcw_wav_eval(d, xwav, index, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

multivariate wavelet Whittle criterion.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

mcw,mcw_wav,mcw_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

res_mww <- mww_wav_eval(d,xwav,index,LU)
res_d <- optim(rep(0,k),mww_wav_eval,xwav=xwav,index=index,LU=LU,
          method='Nelder-Mead',lower=-Inf,upper=Inf)$par

multivariate Fourier Whittle estimators

Description

Computes the multivariate Fourier Whittle estimators of the long-memory parameters and the long-run covariance matrix also called fractal connectivity.

Usage

mfw(x, m)

Arguments

Details

The choice of m determines the range of frequencies used in the computation of the periodogram, \lambda_j = 2\pi j/N, j = 1,... , m. The optimal value depends on the spectral properties of the time series such as the presence of short range dependence. In Shimotsu (2007), m is chosen to be equal to N^{0.65}.

Value

Author(s)

S. Achard and I. Gannaz

References

K. Shimotsu (2007) Gaussian semiparametric estimation of multivariate fractionally integrated processes Journal of Econometrics Vol. 137, N. 2, pages 277-310.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mfw_eval, mfw_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

m <- 57 ## default value of Shimotsu 2007
res_mfw <- mfw(x,m)

multivariate Fourier Whittle estimators

Description

Computes the multivariate Fourier Whittle estimator of the long-run covariance matrix (also called fractal connectivity) for a given value of long-memory parameters d.

Usage

mfw_cov_eval(d, x, m)

Arguments

Details

The choice of m determines the range of frequencies used in the computation of the periodogram, \lambda_j = 2\pi j/N, j = 1,... , m. The optimal value depends on the spectral properties of the time series such as the presence of short range dependence. In Shimotsu (2007), m is chosen to be equal to N^{0.65}.

Value

long-run covariance matrix estimation.

Author(s)

S. Achard and I. Gannaz

References

K. Shimotsu (2007) Gaussian semiparametric estimation of multivariate fractionally integrated processes Journal of Econometrics Vol. 137, N. 2, pages 277-310.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mfw_eval, mfw

Examples

### Simulation of ARFIMA(0,\code{d},0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

m <- 57 ## default value of Shimotsu
G <- mfw_cov_eval(d,x,m) # estimation of the covariance matrix when d is known

evaluation of multivariate Fourier Whittle estimator

Description

Evaluates the multivariate Fourier Whittle criterion at a given long-memory parameter value d.

Usage

mfw_eval(d, x, m)

Arguments

Details

The choice of m determines the range of frequencies used in the computation of the periodogram, \lambda_j = 2\pi j/N, j = 1,... , m. The optimal value depends on the spectral properties of the time series such as the presence of short range dependence. In Shimotsu (2007), m is chosen to be equal to N^{0.65}.

Value

multivariate Fourier Whittle estimator computed at point d.

Author(s)

S. Achard and I. Gannaz

References

K. Shimotsu (2007) Gaussian semiparametric estimation of multivariate fractionally integrated processes Journal of Econometrics Vol. 137, N. 2, pages 277-310.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mfw_cov_eval, mfw

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

m <- 57 ## default value of Shimotsu
res_mfw <- mfw(x,m)
d <- res_mfw$d
G <- mfw_eval(d,x,m)
k <- length(d)
res_d <- optim(rep(0,k),mfw_eval,x=x,m=m,method='Nelder-Mead',lower=-Inf,upper=Inf)$par

Multivariate real wavelet Whittle estimation

Description

Computes the multivariate real wavelet Whittle estimation for the long-memory parameter vector d and the long-run covariance matrix, using DWTexact for the wavelet decomposition.

Usage

mww(x, filter, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww_eval, mww_cov_eval,mww_wav,mww_wav_eval,mww_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

res_mww <- mww(x,filter,LU)

Multivariate real wavelet Whittle estimation of the long-run covariance matrix

Description

Computes the multivariate real wavelet Whittle estimation of the long-run covariance matrix given the long-memory parameter vector d, using DWTexact for the wavelet decomposition.

