Wavelet Canonical Coherence for Nonstationary Signals
Authors: Haibo Wu
Marina I. Knight
Keiland W. Cooper
Norbert J. Fortin
Hernando Ombao
Venue: NeurIPS 2025 spotlight ()
Keywords: Multivariate time series
Neural signal analysis
Nonstationarity
Wavelet canonical coherence
Primary area: neuroscience_and_cognitive_science
TL;DR: A novel method to measure the time-varying global association between two clusters of time series

Abstract: Understanding the evolving dependence between two sets of multivariate signals is fundamental in neuroscience and other domains where sub-networks in a system interact dynamically over time. Despite the growing interest in multivariate time series analysis, existing methods for between-clusters dependence typically rely on the assumption of stationarity and lack the temporal resolution to capture transient, frequency-specific interactions. To overcome this limitation, we propose scale-specific wavelet canonical coherence (WaveCanCoh), a novel framework that extends canonical coherence analysis to the nonstationary setting by leveraging the multivariate locally stationary wavelet model. The proposed WaveCanCoh enables the estimation of time-varying canonical coherence between clusters, providing interpretable insight into scale-specific time-varying interactions between clusters. Through extensive simulation studies, we demonstrate that WaveCanCoh accurately recovers true coherence structures under both locally stationary and general nonstationary conditions. Application to local field potential (LFP) activity data recorded from the hippocampus reveals distinct dynamic coherence patterns between correct and incorrect memory-guided decisions, illustrating capacity of the method to detect behaviorally relevant neural coordination. These results highlight WaveCanCoh as a flexible and principled tool for modeling complex cross-group dependencies in nonstationary multivariate systems. Code for implementing WaveCanCoh is available at https://github.com/mhaibo/WaveCanCoh.git.

Wavelet Canonical Coherence for Nonstationary Signals
Haïbo Wu! Marina I. Knight
Keiland W. Cooper Norbert J. Fortin Hernando Ombao
Statistics Program, King Abdullah University of Science and Technology Department of Mathematics, The University of York DDepartment of Neurobiology and Behavior, University of California, Irvine *Cetter for the Neurobiology of Learning and Memory, University of California, Irvine
Abstract
Understanding the evolving dependence between two sets of multivariate signals is fundamental neuroscience and other domains where sub-networks sys- tem interact dynamically over time. Despite the growing interest in multivariate time series analysis, existing methods for between-clusters dependence typically rely on the assumption of stationarity and lack the temporal resolution capture transient, frequency-specife interactions. To overcome this limitation, we propose scale-specific wavelet canonical coherence (WaveCanCoh), novel framework that extends canonical coherence analysis to the nonstationary setting by leveraging the multivariate locally stationary wavelet model. The proposed WaveCanCofl enables the estimation of time-varying canonical coherence between clusters, pro- viding interpretable insight into scale-specific time-varying interactions between clusters. Through extensive simulation studies, we demonstrate that WaveCanCoh accurately recovers true coherence structures under both locally stationary and general nonstationary conditions. Application to local field potential (LFP) activity data recorded from the hippocampus reveals distinct dynamic coherence patterns between correct and incorrect memory-guided decisions, illustrating the capacity the method detect behaviorally relevant neural coordination. These results highlight WaveCanCoh cross-group dependencies in nonstationary multivariate systems. Code for imple- menting WaveCanCoh is available https://github.c wwwm/maabb//aaeeaant
flexible and principled tool for modeling complex
Introduction
Assessing the dependence structure between node clusters in network is of the most critical aspects of network time series analysis. Many models and frameworks have been developed to capture between-clusters association (e.g., correlation, coherence, and causality). Most existing methods characterize the dependence between two clusters through the dependence between (many) node pairs. However, in many scenarios, the primary interest lies understanding the dependence structure between two groups of multivariate time series rather than individual processes. Figure [ Iillustrates this perspective using brain activity signals. In this experiment, local field potential (LFP) activity was recorded from multiple electrodes implanted in different subregions of the hippocampus of rodents (rats) as they performed a complex sequence memory task. Instead of focusing on coherence between individual channels (electrodes), the main goal is to quantify time-varying functional interactions between groups of electrodes in order to understand how information processing differs in these two
*Corresponding author: haibo. wu@kaust. edu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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subregions. Similar challenges arise in other domains. For instance, in finance, understanding the dependence between entire market sectors (e.g., technology and energy) can be more informative than analyzing associations between individual stocks. These scenarios require framework capable of capturing dynamic coherence between sets of nonstationary multivariate signals. In this paper, we propose a novel framework called scale-specific wavelet canonical coherence (WaveCanCoh) to characterize time-localized and scale-specific coherence between two clusters of multivariate time series. By leveraging the time-frequency localization properties of wavelets, WaveCanCoh is well-suited for analyzing nonstationary multivariate signals in neuroscience, finance, and other fields where transient, cross group interactions of scientific interest.
Odors
Cidor
Tradd
Figure I: Schematic overview of the experimental and analytical motivation. Local field potential (LFP) activity was recorded from multiple electrodes in the hippocampus as rats performed an odor sequence memory task. Rather than focusing on individual electrode pairs, our goal is to characterize the dynamic dependence between two hippocampal subregions, each represented by a group of electrodes, using savelct-based multivariate coherence.
Canonical variate analysis (CVA) (44 provides method for measuring the correlation between two vector variables, its application to time series data having started in the 1950s and 1960s, primarily in the econometrics and signal processing fields. B and [10] extend CVA to time series for forecasting and causality detection. 4 provides a spectral domain formulation of canonical correlation, useful for frequency-domain time series analysis. This approach enables the analysis of canonical coherence between two sets of time series across different frequency bands. Many modern studies have been developed within this framework (e.g.. (24 and (25)), and related methods have been widely applied across various fields, including neuroscience, finance, speech processing, and machine learning.
However, previous methods rely on the assumption that time series are (weakly) stationary, meaning their statistical properties (e.g., expectation and covariance, spectral signature) remain constant over time. In practice, this assumption often does not hold for time series that arise in practice. Moreover, two sets of time series commonly exhibit time- vane-varingg global coherence, which can sometimes be crucial for analysis. Thus, a method capable of handling nonstationary time series is necessary. Wavelet analysis is a widely used tool for studying nonstationary time series, as its localization property allows for the examination of localized correlations between two time series across both time and frequency domains. Wavelets are particularly effective for capturing transient properties of nonstationary signals [3] due to their compact support, which can be compressed or stretched to adapt to the dynamic characteristics of the signal. Wavelet coherence has been well-defined and extensively studied previous research, with applications spanning various fields coherence in multivariate systems or pairwise coherence between univariate channels [5]6], and several wavelet- based connectivity approaches have been proposed for fMRI (23] and MEG [ To the best of our knowledge, no existing work has extended classical canonical coherence to the wavelet domain to measure the time-evolving canonical coherence between two groups of multivariate time series.
[9 [2] However, prior studies primarily focus on
The key novelty of this paper lies in the development of a comprehensive and rigorous framework based on wavelets for measuring canonical coherence between two sets of nonstationary multivariate time series. Specifically, our main contributions include: (1) we define scale-specific wavelet canonical coherence (WaveCanCoh) and introduce its use as a tool to quantify the coherence between two sets of multivariate time series, (2) we provide a complete and theoretically justified algorithm for its estimation, and (3) apply our method to LFP activity data from multiple electrodes to

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quantify dynamic interaction patterns among different subregions in the hippocampus. Multivariate locally stationary wavelet processes (MvLSW) underpin our construction, and the reader is directed to [9 and [21] for details on their construction. Our framework not only captures the time-varying coherence between two sets of signals but also determines the contribution of each individual channel to the global coherence. Compared to previous models, the proposed approach provides a detailed, localized characterization of interactions within the multivariate time series. Our findings on the LFP activity data offer new insights into the functional relationship between hippocampal subregions, demonstrating the potential of our method to advance the study of functional brain connectivity.
The format of the paper is as follows. Section Ditiiviiowervwwww the current methodology for assessing time series canonical coherence. 3 a brief overview of MvLSW processes, supporting a detailed introduction to our proposed WaveCanCoh framework. Its estimation, and that of related parameters, is tackled in Section [ Section 5
validates the proposed framework and demonstrates its performance through simulation. In Section 6 we apply the WaveCanCoh method on LFP activity data collected from rats to investigate the dynamic interactions of different subregions in the hippocampus during memory tasks. Section concludes the paper, with the Appendix offering further theoretical and empirical supporting information, as well as reflections on the method's limitations in Appendix F
Related works
First, we provide a brief introduction to classical canonical correlation analysis for time series, with the primary goal to characterize dependence between two clusters of time series, where each cluster fea- tures several nodes. In particular, we consider two multivariate time series with dimensions p and g re-
spectively, denoted by X, = (x(") .'""....."""") T and Y, = (x1") ...... v.!)) for t = {.,..., T). Typically, } and {Y,) are assumed to be aero-mean weakly stationary time series. Letting Z, ,...., y!"), ...., y-(0)
x. x(0), denote the concatenated (p + q) dimension time series, Ze (x(")...,, its covariance matrix at lag 7 is
Ezz(T)- = ( (xx(T) Exy(7)) Exx(T) 2
where Exx(-), (vr()) are the autocovariance matrices of {X.} and (Y.) respectively, and Exv(). (-), Exx(+) (+) are their cross-covariances. The canonical correlation between {X+} and {Y,) at lag T,
p(T), is defined as
a T (T)
((T) max n,b
(1)
(Txx()aabb ETy(T)b'
where a E IP' and b E IR are called canonical correlation vectors, subject to standardized constraints T Σxxa b. Eyyb [4. Thus, the canonical correlation can be rewritten as
p(T) max a.b (a (T)())).
(2)
Remark 1: In the preceding definitions for the cross-covariance matrices, we have Exy(T) =
Σxx( (-T) G-x(-T) = EE [X,YT=r), which measure the lagged covariance between [X.) and (Y,). i.e past values of may be associated to present values of for O. and vice-versa when T < 0. The case of T = 0 illustrates contemporaneous relationships.
The solution for and b in equation (1)
be obtained from the eigenvectors of the following matrices, respectively (in what follows, 7 is dropped for brevity),
E Exy Eyy Exx. and E Eyx Exx
(3)
(4)
Here, a and b are the eigenvectors corresponding to the largest eigenvalue, X, of matrices 3 and
respectively, and for largest canonical correlation coefficient we have p vā [17. The framework above yields the canonical correlation between (X,) and (Y,). However, in many practical cases,
such EEG analysis, canonical correlation in the spectral domain is more meaningful than in the
3

