Wavelet Spectra for Multivariate Point Processes

Alexander J. Gibberd; Edward A.K. Cohen

REVIEW 3 major objections 3 minor 21 references

Wavelet Spectra for Multivariate Point Processes

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A temporally smoothed wavelet periodogram of a multivariate point process is asymptotically a scaled complex Wishart matrix, giving closed-form coherence distributions and a chi-square test for non-stationarity.

desk verdict The eigen-wavelet representation is a real advance, but the asymptotic Wishart result in Theorem 2 is a moment-matched approximation, not a genuine limiting distribution. read the letter →

arxiv 1908.02634 v3 pith:LIE2IOTO submitted 2019-08-07 stat.ME

Edward A.K. Cohen , Alexander J. Gibberd This is my paper · ORCID

classification stat.ME MSC 62M1560G5542C4062H10

keywords waveletperiodogrammultivariatepointprocessHawkesmultitaperWishartdistributioncoherencestationaritytestneuralspiketrain

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to put wavelet spectral analysis of multivariate point processes on the same statistical footing that Fourier and multitaper methods enjoy for stationary time series. It shows that a wavelet periodogram of event data, smoothed across time with a kernel, is exactly a weighted average of periodograms formed from an orthonormal eigen-wavelet system, and that under stationarity this average is asymptotically a scaled complex Wishart matrix. That distribution gives closed-form thresholds for wavelet coherence and a likelihood-ratio test, with $\chi^2$ asymptotics, for whether the process is stationary at a given scale. The development matters because it makes statements about time-varying correlation in neural spike trains testable from a single trial.

What carries the argument

The load-bearing object is the Hermitian kernel $K(s,t)=\int h_\kappa(u)\psi(s-u)\psi^*(t-u)\,du$ formed from the wavelet $\psi$ and smoothing window $h$, together with its Mercer eigen-decomposition into eigen-wavelets $\{\phi_l\}$ and eigenvalues $\{\eta_l\}$. The identity $\sum_l \eta_l |\Phi_l(f)|^2 = |\Psi(f)|^2$ preserves the wavelet's frequency response across the channel bank. This machinery converts temporal smoothing into a finite, in practice, number of approximately independent wavelet channels, so that the smoothed periodogram becomes a weighted sum of Wishart matrices; the weights $\eta_l$ directly set the effective degrees of freedom $n=1/\sum_l \eta_l^2$.

What would settle it

Run a stationary bivariate Hawkes or Poisson process through the estimator and compare the empirical distribution of the smoothed periodogram, or the coherence, against the claimed $(1/n)\mathrm{WC}_p$ law with $n$ taken from the kernel eigenvalues; a statistically significant deviation, or a failure of the empirical covariance of $w^{(T)}$ and its conjugate to decay to zero as $T$ grows, would refute Theorem 2.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a distributional equivalence: for a $p$-dimensional stationary point process with spectral density matrix $S(f)$, the temporally smoothed wavelet periodogram $\Omega^{(T)}(\tilde a,\tilde b)$ is asymptotically $(1/n)\,\mathrm{WC}_p\{n, S(f_{\tilde a})\}$ as $T\to\infty$, where $n=1/\sum_l \eta_l^2$ is computed from the eigenvalues of the kernel $K(s,t)=\int h_\kappa(u)\psi(s-u)\psi^*(t-u)\,du$. The proof runs through the Mercer expansion $K(s,t)=\sum_l \eta_l \phi_l(s)\phi_l^*(t)$, which makes the smoothed periodogram a weighted sum of eigen-wavelet periodograms, each asymptotically complex normal by a cumulant argument under the paper's mixing condition. The Wishart law then yields a closed-form density for temporally smoothed wavelet coherence (Corollary 1) and, through a likelihood-ratio construction, an asymptotic $\chi^2$ test for stationarity at dyadic scales (Theorem 3).

Load-bearing premise

The argument's load-bearing premise is that the complex wavelet used is analytic, meaning it has no response at negative frequencies; the paper states this as orthogonality to its own complex conjugate, a condition that cannot hold for a unit-norm wavelet, and the Morlet wavelet used in the applications only satisfies the intended condition approximately.