Usage

mww_cov_eval(d, x, filter, LU)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

long-run covariance matrix estimation.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww, mww_eval,mww_wav,mww_wav_eval,mww_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

res_mww <- mww_cov_eval(d,x,filter,LU)

Evaluation of multivariate real wavelet Whittle estimation

Description

Evaluates the multivariate real wavelet Whittle criterion at a given long-memory parameter vector d using DWTexact for the wavelet decomposition.

Usage

mww_eval(d, x, filter, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

multivariate wavelet Whittle criterion.

Author(s)

S. Achard and I. Gannaz

References

E. Moulines, F. Roueff, M. S. Taqqu (2009) A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series. Annals of statistics, vol. 36, N. 4, pages 1925-1956

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww, mww_cov_eval,mww_wav,mww_wav_eval,mww_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

res_mww <- mww_eval(d,x,filter,LU)
k <- length(d)
res_d <- optim(rep(0,k),mww_eval,x=x,filter=filter,
	  	        LU=LU,method='Nelder-Mead',lower=-Inf,upper=Inf)$par

Multivariate real wavelet Whittle estimation for data as wavelet coefficients

Description

Computes the multivariate real wavelet Whittle estimation of the long-memory parameter vector d and the long-run covariance matrix for the already wavelet decomposed data.

Usage

mww_wav(xwav, index, psih, grid_K, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww_eval, mww_cov_eval,mww,mww_wav_eval,mww_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

##### Compute the wavelet functions 
res_psi <- psi_hat_exact(filter,10)
psih <- res_psi$psih
grid <- res_psi$grid

res_mww <- mww_wav(xwav,index, psih, grid,LU)

Multivariate real wavelet Whittle estimation of the long-run covariance matrix

Description

Computes the multivariate real wavelet Whittle estimation of the long-run covariance matrix given the long-memory parameter vector d for the already wavelet decomposed data.

Usage

mww_wav_cov_eval(d, xwav, index,psih,grid_K, LU)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

Long-run covariance matrix estimation.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww, mww_eval,mww_wav,mww_wav_eval,mww_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho<-0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d<-c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

##### Compute the wavelet functions 
res_psi <- psi_hat_exact(filter,10)
psih<-res_psi$psih
grid<-res_psi$grid

res_mww <- mww_wav_cov_eval(d,xwav,index, psih, grid,LU)

Multivariate real wavelet Whittle estimation for data as wavelet coefficients

Description

Evaluates the multivariate real wavelet Whittle objective function at a given long-memory parameter vector d for the already wavelet decomposed data.

Usage

mww_wav_eval(d, xwav, index, LU = NULL)

Arguments

Details

L is fixing the lower limit of wavelet scales. L can be increased to avoid finest frequencies that can be corrupted by the presence of high frequency phenomena.

U is fixing the upper limit of wavelet scales. U can be decreased when highest frequencies have to be discarded.

Value

multivariate wavelet Whittle criterion.

Author(s)

S. Achard and I. Gannaz

References

E. Moulines, F. Roueff, M. S. Taqqu (2009) A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series. Annals of statistics, vol. 36, N. 4, pages 1925-1956

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

mww, mww_cov_eval,mww_wav,mww_eval,mww_wav_cov_eval

Examples

### Simulation of ARFIMA(0,d,0)
rho <- 0.4
cov <- matrix(c(1,rho,rho,1),2,2)
d <- c(0.4,0.2)
J <- 9
N <- 2^J

resp <- fivarma(N, d, cov_matrix=cov)
x <- resp$x
long_run_cov <- resp$long_run_cov

## wavelet coefficients definition
res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h

LU <- c(2,11)

### wavelet decomposition

if(is.matrix(x)){
     N <- dim(x)[1]
     k <- dim(x)[2]
}else{
     N <- length(x)
     k <- 1
}
x <- as.matrix(x,dim=c(N,k))