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time domain, as components at different frequencies (or scales) reveal crucial information about neural dynamics and functional connectivity (0)] 4 extends the time-domain canonical correlation
into the spectral domain). Namely, suppose the spectral matrix of Ze = (x, Y,) is
Fxx(w) Bxy(w) fYx(w) fyr(w)
fzz(w)
where (xx(w) is the p p autospectral matrix of (X,), fYY (w) is the q x q autospectral matrix of (Yr), and (w) is the p q cross-spectral matrix between (X+) and {Y,). Given vectors a E C and b E C such that Fxx(w)a = b
= 1, the canonical coherence at frequency w is
a fFY((W) [xx(w)ay b fyr(w)b
p(w) = max a.b
By solving the maximization problem in equation (5), the canonical coherence vectors and b are determined, leading to the quantification of the canonical coherence at frequency w.
Note the classic canonical coherence in (5) completely ignores temporal dynamics, consequence of the stationarity assumption where dependence between clusters imposed remain constant over time, whilst most real-world data, such as EEG, exhibit nonstationarity [15 [6] 26). Hence the lack of time-localization information in the above approach may result in misleading results and novel method capable of capturing timevarryin canonical coherence is needed.
3 Wavelet canonical coherence (WaveCanCoh)
Our WaveCanCoh framework is built upon the multivariate locally stationary wavelet (MvLSW) process (I19), (21), which is a model based on wavelet analysis for time series. A brief overview of wavelets and highlight their differences from Fourier-based methods are provided in Appendix A
A new representation for discretely sampled nonstationary time series based on discrete non non-decimated wavelets is the locally stationary wavelet process introduced by [19], later extended to multivariate framework in (2I]. A P-variate stochastic process with time evolving second-ordes structure. X, =
(x() x(2)
x(")
can be represented with the MvLSW formulation where 1,..., T, T..,,,T,
06
X, ( j=
where V (k/T) is a P x transfer function matrix assumed to have a lower-triangular form; ( j.k a set of discrete non-decimated wavelets; {zy,A,5, j. is set of P x uncorrelated random vectors with (column) mean vector 0 and P P identity covariance matrix. Since the wavelet basis Vy (t) is localized in both time and frequency. the transfer matrix V, (k/T) provides measure of the time-varying contribution the variance among channels a specific scale j and rescaled time k/T, thus enabling the statistical properties of the process (X,} } to change smoothly over time.
The time- varying statistical properties of {X,) can be captured through the localized, scale-specific local wavelet spectral matrix (LWS, 21)). ), Sy (u), defined at scale j and rescaled time u E (0, 1), as
Sy(w) V (n).
Note (u) is Px P symmetric, positive semi-definite matrix and its (p, q) entry, S(P-0) (u), denotes the cross-spectrum between channels and q. We now extend the LWS matrix construction from a sin-
gle set of multivariate time series to cross-group LWS matrix, between X, (x(1) -xx,",,....
x!!))
and Y, (r, ...., ((
Denoting Z. = (XT YT) T, the LWS matrix of { at scale
and rescaled time u, Sjzzz(() (u), is
Syzz(u) (u) = V Viz(u)VJzzz) = S) S))x(() ()) (

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In equation 7p. V v:z(2) denotes the (P+ Q) (P + Q) transfer function matrix of the MvLSW process {Z, and Jzzz (u) is its corresponding LWS matrix. The main diagonal blocks Sj,x (u) (P x P) and S j: vY (Q x Q) denote the auto-LWS matrices of the (X,) and (Y,) processes, respectively, while x (u) and Sj,Yx (u) denote their cross-LWSS matrices.
Remark 2: In equation (7. S ()) is a P x Q matrix at each time point, and the (p.g) element gives the cross-spectrum between channel p of {X+} and channel q of (Y Moreover, it easy to show that Sy:xy(u) (u) (1))
The LSW matrix quantifies the localized contributions to the process variance for individual and cross-channels, which motivates to next define the localized canonical coherence between two sets of locally stationary time series at specific scale (corresponding to a determined frequency band).
Definition 1 (Localized Scale-specific Wavelet Canonical Coherence)
=(x;") multivariate locally stationary time series. We define the localized scale-specific wavelet canonical coherence (WaveCanCoh) between { and {Y,), at scale j and rescaled time u, as
Let X, (x!") ....., x(P) and Y, - (Y:")) ....., y( where t 1, .,... T, be (jointly)
{a] (w) (u)b;(u))² PJx max (1)
(8)
P
a P vector and bJ (u) (b3 (w) Q p=1
where a7 (u) (a.) (u)
q=1 is 1 x Q vector, rep-
resenting the localized canonical coherence vectors of (X,} and { [Y,). respectively. The constraints here are a. (x)s)xx((x)aa(()) = 1 and bj. ( (u) = 1.
Remark The WaveCanCoh time-dependent trace Pj:x((-) measures the 'global' coherence value close to I indicates strong linear dependence, while value close to 0 shows little to no linear dependence. Furthermore, (p) and b (.) represent the localized contributions from the (p.9) 9) channels to P,xx((-)
between (Xi) and {Y,) at scale j. and takes values belween O and
(-)
The canonical coherence vectors a- ), by(.) can be obtained by maximizing (8) and the solution can be found by solving the cigenvalue and cigenvector problem associated with the following matrices
Mja (w) = S (VS (u),
= (X
(10)
Denote by A (id (u) the k-th largest eigenvalue of matrix (ua(u) in equation (9. and by A A (u)
(1)
the 1-th largest eigenvalue of matrix Mj, (u) in equation (10), for k, = 1,..., min(m,, Q). An
important observation is that Mja (w) and MI (u) share the same eigenvalues (see Appendix B
for details), hence denoting by A (w) (1 (w) (l) (u) their largest eigenvalue, the canonical
coherence between {X,} and { (Y.) at rescaled time u, as defined in equation (8), becomes
PjxXY (u) - (w).
(11)
The eigenvectors of Mjaa (u) and Mj (u) corresponding to A, (u) provide the solutions to the canonical directions of {X,} and {Y,), respectively. Proof: see Appendix B
Af)(w)
An important extension of our framework is the incorporation of leading-lag relationships into the scale-specific wavelet canonical coherence (WaveCanCoh). To account for potential causal effects, we define a lagged joint process, Z, (h) = (x,,, Y xIT)² where h is the value of lag, and t = I....,T h for h > We define the LWS matrix of (h)) at scale j. as
Sy h) S, Sj hi) V (z M (J, = ( + (0/T).-)) SJyY (h/T)) ] (12)
())
where Sj.Y (u, lt) denotes the cross-LWS matrix between X(uz] and Y |u capturing the inter- action between current values of X and future values of Y. Based on this construction, we can define and estimate the lagged version WaveCanCoh, enabling us to infer potential causal relationships between two groups of time series.
5

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Definition 2 (Causal Localized Scale-specific Wavelet Canonical Coherence) The causal localized scale-specific canonical coherence WaveCanCoh) between {X,} and (Yr) with lag h (or, X; - Yeth). at scale j and rescaled time u, is defined
Pj h1,
max (1)
(u)) (ub (h/T)))²
(13)
where the notations and constraints are the same in the standard WaveCanCoh framework. This extension allows for a scale- and time-specific evaluation of causal overall association between two sets of time series, enhancing interpretability in dynamic, multivariate, and nonstationary settings.
The framework above allows us to capture the time-varying overall association between two sets of multivariate time series, well as the time-varying contributions from each individual channel within these sets. However, natural consideration is how to project this time-varying coherence at each scale j into the frequency domain in a manner consistent with the Fourier-based method described in Section 2 as a key concern in many analyses is to interpret the results in the frequency domain. As mentioned earlier, each scale in the wavelet analysis corresponds approximately, but not exactly, to a specific frequency band. This correspondence governed by the unique filtering mechanism of wavelets, and an explanation for this relationship is provided in Appendix A
4 Estimation procedure
we developed rigorous framework that allowed us to introduce the localized, scale- specific wavelet canonical coherence. this section, we propose a well-behaved estimation procedure for quantifying the canonical coherence and corresponding canonical vectors. We start by estimating the local wavelet spectrum (LWS) matrices in equations and (10) the spirit of (21, given by
In Section 3]
M Σ
S SAM=
Σ Az where I
1 the smoothed periodogram, (14)
2M
represents the shift of the wavelet function and is equivalent to time [u]] in our context, and M is the halt-width of the rectangular smoothing kernel, controlling the amount of temporal smoothing. The matrix I & is the raw periodogram at scale and time k. obtained
T
I dr where di
Σ (t,r,,((t the empirical wavelet coefficient vector.
This multistep procedure yields consistent estimators of the LWS matrices under the asymptotic conditions T, MM + oo and M/T (2]]. We propose the following estimator for the scale-specific wavelet canonical coherence (WaveCanCoh)
P (u) (() (u),
(15)
where N("(w)) is the largest eigenvalue of M M (u) and M My,((u). defined as
(u) = 5 (1) Xy S vu), M Mya(1)) = = S (1)
(16)
S YX (x)
(17)
The estimated localized, scale-specific canonical direction vectors ā, and (u(w) are the eigen- vectors of M aa (u) and My (u) respectively, associated with (u). These quantities provide estimates of the time-varying global coherence and the channel-specific contributions at scale j. The proposed estimators
consistent with the true quantities they aim to approximate, provided certain asymptotic conditions are met. These include increasing sample size and appropriate smoothing bandwidth, ensuring reliable estimation in the limit. Proof: See Appendix
Algorithm D summarizes the estimation procedure for WaveCanCoh and results can be further used to investigate the temporal channel contributions to the global association, a particular scale.

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Algorithm 1 Proposed WaveCanCoh estimation algorithm for nonstationary time series
Suppose the observed data are two sets of multivariate locally stationary time series, denoted as and Y, (y y:Q)) observed for / (1,...,77 T)-
X, (x()) x(P))
1. Fuse: fuse the data to new (P+Q) -vaariite time series denoted as Z, with Z, (X, x,) (This fused representation allows for the joint analysis of the two multivariate processes within a unified framework. The Causal- Waaal--WveCanCohh
Y,)"
also be estimated by appropriately incorporating
the leading-lag into the existing WaveCanCoh estimation procedure.)
2. Spectral estimation: estimate the LWS matrix of {Z,] using equation 14). Denote the
estimator as (zz for any rescaled time u E (0, 1). The estimated auto- and cross-LWS
between (X:) and (Yr), denoted as Sy.xx( S (1)r(()), S (u), 5),x(()) can be obtained
by partitioning (,) into four submatrices as illustrated in equation 77.
3. Eigendecompoosiion: compute the matrices Mya (u) and Mj (u) in equations [6 then
perform their eigendecompooitions and obtain their (common) largest eigenvalue A (u). This
will serve as the estimated WaveCanCoh in (1). Pj (w) = A (w), while its corresponding cigenvectors of Mjaa(u (u) and (u) give the canonical direction vectors, äy (u) and (u)
5 Simulation study
In this section, we implement the proposed framework using simulated data under two distinct scenarios, one adhering the MvLSW assumptions underpinning our method, while the other introduces nonstationarity without strictly satisfying the MvL.SW assumptions. These setups allow us validate both the theoretical soundness and empirical performance of the proposed approach, as well as to assess its robustness and practical applicability in real-world scenarios where model assumptions may be violated. To further evaluate performance, we also compare the results with those obtained from the classical based canonical coherence approach.
MvLSW-based simulation. We generate the multivariate time series {Z.} from MvLSW process with P = 6, Q- = 4, observed across T = 1024 time points. The process is constructed using non-decimated Haar wavelets, with non-zero spectral structure specified at scale
2, as detailed in
Appendix C.I We impose a weaker dependence structure between } and {Y,} in the interval
0 < < 0.5, and a stronger dependence in the interval 0.5 < < 1, allowing us to examine the framework's 's sensitivity to changes in cross-group coherence. Using the process realization (Figure 9) we estimate WaveCanCoh using Algorithm [ To assess the estimation accuracy and account for variability, we replicate the simulation and estimation process 1000 times. At each time point, we compute the average of the estimated WaveCanCoh across the replicates and construct a 95% Wald confidence interval using the empirical variance. Figure 2 (left) demonstrates that the proposed WaveCanCoh method accurately tracks the true coherence structure and effectively reflects its time-van varying nature, while the estimated canonical direction vectors in Figure B map the temporal and individual channel heterogeneity in their roles within the multivariate dependence structure.
Mixture of A AR(2)-based processes. To evaluate the robustness and generality of the proposed framework, we investigate WaveCanCoh using synthetic data generated from a mixture of AR(2) processes. Unlike the MyLSW-based simulations, this setting introduces nonstationary dynamics without any wavelet-based structure, providing more flexible and realistic test scenario. To the best of our knowledge, WaveCanCoh is the first framework designed to estimate time-varying canonical coherence between two multivariate time series groups. However, to benchmark its performance, we compare it against a method (henceforth referred to as LSP) based on the time-varying Cramér representation 7 and described in Appendix C.2, with canonical coherence estimated via STFT- based localized spectra []] We simulate 500 replicates of X, E IR and Y, E IR,, with T = 1024, each formed by mixing five latent AR(2) sources tuned to neural frequency bands. In the first half, shared gamma (30- 20H2) bands induce cross-group coherence, while the second half contains no shared structure (see details in Appendix C.2). Figure Figure (right) illustrates that while both methods detect the existence of coherence in the first half, only WaveCanCoh captures its sharp drop and true behaviour in the second half, thus demonstrating its advantage identifying transient, localized changes that global Fourier- based methods fail to detect.
7

