mww_wav function

multivariate wavelet Whittle estimation for data as wavelet coefficients

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

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

Arguments

• xwav: wavelet coefficients matrix (with scales in rows and variables in columns).
• index: vector containing the largest index of each band, i.e. for $j>1$ the wavelet coefficients of scale $j$ are $\code{dwt}(k)$ for $k \in [\code{indmaxband}(j-1)+1,\code{indmaxband}(j)]$ and for $j=1$, $\code{dwt}(k)$ for $k \in [1,\code{indmaxband}(1)]$.
• psih: the Fourier transform of the wavelet mother at values grid_K.
• grid_K: the grid for the approximation of the integral in K.
• LU: bivariate vector (optional) containing L, the lowest resolution in wavelet decomposition U, the maximal resolution in wavelet decomposition. (Default values are set to L=1, and U=Jmax.)

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.

Returns

• d: estimation of the vector of long-memory parameters.

• cov: estimation of the long-run covariance matrix.

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.

Author(s)

S. Achard and I. Gannaz

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)