DWTexact function

Exact discrete wavelet decomposition

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

DWTexact(x, filter)

Arguments

• x: vector of raw data
• filter: Quadrature mirror filter (also called scaling filter, as returned by the scaling_filter function)

Returns

• dwt: computable Wavelet coefficients without taking into account the border effect.

• indmaxband: 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 $k in [1,\code{indmaxband}(1)]$.

• Jmax: largest available scale index (=length of indmaxband).

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.

Author(s)

S. Achard and I. Gannaz

See Also

scaling_filter

Note

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

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