Accordingly, Fourier basis functions approximate global variation well. Nevertheless, local variation, otherwise known as time-variation in time-series, is poorly adapted to cylical functions such as the sinus and cosinus. Accordingly, the trigonometric basis functions are ideally adapted to stationary processes characterized by impulses which wane with time, but are poorly adapted to discontinuous, non-linear, and non-stationary processes whose impulses persist and evolve with time. The Fourier transform is localized well in frequency, but lacks good time resolution. Furthermore, if components captured at a given frequency are non-stationary and time-dependent, then attempting to capture all such components with a single frequency (which is itself time independent), will clearly not work very well. To surmount this time-frequency relationship, a new set of basis functions are needed.Alternatively, the wavelet transform, and therefore the wavelet basis functions, are localized both in scale and time. In particular, the latter form a set of shifts and dilations of a mother wavelet function which is any function satisfying:

for some function . The term is interchangeable with the familiar variance for non-complex functions. Wavelet basis functions assume the form:

where is the location constant, and is the scaling factor (which corresponds to the notion of frequency in Fourier analysis). Wavelet transforms are well-adapted to non-stationary and discontinuous phenomena since both scale and time resolutions are preserved.

In the context of continuous functions, the continuous wavelet transform (CWT) of a time series is defined as

Since continuous functions are rarely observed, a discretized analogue of the CWT is generally employed. The discrete wavelet transform (DWT) is a collection of CWT “slices” at nodes such that , , where , for dyadic observation length for . The resulting discrete wavelet basis function is of the form

Unlike the CWT which is highly redundant in both location and scale, the DWT can be designed as an orthonormal transformation. If the location discretization is restricted to the index , at each scale , half the available observations are lost in exchange for orthonormality. This is the classical DWT framework. Alternatively, if the location index is restricted to the full set of available observations with , the discretized transform is no longer orthonormal, but does not suffer from observation loss. The latter framework is typically referred to as the maximal overlap discrete wavelet transform (MODWT), and sometimes as the non-decimated DWT.

In case of the classical DWT with , let and define the matrix of DWT coefficients . Here, is a vector of wavelet coefficients of length and is associated with changes on a scale of length . Moreover, is a vector of scaling coefficients of length and is associated with averages on a scale of length ‑. Furthermore, where is some orthonormal matrix generating the DWT coefficients.

The algorithm convolves an input signal with wavelet high and low pass filter coefficients to derive the j-th level DWT matrix . In particular, each iteration convolves the scaling coefficients from the preceding iteration, namely , with both the high and low pass filter coefficients. The input signal in the first iteration is . The entire algorithm continues until the -th iteration, although it can be stopped earlier.

The simplest and shortest wavelet filter is of length and is called the Haar wavelet. This is a sequence of rescaled rectangular functions and is therefore ideally suited to analyzing signals with sudden and discontinuous changes. In this regard, it is ideally suited for outlier detection. Unfortunately, the Haar filter is typically too simple for most other applications.

It’s important to note that both the DWT and the MODWT make use of circular filtering. In particular, when a filtering operation reaches the beginning or end of an input series, otherwise known as the boundaries, the filter treats the input time series as periodic with period . In other words, we assume that , are useful surrogates for unobserved values . Those wavelet coefficients which are affected by this circularity are also known as boundary coefficients. Note that the number of boundary coefficients only depends on the filter length and is independent of the input series length . Furthermore, the number of boundary coefficients increases with filter length . Both DWT and MODWT boundary coefficients will appear the beginning of and . You may refer to Percival and Walden (2000) for further details.

Similar to Fourier, spline, and linear approximations, a principal feature of the DWT is the ability approximate an input series as a function of wavelet basis functions. The general idea in wavelet theory is termed multiresolution analysis (MRA) and refers to the approximation of an input series at each scale (and up to all scales) . For each ,

where the and are -dimensional vectors of the j-th level detail and the -th level smooth series. Furthermore, since the low-pass (high-pass) wavelet coefficients are associated with changes (averages) at scale , the detail and smooth series are associated with changes and average at scale , respectively, in the input series .

The orthonormality of the DWT generating matrix has important implications. In particular, the DWT is an energy (variance) preserving transformation. Coupled with this preservation of energy is the decomposition of energy on a scale-by-scale basis. Then,

where is the Euclidean norm. Thus, quantifies the energy of accounted for at scale .

The total variance in can be decomposed as

where is the contribution to due to scale , and is estimated as

since is the contribution to the sample variance of at scale .

Unfortunately, this estimator is biased due to the presence of boundary coefficients. To derive an unbiased estimate, boundary coefficients should be dropped from consideration. Accordingly, an unbiased estimate of the variance contributed at scale is given by

where and represents the number of coefficients affected by boundary conditions.

and is the integral of the squared spectral density function of wavelet coefficients excluding any boundary conditions. As shown in Percival and Walden (2000), can be estimated as the sum of squared serial correlations among excluding boundary coefficients. Priestly (1981) notes that there is no condition that prevents the lower bound of this confidence interval from becoming negative.

where is the -quantile of the distribution and is the equivalent degrees of freedom (EDOF) which may be estimated using one of two suggestions

The first estimator, , relies on large sample theory, and in practice requires a sample of at least to yield a decent approximation. The second estimator, , assumes that the spectral density function of the wavelet coefficients at scale is a band-limited. See Percival and Walden (2000) for detail.

For , the number of wavelet coefficients at scale , the outlier detection algorithm is given by:

A key objective in any empirical work is to discriminate noise from useful information. Suppose that the observed time series where is an unknown signal of interest obscured by the presence of unwanted noise . Traditionally, signal discernment was typically achieved using discrete Fourier transforms. Naturally, this assumes that any signal is an infinite superposition of sinusoidal functions; a strong assumption in empirical econometrics where most data exhibits unit roots, jumps, kinks, and various other non-linearities.

The threshold value is the key to wavelet shrinkage. In particular, optimal thresholding is achieved when , the standard deviation of the noise process . There are several strategies for determining the optimal threshold:

There remains the issue of how to estimate the variance of the wavelet coefficients . If the assumption is that the observed data is obscured by some noise process , the usual estimator of variance will exhibit extreme sensitivity to noisy observations.

Let and denote the mean and median, respectively, of the wavelet coefficients at scale , and let denote its cardinality (total number of coefficients at that scale). Then, several common estimators have been proposed in the literature:

Recall that wavelet filters exhibit boundary conditions due to circular recycling of observations. Although this may be an appropriate assumption for series with naturally exhibiting cyclical effects, it is not appropriate in all circumstances. An alternative approach is to reflect the original series to generate a series of length so that filtering proceeds using observations .