Assume that an appropriate model for our time series is
$\mathbf{X}=\mathbf{D} + \mathbf{\epsilon}$ where
$\mathbf{D}$ represents an unknown
deterministic signal of interest and
$\mathbf{\epsilon}$ is some
undesired stochastic noise that is independent and identically
distributed and has a process mean of zero. Waveshrink seeks to
eliminate the noise component
$\mathbf{\epsilon}$ of $\mathbf{X}$ in hopes of obtaining
(a close approximation to) $\mathbf{D}$. The basic algorithm works as
follows:[object Object],[object Object],[object Object]
This function support different shrinkage methods and threshold estimation
schemes. Let $W$ represent an arbitrary DWT coefficient and
$W^{\mbox{(t)}}$ the correpsonding thresholded coefficient
using a threshold of $\delta$.
The supported shrinkage methods are
[object Object],[object Object],[object Object]
Hard thresholding reduces to zero all coefficients that do not
exceed the threshold. Soft thresholding pushes toward zero any
coefficient whose magnitude exceeds the threshold, and zeros the
coefficient otherwise. Mid thresholding represents a compromise
between hard and soft thresholding such that coefficients whose
magnitude exceeds twice the threshold are not adjusted, those between
the threshold and twice the trhreshold are shrunk, and those
below the threshodl are zeroed.
The threshold is selected based on a model of the noise. The supported
techniques for estimating the noise threshold are
[object Object],[object Object],[object Object]
Finally, the user has the choice of using either a decimated (standard)
form of the discrete wavelet transform (DWT) or an undecimated version
of the DWT (known as the Maximal Overlap DWT (MODWT)).
Unlike the DWT, the MODWT is a (circular) shift-invariant transform so
that a circular shift in the original time series produces an equivalent shift of
the MODWT coefficients. In addition, the MODWT can be interpreted as
a cycle-spun version of the DWT, which is achieved by averaging
over all non-redundant DWTs of shifted versions of the original series. The z
is a smoother version of the DWT at the cost of an increase in computational
complexity (for an N-point series, the DWT requires $O(N)$ multiplications
while the MODWT requires $O(N\log_2N$ multiplications.