     ## Wavelet decomposition
     xwav <- matrix(0,N,k)
     for(j in 1:k){
          xx <- x[,j]
             
          resw <- DWTexact(xx,filter)
          xwav_temp <- resw$dwt
          index <- resw$indmaxband
          Jmax <- resw$Jmax
          xwav[1:index[Jmax],j] <- xwav_temp;
     }
     ## we free some memory
     new_xwav <- matrix(0,min(index[Jmax],N),k)
     if(index[Jmax]<N){
          new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
     }
     xwav <- new_xwav
     index <- c(0,index)

res_mww <- mww_wav_eval(d,xwav,index,LU)
res_d <- optim(rep(0,k),mww_wav_eval,xwav=xwav,index=index,LU=LU,
          method='Nelder-Mead',lower=-Inf,upper=Inf)$par

Discrete Fourier transform of a real wavelet

Description

Computes the discrete Fourier transform of the real wavelet associated to the given filter using scaling_function. The length of the Fourier transform is equal to the length of the grid where the wavelet is evaluated.

Usage

psi_hat_exact(filter,J=10)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

DWTexact, scaling_filter

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
psi_hat_exact(filter,J=6)

Discrete Fourier transform of a complex Common-Factor wavelet

Description

Computes the discrete Fourier transform of the complex Common-Factor wavelet associated to the given filters using scaling_function. The length of the Fourier transform is equal to the length of the grid where the wavelet is evaluated.

Usage

psi_hat_exact_complex(h,g,J=10)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

I.W. Selesnick (2001) Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters, Vol 8, N.6, pp 170-173.

I.W. Selesnick (2002) The design of approximate Hilbert transform pairs of wavelet bases, IEEE Transactions on Signal Processing, Vol 50, N.5, pp 1144-1152.

S. Achard, M. Clausel, I. Gannaz, F. Roueff (2020). New results on approximate Hilbert pairs of wavelet filters with common factor structure. Applied and Computational Harmonic Analysis, Vol 49, N.3, pp 1025-1045.

S. Achard, I. Gannaz (2024). Local Whittle estimation with (quasi-)analytic wavelets. Journal of Time Series Analysis, Vol 45, pp 421-443.

See Also

DWTexact, scaling_filter

Examples

filter_complex <- hwlet(M=4,L=4,type='const')
psi_hat_exact_complex(filter_complex$h,filter_complex$g,J=6)

wavelet scaling filter coefficients

Description

Computes the filter coefficients of the Haar or Daubechies wavelet family with a specific order

Usage

scaling_filter(family, parameter)

Arguments

Value

Author(s)

S. Achard and I. Gannaz

References

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

DWTexact

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
M <- res_filter$M
alpha <- res_filter$alpha

scaling function and the wavelet function

Description

Computes the scaling function and the wavelet function (for compactly supported wavelet) using the cascade algorithm on the grid of dyadic integer 2^{-J}

Usage

scaling_function(filter,J)

Arguments

Value

Note

This function was rewritten from an original matlab version by Fay et al. (2009)

Author(s)

S. Achard and I. Gannaz

References

G. Fay, E. Moulines, F. Roueff, M. S. Taqqu (2009) Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics, vol. 151, N. 2, pages 159-177.

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

DWTexact, scaling_filter

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
scaling_function(filter,J=6)

Evaluation of the roots for spectral factorization

Description

This program is for spectral factorization. The roots on the unit circle must have even degree. Roots with high multiplicity will cause problems, they should be handled by extracting them prior to using this program.

Usage

seprts(p, type='mid')

Arguments

Value

The roots of the polynomial p, separated depending of either they are inside the unit circle or on the unit circle. Technical function for the function sfact.

Author(s)

Matlab codes provided by Selesnick (2001). R code by Achard, Clausel, Gannaz, Roueff (2017).

References

I.W. Selesnick (2001) Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters, Vol 8, N.6, pp 170-173.

I.W. Selesnick (2002) The design of approximate Hilbert transform pairs of wavelet bases, IEEE Transactions on Signal Processing, Vol 50, N.5, pp 1144-1152.