Editorial extensions

If this is right

Wavelet coherence of point processes can be thresholded without bootstrap or permutation: the 95% contour follows from Corollary 1 with $n$ taken from the kernel eigenvalues.
The likelihood-ratio statistic gives a scale-by-scale stationarity test with known $\chi^2$ degrees of freedom, and the scale-specific tests combine into a global test.
The degrees of freedom grow linearly with the smoothing width $\kappa$ for a rectangular window (Proposition 4), so users can trade time resolution against test power explicitly.
Real-valued wavelets such as the Mexican hat inherit the framework with a real Wishart limit and a Beta null coherence distribution, so the same calibration applies there.
In applications to neural spike trains, the test can flag changes in cross-stream correlation over short single-trial recordings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the identity $\Omega=\sum_l \eta_l v_l v_l^H$ suggests that the same $n=1/\sum_l\eta_l^2$ calibration should apply to any non-negative definite, trace-one smoothing kernel, not only the rectangular-window examples; testing a different $h$ would be a direct check.
Extension: the paper tests each dyadic scale separately, but the rejection information across scales could be combined into a change-point localizer for when the dependence structure changes, something the authors do not attempt.
Extension: because the Morlet wavelet is not exactly analytic, the circular-complex-normal step holds only approximately; a finite-sample study of how far the Wishart approximation drifts with the size of $\int\psi(t)^2\,dt$ would quantify the practical gap.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is genuinely nice. Rewriting the temporally smoothed wavelet periodogram as a weighted sum of eigen-wavelet periodograms (S5) is elegant, and it opens a practical route to computation via the Nyström method. The paper is also honest about the machinery: the proofs rely on Brillinger's cumulant conditions, the simulations are serious, and there is code. That is real value.

The problem is Theorem 2. The authors claim the smoothed periodogram is asymptotically (1/n)WC_p(n, S) with n = 1/Ση_l². The proof says this follows from asymptotic independence of the eigen-wavelet coefficients, in analogy to Walden (2000). But it doesn't. For independent v_l ~ CN(0,S), each v_l v_l^H is WC_p(1,S), but Ση_l v_l v_l^H is a weighted sum of Wisharts. A weighted sum of independent Wisharts is only Wishart when all weights are equal. Matching the mean and covariance fixes n = 1/Ση_l², but that is Satterthwaite-style moment matching, not an asymptotic law. The η_l are fixed by the kernel as T→∞, so the discrepancy between the true distribution of the weighted sum and the claimed Wishart does not vanish. The stated asymptotic distribution is therefore not a limit. Corollary 1's coherence density and the likelihood-ratio stationarity test inherit this mismatch.

This is not a nitpick. The Wishart result is load-bearing for the testing framework. The simulations still look reasonable, because the eigenvalues decay quickly and the moment-matched approximation can be decent, but that doesn't make the theorem correct as written.

The other issues are smaller. Assumption 4 as printed requires ∫ψψ* = 0, which is impossible since ||ψ||=1; the intended condition is presumably ∫ψψ = 0. Easy typo, but it appears in the statement of a key assumption. The proof of Theorem 2 is also a sketch: the independence step is asserted through Lemma 2 and a hand-wave to Walden. And calling p=0.032 at one of several scales "strong evidence" ignores the multiple comparisons; minor but should be softened.

Who is this for? People analyzing multivariate event data, especially neural spike trains, will find the framework useful in principle. The paper deserves a serious referee, but not because the theorem is correct as stated. It deserves one because the framework is good enough to fix. The authors should be pushed to either prove a real limit (which I doubt is possible) or, much more realistically, present the Wishart law as an explicit approximation with error bounds, and recalibrate the stationarity test and coherence thresholds accordingly. That is a substantial revision, but the core contribution can survive it. Send it to peer review, but the referee should insist on this change before acceptance.