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Estimated scale =2
Estimated for two different methods
Method
estimoted
LSP WeveCanCoh
02
Figure Left: Estimated wavelet canonical coherence at scale j = over 1000 MvLSW replications. The solid line shows the average estimated coherence, the dashed line is the true coherence computed from the specified spectrum. Right: Estimated canonical coherence over 500 replicates of the AR mixture using WaveCanCoh and LSP at scale j 1 and w E [25, 50) respectively. Shaded areas indicate the corresponding 95% Wald confidence interval.
Estimated à (charnel conti butinn for XI atscale =2
Estimated (channe contibutions for atsaale
61
charnel
62
owneel
channel
mamell
4.11
0amell
ciamel2
Oamell
08
4.
manels
43
-21
08
02
34
00
82
C8
Time
Time
Figure 3: Estimated time-varying canonical direction vectors āz(-) (left) and bz (right).
Local field potential (LFP) data analysis
To demonstrate the practical utility our our proposed WaveCanCoh framework, we analyze LFP activity
recorded from the hippocampus of rats engaged in sequence memory task 2 22]. The data were recorded using a 22-electrode microdrive implanted the CA1 subregion to capture high-resolution LFP signals across all channels at sampling rate of 1000H². In this task, rats were tested on their memory of a sequence of five odors (odors ABCDE). Each odor was presented for 1.2 second (s) and variable delay of ~5 separated each odor (see Figure [ For each trial (i.e., each odor presentation), the rat had to judge whether the odor was presented "in sequence" (e.g.. ABC...) or "out of sequence" (c.g., ABD...) and indicate their decision by holding their nosepoke response until a tone signal (at 1.2s) or withdrawing before the signal, respectively. Correct-response trials (i.c., correct "in sequence" over 4s period (I' = 4000 time points) per trial, with t = O marking the moment the rat initiated a nosepoke to receive the odor stimulus. This paradigm provides a well-controlled setting to investigate dynamic, time-varying functional interactions in the hippocampus during memory- -uidddd decisions. We employ Wav framework with Haar wavelets quantify reequency-spectie functional coherence between two groups of hippocampal electrodes (T1, T2, T4, T5 and T13-T17), and to examine how coherence patterns differ between correct- and incorrect-repponse trials ("in sequence" trials only). Specifically, we analyze LFP data from the rat Mitt, which included 40 correct-response trials and incorreet-response trials. Figure 4 presents the estimated wavelet canonical coherence at scale 5, corresponding the 15.625 31. 25.25 frequency band. The results, averaged across trials for each condition, reveal dynamic changes in inter-regional coherence, with a pronounced peak around the time of odor stimulus delivery (t 0). Notably, distinct patterns emerge between correct and incorrect trials, suggesting that coherent activity in this frequency band may play a role in supporting successful memory retrieval and decision making. More results for several other scales can be found in Figure 10 in Appendix D.I
"out of sequence" decisions) were rewarded. LFP activity data were recorded
To further interpret the coherence patterns, Figure provides a spatial summary of the canonical coherence between the two electrode groups at several selected time points. The double-headed arrows represent the magnitude of estimated coherence between the two regions, while the numbers

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Wavelet coherence at scale =
0.56
Door presentation
Correct Incorrect
0.35
Time (s)
Figure 4: Estimated wavelet canonical coherence between two hippocampal subregions (TI, T2, T4, T5 vs. T13-T17) in one subject (Mitt) at scale j = 5 (15.623 115.625-31.2 2)). 'hee estimates are averaged across 40 correct- and 32 incorrect-repponse trials, using a rectangular smoothing window of 0.2s.
in the circles reflect the individual channel contributions to the global coherence, derived from the elements of the canonical vectors a. (-) and by(.).
-1.0s
ts -0.55
t= Ds
1- 0.5s
0.40
0.51
0,55
0.36
Correct
0.50
0.49
0.41
Incorrect
1B
Figure 5: Spatio-temporal illustration of WaveCanCoh between hippocampal regions (TI, T2, T4, T5 and TI3-T17) in rat Mitt at scale j = 5 (15.625-31.2)) 25/ Arrows indicate the magnitude of inter-region coherence at selected time points. Numbers inside circles represent the channel-wise relative contributions to the canonical coherence for each region, for correct- and incorrect-responess. Negative values indicate that the corresponding channels contribute to coherence in the negatively correlated direction between the two regions.
The results highlight that both the strength and structure of inter-regional coherence vary dynamically over time and differ across trial outcomes, reflecting the nonstationary nature of neural interactions during task performance. Specifically, incorrect-response trials exhibit lower coherence at time points immediately following the odor stimulus delivery and are often driven by few dominant channels, while correct trials show higher coherence values and relatively balanced contributions across channels. These findings illustrate the necessity of a framework like WaveCanCoh, which simultaneously identifies time-varying and scale-specific dependence structures between multivariate time series. Traditional stationary or pairwise approaches would fail to capture such nuanced dynamics, highlighting the need for method like WaveCanCoh to extract such complex relationships in brain activity. The directed interactions between brain regions are also explored Appendix with the proposed Causal- CaneCaaCoCn framework.
To further validate the existence of significant differences the activity between correct- and incorrect- response trials, we propose time-specific detection procedure based on the permutation test to determine temporally localized differences in the wavelet canonical coherence at a given scale between conditions, while maintaining the nonparametrie nature of the statistical inference. The detailed inference steps are shown in Algorithm 2 (Appendix D.2. Table T reports the permutation test results on the LFP data, with the value in each cell representing the difference in the median wavelet coherence between correct and incorrect trials at the corresponding scale j and time 1* = 1.0,, -0.5, 0.5, and 1.0 seconds. The values in parentheses denote the permutation p-values obtained using the windowed test procedure (window size = 0.2s, Tperm 1000). Detailed test results are shown in Figure []] (Appendix D.3). revealing significant differences between correct- and incorrect-

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response trials at scales 4 to 7 following the odor stimulation. Statistically significant differences in canonical coherence between correct and incorrect trials emerge at time points following odor sampling (t 0), predominantly at intermediate wavelet scales corresponding approximately to the 8-62 Hz frequency range. On the other hand, no significant differences are observed prior to stimulus onset, indicating that the coherence patterns distinguishing inter-regional communication among trial types are tightly linked to task engagement. In comparison, we implement the LSP algorithm on the same data and conduct the same permutation test. The results (see Table 2 show that this approach lacks the sensitivity to distinguish between correct and incorrect trials. A likely explanation is that the smoothed approximation may have masked the true differences. These findings demonstrate the effectiveness of the proposed WaveCanCoh framework and associated permutation test in capturing localized, requeney-specinn differences in neural coordination between behavioral conditions, thus offering a powerful tool for analyzing complex brain interactions.
Table I: Differences in median wavelet canonical coherence (WaveCanCoh) between correct- and incorrect-response trials across time points and scales. Each cell reports the median difference scale j and time t", with p-values obtained from the time-specific permutation test shown in parentheses.
1" (s)
-1.0
-0.5
0.5
1.0
3 (62.5 125Hz)
0.222 (0.786)
0.068 (0.704)
-0.178 (0.355)
0.106 (0.886)
4 (31.25 - 2)
-0.090 (0.180)
-0.021 (0.506)
0.023 (0.001** **)
0.027 (0.691)
5 2)
0.006 (0.977)
0.229 (0.999)
0.334 (0.002**)
0.016 (0.079)
6 (7.81 - 25
-0.025 (0.239)
-0.049(0.11))
0,039 (0.001 **)
0.002 (0.025**)
7 (< 7.81/12)
-0.014 (0.999)
0.058 (0.640)
0.012 (0.059)
-0.030 (0.489)
Table 2: Differences median Fouricr-base canonical coherence (LSP) between correct- and incorrectrresoonse trials actoss time points and scales. Each cell reports the median difference at scale j and time t", with p-values obtained from the time-specific permutation test shown in parentheses.
(s)
-1.0
-0.5
0.5
1.0
j
3 (62.5-125 22)
-0.065 (0.886)
-0,002 (0.901)
-0.009 (0.991)
-0.060 (0.471)
4
(31.25-62.5 22)
0.001 (0.128)
-0.044 (0.141)
0,010 (0.056*)
0.010 (0.991)
5
1..63-31.25HH Hz)
0.009 (0.470)
0.003 (0.970)
0.008 (0.999)
0.003 (0.983)
6 (7.81-15.63
-0.001 (0.512)
-0.002 (0.052*)
0.002 (0.901)
0.000 (0.842)
7
(< 7.81122
0.001 (0.094)
0.001 (0.121)
-0.004 (0.754)
-0.002 (0.901)
7
Conclusions
We introduced novel methodological framework, scale-specific wavelet canonical coherence (Wave- CanCoh), designed to quantify the dynamic multiscale coherence between two sets of nonstationary multivariate time series. Our primary contributions include the rigorous definition of WaveCanCoh within the multivariate locally stationary wavelet framework and the development of comprehensive estimation and theoretically-backed inference procedure based on wavelet analysis. We validated our proposed methodology through simulation studies, demonstrating its accuracy in tracking true coherence structures. The application to local field potential activity recorded from subregions of the hippocampus effectively showcased the WaveCanCoh capability to identify nuanced spatio-tempore coherence patterns associated with cognitive performance, while our permutation-bia based inference procedure provided robust, nonparametric approach for detecting significant coherence differences between conditions. Compared to existing stationary and Fourier-based canonical coherence methods, WaveCanCoh is shown to offer significant advantages, particularly through its ability to adaptively capture transient and loceliecli interactions, rendering it highly suitable for analyzing signals in neuroscience and other fields with data exhibiting dynamic gross-group interactions.
10

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Acknowledgments and Disclosure of Funding
Knight gratefully acknowledges support from UKRI EPSRC NeST Programme Grant EP/X002195/1. Fortin gratefully acknowledges support from NIH (R01-MHI 5697 and R01-DC017687), NSF and the Whitehall
(CAREER IOS- SSS--15000 1150222, BCS--1432200 DGE-183928S, and MCS-FR-2311611 Foundation (2010-05-84), Ombao gratefully acknowledges support from KAUST research fund. We sincerely appreciate the time and effort that the anonymous reviewers and area chairs devoted to evaluating our work and are grateful for their valuable, insightful feedback.
References
[1] J.B. Allen and L.R. Rabiner. A unified approach to short-time fourier analysis and synthesis. Proceedings of the IEEE, 65(11))1558-155 1977.
[2] Timothy A. Allen, Daniel M. Salz, Sam McKenzie, and Norbert Fortin. Nonspatial sequence coding in cal neurons, The Journal of Neuroscience, 36(5):1547-1563 2016.
[3] G. E. P. Box and C. Tiao. canonical analysis of multiple time series. Biometrika, 64(2):
355-365, August 1977.
[4] D. R. Brillinger. Time Series: Data Analysis and Theory. SIAM, 2001.
[5] Andreas Bruns. Fourier-, hilbert- and wavelet-based signal analysis: are they really different approaches? Journal of Neuroscience Methods, 137(2):321-332, August 2004.
[6] Catie Chang and Gary H. Glover. Time-frequency dynamics of resting-state brain connectivity measured with fmri. Neurolmage, 50(1):81 March 2010.
[7] Rainer Dahlhaus. Fitting time series models to nonstationary processes. The Annals of Statistics, 25(1) February 1997.
[8] Ingrid Daubechies. Ten lectures on wavelets. SIAM, 1992.
[9] Jonathan Embleton, Marina Knight, and Hernando Ombao. Multiscale spectral modelling for nonstationary time series within ordered multiple-trial experiment. Annals of Applied Statistics, 16(4):274-28033 2022.
[10] John Geweke. Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association, 77(378):304-31 -33,, June 1982.
[11] Avniel S. Ghuman, Jonathan R. McDaniel, and Alex Martin. A wavelet-based method for measuring the oscillatory dynamics of resting-state functional connectivity in meg. Neurolmage, 56(1):69-77, May 2011.
[12] A. Grinsted, J. C. Moore. and S. Jevrejeva, Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 11(5-6): 561-566, 2004.
[13] Jessica K. Hargreaves, Marina I. Knight, Jon W. Pitchford, Rachael J. Oakenfull, and Seth J. Davis. Clustering nonstationary circadian rhythms using locally stationary wavelet representa- tions. Multiscale Modeling & Simulation, 16(1):184 214, 2018.
[14] Harold Hotelling. Relations between two sets of variates. Biometrika, 28(3/4):321-33 -377, 1936.
[15] Hsiao-Yun Huang, Hernando Ombao, and David S. Stoffer. Discrimination and classification of nonstationary time series using the slex model. Journal of the American Statistical Association, 99(467):73--77, 2004.
[16] Marina I. Knight, Matthew A. Nunes, and Jessica K. Hargreaves. Adaptive wavelet domain principal component analysis for nonstationary time series. Journal of Computational and Graphical Statistics, 33(3):941-954, 2024.
[17] K. V. Mardia, J. T. Kent, and J. M. Bibby. Multivariate Analysis. Academic Press, 1979.

