Regularised Spectral Estimation for High-Dimensional Point Processes

Summary

Advances in modern technology have enabled the simultaneous recording of neural spiking activity across large numbers of neurons, which statistically can be represented by a multivariate point process. We characterise the second order structure of this process via the spectral density matrix, a frequency domain equivalent of the covariance matrix. In the context of neuronal analysis, statistics based on the spectral density matrix can be used to infer connectivity in the brain network between individual neurons. However, the high-dimensional nature of spike train data mean that it is often difficult, or at times impossible, to compute these statistics. To improve the efficiency of spectral estimation for point processes, we propose methodology that combines a Whittle pseudo-likelihood with ridge or lasso style penalties. We establish asymptotic and large sample properties for our proposed estimators and evaluate their performance on synthetic data simulated from multivariate Hawkes processes. Finally, we apply our methodology to neuroscience spike train data in order to illustrate its ability to infer brain connectivity.

1. Introduction

We consider the $p$-dimensional point process $N(t)≔{\left\{N_q(t)\right\}}_{q\in \{1,\dots ,p\}}$ whose $q^{\mathit{\text{th}}}$ component gives the number of events of type $q$ that have occurred in the time interval $(0,t],t\le T$. As in Bartlett (1963), we denote by $d{N}_q(t)$ the number of events observed in process $q$ in some small time interval $dt$, whereby $d{N}_q(t)=N_q(t+dt)-N_q(t)$. The first order properties of $N(t)$ are characterised via the intensity function $\Lambda (t)\in {\mathbb{R}}^p$, defined for each unit $q$ as ${\Lambda }_q(t)dt=\text{Pr}\left(d{N}_q(t)=1\right)$, or equivalently $\Lambda (t)≔E\{dN(t)\}/dt$.

Hawkes (1971a) introduced an important class of point process models in which past events of $N(t)$ can affect the evolution of the process. In particular, the intensity function for each unit $q$ takes the form

$${\Lambda }_q(t)=v_q+\sum _{r=1}^p{\int }_{-\infty }^t{\gamma }_{qr}(t-u)d{N}_r(u),$$

where $v_q>0$ and ${\gamma }_{qr}(\cdot )$ represent the background intensity and excitation functions respectively. The excitation function, ${\gamma }_{qr}(t-u)$ represents the effect on the $q^{\mathit{\text{th}}}$ type at time $t$ due to an event in the $r^{\mathit{\text{th}}}$ type at a previous time point, $u$. Motivated by neuroscience applications, the multivariate Hawkes process will serve as an illustrative throughout this paper.

The second order structure of $N(\cdot )$ at times $t$ and $u$ is given by the covariance density matrix,

$$\mu (t,u)=\frac{E\left\{dN(u)d{N}^{\top }(t)\right\}}{\mathit{\text{dtdu}}}-\Lambda (u){\Lambda }^{\top }(t),$$

where the superscript $(\cdot {)}^{\top }$ denotes transpose. When $\Lambda (t)$ is constant for all $t$, and $\mu (t,u)$ depends only on lag $\tau =u-t$, we refer to process $N(t)$ as being second-order stationary. In this case, we denote the intensity function as $\Lambda (t)=\Lambda$ and the covariance density matrix simply as $\mu (\tau )$.

In this work, we characterise the second order structure of $N(t)$ via the spectral density matrix, building on the frequency domain representation of the stochastic process (Bartlett, 1963; Brillinger, 1972). The spectral density matrix $S(\omega )$ of a stationary point process is defined as the Fourier transform of its complete covariance density matrix, namely

$$S(\omega )=\frac{1}{2\pi }\left\{\text{diag}(\Lambda )+{\int }_{-\infty }^{\infty }\text{exp}(-i\omega \tau )\mu (\tau )d\tau \right\},$$

where $S(\omega )$ is a $p\times p$ Hermitian positive definite matrix. Analogous to the covariance matrix in the time domain, the spectral density matrix captures both the within and between type dynamics of the multivariate point process. More specifically, it describes the variance in each process or the covariance between processes that is attributable to oscillations in the data at a particular frequency $\omega \in \mathbb{R}$.

