10
where SX(f) is the SDF at the frequency f ∈ [-1/2, 1/2].
The SDF for a stationary process decomposes the variance across different
frequencies, whereas the wavelet variance decomposes it across different scales. Given that
the scale τj can be related to range of frequencies in the interval [1/2j, 1/2j-1], the wavelet
variance usually leads to a more succinct decomposition. Moreover, unlike the SDF, the
square root of the wavelet variance is expressed in the same units as the original data.
Another advantage of the wavelet variance is that it replaces the sample variance
with a sequence of variances over given scales. That is, it offers a scale-by-scale
decomposition of variability, which makes it possible to analyze a process that exhibits
fluctuations over a range of different scales.
Let n ′j = n/2 j be the number of discrete wavelet transform (DWT) coefficients at
level j, where n is the sample size, and L' ≡ (L-2)(1 -2-j) be the number of DWT
boundary coefficients2 at level j (provided that n' > L' ), where L is the width of the
wavelet filter3. An unbiased estimator of the wavelet variance is defined as
UX(Tj) ≡
n 'j -1
--------- Σ d2t
(nj-L')2j t∑- 1 j,t
(11)
Given that the DWT decorrelates the data, the non-boundary wavelet coefficients in
a given level (dj) are zero-mean Gaussian white noise process. For a homogeneous
distribution of dj,t, there is an expected linear increase in the cumulative energy as a
function of time. The so-called D-statistic denotes the maximum deviation of dj,t from a
2 Boundary coefficients are those that are formed by combining together some values from the beginning of
the sequence of scaling coefficients with some values from the end.
3 In practical applications, we deal with sequences of values (i.e., time series) rather than functions defined
over the entire real axis. Therefore, instead of using actual wavelets, we work with short sequences of values
named wavelet filters. The number of values in the sequence is called the width of the wavelet filter, and it is
denoted by L.