waveslim (version 1.12)

dwt: Discrete Wavelet Transform (DWT)

Description

This function performs a level $J$ decomposition of the input vector or time series using the pyramid algorithm (Mallat 1989).

Usage

dwt(x, wf="la8", n.levels=4, boundary="periodic")
dwt.nondyadic(x)

Arguments

x

a vector or time series containing the data be to decomposed. This must be a dyadic length vector (power of 2).

wf

Name of the wavelet filter to use in the decomposition. By default this is set to "la8", the Daubechies orthonormal compactly supported wavelet of length $L=8$ (Daubechies, 1992), least asymmetric family.

n.levels

Specifies the depth of the decomposition. This must be a number less than or equal to $\log(\mbox{length}(x),2)$.

boundary

Character string specifying the boundary condition. If boundary=="periodic" the default, then the vector you decompose is assumed to be periodic on its defined interval, if boundary=="reflection", the vector beyond its

Value

• Basically, a list with the following components
• d?
  Wavelet coefficient vectors.
• s?
  Scaling coefficient vector.
• wavelet
  Name of the wavelet filter used.
• boundary
  How the boundaries were handled.

Details

The code implements the one-dimensional DWT using the pyramid algorithm (Mallat, 1989). The actual transform is performed in C using pseudocode from Percival and Walden (2001). That means convolutions, not inner products, are used to apply the wavelet filters.

For a non-dyadic length vector or time series, dwt.nondyadic pads with zeros, performs the orthonormal DWT on this dyadic length series and then truncates the wavelet coefficient vectors appropriately.

References

Daubechies, I. (1992) Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM: Philadelphia.

Gencay, R., F. Selcuk and B. Whitcher (2001) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press. Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, No. 7, 674-693.

Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.

See Also

modwt, mra.

Examples

## Figures 4.17 and 4.18 in Gencay, Selcuk and Whitcher (2001).
data(ibm)     
ibm.returns <- diff(log(ibm))
## Haar
ibmr.haar <- dwt(ibm.returns, "haar")
names(ibmr.haar) <- c("w1", "w2", "w3", "w4", "v4")
## plot partial Haar DWT for IBM data
par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.returns, axes=FALSE, ylab="", main="(a)")
for(i in 1:4)
  plot.ts(up.sample(ibmr.haar[[i]], 2^i), type="h", axes=FALSE,
          ylab=names(ibmr.haar)[i])
plot.ts(up.sample(ibmr.haar$v4, 2^4), type="h", axes=FALSE,
        ylab=names(ibmr.haar)[5])
axis(side=1, at=seq(0,368,by=23), 
     labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))
## LA(8)
ibmr.la8 <- dwt(ibm.returns, "la8")
names(ibmr.la8) <- c("w1", "w2", "w3", "w4", "v4")
## must shift LA(8) coefficients
ibmr.la8$w1 <- c(ibmr.la8$w1[-c(1:2)], ibmr.la8$w1[1:2])
ibmr.la8$w2 <- c(ibmr.la8$w2[-c(1:2)], ibmr.la8$w2[1:2])
for(i in names(ibmr.la8)[3:4])
  ibmr.la8[[i]] <- c(ibmr.la8[[i]][-c(1:3)], ibmr.la8[[i]][1:3])
ibmr.la8$v4 <- c(ibmr.la8$v4[-c(1:2)], ibmr.la8$v4[1:2])
## plot partial LA(8) DWT for IBM data
par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.returns, axes=FALSE, ylab="", main="(b)")
for(i in 1:4)
  plot.ts(up.sample(ibmr.la8[[i]], 2^i), type="h", axes=FALSE,
          ylab=names(ibmr.la8)[i])
plot.ts(up.sample(ibmr.la8$v4, 2^4), type="h", axes=FALSE,
        ylab=names(ibmr.la8)[5])
axis(side=1, at=seq(0,368,by=23), 
  labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))