---

[18] G. P. Nason. Wavelet Methods in Statistics with R. Springer New York, NY, 2008.
[19] Guy P. Nason, Rainer Sachs, and Gerald Kroisandt. Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Journal the Royal Statistical Society: Series B (Statistical Methodology), 62(2):271-292, 2000.
[20] Hernando Ombao and Marco Pinto. Spectral dependence. Econometries and Statistics, 32: 122-159, 2024.
[21] Timothy Park, Idris Eckley, and Hernando Ombao. Estimating time-evolving partial coherence between signals via multivariate locally stationary wavelet processes. IEEE Transactions on Signal Processing, 62:5240-5250, 2014.
[22] Babak Shahbaba, Lu Li, Francesco Agostinelli, Meenakshi Saraf, Keiland W. Cooper, Daniel Haghverdian, George A. Elias, Pierre Baldi, and Norbert J. Fortin. Hippocampal ensembles represent sequential relationships among an extended sequence of nonspatial events. Nature Communications, (3(:)787, February 2022.
[23] Frank Skidmore, Dmitry Korenkevych, Yilu Liu, Guolin He, Edward Bullmore, and Panos M. Pardalos. Connectivity brain networks based wavelet correlation analysis in parkinson fimi data. Neuroscience Letters, 499(():47-51)) July 2011.
[24] Mara Sherlin D. Talento, Sarbojit Roy, and Hernando C. Ombao. Kencoh: A ranked-based canonical coherence. arXiv preprint arXiv:2412.10521, December 2024.
[25] C. Viduarre, G. Nolte, I.E.. de Vries, M. Gómez, T.W. Boonstra, K.-R. Müller, A. Villringer, and V.V. Nikulin. Canonical maximization of coherence: A novel tool for investigation of neuronal interactions between two datasets. Neurolmage, 201:1 201:116009, November 2019.
[26] Mike West, Raquel Prado, and Andrew D. Krystal. Evaluation and comparison of eeg traces: Latent structure in non-stationary time series. Journal of the American Statistical Association. 94(446)118331009) 1999.
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Appendix
A Basic introduction to wavelets
Wavelet analysis provides powerful framework for studying signals with both time- and scale- varying structure, making particularly well-suited for nonstationary data. Unlike traditional Fourier-based methods, which decompose signals into global sinusoidal bases and therefore assume stationarity, wavelets enable localized, adaptive decompositions by projecting signals onto functions that are compact in both time and scale. This localization is achieved through two core operations: scaling, which adjusts the width of the wavelet to analyze different resolution levels, and shifting, which moves the wavelet across time to detect when features occur. Specifically, the wavelet functions scale and shift k, denoted by $1,,k. are derived from mother wavelet 1/ and defined as
t 22 (t-22k) 2i
()) = 2/2
j= j=
where represents the number of scales. Smaller scale j corresponds to finer hhigh-frequency resolution, and larger j captures coarser (low-frequency) trends. A similar construction applies to the father wavelet, denoted by Φy.A. which serves scaling function (see 8 and [8] for more details on wavelets). Figure iillustrates the effect of scaling and shifting operations on the wavelet function. This ability isolate both short-lived and long-term features distinguishes wavelet methods from Fourier analysis, allowing for nuanced investigation of nonstationary signals such neural activity, where structure evolves dynamically across time and scale.
Signal
|
+
scaling
1/1 1
shifting
Figure 6: Illustration of scaling and shifting of wavelet functions. Smaller scales (top) capture high- frequency, localized features, while larger scales (bottom) capture broader, low-frequency structures.
The correspondence between wavelet scale and signal frequency arises from the principle of multi- resolution analysis, which provides the foundational framework for wavelet construction and signal representation. Briefly, the discrete wavelet transform (DWTT decomposes signal into frequency subbands via a dyadic filter bank architecture. At each level, the signal is passed through a pair of conjugate quadrature filters: low-pass filter h(n)) and high-pass filter g[n]. followed by down- sampling by factor of two. The low-pass branch yields approximation coefficients that retain coarse-scale information, while the high-pass branch produces detail coefficients that capture lo- calized figh-frreuency variations. Specifically, given a discrete signal x /y].. the approximation and detail coefficients at level j are obtained by
ay(1]] = Σ hikka, k), dr(1o] Σ h
Cakkaa--122 k].
20

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where aj-1 is the approximation from the previous level (with z). This process is iterated on the low-pass output, producing a multiscale representation which each level isolates specific frequency band. For a signal sampled at rate Js. the detail coefficients at level j correspond approximately to the
frequency interval g 23 The orthogonality between subbands ensures perfect reconstruction and energy preservation, and the hierarchical filter bank provides localized time-frequency analysis with increasingly coarse temporal resolution at lower frequencies. The explanation above offers an intuitive understanding of how components each wavelet scale can be approximated to true frequency bands (see Figure 7 for an intuitive illustration). This approximation enables our framework capture the overall association between two sets of locally stationary time series both the temporal and frequency domains (more details can be found in 8)
fs the sampling rate
w/Hz
f0/2
MWM0
scale j=1
fs/4
scale j=2
13/8
scale 3+3
///
time/s
Figure 7: Illustration of mapping relationship between wavelet scales and their approximated fre- quency bands.
B Theoretical proofs
Proof of solution to equation (8) Suppose that X, and Y, are two multivariate time series and for each time point f, the goal is to find the vectors a, (u) and (u) that maximize the coherence,
P PyxY (u) = {a, (w) (w)}
subject the normalization constraints described in the main paper
a1 xx 1, bJ (w) (u) 1.
Assuming rescaled time and scale are fixed, suppress them for clarity. To solve the above optimization problem, we set up the Lagrangian as
C (a. b. )1. 22) = 2 (a Sxrb)²
d (a Sa_Sxxa-1y ( 2
A2 2 (bJSyrb- b-ybb1)
Differentiating the Lagrangian with respect to and b, respectively and requiring the partial deriva- tives to equal zero, we obtain
ac
da = (Sxyb) (Sxy b) T a - 11Sxxa=0
(Sxrb) (Sxyb) = A,Sxxa,
(i)
dc (Sxxa): (Syxa) b 12Syyb=
Øb
(Syxa) (Syxa) b = XySyyb.
(ii)
21

---

With (i), (ii) and since Sxyb) is real-valued quantity, we further obtain
(a Sxy b) (a (aTS (XTSx) A A Sxxa, and (b Syxaaa (a (11Sbr)) >2b Syyb.
Recalling the constraints a axxxa = and b Syy b 1, it immediately follows that
A1 2 = (a Sxy : X.
By substituting the above back into (i), (ii) and assuming À to be non-zero, we obtain
1 S 5xxSxx' and
V
V SyrSa
which plugged into (ii). (i), respectively, yield
SxySy Syx xX Sxyb = ASyy b or, equivalently,
xx
S S da
S Syr Syx XX Sxyb = Ab.
Hence = (a (xrb) an eigenvalue for both matrices
Sxy S Syx. and Syx
whose corresponding eigenvectors and b, respectively. Thus, the defined canonical coherence p max (a b) is the largest eigenvalue of above matrices. The case when À illustrates the canonical coherence between X and Y is 0, which is not meaningful scenario for this problem. We recall the above equations hold for every time u and scale j.
Proof of consistency of WaveCanCoh estimator. We aim establish the consistency of the matrix estimators
Maa- xx M, = ,,
According
19 and (2I), the smoothed periodogram- based estimators of the local wavelet spectral (LWS) matrices are consistent. Specifically, as the number of time points T and the smoothing parameter M - oc with M/T - have
xx P Syxxx
, P SyXY, S, P S,,x. ,,YY P. S,YY
Assuming the spectral matrices and their estimators are non-singular, it follows by the continuous mapping theorem that
x P Sjxx
S Y
J
Since matrix multiplication is continuous with respect convergence in probability, we obtain the consistency of the matrix estimators by Slutsky's 's theorem, namely
M P. Mja: MD. P, Mj
Consequently, the estimated wavelet canonical coherence and associated canonical vectors, derived from the largest eigenvalue and corresponding eigenvectors of M Mj and M also converge in probability to their population counterparts, following arguments akin those in [16].
C Details of the simulation setup
This appendix provides the full specification of the simulation experiments described in Section 5 of the main text. The code is provided in the supplementary materials for results reproducibility. and here we firstly provide a brief discussion on the computational complexity of the method. The estimation of the spectrum step leads to time complexity O (JT(P + Q)?) (where the number of scales J is most commonly used J (T))(T)). The eigenvalue decomposition step takes
22

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the total time complexity to O(JTd") (let P Q= d). to be compared to that of the standard CCA, O + ) The proposed algorithm naturally more expensive than standard CCA, since it is designed to obtain time-localized and seale-specific results. In our practical experience with WaveCanCoh, the spectral estimation step is extremely fast, with virtually all of the computational time being spent the eigenvalue decompasition step. Although compared to standard CCA, our method produces a set results at each time point, resulting in increased computational burden, in standard
the experiments with 7 1024, single replicate is completed within 2.5 seconds personal computer (Apple Mac, 16GB RAM, 6-core CPU) without resorting to parallel computing or to the of cluster, and all results reported in the paper can be obtained within 3 hours. Moreover, computations for WaveCanCoh can streamlined by storing canonical vectors and coherence values each time and scale, with the requirement being (JTdd), and the memory complexity O(d²) for standard, one time point global estimate.
C.1 MvLSW-based simulation
We simulate P Q 6 + dimensional multivariate time series Z, X YJJT E R from
multivariate locally stationary wavelet (MvLSW) process over 7 = 1024 time points. The wavelet spectrum Syzzz (u) is non-zero only at scale j = and is structured as 10 x 10 block matrix
SjXX
S S
t
2
())
Sj.YY
T (0.
Auto-spectral block: Sjxx E R6x6
8
0
0
07
0
0
0
0
0
Sjxx
0
0
8
1
0
0
8
0
0
0
8)
Auto-spectral block: SyxY E R4x4
1
07 1
SJYY
1
6
0
Cross-spectral block: (u) 16x4 This block is time-varying. For 0.5, each specified cross-group pair has spectrum value I; for 2 0.5, the same entries increase to 2.
Let
if < 0.5
c(u)
2
u 2 0.5
then
c((u)
0
0
c(u)
0
cuu)
0
0
0
0
cu))
S (w)
S SJxx(())
c(u)
0
0
0
0
0
0
0
The entries of the cross-spectral matrix determine how individual channels contribute to the global coherence structure between }X,} and (Y,). Figure visualizes the spectral structure S)=2:zz(1) and Figure 9 shows example realizations of the simulated processes {X,) and (Y.).
C.2 Mixture of AR(2)-based simulation
Fourier-based L.PP method for comparison benchmark, we implement a method based on the time-varying Cramér representation for locally stationary processes (LSP), introduced in [7] A
23

