This text presents several single-variable statistical distributions that engineers and scientists use to describe the uncertainty and variation inherent in measured information. It lists significant properties of these distributions and describes methods for estimating their parameters, constructing confidence intervals, and testing hypotheses. Each distribution is illustrated by working through typical applications including some of the special methods associated with them. The intention is to provide the professional with a ready source of information on a useful range of distribution models and the techniques of analysis specific to each.

The need to deal rationally with the uncertainties that enter engineering analysis and design appears now well recognized by the engineering profession. This need is driven, on the one hand, by the competitive pressure to optimize designs and, on the other hand, by market demand for reliable products. Hence, engineers design their products closer to the limits of the materials used, while improving product durability. The result of these opposing pressures is that the engineer needs to replace traditional “contingency factors” by careful uncertainty analysis.

What is perhaps less well understood by the professional is the need to choose a distribution model that closely represents the entire range of measured values. This need arises from the skewness typical of the frequency functions of engineering data, coupled with the usual focus of engineering decisions on the location of distribution tails.

In many areas of the physical sciences, spectral analysis finds frequent and extensive use. Examples abound in oceanography, electrical engineering, geophysics, astronomy and hydrology. Spectral analysis is a well-established standard – its use in so many areas in fact facilitates the exchange of ideas across a broad array of scientific endeavors. As old and as well-established as it is, however, spectral analysis is still an area of active on-going research (indeed, one of the problems for both practitioners and theorists alike is that important methodological advances are spread out over many different disciplines and literally dozens of scientific journals). Since the 1960s, developments in spectral analysis have had to take into account three new factors:

1. Digital data explosion. The amount of data routinely collected in the form of time series is staggering by 1960s standards. Examples include exploration seismic data, continuous recordings of the earth's magnetic field, large seismology networks, real-time processing of sonar signals and remote sensing from satellites. The impact on spectral analysis is that fast and digital processing has become very important.

2. Enormous increase in computational power. In retrospect, it is clear that much of the spectral analysis methodology in vogue in the 1960s was influenced heavily by what could be calculated with commonly available computers. With the computational power now widely available on today's computers, we are finally in a position to concentrate on what should be calculated.

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Introduction

If a stationary process has a purely continuous spectrum, it is natural to estimate its spectral density function (sdf) since this function is easier to interpret than the integrated spectrum. Estimation of the sdf has occupied our attention in Chapters 6, 7 and 9. However, if we are given a sample of a time series drawn from a process with a purely discrete spectrum (i.e., a ‘line’ spectrum for which the integrated spectrum is a step function), our estimation problem is quite different: we must estimate the location and magnitude of the jumps in the integrated spectrum. This requires estimation techniques that differ – to some degree at least – from what we have already studied. It is more common, however, to come across processes whose spectra are a mixture of lines and an sdf stemming from a so-called ‘background continuum.’ In Section 4.4 we distinguished two cases. If the sdf for the continuum is that of white noise, we said that the process has a discrete spectrum – as opposed to a purely discrete spectrum, which has only a line component; on the other hand, if the sdf for the continuum differs from that of white noise (sometimes called ‘colored’ noise), we said that the process has a mixed spectrum (see Figures 142 and 143).

In this chapter we shall use some standard concepts from tidal analysis to motivate and illustrate these models. We shall begin with a discrete parameter harmonic process that has a purely discrete spectrum.

Introduction

In Chapter 6 we introduced the important concept of tapering a time series as a way of obtaining a spectral estimator with acceptable bias properties. While tapering does reduce bias due to leakage, there is a price to pay in that the sample size is effectively reduced. When we also smooth across frequencies, this reduction translates into a loss of information in the form of an increase in variance (recall the Ch factor in Equation (248b) and Table 248). This inflated variance is acceptable in some practical applications, but in other cases it is not. The loss of information inherent in tapering can often be avoided either by prewhitening (see Sections 6.5 and 9.10) or by using Welch's overlapped segment averaging (WOSA – see Section 6.17).

