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This chapter discusses time series analysis. Topics covered include method of moments, evoked potentials and peristimulus time histogram, univariate spectral analysis, bivariate spectral analysis, prediction, point process spectral estimation, and higher order correlations.

This chapter suggests a general working framework for experimental study of large-scale dynamics of EEG, including interpretation of data recorded at different spatial and temporal scales. It views source dynamics as a stochastic (random) process having statistical properties that change with changes in behavior and cognition. It avoids any attempts to choose the “best” methods of EEG or SSVEP data analysis because any such optimization requires prior knowledge of the underlying source dynamics. Fourier-based methods are emphasized to estimate power, phase, coherence, and closely related dynamic measures in distinct frequency bands. Additional methods include estimates of phase velocity across the scalp and frequency-wavenumber spectral analysis. The latter is expressed in terms of spherical harmonics, the natural spatial functions for dynamics on a spherical surface. This approach is used to pick out individual Schumann resonances as an example, in preparation for a similar application to EEG and SSVEP presented in Chapter 10.

This chapter looks at methods used for spectral analysis of economic time series and other forms including microwave devices and global warming. It examines how the spectrum of economic time series can be evaluated to detect and separate seasonal and long-term trends, whether one can devise a trading strategy using this information, or how one can determine the presence of a long-term trend such as global warming from climate statistics. A study of economic time series by David J. Thomson is reviewed. For example, studies of global warming are sensitive to whether one uses the solar year, sidereal year, the equatorial year, or any of several additional choices. The Wiener–Khinchine and Wold theorems are examined, along with means, correlations, and the Karhunen–Loeve theorem, Slepian functions, discrete prolate spheroidal sequence, Thomson's procedure, adaptive weighting, and trend removal and seasonal adjustment.

Spectral analysis provides an alternative to the time domain approach to time series analysis. This approach views a stochastic process as a weighted sum of the periodic functions sin(·) and cos(·) with different frequencies. This chapter discusses the spectral representation theorem; relates the spectral density function to the autocovariance generating function, properties of the spectral density function; and spectral density of distributed lag models. Exercises are provided at the end of the chapter.

Old Regular Baptist traditions from the Appalachian Mountains of the Southern United States rely on seemingly non-metric, non-liturgical practices of song, as opposed to hymns. This chapter uses ethnography, interviews, and spectral analysis to question assumptions about the general freedom Old Regular Baptist singing seemingly exhibits. The analytical interpretation is contextualized by historical examination of singing as developing over the course of centuries and the broadly inclusive practice of absorbing sacred songs from numerous sources. The close analyses of individual performances clarify the ways in which melodic codes and performance gestures bring Old Regular Baptists together in song to express their common belief in being tuned up with the grace of God.

Monitoring brain activity requires methods with appropriate resolutions. Field potential analysis (EEG and MEG), imaging of energy production (fMRI), optical recording methods, and single-cell recording techniques are dominant in contemporary cognitive-behavioral neuroscience. Unfortunately, even combined they fall short of explaining how neuronal groups generate representations of environments and appropriate responses. In the brain, behaviors emerge from the interaction of neurons and neuronal pools. Studying these processes requires the simultaneous monitoring of large numbers of neurons in multiple areas. Large-scale recording from multiple single neurons with tetrodes or silicon probes is an attempt at this. However, these methods are invasive and cannot be used to investigate the healthy human brain. Many other methods, such as pharmacological manipulations, macroscopic and microscopic imaging, and molecular biological tools, can provide insights, but all these indirect observations should be reconverted into neuronal spike trains to understand the brain's control of behavior.

Local dynamical processes are a key factor determining the microphysical characteristics and typically heterogeneous macroscopic structure of cirrus cloud fields. The internal and background flow fields are correspondingly heterogeneous, albeit only weakly turbulent in most instances, as is discussed here. Nucleation processes and ice crystal growth and habit are intrinsically governed by the local temperature and humidity (saturation ratio) conditions that, in turn, are strongly regulated by the intensity and duration of local updrafts and downdrafts. The microphysical result of equivalent lift by a 50cm/s updraft over a cell width of 200m is quite different from that by a 0.5 cm/s updraft over a 2-km width, even though the overall mass fluxes are equivalent. The great degree of horizontal structure seen in fallstreaks emanating from cirrus likely reflects corresponding variability in microphysical properties, primarily ice crystal size, resulting from variability in the dynamical conditions in the ice-crystal-generating layer. The ice fallout process is a first-order effect in determining overall cloud ice water path. Entrainment of noncloudy environmental air and internal mixing processes are other dynamical aspects that likely play a significant role in cloud life cycle. Dynamical processes provide an important coupling between cirrus cloud microphysical and radiative processes, as described in chapter 18 and illustrated in figure 17.1. Cirrus cloud microphysical properties and macroscopic structure strongly affect the overall radiative properties of a cirrus cloud field and thus the important radiative effect of cirrus in the climate system. Knowledge of the dynamical processes influencing cloud macrophysical properties and microphysical structure is important to understanding the origin of these characteristics. Moreover, cloud-resolving models of cirrus cloud systems must be evaluated in these respects due to the importance of cloud dynamical processes in determining overall cloud properties. Dynamical processes in cirrus are linked to the state of the background flow field that, in general, is characterized by significant wind shear and a stable thermal stratification. Gravity waves are ubiquitous and occur over a range of scales. Upper tropospheric turbulence tends to occur intermittently in patches, likely a result of sporadic shear generation (Kelvin-Helmholtz instabilities) or breaking gravity waves. Turbulent mixing in stratified shear flows is a notoriously difficult subject, and advances in its description have been obtained only recently (e.g., Fernando 1991; Schumann and Gerz 1995; Vanneste and Haynes 2000).