---

X1
Level
X2
X3
Time
Ym
Figure 8: Visualization of the specified block structure of S2,z((n).
Aeyle pornett wi ti bo
Figure 9: Example realization of {X+} and {Y,) generated from stated MvLSW process {Z,}.
locally stationary process
[X,7] can be expressed as
X [
_ A A(u, 2niot dZ(w), u = I/T,
where A(u, w) is a smoothly varying transfer function and Z(w) is complex orthogonal increment process with (w)d dw. The local spectral density then defined f(u, w) |A (u, w) / We estimate f(u, w) via the Short-Tima Fourier Transform (STFT) with Gaussian smoothing kernel and compute canonical coherence over w [25, 50) H
Simulation setting We simulate 500 independent replicates of two multivariate processes, (X,) I+ and (Y,) 3 for t I....,II T. with T 1024 and sampling rate fa 100Hz. Each process is generated from mixture of K = 5 latent AR(2) sources, z (x)) .2(x) E K peaking at different frequency bands (delta, theta, alpha, beta, gamma), respectively. For the first half the time series < T/2), the observed processes given by:
X, (x)
Y, Y, (()
24

---

and for the second half (t > T/2):
X; x B(217
where the mixing matrices are:
0
0.95
1122
2
6
B
0
0
0
0.90
(
0
0.80
3.1
(1)
3.2
(1)
0
0
0.90
0
4,,1
(1)
04,2
(1)
4,3
(1)
0
0
012
(2)
043
(2)
TO
0
0
0
0.95
0
0
0.90
0
CO
0.90
((22
0
2,3
3,3
0 0
For each row of the mixing matrices, a selected subset of frequency bands is assigned non-zero weights that are randomly drawn to sum to a predefined total (e.g., 0.95, 0.90, or 1.0), introducing controlled variability across replicates while preserving the intended contribution structure. The
latent sources z₂x) and ( are partially shared in the gamma band (component 5) during the first regime, with mixing weights a 0.7 and B = 0.6, respectively:
az
(shared) + (1 - a) Z
XX private)
z 1,5 = 3Z (Shaaardd) + (1 - 2) private) 2/2
1,5
t,5
(shared) is a common latent process. Specifically, each channel, say p. of (Z (Zharee))) }. where 7,3
( (--iivatt)) is generated from the following AR(2) process independently, {z! x. private) and {Z}
Ze. 7 + 62741-20 + Wo
where {w:} is white noise and the coefficients are Φ1 2cos(2mn)/e² 02 -1/620
For each component p 5, we use the frequency vector, n (1) 15) (0.02, 0.06, 0.10, 0.175, 0.375). and the sharpness parameter (0.03, 0.03, 0.03, 0.05, 0.05). where smaller yields narrower frequency bands.
{1(1) (5))
This design induces time-narying coherence between the first two channels of X and Y during the first regime, primarily through the shared gamma component, while maintaining independence during the second regime. The abrupt transition in mixing structure at T/2 provides controlled setting for evaluating the sensitivity of coherence estimation methods to sudden changes in cross-dependence.
D
Additional analysis for the LFP data in Section
D.1 Additional results
We display the estimated wavelet canonical coherence between two electrode clusters across additional scales. Figure shows that gooss-grupp associations vary significantly time, with the stimulus clearly eliciting a scale-dependent response, indicative of the heterogeneous brain activity across different frequency bands. These results emphasize the importance of capturing scale-specific, lime-varying coherence.
Wavolet coherence scale
Wavolet coheronce
Wavvllet coherence scale
02H
0.24 028 6ap
0:30
B3P 0.35
Incorrect
Incomeed
Time (6)
Time (8)
Time (6)
Figure 10: Estimated wavelet canonical coherence between two hippocampal regions (electrodes T1. T2, T4, T5 vs. T13-T17) in rat Mitt at scale j = 4 131.25-62.55FF j = 6 (7.81-155663FF and
j = 7 (3.90-7.81 Hz). The estimates are averaged across 40 correct- and 32 incorrect-response trials,
using a rectangular smoothing window of 0.2 seconds.
25

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D.2 Permutation test: Algorithm
Algorithm 2 below gives the detailed procedure used in Section for detecting significant differences in canonical coherence between correct- and incorrect-responss trials, at selected time points.
Algorithm 2 Time-specific window permutation test for wavelet coherence analysis
Assume the time-localized, scale-specific WaveCanCoh Peorrecu incorrect (j. u) available for each trial r, along with the trial label 'correct/ incorrect'. We want to test for significant differences between correct- and incorrect-response trial groups at specific time points t" and given scale j. 1.Window definition: For each time point t", define a window of size w centered at t*:
start l" w/2, lend = l" + w/2
2. Rest statistic: For each time point t* and given scale j. compute:
Tobs (i,t))
E
median,
(péoncc (i.t/T)) median, (Pisouree (i. (ittTT)))
3. 3.Permutation: For each t* and scale j:
Combine all (i,(T) values from both correct- and incorrect-response trials across the time window.
Perform Peem random permutations. For each permutation i:
RRndomly assign trials into two groups of the same sizes the original groups.
Compute the permuted statistic:
T perm (i,tt)
mmdiaan, (ppri ( median, (p.mm.
t= Eoon
where Ppem.1 and Ppemm. are the permuted trial groups.
(r)
(r)
4.Calculate p-value:
Opperm
P (i.tt)=
(Ti))
(j,t") 2 Tobs (1,t*)
perm
uperm
im
D.3 Permutation test: LFP results
Table summariizs permutation test results on the LFP data, where each cell reports the difference in
median wavelet coherence between correct and incorrect trials at scale and time t" = -1.0, - 00.5 0.5, and 1.0s. Figure I shows the full distributions, highlighting significant differences at scales j = 4 to 7 after odor stimulation.
E
Causal-WaveCanCaa analysis
To further explore directed interactions between brain regions, we implement the Causal- WaveCanCoh framework (equation [1 LFP activity data, which extends WaveCanCoh by introducing lead-lag structure to evaluate time-lagged canonical coherence. Specifically, we define X, = (X(I)) (T2), (T4), (15) and Y, = (TI3), .y.... y(T17)
as the multivariate signals correspond- ing to the two investigated distinct hippocampal regions, and we conduct the analysis in both directions, namely X, + Yith and Y, + X,+h, for lags ht = 0, 10, 20, 30, 40, 50, corresponding to time shifts from 0 (contemporaneouss dependence, used as reference point) to 0.05 seconds.
Figure shows average scale-specific estimated causal canonical coherence for both directions across the odor presentation time (0s - 1.28) and trial type (correct- vs incorrect-respone trials). The results reveal lag- deg-deenddntt and scale-driven patterns of directed coherence that identify stronger association between the two hippocampal regions scale j 5 (corresponding the frequency band
26

---

Permutation densiles stj=3
(m-1s
m-0555
('m05s
(=1s
Denshy
0.6
0.4
Test statistic
Permulation densities jlj
E' mm-1
m-0.55
1-0.55
f-15
10.0
Density
3.0
2.5
1.2
0.5
0.8
1.00
1.25
1.50
Test
Permutation densities at j45
5 1--11
0.5
-0.5
E-15
1.00
00999
0-0
2-0022
Densly
0.75
0.50
0.25
0.00
2.5
Permutation densities st
1m-15
m-055
C==055
(=11
10.0
p-0000
pwolll
p-000
Denshy
3.0
2.5
0.0
statistic
Permuletion densittes atj=?
(=mls
-0.5
I' 70.55
f [m]
p-0055
p.0000
2-0.433
Deesiity
15
10
0.7
0.3
0.4
0.20
0.25
0.30
0.35
0.40
0.125 0.150 0.135 0.00 0.235.100
0.15
0.20
0.25
Teststabstic
Figure 11: Permutation test distributions (Tperm) obtained using Algorithm 2 based on WaveCanCoh for different time points across scales j = 3 to j 7. Red vertical lines indicate observed test statisties (Toba): valuee shown each panel, corresponding the results reported in Table T
27

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15.625 31.25H2). This notably occurs in both directions, with the activity in T13-T17 leading that of T1-T5 in correct-response trials after 10. The strength and behavior of coherence vary across scales, as well as across correct- and incorrect-response trial groups, which can be captured by our framework.
MannCaial Coheence
020
6.00
Yeum Row atscalej
Rea acale
Y-1 absicale w-5
Tumn Toa atscolaj
Hooe Cascal Coneeence
0.85 0,36 0:00 0,20
U.C 0.50
22
Time
Times log
Timn ug
Figure 12: Average causal wavelet canonical coherence across time lags h = 0 to 50 at scales j = 4 7, comparing correct- and incorrect-response trials. Top: X, + Yith. Bottom: Y, X,th-
Limitations
While our proposed WaveCanCoh framework provides robust, nonparametric approach for quanti- fying scale-specific time-varying canonical coherence between two sets of nonstationary multivariate time series, inherently limited fixed of wavelet scales. This restricts its flexibility in applications requiring precise frequency localization or alignment with arbitrary bands. As shown in Figure 7 the mapping between scales and true frequency depends the sampling rate and signal spectrum. For signals with broad frequency content or low sampling rates, certain bands may be poorly resolved, for example, the sampling rate 50/ 2, becomes infeasible to resolve components in the 30 - 40/ range. One practical solution is downsampling when a high sampling rate available, allowing better alignment between scales and target frequency bands. However, due to time-frequenc trade-offs in wavelet analysis, a fully flexible frequency resolution remains challenging. Future work may consider adaptive or overcomplete wavelets improve frequency targeting while retaining nonstationary modeling capabilities.
28


