scale coefficients describe the overall behaviour of the signal.

#### The Discrete Wavelet Transform (DWT).

Computation of wavelet coefficients resulting from traditional wavelet transforms is performed using the so-called [*Discrete Wavelet Transform*]{} (DWT), first introduced by [@Mallat]. The algorithm proceeds by alternately applying high- and low-pass filtering and decimation (subsampling) operations to the observed data.

Let $\mathcal{H}:=\{h_{k}\}$ and $\mathcal{G}:=\{g_{k}\}$ be a low- and high-pass filter pair associated with a given wavelet, such as the quadrature mirror filters used in the construction of compactly supported wavelets introduced by [@daubechies]. Following [@NSDWT], let $ \mathcal{D}_{0} $ denote the even decimation operator that selects every even-indexed element in a sequence, in other words $(\mathcal{D}_{0}x)_{l}=x_{2l}$. The detail coefficients of the DWT of a time series $X=\{x_{t}\}_{t=1}^{T}$ (where $T=2^J~\text{for some positive integer}~J$) can be found using $$\mathbf{d}_{j} = \mathcal{D}_{0}\mathcal{G}(\mathcal{D}_{0}\mathcal{H})^{J-j-1}X,$$ for $j=0,\hdots,J-1$. Similarly, the scaling or smooth coefficients of the DWT are given by $$\mathbf{c}_{j} = (\mathcal{D}_{0}\mathcal{H})^{J-j}X,$$ for $j=0,\hdots, J$.

The information contained in the original time series $X$ can thus be fully described by the set of coefficients $ \{\mathbf{d}_{J-1}, \mathbf{d}_{J-2},\hdots,\mathbf{d}_{0},\mathbf{c}_{0}\} $.

#### The Nondecimated Discrete Wavelet Transform (NDWT).

The [*nondecimated*]{} wavelet transform (NDWT) is a modification of the DWT outlined above in which the 