This chapter examines vowel differentiation and modification in a professional tenor voice using a spectrograph. Spectral analysis supplies information regarding technical maneuvers during singing. What the ear can hear, the eye then simultaneously verifies. Comparative studies of phonations can provide clues to common practices among singers of similar vocal category, and can point out individual differences that contribute to the unique characteristic of each singing voice. This study considers phonations from Benjamin Britten's Sonnets of Michelangelo, sung in the original Italian. Spectral analysis confirms what the practical pedagogical ear discerns: in a male voice, vowel modification is possible for registration purposes in upper range, so as to maintain desirable harmonic balance in a mounting scale without destroying vowel integrity. This is in accordance with vocal pedagogy based on the Italian School model.

The reception of Mendeleev’s periodic system in Sweden was not a dramatic episode. The system was accepted almost without discussion, but at the same time with no exclamation marks or any other outbursts of enthusiasm. There are but a few weak short-lived critical remarks. That was all. I will argue that the acceptance of the system had no overwhelming effect on chemical practice in Sweden. At most, it strengthened its characteristics. It is actually possible to argue that chemistry in Sweden was more essential for the periodic system than the other way around. My results might therefore suggest that we perhaps have to reevaluate the role of Mendeleev’s system in the history of chemistry. Chemistry in Sweden at the end of the nineteenth century can be characterized as a classifying science, with chemists very skilled in analysis, and as mainly an atheoretical science, which treated theories at most only as hypothesis—the slogan of many chemists being “facts persist, theories vanish.” Thanks to these characteristics, by the end of the nineteenth century, chemistry in Sweden had developed into, it must be said, a rather boring chemistry. This is obviously not to say that it is boring to study such a chemistry. Rather, it gives us an example of how everyday science, a part of science too often neglected but a part that constitutes the bulk of all science done, is carried out. One purpose of this study is to see how a theory, considered to be important in the history of chemistry, influenced everyday science. One might ask what happened when a daring chemistry met a boring chemistry. What happened when a theory, which had been created by a chemist who has been described as “not a laboratory chemist,” met an atheoretical experimental science of hard laboratory work and, as was said, the establishment of facts? Furthermore, could we learn something about the role of the periodic system per se from the study of such a meeting? Mendeleev’s system has often been considered important for teaching, and his attempts to write a textbook are often taken as the initial step in the chain of thoughts that led to the periodic system.

The line spectrum can be used to monitor and even assist in tuning a piano. In fact, for those like the author, who do not have the experience or expertise to tune by ear, the line spectrum approach is a handy aid. The chapter initially examines beating, since aural tuning is based on being able to analyze beating sounds resulting from striking two notes simultaneously. Next, some time domain and line spectrum plots associated with an amateurish attempt to tune a three-string unison are displayed. Because the aim is to discern differences in frequencies, a short discussion on the limits of detectability is presented next. A spectral analysis of the results of an intervallic tuning completes the chapter.

The circular flexible membrane can be considered an idealization of a drum head. This chapter looks at the response of the membrane surface to a variety of initial conditions and compares it to the bar and string. An actual drum is spectrally analyzed.

This chapter presents a letter that the author wrote on the occasion of Leonid Hurwicz’s 90th birthday in 2007. The author talks about working with Hurwicz: they did early spectral analysis of Frickey’s aggregate U.S. output for the time slot 1865–1935. Even more melodramatic was the new Hurwicz-Samuelson grading system for the author’s first regular statistics course. MIT engineers have always been grade chasers. Thus, all hell broke loose when they learned that their exam mark of 115 put them below the median of the class grades. It did not help when Hurwicz explained that this was the famous Chicago grading system. Ultimately, the author considers the Leo MIT year a fun year.

Here we present our attempt to characterize a time series of drop-to-drop intervals from a dripping faucet as a nonextensive system. We found a long-range anticorrelated behavior as evidence of memory in the dynamics of our system. The hypothesis of faucets dripping at the edge of chaos is reinforced by results of the linear rate of the increase of the nonextensive Tsallis statistics. We also present some similarities between dripping faucets and healthy hearts…. Many systems in Nature exhibit complex or chaotic behaviors. Chaotic behavior is characterized by short-range correlations and strong sensitivity to small changes of the initial conditions. Complex behavior is characterized by the presence of long-range power-law correlations in its dynamics. In the latter, the sensitivity to a perturbation of the initial condition is weaker than in the former. Because the probability densities are frequently described as inverse power laws, the variance and the mean often diverge. Although it is hard to predict the long-term behavior of such systems, it is still possible to get some information from them and even to find similarities between two apparently very distinct systems. Tools from statistical physics are frequently used because the main task here is to deal with diverse macroscopic phenomena and to try to explain them, starting with the microscopic interactions among many individual components. The microscopic interactions are not necessarily complicated, but the collective behavior can determine a rather intricate macroscopic description. Nonextensive statistical mechanics, since its proposal in 1988 [27], has been applied to an impressive collection of systems in which spatial or temporal longrange correlations appear. Hence, it can also become a useful tool to characterize such systems. Here, we present an attempt of using such formalism to try to understand the intriguing behavior of an apparently simple system: a dripping faucet.