Reviews:
[{"review_id": "0MMtfmMZGk", "reviewer": "Reviewer_RXsG", "summary": "The authors present a method for quantifying multi-channel correlations at different time scales. The method produces scale-specific components for each channel via wavelet decomposition and then applies CCA to the resulting components (with some additional pre-processing). The method is presented in the context of canonical coherence for stationary processes and outlines both the model and estimation procedure. The model is then validated with simulated data using both wavelet-based and non-wavelet based nonstationary generative models and is then used on LFP data from mice performing a 2AF task at a pre-selected number of scales, and lags.", "questions": "- I would recommend changing the word “clusters” used throughout the manuscript to refer to distinct sets of channels since “clustering” is unrelated to the present paper.\n- The a and b elements in the inline text under equation (8) are bolded but the authors previously used the convention of non-bold symbols for scalars earlier in th paper. The authors should maintain a single convention throughout the paper.\n- Please explain A_{jl} in equation (14). It is not mentioned anywhere in the text\n- What kind of wavelet was used for the analysis in section 6?\n- Could the authors include the benchmark I mentioned in S&W?\n- Could the authors independently validate the results with another approach and/or give a deeper explanation for what the results mean and why they demonstrate the validity or value of their method? Alternatively, could the authors demonstrate that another method could not be used? This is perhaps the most important point to address for me to increase my score.\n- It is not stated if the authors controlled for multiple comparisons in their LFP data analysis. They should control for multiple comparisons in this case.", "limitations": "The authors didn’t list any limitations. They could provide ways of further improving on their method and suggest scenarios where their method would not work. They could also comment on the scalability of their method and the suitability of their method for exploratory or discovery-based analysis.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The paper is organized clearly and logically developed and is mostly well-written, with a few omissions that could be corrected rather easily. I would imagine the method being useful to some researchers but could require some considerable guidance on how to choose wavelets, scales, and averaging window sizes M. Using specific windows of analysis to deal with nonstationary data is a pretty intuitive step for many neuroscientists but developing an approach explicitly in a wavelet setting is certainly more rigorous. However, it is not entirely clear from the text what impact it could have, however. First, the analysis was not compared against the most obvious benchmark, which would have been to apply canonical coherence on a moving window and evaluate the SNR against their method. Second, the analysis of LFP data was extremely limited with ambiguous ROI. Specifically, the results were not independently validated or interpreted either physiologically or algorithmically.", "was_revised": true, "final_justification": "the authors have satisfied all of my concerns. As such I think an increase in score is warranted.", "rebuttal": "Thank you for your insightful review. We appreciate your time and believe that addressing your comments will greatly improve the clarity and quality of our work. We're also grateful for your recognition of WaveCanCoh's novelty and practical value. Below are our detailed responses. \n\n## Responses to the weakness\n### W1. Choice of wavelets, scales and kernel size, as well as their impact\nThank you for highlighting this. We agree that a clearer explanation will better guide researchers applying the method.\n\nThe choice of wavelets primarily depends on the characteristics of the signal, e.g., Haar wavelets are suitable for signals with sharp transitions, while Daubechies wavelets are suggestted for smoother signals. In practice, unless the signal shows very distinct features, results tend not to vary much across different wavelets. In our LFP analysis, we used Daubechies wavelets with 4 vanishing moments (Db-4), which offers a balanced choice for diverse features. To ensure robustness, we compared results using Haar and Db-8 for LFP data analysis. The outcomes were largely consistent, supporting the reliability of the findings obtained with Db-4.\n\nThe choice of scale depends on the frequency range of interest. In our study, we focused on scales $j = 3$ to $j = 7$, as they cover the primary frequency bands where neural activity is most prominent. WaveCanCoh makes no assumptions on coherence strength at any scale but aims to capture true scale-specific (frequency) dependence driven by data, \n\nFor window size $M$, the recommended value in wavelet literature is $\\lfloor \\sqrt{T} \\rfloor$, which offers a good trade-off between bias and variance in spectral estimation. Large deviations from this may introduce bias and reduce estimation accuracy. We have verified this behaviour through extensive simulations for our method.\n\n### W2. Comparison benchmark\nThank you for this important point. In our study, we implemented a Fourier-based coherence benchmark using a moving window, which referred to in text under the acronym LSP (Locally Stationary Fourier Process), as described in Appendix C.2 in the paper. This benchmark was evaluated using simulated data generated from a mixture of AR(2) processes. We would like to emphasize the key differences between the LSP and WaveCanCoh. Specifically, in LSP, coherence at each time point is approximated using the surrounding window, making the results sensitive to e.g., window size and kernel choice. More importantly, due to the global nature of the Fourier basis, LSP struggles to capture localized or transient features—especially in the presence of sharp transitions, a point illustrated in Fig. 2 right panel. \n\nWe agree that including this benchmark in LFP data analysis would strengthen the comparison. In the revised version, we have incorporated LSP into the analysis and produced results analogous to Fig. 4 and Table 1. As anticipated, the LSP approach fails to effectively capture the sharp trend change at the onset of odor presentation (much smoother). We are fully committed to updating the manuscript with the revised results. \n\n### W3. Validation and interpretation of the results\nVery good concerns! We'd like to address them from both algorithmic and neuroscientific perspectives.\n\nFirstly, we clarify about the 'ROI'. The selection of the two brain regions was not outcome-driven, but based on prior knowledge: the hippocampus is critical for learning, memory, and decision-making. Despite extensive study, how its neural activity differs between correct and incorrect decisions remains unclear. This dataset offers millisecond-level resolution in neural and behavioral events under a decision task, making it well-suited to explore this. We have revised Fig. 1 to include a panel showing the ROI and electrode locations, clarifying the recording sites. The two selected regions are spatially distant, our aim is to examine whether they communicate, and whether this communication differs by behavioral outcome.\n\nWaveCanCoh was developed to get detailed insights into dependence structure across time and frequency, enabling accurate detection of neural activity differences across behaviors (correct vs. incorrect). As shown in Table 1, following odor presentation ($t = 0$), clear differences emerge between behaviours at specific time points and frequency bands, while no significant differences are seen before $t = 0$, as expected.\n\nOur interpretation is that if significant differences were observed across all frequency bands after $t = 0$, the use of a frequency-domain (wavelet) method would not be particularly necessary. Similarly, if significant differences appeared at all time points within a specific frequency band, the value of a method with fine temporal localization would be diminished (which means classic Fourier based method can do this). The results demonstrate that it is necessary to have a method like WaveCanCoh that can capture detailed information, specifically, which frequency components differ and when they differ between behavioral conditions. We believe this capability is essential for advancing our understanding of the neural mechanisms underlying behavior and task performance. For example, as shown in Table 1, at scale $j = 5$ and $t=0.5$, the canonical coherence between the selected brain regions is significantly higher in correct trials compared to incorrect trials. This suggests that weaker communication between these regions may contribute to incorrect decisions. A reasonable interpretation is that the interaction between these regions may play a crucial role in decision-related processes involved in the task. \n\nTo further validate our findings and assess the reliability of the algorithm, we conducted a permutation test on simulated data. Two datasets with different spectral structures were generated: one matching Simulation 1 (Figure 2, left) and another with a slight modification ($c(u) = 2$ for $u \\in (0, 1)$; see Appendix C.1), introducing coherence differences in the first half of the time period while keeping the second half identical. We applied the permutation test (Algorithm 2) at several time points to test $H_0$: no difference in canonical coherence, versus $H_1$: significant difference. Based on 1000 replicates, the test yielded a Type I error rate of 2.1\\% and a Type II error rate of 1.8\\% (i.e., 98.2\\% power), confirming its validity and supporting the reliability of the real-data results. Moreover, we also applied  the permutation test on proposed Fourier-based benchmark . We found that it was mostly unable to detect differences between correct and incorrect trials. This suggests that the low temporal resolution inherent in the moving-window approach may obscure meaningful temporal dynamics, potentially leading to incorrect interpretations of the underlying neural mechanisms.\n\nFinally, from a neuroscience perspective, the results are consistent with prior research using the same dataset in which correct and incorrect decisions could be differentiated using the spiking activity data (Allen et al., 2016; Shahbaba et al., 2022). We extend prior results by achieving similar differentiation using LFP data alone—a noisier and more challenging signal to classify. Notably, the distinction between trial types discovered by the model within the 15.625 - 31.25 Hz frequency range is in line with previous work showing learning-related signals in a similar band (Gattas et al., 2022). Further, a key innovation of the present method is that it can take the spatial (anatomical) location of LFP electrodes into account. This advance can provide novel insights into the specific contributions of sub-networks within a brain region or circuit, as demonstrated here for the hippocampus, which can then be mapped onto known characteristics of the network such as anatomical gradients.\n\n### W4. Limitations of the proposed method\nPlease refer to W1. of responses to the reviewer HjCP, thanks for your understanding.\n\n## Question\n### Q1. The use of the term 'clusters’\nThank you for your suggestion, we will revise this to 'sets' to make it clearer.\n### Q2. Bold symbol inconsistency\nThanks for pointing this out, it will be fixed.\n### Q3. Interpretation for $A_{jl}$\nSorry for the lack of further information. The wavelet periodogram provides an asymptotically unbiased estimator for a linear combination of the true spectral quantities across scales. The matrix ${A}$, whose $(j,l)$ entry is $A_{jl}$, encodes the bias correction needed to recover a well-behaved estimate of the true spectra at scale $j$ from the set of smoothed raw periodograms across scales $\\{\\tilde{\\mathbf{I}}_{l, k}\\}$, at each time point ($k/T$). This leads to an asymptotically unbiased spectral estimator as given in equation (14).  \n\nSpecifically, $A_{j l}=<\\Psi_j, \\Psi_l>=\\sum_\\tau \\Psi_j(\\tau) \\Psi_l(\\tau)$ for $j, l \\in \\mathbb{N}$ where $\\Psi_j(\\tau) = \\sum_k \\psi_{j, k} \\psi_{j, k-\\tau}$ is the discrete autocorrelation wavelet. \n\n### Q4. Wavelet used for Section 6\nSorry for this omission. The wavelet used here is the Db-4,  which is a robust choice for extracting varying signal features (refer to W1). \n\n### Q5. Multiple comparisons in data analysis\nWe fully agree that accounting for multiple comparisons is an important aspect, with one being potentially interested in more or less conservative testing, as dictated by the context and analysis aims. Our permutation test results in Table 1 report the p-values, and the significance ($^\\star$) follows standard convention. In the light of your suggestion, we will also comment on the results associated to having applied a Bonferroni correction, with the number of tests determined by the 5 scales and 4 time points of interest. We note that with one exception ($t^\\star=1.0$ at scale 6), all other significant differences still hold under this conservative regime. \n\n***Thank you again*** for your thoughtful feedback. We hope our responses provide clarity and welcome further discussion."}, {"review_id": "IPSBhlq6mJ", "reviewer": "Reviewer_HjCP", "summary": "In this work, the authors describe a new framework which uses wavelets to measure canonical coherence between sets of multivariate time series. Their approach, called wavelet canonical coherence (WaveCanCoh) builds from the local wavelet spectral (LWS) matrix along with eigenvector analysis typically seen in canonical correlation analysis (CCA) in order to quantify associations given in terms of spectral associations, rather than standard temporal associations.  The authors demonstrate their method on simulated time series and real local field potential (LFP) data.", "questions": "1. CCA has seen quite a few applications in fusion across imaging modalities (e.g. fMRI+EEG). Could the proposed method be used for multi-modal fusion of time-series data with different sampling rates? I fear the scale-specifity of the method might limit this potential?\n2. How does the provided estimation procedure scale in terms of time and memory complexity?  How does that compare to standard CCA? Eigenvalue decompositions in particular are notoriously intensive both in time and memory. \n3. How do the asymptotic guarantees given for the WaveCanCoh estimator apply in practice? Time time points and smoothing needing to approach infinity doesn’t provide a good idea of the practical upshot of longer sequences or increased smoothing.", "limitations": "As stated earlier, I cannot find any discussion of the limitations in this work except for one sentence in the appendix. My questions regarding computational complexity and scale-specificity hint at some possible limitations that may arise, but a more thorough discussion of limitations in the main body of the text is required.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "# Quality\nOverall, the quality of this work is good. The work is technically sound, rigorous, and correct as far as I have found. Limitations are not provided, which lowers this to a good for me, but I am willing to change this if I missed them or they are included in a revision. \n## Strengths\nTo the best of my knowledge, the theoretical framework and its associated proofs are all technically sound. The provided experiments are rigorous and demonstrate the benefits of the proposed method.