Referee Report

3 major / 3 minor

Summary. The paper develops a temporally smoothed wavelet periodogram for multivariate point processes, expresses it as a weighted sum of eigen-wavelet periodograms, and derives asymptotic distributional results. Under stationarity, Theorem 1 claims the wavelet transform is asymptotically circular complex normal, and Theorem 2 claims the smoothed periodogram is asymptotically a scaled complex Wishart with degrees of freedom n = 1/Σ_l η_l² determined by the eigenvalues of the smoothing kernel. From Theorem 2 the authors derive the distribution of wavelet coherence (Corollary 1) and construct a likelihood-ratio test for stationarity across scales (Theorem 3). The paper includes simulation support, a real neural spike-train analysis, and a GitHub implementation. The main novelty is the eigen-wavelet representation and the attempt to extend multitaper-style Wishart asymptotics to point-process wavelet spectra.

Significance. If Theorem 2 and its corollaries were correct, the paper would provide a practically useful and theoretically motivated inferential framework for wavelet coherence and stationarity testing in point-process data, extending earlier work by Cohen and Walden from time series to point processes. The eigen-wavelet representation in Section 3 is elegant and Proposition 2 is a clean and useful result. The authors also provide simulations, a real-data example, and reproducible code, which are strengths. However, the central distributional claim is not valid as stated: the weighted sum of independent complex Wisharts that arises in the eigen-wavelet representation is not itself a scaled complex Wishart when the eigenvalue weights are unequal, and the weights are fixed as T→∞. Since Corollary 1 and Theorem 3 inherit this claim, the paper's main inferential results do not follow. The manuscript is therefore not publishable in its current form, although the underlying ideas may be salvageable if reframed as an approximation with explicit error control.

major comments (3)

[Section 4.2, Theorem 2 and its proof in Supplementary Material Section 1] The proof of Theorem 2 does not establish the stated Wishart limit. From Eq. (S5) and Lemma 3, the asymptotics give Ω(T)(ã,b̃) ≈ Σ_l η_l v_l v_l^H with the v_l asymptotically independent CN_p(0,S). Each term v_l v_l^H is WC_p(1,S), but a weighted sum of independent complex Wisharts with unequal weights is not a scaled WC_p(n,S); a WC_p(n,S) distribution corresponds to an unweighted average of n i.i.d. rank-one terms. Choosing n = 1/Σ_l η_l² matches the mean and a certain covariance adjustment, but this is the Satterthwaite/Box equivalent-degrees-of-freedom approximation, not an exact asymptotic distribution. The mismatch does not vanish as T→∞ because the eigenvalues η_l are fixed. For example, for p=1 with weights (0.9,0.1), the MGF of Σ η_l Exp(1) at t=1 is 1/{(1−0.9)(1−0.1)} ≈ 11.1, while the claimed (1/n)WC_p(n,S) MGF is (1 − (1/n)t)^{-n} ≈ 8.0 at t=1. The citation to Walden (2000, p. 776) concerns unweighted multitaper averaging and does not cover unequal eigenvalues. This issue directly invalidates Corollary 1 and the distributional statements used in Theorem 3 and Proposition 7.
[Section 4.1, Assumption 4; also Lemma 2 and Theorem 1 proof in Supplementary Material] Assumption 4 as printed is internally inconsistent: it states that ψ is orthogonal to its complex conjugate, i.e. ∫ ψ(t)ψ*(t)dt = 0, but Assumption 1 requires ∫|ψ(t)|²dt = 1, so this condition is impossible. The proofs actually require the different condition ∫ ψ(t)ψ(t)dt = 0; see Lemma 2, where ψ1 = ψ*(T) and ψ2 = ψ(T) are used to conclude ∫ ψ*(T)(t)ψ*(T)(t)dt = 0. This stronger condition is not satisfied exactly by the Morlet wavelet used throughout the paper. If it fails, cov{w(T), w*(T)} does not vanish asymptotically, so the limit is not circular complex normal and the complex Wishart claim in Theorem 2 does not follow. The authors should either restrict to wavelets for which the intended condition holds exactly, or quantify the effect of its violation on the limiting distribution.
[Theorem 2 proof and Proposition 5 proof, Supplementary Material Section 1] The statement that the eigen-wavelet coefficients v_l are asymptotically independent is only sketched. Lemma 2 bounds a second-order cumulant between two wavelet transforms with orthogonal wavelets, but asymptotic joint normality of the infinite family {v_l} requires control of all higher-order joint cumulants, or a suitable infinite-dimensional CLT. The phrase 'Lemma 2 combined with a similar argument to Theorem 1' does not supply this argument. This matters for Theorem 2 because the weighted-sum law in the preceding comment presupposes independence of the v_l; it also matters for Proposition 5, where independence of the periodograms at different dyadic locations is asserted from non-overlap of kernels. An explicit proof, or a reference that covers joint asymptotic normality of wavelet coefficients in point-process settings, is needed.

