Multiresolution Analysis of Time Series

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

mra(x, wf = "la8", J = 4, method = "modwt", boundary = "periodic")

Arguments

x	A vector or time series containing the data be to decomposed. This must be a dyadic length vector (power of 2) for method="dwt".
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 \(L=8\) least asymmetric family.
J	Specifies the depth of the decomposition. This must be a number less than or equal to \(\log(\mbox{length}(x),2)\).
method	Either "dwt" or "modwt".
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 boundaries is assumed to be a symmetric reflection of itself.

Value

Basically, a list with the following components

D?

Wavelet detail vectors.

S?

Wavelet smooth vector.

wavelet

Name of the wavelet filter used.

boundary

How the boundaries were handled.

Details

This code implements a one-dimensional multiresolution analysis introduced by Mallat (1989). Either the DWT or MODWT may be used to compute the multiresolution analysis, which is an additive decomposition of the original time series.

References

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

dwt, modwt.

Author

B. Whitcher

Examples

## Easy check to see if it works...
x <- rnorm(32)
x.mra <- mra(x)
sum(x - apply(matrix(unlist(x.mra), nrow=32), 1, sum))^2
#> [1] 8.491348e-29

## Figure 4.19 in Gencay, Selcuk and Whitcher (2001)
data(ibm)     
ibm.returns <- diff(log(ibm))
ibm.volatility <- abs(ibm.returns)
## Haar
ibmv.haar <- mra(ibm.volatility, "haar", 4, "dwt")
names(ibmv.haar) <- c("d1", "d2", "d3", "d4", "s4")
## LA(8)
ibmv.la8 <- mra(ibm.volatility, "la8", 4, "dwt")
names(ibmv.la8) <- c("d1", "d2", "d3", "d4", "s4")
## plot multiresolution analysis of IBM data
par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.volatility, axes=FALSE, ylab="", main="(a)")
for(i in 1:5)
  plot.ts(ibmv.haar[[i]], axes=FALSE, ylab=names(ibmv.haar)[i])
axis(side=1, at=seq(0,368,by=23), 
  labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))

par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.volatility, axes=FALSE, ylab="", main="(b)")
for(i in 1:5)
  plot.ts(ibmv.la8[[i]], axes=FALSE, ylab=names(ibmv.la8)[i])
axis(side=1, at=seq(0,368,by=23), 
  labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))