\n## Weaknesses\nAlthough the authors claim to discuss limitations, I could not find the referenced material anywhere in the text or appendix. \n\nAdditionally, the related work section is limited in its discussion of actual related literature. Although the authors are right and I know of no exactly analogous methods which do CCA as such with Wavelets, there are plenty of techniques that use the frequency domain or wavelet-based transformations to study associations between time-series, especially in neuroimaging e.g. [1,2,3,4]. These methods (and others) are indeed related—at least in spirit—and ought to be mentioned. I think using the related work section to introduce the theoretical background is an unconventional choice. \n\n[1] Skidmore, F., et al. \"Connectivity brain networks based on wavelet correlation analysis in Parkinson fMRI data.\" Neuroscience letters 499.1 (2011): 47-51.\n\n[2] Ghuman, Avniel Singh, Jonathan R. McDaniel, and Alex Martin. \"A wavelet-based method for measuring the oscillatory dynamics of resting-state functional connectivity in MEG.\" Neuroimage 56.1 (2011): 69-77.\n\n[3] Sato, Joao Ricardo, et al. \"A method to produce evolving functional connectivity maps during the course of an fMRI experiment using wavelet-based time-varying Granger causality.\" Neuroimage 31.1 (2006): 187-196.\n\n[4] Tan, Qitao, et al. \"Frequency‐specific functional connectivity revealed by wavelet‐based coherence analysis in elderly subjects with cerebral infarction using NIRS method.\" Medical physics 42.9 (2015): 5391-5403.\n\n# Clarity\nOverall, the clarity of the work is good. I am willing to raise this score if my concerns are addressed.\n## Strengths\nOverall, this work is well-structured and well-written. I have no issues with the structure of the paper or the writing quality.\n## Weaknesses\nMany of the figures have been scaled down to fit the space. While this is fine, much of the text in the figures (especially in the legends) is quite small and difficult to read when printed. For example: \n* The text of the legend in figures 2 and 3 is especially hard to read. \n* Figure 4 is clearer, but larger text or scaling up the figure would be an improvement.  \n* Figure 10 in the appendix also has quite small text, especially “Odor presentation”.\n# Significance\nOverall, the significance of this work is excellent. Good work!\n## Strengths\nI believe this method holds great promise for applications in M/EEG and other neuroimaging analyses. Upon reading this paper, I immediately want to try to compare this method to other metrics for studying associations in EEG/MEG, such as wPLI. I am especially excited to see if this approach could be used for multi-modal fusion of time-series from different acquisitions.\n## Weaknesses\nI have no weaknesses to discuss in terms of significance.\n# Originality\nOverall, I would call the originality of this work fair to good (I am going to put fair on my score, but I am leaning toward good). I am willing to improve this score if the authors ground their work in the literature and provide a convincing argument for why this method stands out from other wavelet-based connectivity approaches.\n## Strengths\nAs far as I have found, this work is original in terms of performing CCA to find multivariate time-series associations via wavelets.\n## Weaknesses\nDespite the apparent originality, the authors have neglected to ground the approach in other wavelet-based measures for brain connectivity. This work would be significantly improved by a more thorough grounding in the literature.", "was_revised": true, "final_justification": "I would recommend this work for acceptance. I was initially quite happy with the quality of this work, and all of my concerns have been adequately addressed. The author responses have been more than sufficient, and I would recommend that this work be accepted. I do agree with some of the negative reviews that larger-scale experiments would improve the quality of the paper; however, I do believe the quality of the experiments and other work is sufficient and not a significant detractor from the overall impact of the work.", "rebuttal": "Thank you for your detailed and constructive review. We truly appreciate your deep understanding of the field, and many of your comments have been very insightful for us. We are grateful for your recognition of the novelty and practical relevance of our work. Our intention was to develop a framework that is both simple and novel, yet highly useful in practice. In the following, please allow us to respond and clarify the concerns you have raised.  \n\n## Responses to the weakness\n### W1. Discussion on the limitations\nWe stated the limitation of our approach in Appendix D.3 (lines 497–500) as you pointed out, although we acknowledge that due to our concern for the strict space constraints, the discussion was very succinct leading to a lack of clarity for which we apologise. We will revise this part and include a dedicated paragraph on method limitations in the main text. Below, we outline the discussion we plan to further incorporate. \n\nWhile our WaveCanCoh framework provides a robust, nonparametric approach for quantifying scale-specific time-varying canonical coherence between two sets of nonstationary multivariate time series, it is inherently limited to a fixed set of wavelet scales. This restricts its flexibility in applications requiring precise frequency localization or alignment with arbitrary bands. As shown in Figure 7, the mapping between scales and true frequency depends on the sampling rate and signal spectrum. For signals with broad frequency content or low sampling rates, certain bands may be poorly resolved. One practical solution is downsampling when a high sampling rate is available, allowing better alignment between scales and target frequency bands. However, due to time-frequency trade-offs in wavelet analysis, fully flexible frequency resolution remains challenging. Future work may consider adaptive or overcomplete wavelets to improve frequency targeting while retaining nonstationary modeling capabilities.\n\n### W2. Related literature review\nWe agree with your suggestion that more background on previous time series studies using wavelets, as well as a clearer articulation of the importance and novelty of our contribution, would strengthen the paper. In the revised version, we will provide a more comprehensive and structured review of relevant literature to better position our work in the existing body of research. To briefly clarify the key novelty of our work: while prior methods such as wavelet coherence focus on dependencies between pairs of univariate signals, our framework—Wavelet Canonical Coherence (WaveCanCoh)—is the first to define and estimate canonical coherence between two sets of nonstationary multivariate time series in a  ***joint*** time- and scale-localized manner. This enables us to capture interpretable, global inter-group interactions and their dynamics, rather than just localized pairwise associations. Our approach extends canonical coherence analysis into the wavelet domain, providing a principled framework for nonstationary dependence modeling at the group level—something that existing methods, to our knowledge, do not address. As you suggested, we will discuss a more complete set of references to related studies in the revised version of the paper to better place our contribution within the existing literature. \n\n### W3. Label visibility for some figures\nWe acknowledge that the font size in several of the figures you mentioned is a bit small and may be difficult to read especially when relying on a printed version. We sincerely apologize for this and fully commit to increasing the font size in the revised version to improve legibility. We will also ensure that the quality of all figures throughout the paper meets a consistent and readable standard.\n\n## Questions\n### Q1. Application to multi-modal fusion\nDue to its flexibility, this framework can be easily applied on the signals obtained from other modalities, e.g., fMRI, MEG and EEG. In fact, we have already implemented this method on EEG data from subjects with and without ADHD data, and obtained neurologically promising results. In this paper, we chose to use the LFP data because it has typical nonstationary features, which allow us to illustrate both the motivation and novelty of our proposed approach. Although we have not experimented with it ourselves, we believe the method can also be used for multi-modal fusion of time series data with different sampling rates, as these will be appropriately captured by the spectral representation and subsequently formed quantities.\n\nSpecifically, when applied to multi-modal data, the WaveCanCoh framework can be adapted to analyze cross-modal interactions by treating each modality as a component in the multivariate signal sets. Suppose we have $\\mathbf{X}_t = (X^{(1)}_t, \\ldots, X^{(P)}_t)^\\top$ denoting signals from modality 1 (e.g., EEG) and $\\mathbf{Y}_t = (Y^{(1)}_t, \\ldots, Y^{(Q)}_t)^\\top$ denoting signals from modality 2 (e.g., MEG or fMRI). Scale-specificity would not be a problem here, but we should be careful to match the scales and frequency bands when the aim is to draw conclusions in the Fourier frequency domain. For example, if the sampling rate of $\\mathbf{X}_t$ is $20Hz$ and that of $\\mathbf{Y}_t$ is $10Hz$, then WaveCanCoh between $\\mathbf{X}_t$ and $\\mathbf{Y}_t$ at scale $j=1$, $\\boldsymbol{\\rho}_j (t)$ represents the time dependent coherence between the components of $\\mathbf{X}_t$ at $5-10Hz$, and the components of $\\mathbf{Y}_t$ at $2.5-5Hz$. Furthermore, this may also provide a way to identify how fluctuations in longer-term dynamics (low frequency components) of signal from modality 1 may have an impact on the amplitude of the shorter-term dynamics (high frequency components) of modality 2. The canonical directions obtained through WaveCanCoh estimation will therefore highlight maximally correlated linear combinations of the channels from each modality, thus allowing the framework to capture joint dynamics that may not be apparent in direct pairwise analysis. In summary, we believe multi-modal fusion of time series data with WaveCanCoh is feasible, easy to implement and may provide subtle information on joint dynamics. \n\n### Q2. Computational complexity of the method\nThe estimation of spectrum step leads to a time complexity $\\mathcal{O}(JT(P+Q)^2)$ (where the number of scales $J$ is used most commonly as $J=\\log_2(T)$). The eigenvalue decomposition computation step takes the total time complexity to $\\mathcal{O}(JTd^3 )$ (let $P \\approx Q =d$), to be compared to that of the standard CCA,  $\\mathcal{O}(T d^2 + d^3)$. The proposed algorithm is naturally more expensive than standard CCA, since it is designed to obtain time-localized and scale-specific results. In our practical experience with WaveCanCoh, the spectral estimation step is extremely fast, with virtually all of the computational time being spent on the eigenvalue decomposition step. Although compared to standard CCA, our method produces a set of results at each time point, resulting in increased computational burden, in our experiments with $T = 1024$, a single replicate is completed within 2.5 seconds on a standard personal computer (Apple Mac, 16GB RAM, 6-core CPU) without resourcing to parallel computing or to use a cluster. Therefore, we believe the computational expense would not be a major concern, even in large-scale experiments. \n\nMoreover, computations for WaveCanCoh can be streamlined by storing canonical vectors and coherence values at each time and scale, with the requirement being $\\mathcal{O}(JTd^2)$, and the memory complexity $\\mathcal{O}(d^2)$ for standard, one time point global estimate.\n\n### Q3. Asymptotic guarantees for the WaveCanCoh estimator\nFirstly, we would like to clarify that the conditions assuming $M, T \\to \\infty$ with $M/T \\to 0$ are theoretical requirements to rigorously guarantee the consistency of the estimator towards the true, unknown quantity. In practice, a commonly recommended choice for the smoothing parameter is $M = \\lfloor \\sqrt{T} \\rfloor$, where $\\lfloor \\cdot \\rfloor$ denotes the floor function. For example, in our experiments with $T = 1024$, we used $M = 32$, a value is that is in line with those reported in the related wavelet process literature.\n\nWe understand your could concern regarding the convergence of the estimator for low temporal resolution data (e.g., fMRI). As a response, we would like to highlight the results shown in Figure 2 (left), where the convergence of the estimator was empirically verified. In our view, both $T = 1024$ and $M = 32$ are moderate in size, and the estimator performs reliably under these conditions. Moreover, based on our prior experience with this type of wavelet spectral estimation, we have observed stable convergence even with much (8-fold) smaller values of $T$ and $M$. Thus, based on both theoretical and practical arguments, the WaveCanCoh estimator remains practically effective unless the time series is extremely short (e.g., $T = 20, 30$).\n\nIf further improvements in convergence are desired for shorter time series, a practical strategy is to apply periodic extension to the original time series to estimate the local spectral matrix, and then to perform the eigenvalue decomposition step using only the original time segment. Since the computational cost of spectral estimation is low, this approach offers a feasible way to improve performance without incurring significant computational overhead.\n\n***Once again***, we sincerely thank you for your insightful comments and the time you dedicated to reviewing our work. We hope that our responses above have fully addressed your concerns. If you have any further questions or suggestions, please do not hesitate to let us know—we would be more than happy to continue the discussion in the next stage."}, {"review_id": "jOo8tfwoR4", "reviewer": "Reviewer_Jb5K", "summary": "This paper introduces scale-specific wavelet canonical coherence (WaveCanCoh), a new framework to measure time-varying coherence between two clusters of multivariate time series. The authors target the problem of analyzing dynamic dependencies between groups of signals in a nonstationary setting. The main contributions of this paper include defining this new measure of coherence, developing a theoretically grounded algorithm for its estimation, and demonstrating its utility on simulated data. Overall, this paper will have broad potential in certain specific time series applications, especially in non-stationary time series data such as EEG. Since the Fourier transform is widely used on some time series, the wavelet-based approach proposed by the author is interesting.", "questions": "Does the effect of WaveCanCoh depend on frequency band design? For example, does the high-frequency part lead to low correlation?", "limitations": "The authors have addressed potential limitations.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength\n\n1/ The paper is generally well-written and organized. The paper thoughtfully provides a detailed introduction to the technical background used, making this paper highly readable. \n\n2/ The use of wavelet models to calculate the correlation of time series is relatively rare in ML research. Therefore, this work may provide new insights and vitality to time series researchers in the ML community.\n\n3/ WaveCanCoh has strong innovation, extending Canonical Coherence to both time and frequency domains for the first time. This enables the model to extract non-stationary dependencies from some EEG signals.\n\n4/ The paper’s technical quality is excellent. It provides a rigorous theoretical development and backs it up with thorough experiments.\n\nWeakness\n\n1/ I personally is not entirely sure how relevant this topic is to the NeurIPS audience. But I am open to be convinced otherwise.\n\n2/ The manuscript includes a large amount of notation in the paper, which may not be friendly for the researchers to follow.