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Spectral analysis is widely used to interpret time series collected in diverse areas. This book covers the statistical theory behind spectral analysis and provides data analysts with the tools needed to transition theory into practice. Actual time series from oceanography, metrology, atmospheric science and other areas are used in running examples throughout, to allow clear comparison of how the various methods address questions of interest. All major nonparametric and parametric spectral analysis techniques are discussed, with emphasis on the multitaper method, both in its original formulation involving Slepian tapers and in a popular alternative using sinusoidal tapers. The authors take a unified approach to quantifying the bandwidth of nonparametric spectral estimate. An extensive set of exercises allows readers to test their understanding of theory and practical analysis. The time series used as examples and R language code for recreating the analyses of the series are available from the book's website.
The wavelet analysis of a time series can be defined in terms of an orthonormal transform, so
here we briefly review the key ideas behind such transforms. We first review the basic theory for
orthonormal transforms in Section 3.1. Section 3.2 discusses the important projection theorem, while
3.3 considers complex-valued transforms. Prior to introducing the discrete wavelet transform (DWT)
in Chapter 4, we discuss the orthonormal discrete Fourier transform (ODFT) in Section 3.4 because it
parallels and contrasts the DWT in a number of interesting ways. We summarize the key points of this
chapter in Section 3.5 - readers who are already comfortable with orthonormal transforms can read
this section simply to become familiar with our notation and conventions.
Basic Theory for Orthonormal Transforms
Orthonormal transforms are of interest because they can be used to re-express a time series in
such a way that we can easily reconstruct the series from its transform. In a loose sense, the
‘information’ in the transform is thus equivalent to the ‘information’
in the original series; to put it another way, the series and its transform can be considered to be
two representations of the same mathematical entity. Orthonormal transforms can be used to
re-express a series in a standardized form (e.g., a Fourier series) for further manipulation, to
reduce a series to a few values summarizing its salient features (compression), and to analyze a
series to search for particular patterns of interest (e.g., analysis of variance).
As we saw in Chapters 4 and 5, one important use for the discrete wavelet transform (DWT) and its variant, the maximal overlap DWT (MODWT), is to decompose the sample variance of a time series on a scale-by-scale basis. In this chapter we explore wavelet-based analysis of variance (ANOVA) in more depth by defining a theoretical quantity known as the wavelet variance (sometimes called the wavelet spectrum). This theoretical variance can be readily estimated based upon the DWT or MODWT and has been successfully used in a number of applications; see, for example, Gamage (1990), Bradshaw and Spies (1992), Flandrin (1992), Gao and Li (1993), Hudgins et al. (1993), Kumar and Foufoula-Georgiou (1993, 1997), Tewfik et al. (1993), Wornell (1993), Scargle (1997), Torrence and Compo (1998) and Carmona et al. (1998). The definition for the wavelet variance and rationales for considering it are given in Section 8.1, after which we discuss a few of its basic properties in Section 8.2. We consider in Section 8.3 how to estimate the wavelet variance given a time series that can be regarded as a realization of a portion of length N of a stochastic process with stationary backward differences. We investigate the large sample statistical properties of wavelet variance estimators and discuss methods for determining an approximate confidence interval for the true wavelet variance based upon the estimated wavelet variance (Section 8.4).
In subsequent chapters we will make substantial use of some basic results from the Fourier theory
of sequences and – to a lesser extent – functions, and we will find that filters play
a central role in the application of wavelets. This chapter is intended as a self-contained guide to
some key results from Fourier and filtering theory. Our selection of material is intentionally
limited to just what we will use later on. For a more thorough discussion employing the same
notation and conventions adopted here, see Percival and Walden (1993). We also recommend Briggs and
Henson (1995) and Hamming (1989) as complementary sources for further study.
Readers who have extensive experience with Fourier analysis and filters can just quickly scan
this chapter to become familiar with our notation and conventions. We encourage others to study the
material carefully and to work through as many of the embedded exercises as possible (answers are
provided in the appendix). It is particularly important that readers understand the concept of
periodized filters presented in Section 2.6 since we use this idea repeatedly in Chapters 4 and
5.
Complex Variables and Complex Exponentials
The most elegant version of Fourier theory for sequences and functions involves the use of
complex variables, so here we review a few key concepts regarding them (see, for example, Brown and
Churchill, 1995, for a thorough treatment). Let i ≡ √–1 so
that i2 = –1 (throughout the book, we take
‘≡’ to mean ‘equal by definition’).
This introduction to wavelet analysis 'from the ground level and up', and to wavelet-based statistical analysis of time series focuses on practical discrete time techniques, with detailed descriptions of the theory and algorithms needed to understand and implement the discrete wavelet transforms. Numerous examples illustrate the techniques on actual time series. The many embedded exercises - with complete solutions provided in the Appendix - allow readers to use the book for self-guided study. Additional exercises can be used in a classroom setting. A Web site offers access to the time series and wavelets used in the book, as well as information on accessing software in S-Plus and other languages. Students and researchers wishing to use wavelet methods to analyze time series will find this book essential.
