costat (version 1.1-1)

lacv: Computes localized (wavelet) autocovariance function

Description

Compute the LACV function for a locally stationary wavelet process.

Usage

lacv(x, filter.number = 1, family = "DaubExPhase", ...)

Arguments

x

The time series you want to compute the LACV for

filter.number

The wavelet that you wish to compute the LACV with respect to

family

The wavelet family

...

Additional arguments to the spectrum computation contained within

Value

• A matrix that contains the localized autocovariance. If the original time series was of length T, then the number of rows of the returned matrix is also T, one row for each time point. The columns of the array correspond to the lag. The number of columns, 2K+1, depends both on the length of the time series and also the order of the wavelet (smoother wavelets return lacv matrices with larger number of lags). Lag 0 is always the centre column, with negative lags from -K to -1 are the leftmost columns, lags from 1 to K are the rightmost columns.

Details

A locally stationary wavelet process is a particular kind of non-stationary time series constructed out of wavelet atoms, with a time-varying spectrum (slowly varying). This kind of model is useful for time series whose spectral properties change over time.

The time-varying spectrum can be computed from within the WaveThresh library by the ewspec function. However, just as in the classical stationary case, where the spectrum and autocovariance are a Fourier transform pair, the paper Nason, von Sachs, Kroisandt (2000) [NvSK2000] shows that the evolutionary wavelet spectrum is paired to a localized autocovariance function using a wavelet-like transform. This is expressed in formula (14) of the NvSK2000 paper.

This function computes the localized autocovariance by first computing the estimate of the evolutionary spectrum, and then directly transforming it using formula (14) via the autocorrelation wavelet transform.

References

`Costationary and stationarity tests for stock index returns' by Car dinali and Nason

See Also

ewspec

Examples

#
# Generate an AR(1) time series
#
vv <- arima.sim(model=list(ar=0.8), n=1024)
#
# Compute the ACF of this stationary series
#
vv.acf <- acf(vv, plot=FALSE)
#
# Compute the localized autocovariance. We'll use
# a reasonably smooth wavelet.
#
vv.lacv <- lacv(vv, filter.number=4)
#
# Various plots of the lacv
#
# The whole lacv at time t=100. First work out sizes:
#
dim(vv.lacv)
# [1] 1024 14323
#
(14323-1)/2
# [1] 7161
#
# So, rows of vv.lacv index time ranging from 1 to 1024
# Cols index lag ranging from -7161 through to 7161.
#
# Now the plot
#
plot( -7161:7161, vv.lacv[100,], type="l", xlab="lag", ylab="Autocovariance")
#
# Most of the action, for this time series especially, is near to the zero
# lag. So, let's zoom in on the action. And let's plot autocorrelation,
# rather than autocovariance. We'll use the estimate of the autocovariance
# at lag zero at this time for the denominator.
#
# The following commands plots the lacv for lags ranging from -10 to 10 at
# time t=100. Change the 100 to something else for different times
#
plot(-10:10,vv.lacv[100,(7162-10):(7162+10)]/vv.lacv[100,7162], type="l", xlab="lag")
#
# Let us get an indication of how accurate the lacv is, we shall superimpose
# some of the regular acf values on this plot. Note that vv.acf[1] contains
# the acf at lag zero, which is 1.
# 
points(1, vv.acf[2])
points(2, vv.acf[3])
points(3, vv.acf[4])
points(4, vv.acf[5])
#
# You should find that the lacv is pretty accurate, which is good
# seeing as it is a local quantity.
#