\n\n3/ Please add a citation where LSP first appears in the paper, or at least let readers know that the Fourier-based approach you mentioned above is LSP.", "was_revised": true, "final_justification": "The authors have addressed my questions. But I think my original rating is a fair grading.", "rebuttal": "Thank you very much for your detailed and professional review. Your comments have been extremely helpful in improving the quality of our paper. We also sincerely appreciate your recognition of the novelty of our work and its potential to inspire research in the machine learning community. Below, please allow us to provide some clarifications and responses to the concerns you have raised. \n\n## Responses to the weakness \n\n### W1.  Relevance to the NeurIPS program and audience\nAs you rightly pointed out, the proposed method is effective in capturing the dependence structure between nonstationary signals. Nonstationarity is a typical characteristic of many types of multi-modal brain signals. Therefore, although the current paper only focuses on LFP data, our method can be directly applied to a wide range of neuroscience related studies, such as brain connectivity and brain-computer interfaces, topics that are central to the NeurIPS community.\n\nWhile we understand that much of the work at NeurIPS is closely tied to deep learning methodologies, we would like to emphasize that the existence of a theoretically justified statistical framework for capturing the dependence structure among multivariate time series is crucial for the success of many deep learning-based approaches. Our method is particularly well-suited for detecting time-varying dependence structures, which are often present in real world data and yet current methodologies are often unable to capture this dynamic aspect.\n\nAs such, we believe our work has the potential to inspire and support the development of deep learning models for tasks such as time series classification, anomaly detection, and forecasting. Based on these reasons, we believe that this work aligns well with the scope and interests of the NeurIPS community in several important ways. \n\n### W2.  Notations in the paper\nThank you for pointing out this issue, you are absolutely right in pointing out that the notation is quite heavy. We have carefully reflected on your point, and while we believe that when we first introduce these quantities we need to preserve all dependencies (such as on the rescaled time $u$ and scale $j$) for mathematical clarity, however we agree that we can subsequently drop some of this cumbersome notation upon clear explanations. We will therefore review the notation used throughout the paper to ensure that each term is clearly defined and properly explained, in order to further improve the readability of the manuscript.\n\n### W3.  About LSP method in the paper\nThank you for raising this important point. We acknowledge that due to our efforts to stick within the mandatory page limit, the presentation of the LSP method in the manuscript has been very succinct and relegated to the Appendix C.2, which may have rightly caused some confusion, hence we appreciate the opportunity to clarify here.\n\nTo the best of our knowledge, although a natural starting point as you have also observed, the LSP-based method used as a benchmark in our paper has not been previously formally developed in the literature for measuring time-varying canonical coherence between two sets of multivariate time series. This approach was specifically designed by us as a natural extension of classical Fourier-based canonical coherence methods, which assume stationarity and yield a global frequency-domain measure that discards temporal information. The absence of a method to track both time and frequency-varying canonical dependence motivated our development of a new, theoretically grounded framework.\n\nOur proposed WaveCanCoh addresses this gap by enabling the estimation of canonical coherence at each individual time point and across wavelet (i.e., frequency) scales. This dual localization contrasts with classical canonical correlation analysis, which provides a single global measure in the time domain, and Fourier-based coherence, which captures only frequency-domain information without temporal resolution.\n\nTo provide a baseline for comparison, we introduce this Fourier-based Locally Stationary Process (LSP) method, which approximates time-varying canonical coherence by segmenting the time series into overlapping blocks and applying smoothing across the estimated canonical coherence values in the frequency domain. We will revise the manuscript to explicitly state that the LSP method is proposed in this work and clarify its role and novelty when it is first introduced.\n\n## Questions\n\n### Q1. Does the effect of WaveCanCoh depend on frequency band design? For example, does the high-frequency part lead to low correlation?\nThank you for this insightful question. We would like to clarify that the effectiveness of WaveCanCoh does not inherently depend on the choice of frequency bands. Instead, the correspondence between wavelet scale $j$ and the actual frequency bands is determined by the sampling rate of the data, as illustrated in Figure 7 of the paper. Once the sampling rate is fixed, the relationship between scale and frequency is uniquely determined and remains unchanged—unless the data is explicitly resampled, which we also discuss as a minor limitation in Appendix D.3 (lines 497–500).\n\nRegarding the correlation in high or low frequency bands, WaveCanCoh does not assume stronger or weaker coherence at any particular frequency range. Rather, it is designed to reveal the scale-specific (i.e., frequency-specific) dependence structure that truly exists in the data. Whether stronger coherence appears in high-frequency or low-frequency components entirely depends on the underlying dynamics of the signals being studied.\n\nFor example, although outside of the NeurIPS scope but still relevant for our methodology, in the context of financial time series, if two markets react rapidly to global news, their hourly price movements may show high coherence—this would be captured in the high-frequency bands. On the other hand, if two assets exhibit similar long-term trends, WaveCanCoh will detect stronger coherence in the low-frequency bands. Similarly, in neuroscience applications, certain types of brain activity or connectivity may manifest predominantly in either the theta or gamma bands, and WaveCanCoh provides a natural way to analyze such dependencies over both time and scale.\n\nIn summary, WaveCanCoh is data-adaptive: it does not bias the narrative towards any particular frequency range, but instead captures whichever scale-specific temporal dependence is present in the signals. This makes it a powerful tool for analyzing a wide range of dynamic multivariate systems.\n\n**Once again**, we sincerely thank you for your positive evaluation of our work. We hope that our responses have helped to clarify the manuscript, and we would be very happy to engage in further discussions with you in the next stage."}, {"review_id": "UADqVpBaZU", "reviewer": "Reviewer_XB1w", "summary": "This paper studies the dependence between two clusters of multivariate time series with canonical coherence in the spectral domain. The authors specifically focus on brain activity signals which are nonstationary. Authors provide experiments and analysis to demonstrate the effectiveness of the proposed work.", "questions": "Please see above", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The topic of studying dependency between multivariate signals is of great interest.\n- Authors provide simulation study as supportive evidence for the proposed work.\n- The paper is well-organized and easy to follow. \n\nWeaknesses:\n- The proposed methodology is not quite innovative in my own opinion. Would be great if author can further emphasize the differences between the proposed method and existing techniques, and highlight the original work. \n- The experiment section lacks supportive evidence from more large-scale experiments. It is suggested to do validations on more datasets especially on real data.", "was_revised": true, "final_justification": "Thanks authors for addressing the questions. After reading comments and justifications, I decide to raise final rating.", "rebuttal": "We sincerely thank you for your thoughtful and detailed comments on our paper，many of your suggestions are highly valuable and have been very helpful in improving the quality of our paper. Please allow us to provide clarifications and responses to the concerns you raised.\n## Responses for raised weakness\n### W1. Innovations of WaveCanCoh\nWe sincerely thank you for raising the concern regarding the novelty of our proposed method. Below, we clarify the key differences between our proposed WaveCanCoh framework and existing methods, emphasizing its theoretical and practical contributions beyond prior wavelet- or CCA-based approaches.\n\n1. Beyond Traditional Canonical Coherence Methods.\n\nClassical canonical coherence analysis (CCA), including its spectral domain formulation [Brillinger, 2001], has been extensively studied under the assumption of stationarity. These approaches inherently disregard temporal variation in the dependence structure, which is particularly limiting in domains like neuroscience or finance where interactions evolve over time. WaveCanCoh directly addresses this limitation by modeling {\\em time-varying} canonical coherence between two sets of nonstationary multivariate time series. It achieves this by embedding CCA in the multivariate locally stationary wavelet (MvLSW) framework, thus subsequently offering scale-specific and temporally localized inference.\n\n2. Distinction from Existing Wavelet-Based Dependence Measures.\n\nWhile wavelet coherence (e.g., Grinsted et al., 2004) has been widely used for pairwise time-frequency analysis, existing work does not extend this to between-group canonical coherence. Previous wavelet-based methods are typically limited to univariate or bivariate signal analysis and do not generalize to multivariate set-to-set dependence. To the best of our knowledge, WaveCanCoh is the first method to combine wavelet-domain multivariate analysis with canonical coherence, enabling estimation of scale-specific, time-varying dependence between signal groups.\n\n3. Advantages Over Fourier-Based CCA Extensions.\n\nSome studies have extended canonical coherence to the spectral domain via Fourier-based techniques. However, such approaches rely on global basis functions (sinusoids), which are not localized in time, making them unsuitable for detecting transient phenomena. Our WaveCanCoh, grounded in compactly supported wavelets, captures local, frequency-specific dependencies with high temporal resolution, which is clearly demonstrated in both simulation and LFP applications (e.g., Figures 2–5). As shown in our simulations, WaveCanCoh successfully detects sharp transitions in coherence that Fourier-based methods fail to resolve (e.g., Figure 2 right panel).\n\n4. Theoretical Innovation and Practical Utility.\n\nOur work rigorously defines localized canonical coherence within the MvLSW framework (Definition 1, equation (8)) and provides consistent estimation and inference procedures (Algorithm 1; Theorem in Appendix B). The approach further extends naturally to causal analysis via lag-specific coherence (Causal-WaveCanCoh), allowing exploration of directed inter-group associations. This extension is also novel and absent in existing CCA or wavelet coherence literature.\n\n5. Applicability to high-dimensional, real-world data.\n\nFinally, our application to LFP data showcases the ability of WaveCanCoh to reveal interpretable and behaviorally relevant coherence patterns across trials, scales, and time (Figure 4–5). Notably, the method identifies distinct coherence structures between correct and incorrect decisions that are missed by traditional methods using moving windows or global spectra.\n\nIn summary, WaveCanCoh represents a novel and substantial advancement over existing methods by:\n\n(1) Extending canonical coherence to the nonstationary, multivariate, time-frequency domain; (2) Providing localized, scale-specific inference within the rigorous MvLSW framework; (3) Enabling causal (lag-specific) coherence analysis; (4) Demonstrating both theoretical soundness and empirical superiority in detecting dynamic group-level dependencies.\n\nWe hope this clarifies the unique contributions of our work. In the light of your comments, we will improve our literature review in the revised version to highlight the novelties of our work.\n\n### W2. Large-scale experiments with real data\nThank you for the thoughtful suggestion regarding the need for larger-scale experiments and additional real-world validations. We fully agree that broader empirical evaluation can further strengthen the practical impact of our proposed method. Below, we would like to respectfully reclarify the experiment setting in our work.\n\nOur experimental design was guided by the goal of providing a comprehensive yet focused validation of WaveCanCoh from both theoretical and practical standpoints. The simulation section includes two carefully constructed scenarios: one that adheres to the assumptions of the multivariate locally stationary wavelet (MvLSW) framework and another based on a more general class of nonstationary AR(2) mixtures without any wavelet structure. This dual setup demonstrates not only that the method performs as expected under ideal conditions, but also that it remains robust when the assumptions are not adhered to, which is critical for assessing real-world applicability.\n\nOn the real data side, we applied WaveCanCoh to high dimensional hippocampal LFP recordings during a memory guided behavior task. This dataset was chosen because it presents a prototypical example of multivariate nonstationary brain signals with complex temporal dynamics and behavioral variation. The analysis includes multiple channels grouped into anatomical subregions, comparisons across experimental conditions (correct vs. incorrect decisions), and multiscale frequency-resolved coherence patterns. As shown in Figures 4–5, the method was able to extract interpretable, scale-specific coherence signatures linked to cognitive performance, which would be difficult to detect using traditional approaches.\n\nWe fully acknowledge the value of validating our method across a broader range of real-world datasets and plan to extend our work in this direction. Nevertheless, we believe that the current experimental results already provide strong evidence for the effectiveness, robustness, and interpretability of the proposed framework across both synthetic and real data settings.\n\nWe hope this clarifies our rationale and appreciate the opportunity to discuss potential extensions.\n\n***Thank you again*** for your valuable feedback. We look forward to continuing the discussion with you in the next stage, and please do not hesitate to let us know if you have any further concerns."}]

Author rebuttal:

Meta-review: This submission proposes a novel and theoretically sound method, Wavelet Canonical Coherence (WaveCanCoh), for analyzing dependencies in non-stationary multivariate time series, offering a significant advancement over existing techniques. While initial reviews raised valid questions regarding novelty and experimental breadth, the authors have convincingly addressed these concerns through detailed rebuttals and clarifications, resulting in significantly raised scores from multiple reviewers (a shift from borderline to positive recommendations). The work demonstrates a solid theoretical foundation, validated through both simulation and real-world data, and presents a promising approach for fields like neuroscience where analyzing dynamic signal relationships is crucial. Given the robust methodology, thoughtful responses to reviewer feedback, and potential impact, I recommend acceptance.
Decision: Accept (spotlight)