minor comments (3)

[Main text Theorem 3 vs. Supplementary Material Theorem 3] The main text states 0 < c < 1/2, while the supplementary statement of Theorem 3 says 0 < c < 1. Since β = min{c, 1/2 − c} is used, the condition c < 1/2 is required; the supplementary statement should be corrected for consistency.
[Supplementary Material, proof of Proposition 1] In the proof of Proposition 1, the sentence 'Property (iii) is immediate from Proposition 1 and the fact that ψ(t) itself obeys the admissibility condition' appears to contain a typo: it should cite Proposition 2, not Proposition 1.
[Section 2.2, approximating wavelet] The paper states that ψ(t) is used to denote both the original and the approximating wavelet with support (−α/2, α/2). Since several theoretical results rely on the approximating support, the dependence of the asymptotics on α should be made explicit in the statements of Proposition 4 and Theorem 2, rather than only in the informal discussion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: derivation is self-contained; the cited self-work is contextual and not load-bearing.

full rationale

No circular step is exhibited. The temporally smoothed wavelet periodogram is expanded via Mercer's theorem into a weighted eigen-wavelet sum (Eq. S5), and the asymptotic normality of each eigen-wavelet transform (Theorem 1) is proved from Brillinger's cumulant mixing conditions (Assumption 3) and an external CLT argument; the proof of Theorem 2 then combines these with a Wishart argument cited to Walden (2000). The degrees of freedom n = 1/Ση_l^2 is computed from the kernel eigenvalues, not fitted to data, and the centrality matrix S(f̃_a) is a parameter of the assumed stationary model, not estimated from the data being predicted. The self-citations to Cohen and Walden (2010a,b) appear only in the introduction as background and in the definition of central frequency; they do not carry the load of Theorem 2, Corollary 1, or Theorem 3. The Assumption 4 orthogonality condition is misprinted (∫ψψ* dt = 0 contradicts ||ψ||=1) and the intended analytic-wavelet condition is used in the proof; this is a correctness issue, not a circularity. The skeptic's point that a weighted sum of independent complex Wisharts is not exactly Wishart with unequal weights is also a correctness/approximation concern (the n chosen by moment matching gives only an equivalent-dof approximation), not a circularity, because the claimed distribution is not used as an input to derive itself.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard wavelet assumptions, a strong but standard mixing condition for point processes, and an orthogonality condition that is mistyped and only approximate for the wavelet used. No free parameters are fitted to data; κ, α and c are user choices.

free parameters (3)

Smoothing width scaling κ = κ = 10 in the real data example; no fitted value
User-chosen smoothing parameter. The Wishart degrees of freedom n = 1/Σ η_l^2 depends on κ through the kernel, so all downstream distributional thresholds depend on this choice.
Wavelet approximating support α = α = 4 (real data example); α = 8 suggested for Morlet; no fitted value
Truncation of the infinite-support wavelet. Affects the valid region T_{α,κ,T} and the condition κ > α in Proposition 4. Chosen by hand.
Stationarity test scale parameter c = c = 1/4 optimal in Theorem 3; no fitted value
Controls the smoothing width κ̃ = κT^c in the stationarity test. The convergence rate β = min{c, 1/2−c} depends on it.

assumptions (8)