As discussed in Chapter 4, the discrete wavelet transform (DWT) allows us to analyze (decompose) a time series X into DWT coefficients W, from which we can then synthesize (reconstruct) our original series. We have already noted that the synthesis phase can be used, for example, to construct a multiresolution analysis of a time series (see Equation (64) or (104a)) and to simulate long memory processes (see Section 9.2). In this chapter we study another important use for the synthesis phase that provides an answer to the signal estimation (or function estimation, or denoising) problem, in which we want to estimate a signal hidden by noise within an observed time series. The basic idea here is to modify the elements of W to produce, say, W′, from which an estimate of the signal can be synthesized. With the exception of methods briefly discussed in Section 10.8, once certain parameters have been estimated, the elements Wn of W are treated one at a time; i.e., how we modify Wn is not directly influenced by the remaining DWT coefficients. The wavelet-based techniques that we concentrate on here are thus conceptually very simple, yet they are remarkably adaptive to a wide variety of signals.
Here we introduce the discrete wavelet transform (DWT), which is the basic tool needed for
studying time series via wavelets and plays a role analogous to that of the discrete Fourier
transform in spectral analysis. We assume only that the reader is familiar with the basic ideas from
linear filtering theory and linear algebra presented in Chapters 2 and 3. Our exposition builds
slowly upon these ideas and hence is more detailed than necessary for readers with strong
backgrounds in these areas. We encourage such readers just to use the Key Facts and Definitions in
each section or to skip directly to Section 4.12 – this has a concise self-contained
development of the DWT. For complementary introductions to the DWT, see Strang (1989, 1993), Rioul
and Vetterli (1991), Press et al. (1992) and Mulcahy (1996).
The remainder of this chapter is organized as follows. Section 4.1 gives a qualitative
description of the DWT using primarily the Haar and D(4) wavelets as examples. The formal
mathematical development of the DWT begins in Section 4.2, which defines the wavelet filter and
discusses some basic conditions that a filter must satisfy to qualify as a wavelet filter. Section
4.3 presents the scaling filter, which is constructed in a simple manner from the wavelet filter.
The wavelet and scaling filters are used in parallel to define the pyramid algorithm for computing
(and precisely defining) the DWT – various aspects of this algorithm are presented in
Sections 4.4, 4.5 and 4.6.
The continuous time wavelet transform is becoming a well-established tool for multiple scale representation of a continuous time ‘signal,’ which by definition is a finite energy function denned over the entire real axis. This transform essentially correlates a signal with ‘stretched’ versions of a wavelet function (in essence a continuous time band-pass filter) and yields a multiresolution representation of the signal. In this chapter we summarize the important ideas and results for the multiresolution view of the continuous time wavelet transform. Our primary intent is to demonstrate the close relationship between continuous time wavelet analysis and the discrete time wavelet analysis presented in Chapter 4. To make this connection, we adopt a formalism that allows us to bridge the gap between the inner product convention used in mathematical discussions on wavelets and the filtering convention favored by engineers. For simplicity we deal only with signals, scaling functions and wavelet functions that are all taken to be real-valued. Only the case of dyadic wavelet analysis (where the scaling factor in the dilation of the basis function takes the value of two) is considered here.
In Chapter 4 we discussed the discrete wavelet transform (DWT), which essentially decomposes a
time series X into coefficients that can be associated with different scales and times. We can thus
regard the DWT of X as a ‘time/scale’ decomposition. The wavelet coefficients for a
given scale Tj ≡ 2J−1 tell us
how localized weighted averages of X vary from one averaging period to the next. The scale
Tj gives us the effective width in time (i.e., degree of localization)
of the weighted averages. Because the DWT can be formulated in terms of filters, we can relate the
notion of scale to certain bands of frequencies. The equivalent filter that yields the wavelet
coefficients for scale Tj is approximately a band-pass filter with a
pass-band given by [l/2j+1, 1/2j].
For a sample size N = 2J, the N - 1
wavelet coefficients constitute - when taken together - an octave band decomposition of the
frequency interval [1/2J+1, 1/2], while the single scaling
coefficient is associated with the interval [0, 1/2J+1]. Taken as
a whole, the DWT coefficients thus decompose the frequency interval [0, 1/2] into adjacent
individual intervals.
In this chapter we consider the discrete wavelet packet transform (DWPT), which
can be regarded as any one of a collection of orthonormal transforms, each of which can be readily
computed using a very simple modification of the pyramid algorithm for the DWT.
Wavelets are mathematical tools for analyzing time series or images (although not exclusively so:
for examples of usage in other applications, see Stollnitz et al., 1996, and
Sweldens, 1996). Our discussion of wavelets in this book focuses on their use with time series,
which we take to be any sequence of observations associated with an ordered independent variable
t (the variable t can assume either a discrete set of values such
as the integers or a continuum of values such as the entire real axis - examples of both types
include time, depth or distance along a line, so a time series need not actually involve time).
Wavelets are a relatively new way of analyzing time series in that the formal subject dates back to
the 1980s, but in many aspects wavelets are a synthesis of older ideas with new elegant mathematical
results and efficient computational algorithms. Wavelet analysis is in some cases complementary to
existing analysis techniques (e.g., correlation and spectral analysis) and in other cases capable of
solving problems for which little progress had been made prior to the introduction of wavelets.
Broadly speaking (and with apologies for the play on words!), there have been two main waves of
wavelets. The first wave resulted in what is known as the continuous wavelet transform (CWT), which
is designed to work with time series defined over the entire real axis; the second, in the discrete
wavelet transform (DWT), which deals with series defined essentially over a range of integers
(usually t = 0, 1,…,N – 1, where N
denotes the number of values in the time series). In this chapter we introduce and motivate wavelets
via the CWT.