Abstract
In the analysis of neural data, time is important. We experience life as evolving, and neurophysiological investigations focus increasingly on dynamic features of brain activity.
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- 1.
The term “spectral analysis” sometimes connotes statistical analysis, rather than purely mathematical analysis, but for now we are ignoring any noise considerations.
- 2.
Feynman et al. (1963 Volume I, p. 49–1).
- 3.
The argument we sketch here makes the most sense for functions that are periodic on \([0,1]\), meaning that they satisfy \(f(0)=f(1)\). In Section 18.3.6 we discuss what happens when this condition fails to hold.
- 4.
The first requires the notion of function, which emerged roughly in the 1700s, especially in the work of Euler (the notation \(f(x)\) apparently being introduced in 1735). The second may be considered intuitively obvious, but a detailed rigorous understanding of the situation did not come until the 1800s, particularly in the work of Cauchy (represented by a publication in 1821) and Weierstrass (in 1872). The notion of harmonics was one of the greatest discoveries of antiquity, and is associated with Pythagoras. The third and fourth steps emerged in work by D’Alembert in the mid-1700s, and by Fourier in 1807. Along the way, representations using complex numbers were used by Euler (his famous formula, given below, appeared in 1748), but they were considered quite mysterious until their geometric interpretation was given by Wessel, Argand, and Gauss, the latter in an influential 1832 exposition. A complete understanding of basic Fourier analysis was achieved by the early 1900s with the development of the Lebesgue integral. Recommended general discussions may be found in Courant and Robbins (1996), Lanczos (1966), and Hawkins (2001).
- 5.
The approximation becomes exact when \(f(t)\) is periodic, \(f(t)^2\) has a finite integral, and the expansion involves all of the infinitely many harmonics.
- 6.
With appropriate mathematics (especially the theory of Lebesgue integration) it may be shown that every square-integrable function on [0,1] may be represented, equivalently, by its set of Fourier coefficients, and its integrated squared magnitude is equal to the sum of squares of the coefficients.
- 7.
\(W\) is often used to represent time series noise out of deference to Norbert Wiener, a major figure in the development of time series theory.
- 8.
The fit in Fig. 18.8 avoided step 3, and would not change very much if step 3 were included, but the statistical inferences involving confidence intervals and significance tests do require step 3.
- 9.
To get a sequence of Fourier frequencies \(\omega _j\) that converge to \(\omega \), define \(\omega _{j_n}=j_n/n\) with \(j_n\) a sequence of integers for which \(j_n/n \rightarrow \omega \).
- 10.
A reference advocating methods three and four, above, is Fan and Kreutzberger (1998).
- 11.
This assumes the data are real numbers. It is occasionally useful, instead, to examine data that consist of complex numbers.
- 12.
See Mitra and Pesaran (1999), Percival and Walden (1993), and Thomson (1982).
- 13.
By pointwise we mean that at any given frequency \(\omega \) the bands would provide an approximate 95 % confidence interval. An alternative is to compute approximate simultaneous confidence bands, meaning bands that provide approximate 95 % confidence simultaneously for all \(\omega \). This may be accomplished with a suitable adaptation of the algorithm.
- 14.
One helpful fact is that an average coherence across a given frequency band may be shown to be equal to the complex-valued correlation between band-pass filtered versions of the two series; see Ombao and Vanbellegem (2008).
- 15.
Actually, they reported \(R^2\) between stimulus-based impulse response functions (see p. 544) found from the LFP and BOLD signals.
- 16.
Here \(\sigma _{Y|Y}^2\) is a constant; the notation is intended only to indicate that it is the error variance when \(Y\) appears on both the left-hand side and the right-hand side of the model.
- 17.
In addition, Geweke (1982) defined a spectral measure \(f_{X \rightarrow Y}(\omega )\) representing the \(\omega \)-component of Granger causality in the sense that
$$ F_{X\rightarrow Y} =\int _{-\frac{1}{2}}^{\frac{1}{2}} f_{X \rightarrow Y}(\omega )d\omega . $$ - 18.
They used the spectral decomposition mentioned in the footnote on p. 559 to plot the frequency representation of Granger causality, found its peak, and performed a permutation test analogously to what they had done in analyzing coherence.
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Time Series. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_18
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