domain assumption Assumption 1: wavelet ψ satisfies ∫ψ = 0, ||ψ|| = 1, admissibility ∫ f^{-1}|Ψ(f)|^2 df < ∞.
Standard wavelet conditions invoked throughout Section 2.
domain assumption Assumption 2: smoothing function h(t) is non-negative, symmetric, continuous on (−1/2,1/2), normalized to integrate to 1.
Defines the temporal smoothing in (S3) and the kernel K in (S4).
domain assumption Assumption 3: the point process is strictly stationary with all moments, jointly integrable cumulants, and ∫|u||r(u)|du < ∞ (Brillinger mixing condition).
Sufficient for the central limit theorem underlying Theorem 1 and hence the Wishart result in Theorem 2.
ad hoc to paper Assumption 4: complex wavelet orthogonal to its own conjugate (∫ψ(t)ψ(t)dt = 0 as used in the proof; printed with a typo as ∫ψ(t)ψ*(t)dt = 0).
This condition forces the limiting normal to be circular, which is required for the complex Wishart law. As printed the assumption is false; the intended condition holds only approximately for the Morlet wavelet.
domain assumption Assumption 5: h satisfies a Lipschitz-type condition ∫|h(t+u)−h(t)|dt ≤ C'|u|.
Used in Lemma 3 and Proposition 5 for the eigen-wavelets and independence of dyadic blocks.
standard math Mercer's theorem for the Hermitian kernel K(s,t).
Gives the eigen-wavelet expansion (S5) and Proposition 2.
standard math Brillinger (1972) cumulant expansion for stochastic integrals (Lemma 1).
Used in the proof of Theorem 1 to bound higher-order cumulants.
standard math Goodman (1963) distribution of coherence of complex Wishart matrices.
Needed for Corollary 1.

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Cite this review

Pith. "Pith review of Wavelet Spectra for Multivariate Point Processes." pith.science (2026). https://pith.science/paper/LIE2IOTO

@misc{pith2026190802634,
  author       = {Pith},
  title        = {Pith review of: Wavelet Spectra for Multivariate Point Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LIE2IOTO}},
  note         = {Machine review of arXiv:1908.02634}
}

read the original abstract

Wavelets provide the flexibility to analyse stochastic processes at different scales. Here, we apply them to multivariate point processes as a means of detecting and analysing unknown non-stationarity, both within and across data streams. To provide statistical tractability, a temporally smoothed wavelet periodogram is developed and shown to be equivalent to a multi-wavelet periodogram. Under a stationary assumption, the distribution of the temporally smoothed wavelet periodogram is demonstrated to be asymptotically Wishart, with the centrality matrix and degrees of freedom readily computable from the multi-wavelet formulation. Distributional results extend to wavelet coherence; a time-scale measure of inter-process correlation. This statistical framework is used to construct a test for stationarity in multivariate point-processes. The methodology is applied to neural spike train data, where it is shown to detect and characterise time-varying dependency patterns.

Figures

Figures reproduced from arXiv: 1908.02634 by the authors.

**Figure 2.** The kernel and first five eigen-wavelets for the Morlet wavelet (panels a,b) and [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Wavelet coherence estimated from the mouse neuron firing data with the Morlet [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 1.** Figure 1: Asymptotic probability density functions for temporally smoothed wavelet co [PITH_FULL_IMAGE:figures/full_fig_p032_1.png]

**Figure 2.** Figure 2: QQ plots for empirical quantiles of the wavelet transform against the theoretical [PITH_FULL_IMAGE:figures/full_fig_p036_2.png]

**Figure 3.** Figure 3: Spectral coherence ρ 2 (f) for the bivariate mutually exciting Hawkes process described in the text. 18 [PITH_FULL_IMAGE:figures/full_fig_p037_3.png]

**Figure 4.** Figure 4: QQ plots for empirical quantiles of the temporally smoothed wavelet coherences [PITH_FULL_IMAGE:figures/full_fig_p038_4.png]

**Figure 5.** Figure 5: a. Temporally smoothed wavelet coherence estimated from a single realisation [PITH_FULL_IMAGE:figures/full_fig_p039_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Bartlett, M. S. (1963). The spectral analysis of point processes. Journal of the Royal Statistical Society. Series B\/ 25\/ (2), 264--296

work page 1963
[2]

Brillinger, D. R. (1972). The spectral analysis of stationary interval functions . Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics\/ , 483--513

work page 1972
[3]

Brillinger, D. R. (1996). Some uses of cumulants in wavelet analysis . Journal of Nonparametric Statistics\/ 6 , 93--114

work page 1996
[4]

Carter, G. (1987). Coherence and time delay estimation . Proceedings of the IEEE\/ 75\/ (2), 236--255

work page 1987
[5]