In this chapter we discuss another approach to recovering information lost due to tapering. This approach was introduced in a seminal paper by Thomson (1982) and involves the use of multiple orthogonal tapers. As we shall see, multitaper spectral estimation has a number of interesting points in its favor:

1. In contrast to either prewhitening which typically requires the careful design of a prewhitening filter or the conventional use of WOSA (i.e., a Hanning data taper with 50% overlap of blocks) which can still suffer from leakage for spectra with very high dynamic ranges, the multitaper scheme can be used in a fairly ‘automatic’ fashion. Hence it is useful in situations where thousands – or millions – of individual time series must be processed so that the pure volume of data precludes a careful analysis of individual series (this occurs routinely in exploration geophysics).

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Introduction

In the previous chapter we produced representations for various deterministic functions and sequences in terms of linear combinations of sinusoids with different frequencies (for mathematical convenience we actually used complex exponentials instead of sinusoids directly). These representations allow us to easily define various energy and power spectra and to attach a physical meaning to them. For example, subject to square integrability conditions, we found that periodic functions are representable (in the mean square sense) by sums of sinusoids over a discrete set of frequency components, while nonperiodic functions are representable (also in the mean square sense) by an integral of sinusoids over a continuous range of frequencies. For periodic functions, the energy from ∞ to ∞ is infinite, so we can define their spectral properties in terms of distributions of power over a discrete set of frequencies. For nonperiodic functions, the energy from –∞ to ∞ is finite, so we can define their properties in terms of an energy distribution over a continuous range of frequencies.

We now want to find some way of representing a stationary process in terms of a ‘sum’ of sinusoids so that we can meaningfully define an appropriate spectrum for it; i.e., we want to be able to directly relate our representation for a stationary process to its spectrum in much the same way we did for deterministic functions. Now a stationary process has associated with it an ensemble of realizations that describe the possible outcomes of a random experiment.

Introduction

This chapter provides a quick introduction to the subject of spectral analysis. Except for some later references to the exercises of Section 1.6, this material is independent of the rest of the book and can be skipped without loss of continuity. Our intent is to use some simple examples to motivate the key ideas. Since our purpose is to view the forest before we get lost in the trees, the particular analysis techniques we use here have been chosen for their simplicity rather than their appropriateness.

Some Aspects of Time Series Analysis

Spectral analysis is part of time series analysis, so the natural place to start our discussion is with the notion of a time series. The quip (attributed to R. A. Fisher) that a time series is ‘one damned thing after another’ is not far from the truth: loosely speaking, a time series is a set of observations made sequentially in time. Examples abound in the real world, and Figures 2 and 3 show plots of small portions of four actual time series:

1. the speed of the wind in a certain direction at a certain location, measured every 0.025 second;

2. the monthly average measurements related to the flow of water in the Willamette River at Salem, Oregon;

3. the daily record of a quantity (to be precise, the change in average daily frequency) that tells how well an atomic clock keeps time on a day to day basis (a constant value of 0 would indicate that the clock agreed perfectly with a time scale maintained by the U. S. Naval Observatory); and

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Introduction

Spectral analysis almost invariably deals with a class of models called stationary stochastic processes. The material in this chapter is a brief review of the theory behind such processes. The reader is referred to Chapter 3 of Priestley (1981), Chapter 10 of Papoulis (1991) or Chapter 1 of Yaglom (1987) for complementary discussions.

Stochastic Processes

Consider the following experiment (see Figure 31): we hook up a resistor to an oscilloscope in such a way that we can examine the voltage variations across the resistor as a function of time. Every time we press a ‘reset’ button on the oscilloscope, it displays the voltage variations for the 1 second interval following the ‘reset.’ Since the voltage variations are presumably caused by such factors as small temperature variations in the resistor, each time we press the ‘reset’ button, we will observe a different display on the oscilloscope. Owing to the complexity of the factors that influence the display, there is no way that we can use the laws of physics to predict what will appear on the oscilloscope. However, if we repeat this experiment over and over, we soon see that, although we view a different display each time we press the ‘reset’ button, the displays resemble each other: there is a characteristic ‘bumpiness’ shared by all the displays.

We can model this experiment by considering a large bowl in which we have placed pictures of all the oscilloscope displays that we could possibly observe. Pushing the ‘reset’ button corresponds to reaching into the bowl and choosing ‘at random’ one of the pictures.