SimulatedSNR9Series: Simulated Wavelet Signals with SNR=9
In WiSEBoot: Wild Scale-Enhanced Bootstrap

Description Usage Format Details References Examples

A matrix containing simulated signals with noise added such that the signal-to-noise ratio (SNR) is 9. If the signal vector is called Y, and Y is of length 2^J, we have defined the SNR within the J0=j threshold as

SNR=(1/2^J)( (<Y, Y>/(2^j-1)) / (σ^2/(2^J - 2^j -1)))

where σ^2 is the variance of the noise. These series are obtained by adding white noise to the smooth data in SimulatedSmoothSeries. Each column contains a wavelet coefficient threshold level and rows contain observations.

1	data("SimulatedSNR9Series")

The format is:

num [1:1024, 1:9] -0.01879 -0.01635 -0.01096 -0.00814 -0.00423 ...

- attr(*, "dimnames")=List of 2

..$ : NULL

..$ : chr [1:9] "J0.0" "J0.1" "J0.2" "J0.3" ...

The columns contain noisy series with signals in different wavelet coefficient threshold levels. Series are available for signal thresholds of J0 in {0, 1, 2, 3, 4, 5, 6, 7, 8}. The rows are the data observations. Thus, each smooth series is of length 2^{10}=1024.

The names of each column indicate the threshold (J0) in the smooth series. For example, the 3rd column, named J0.2, has a threshold of J0=2 in the signal, and thus 0-valued wavelet coefficients for all mother wavelet coefficients finer than level 2 in the signal. Notice, the white noise added to the signal creates non-zero coefficients above the threshold.

The original smooth series were generated using 'wd' and 'wr' with the family="DaubLeAsymm", filter.number=8, bc="periodic" options in the 'wavethresh' package.

wavethresh-package, SimulatedSmoothSeries

data(SimulatedSNR9Series)

##See if WiSEBoot selects the correct threshold for this data (J0=3)
## R=10 bootstrap samples is not recommended.  For demonstration only.
bootObj <- WiSEBoot(SimulatedSNR9Series[,4], R=10)
bootObj$MSECriteria

##Look at the noisy data compared to the true smooth
data(SimulatedSmoothSeries)
plot(seq(1, 2^10), SimulatedSNR9Series[ , 6], main="Threshold of J0=5", 
     col="lightgray", xlab="Time", ylab="Observations", type="l")
lines(seq(1, 2^10), SimulatedSmoothSeries[ ,6], col="red", lwd=2)

      J0plusOne  meanOfMSE
 [1,]         9 0.03771718
 [2,]         8 0.03492006
 [3,]         7 0.02749739
 [4,]         6 0.02274555
 [5,]         5 0.01907403
 [6,]         4 0.01829780
 [7,]         3 0.01911155
 [8,]         2 0.01875261
 [9,]         1 0.01892079
[10,]         0 0.01918460