Cohen, E. A. K. and A. T. Walden (2010a). A statistical analysis of Morse wavelet coherence . IEEE Transactions on Signal Processing\/ 58\/ (3 PART 1), 980--989

work page 2010
[6]

Cohen, E. A. K. and A. T. Walden (2010b). A statistical study of temporally smoothed wavelet coherence. IEEE Transactions on Signal Processing\/ 58\/ (6), 2964--2973

work page 2010
[7]

Goodman, N. (1963). Statistical analysis based on a certain multivariate complex G aussian distribution (an introduction). Annals of Mathematical Statistics\/ 34\/ (1), 152--177

work page 1963
[8]

Moore, and S

Grinsted, A, J., C. Moore, and S. Jevrejeva (2004). Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics\/ 11\/ (5), 561--566

work page 2004

Show all 21 references

[9]

Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika\/ 58\/ (1), 83--90

1971
[10]

Kythe, P. K. and P. Puri (2001). Computational Methods for Linear Integral Equations . Springer

2001
[11]

Mallat, S. and G. Peyr\' e (2008). A Wavelet Tour of Signal Processing \/ (3 ed.). Elsevier Science and Technology

2008
[12]

Mercer, J. (1909). Functions of positive and negative type, and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences\/ 209\/ (441-458), 415--446

1909
[13]

Olhede, S. C. and A. T. Walden (2002). Generalized M orse wavelets. IEEE Transactions on Signal Processing\/ 50\/ (11), 2661--2670

2002
[14]

Tang, S., S. C. Ardila Jimenez , S. Chakraborty, and S. R. Schultz (2015). Visual receptive field properties of neurons in the mouse lateral geniculate nucleus . Plos One\/ 11\/ (1), 1--34

2015
[15]

Thomson, D. J. (1982). Spectrum estimation and harmonic analysis. Proceedings of the IEEE\/ 70\/ (9)

1982
[16]

Torrence, C. and P. Webster (1999). Interdecadal changes in the ESNO -monsoon system. Journal of Climate\/ 12 , 2679--2690

1999
[17]

Walden, A. T. (2000). A unified view of multitaper multivariate spectral estimation . Biometrika\/ 87\/ (4), 767--788

2000
[18]

Welch, P. (1967). The use of fast F ourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Transactions on Audio Electroacoustics\/ 15 , 70--73

1967
[19]

Barndorff-Nielsen, O. E. and D. R. Cox (1989). Asymptotic Techniques for Use in Statistics . Chapman and Hall

1989
[20]

Fisher, R. A. (1928). The general sampling distribution of the multiple correlation coefficient. Proceedings of the Royal Society Series A\/ 121\/ (788)

1928
[21]

Muirhead, R. J. (1985). Aspects of Multivariate Statistical Theory \/ (2 ed.). John Wiley & Sons, Inc

1985

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bartlett, M. S. (1963). The spectral analysis of point processes. Journal of the Royal Statistical Society. Series B\/ 25\/ (2), 264--296

work page 1963

[2] [2]

Brillinger, D. R. (1972). The spectral analysis of stationary interval functions . Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics\/ , 483--513

work page 1972

[3] [3]

Brillinger, D. R. (1996). Some uses of cumulants in wavelet analysis . Journal of Nonparametric Statistics\/ 6 , 93--114

work page 1996

[4] [4]

Carter, G. (1987). Coherence and time delay estimation . Proceedings of the IEEE\/ 75\/ (2), 236--255

work page 1987

[5] [5]

Cohen, E. A. K. and A. T. Walden (2010a). A statistical analysis of Morse wavelet coherence . IEEE Transactions on Signal Processing\/ 58\/ (3 PART 1), 980--989

work page 2010

[6] [6]

Cohen, E. A. K. and A. T. Walden (2010b). A statistical study of temporally smoothed wavelet coherence. IEEE Transactions on Signal Processing\/ 58\/ (6), 2964--2973

work page 2010

[7] [7]

Goodman, N. (1963). Statistical analysis based on a certain multivariate complex G aussian distribution (an introduction). Annals of Mathematical Statistics\/ 34\/ (1), 152--177

work page 1963

[8] [8]

Moore, and S

Grinsted, A, J., C. Moore, and S. Jevrejeva (2004). Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics\/ 11\/ (5), 561--566

work page 2004