Description Usage Arguments Details Value Author(s) References See Also Examples
Calculate the WiSE bootstrap sample of the linear parameters describing a set of wavelet coefficients from two WiSEBoot objects. See Details for a precise description.
1  WiSEConfidenceRegion(X, Y, plot = TRUE, ...)

X 

Y 

plot 
logical. If 
... 
additional graphical arguments. See 
Given 2 vectors of equallyspaced data of length T=2^J for a positive integer, J, we assume the following models:
X = γ_{x0} 1 + γ_{x1} t + Wγ_x + e_x
Y = γ_{y0} 1 + γ_{y1} t + Wγ_y + e_y
The WiSE bootstrap is performed on each data series with the same threshold level, J0=j, using WiSEBoot
. Bootstrap estimates of γ_x and γ_y are obtained from these WiSEBoot
objects, called g^*_{xj} and g^*_{yj}. The estimates of γ_x and γ_y from the input data may be noted as g_{xj} and g_{yj}.
This function allows the user to examine a linear relationship between the two sets of wavelet coefficients:
γ_y = α 1 + β γ_x
Thus, we use g^*_{xj} and g^*_{yj} to obtain estimates of α and β, called a^* and b^*. Likewise, the original coefficients from the data, g_{xj} and g_{yj}, yield estimates, a and b. The bootstrap parameter estimates allow the user to visualize the distribution of the linear parameters, α and β. Generally, these parameters give us an idea about the relationship between the two data series signals.
Currently, this function does not calculate a (1α)% 2dimensional region in the parameters. It allows a qualitative visualization of the bootstrap sample distribution for these linear parameters. Asymptotically, the linear parameters follow a multivariate normal distribution when assumptions are met.
Note, the slope and intercept parameters here (α and β) are different from the linear parameters in time within our data series (γ_{x0}, γ_{y0}, γ_{x1}, γ_{y1}). The BootSlope
and BootIntercept
output from WiSEBoot
represent the linear parameters in time within the data series.
dataIntercept 
point estimate of intercept of the wavelet coefficients from the data. Using the notation in the Details section, a. 
dataSlope 
point estimate of slope of the wavelet coefficients from the data. Using the notation in the Details section, b. 
bootIntercept 
bootstrap estimates of the intercept of the wavelet coefficients. Using the notation in the Details section, a^*. This is a vector. 
bootSlope 
bootstrap estimates of the slope of the wavelet coefficients. Using the notation in the Details section, b^*. This is a vector. 
Megan Heyman
The WiSE bootstrap hypothesis test is implemented as an analysis tool in Braverman, A. et al. "Probabilistic Climate Model Evaluation" (in progress). This is the corresponding confidence region to the calculations presented there.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  ## R=10 bootstrap samples is not recommended. For demonstration only.
###Example with random data
x < rnorm(2^8)
y < x + rnorm(2^8, sd=0.001) #y has similar structure to x
xWise < WiSEBoot(x, R=10, J0=4)
yWise < WiSEBoot(y, R=10, J0=4)
xyConf < WiSEConfidenceRegion(xWise, yWise) #does the region contain (0, 1)?
###Example with AIRS and IPSL data
data(CM20N20S60E)
padCM < padMatrix(CM20N20S60E) #pad data so we can use wavelet methodology
AIRS < WiSEBoot(padCM$xPad[,1], R=10, J0=5, XParam=padCM$linearParam[,1])
IPSL1 < WiSEBoot(padCM$xPad[,2], R=10, J0=5, XParam=padCM$linearParam[,2])
AIRS_IPSL1Conf < WiSEConfidenceRegion(AIRS, IPSL1) #how are these signals related?

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