See Also

sfact, leja,hwlet

Spectral factorization of a polynomial.

Description

Spectral factorization of a polynomial h.

Usage

sfact(h, type='mid')

Arguments

Value

Author(s)

Matlab codes provided by Markus Lang : <lang@dsp.rice.edu> in 1993, Rice University. R code by Achard, Clausel, Gannaz, Roueff (2017).

References

I.W. Selesnick (2001) Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters, Vol 8, N.6, pp 170-173.

I.W. Selesnick (2002) The design of approximate Hilbert transform pairs of wavelet bases, IEEE Transactions on Signal Processing, Vol 50, N.5, pp 1144-1152.

S. Achard, M. Clausel, I. Gannaz, F. Roueff (2020). New results on approximate Hilbert pairs of wavelet filters with common factor structure. Applied and Computational Harmonic Analysis, Vol 49, N.3, pp 1025-1045.

See Also

seprts, leja,hwlet

Examples

g = runif(10)
h = conv(g,rev(g))
b = sfact(h)$poly
h - conv(b,rev(b)) ## should be zeros

Transform a vector in a non symmetric Toeplitz matrix

Description

Transform a vector in a non symmetric Toeplitz matrix

Usage

toeplitz_nonsym(vec)

Arguments

Value

the corresponding matrix.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

scaling_function

Examples

res_filter <- scaling_filter('Daubechies',8);
filter <- res_filter$h
Htmp <- toeplitz_nonsym(filter)

simulation of multivariate ARMA process

Description

generates N observations of a k-vector ARMA process

Usage

varma(N, k = 1, VAR = NULL, VMA = NULL, cov_matrix = diag(k), innov=NULL)

Arguments

Value

vector containing the N observations of the k-vector ARMA process.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

fivarma, vfracdiff

Examples

rho1 <- 0.3
rho2 <- 0.8
cov <- matrix(c(1,rho1,rho2,rho1,1,rho1,rho2,rho1,1),3,3)

J <- 9
N <- 2^J
VMA <- diag(c(0.4,0.1,0))
### or another example VAR <- array(c(0.8,0,0,0,0.6,0,0,0,0.2,0,0,0,0,0.4,0,0,0,0.5),dim=c(3,3,2))
VAR <- diag(c(0.8,0.6,0))
x <- varma(N, k=3, cov_matrix=cov, VAR=VAR, VMA=VMA)

simulation of vector fractional differencing process

Description

Given a vector process x and a vector of long memory parameters d, this function is producing the corresponding fractional differencing process.

Usage

vfracdiff(x, d)

Arguments

Details

Given a process x, this function applied a fractional difference procedure using the formula:

diag((1-L)^d) x,

where L is the lag operator.

Value

vector fractional differencing of x.

Author(s)

S. Achard and I. Gannaz

References

S. Achard, I. Gannaz (2016) Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Vol 37, N. 4, pages 476-512. http://arxiv.org/abs/1412.0391.

K. Shimotsu (2007) Gaussian semiparametric estimation of multivariate fractionally integrated processes Journal of Econometrics Vol. 137, N. 2, pages 277-310.

S. Achard, I Gannaz (2019) Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave. Journal of Statistical Software, Vol 89, N. 6, pages 1-31.

See Also

varma, fivarma

Examples

rho1 <- 0.3
rho2 <- 0.8
cov <- matrix(c(1,rho1,rho2,rho1,1,rho1,rho2,rho1,1),3,3)
d <- c(0.2,0.3,0.4)


J <- 9
N <- 2^J
VMA <- diag(c(0.4,0.1,0))
### or another example VAR <- array(c(0.8,0,0,0,0.6,0,0,0,0.2,0,0,0,0,0.4,0,0,0,0.5),dim=c(3,3,2))
VAR <- diag(c(0.8,0.6,0))
x <- varma(N, k=3, cov_matrix=cov, VAR=VAR, VMA=VMA)
vx<-vfracdiff(x,d)