Given the complex nature of the spectrum, a common measure of association between two processes $N_q(t)$ and $N_r(t)$ is the squared spectral coherence, $R_{qr}^2(\omega )≔{\left|S_{qr}\left(\omega \right)\right|}^2/S_{qq}(\omega )S_{rr}(\omega )\in [0,1]$. If $R_{qr}^2(\omega )=0$ for all $\omega$, then the two processes are linearly independent. Conversely, a value of 1 indicates a perfect linear relationship. However, in the context of high-dimensional point processes, it is important to assess whether the association between $N_q(t)$ and $N_r(t)$ is due to a direct connection between them, or whether their correlation can be attributed to an indirect connection involving other processes, i.e. whether this correlation could potentially be explained away through conditioning on other streams. In this work, we address this problem by studying the inverse spectral density matrix $\Theta (\omega )≔S(\omega {)}^{-1}$, whose $(q,r{)}^{\mathit{\text{th}}}$ entry encodes a measure of association between $N_q(t)$ and $N_r(t)$ after the linear effects of the multivariate process $N_{\setminus qr}(t)$ have been subtracted (Dahlhaus, 2000; Eichler et al., 2003). Specifically, the squared partial coherence between two processes $N_q(t)$ and $N_r(t)$ at a particular frequency $\omega {\rho }_{qr}^2(\omega )≔{\left|{\Theta }_{qr}(\omega )\right|}^2/{\Theta }_{qq}(\omega ){\Theta }_{rr}(\omega )$, provides a normalised measure on [0, 1] of the partial correlation between pairs of processes in the frequency domain.

In the time series literature, partial coherence has been used in a variety of application areas, from medicine and oceanography to climatology (Kocsis et al., 1999; Koutitonsky et al., 2002; Song et al., 2020). In the point process setting, partial coherence has been used to identify functional neuronal connectivity from simultaneously recorded neural spike trains (see, for example, Dahlhaus et al., 1997; Eichler et al., 2003). However, these existing spectral estimation methods for multivariate point processes are not suitable for modern high-dimensional spike train data. In particular, the number of recorded neurons $p$ is often large in comparison to the number of experimental trials, which we consider as a monotonically increasing sequence $m_T$. In this setting, see Figure 1c as an example, classical periodogram based estimators exhibit a very large condition number and thus the estimation of the inverse can be highly susceptible to sampling noise. Our main goal in this paper is to better understand the behaviour of the periodogram in high-dimesional settings (for point processes), and then apply regualrisation methods to strike a bias-variance trade-off enabling more efficient estimation of $\Theta (\omega )$, even when $p\ge m_T$.

Fig. 1:

(a) Empirical (histogram) vs theoretical (solid line) distribution of coherence for $p=7$. (b) Plot of the $l_{\infty }$ norm of spectral matrix estimation error, the dashed line indicates when $p=m$. (c) Plot of the condition number of the multivariate spectrum as a function of dimensionality $p$.

The estimation of high-dimensional (inverse) spectral density matrices for time series data has received significant attention in the statistics literature. For example, Böhm & von Sachs (2009); Fiecas & Ombao (2011) and Fiecas & von Sachs (2014) explore shrinkage estimators for the spectral density matrix, leading to numerically stable and therefore invertible estimates. Comparatively, Jung et al. (2015); Tugnait (2022); Dallakyan et al. (2022) obtained estimates of the inverse spectral density matrix directly by optimising a penalised Whittle likelihood. There, a group lasso penalty is used to capture shared zero patterns across neighbouring frequencies in the spectral domain. More recently, Deb et al. (2024) used a penalised Whittle likelihood to estimate the inverse spectral density matrix at a single frequency, and developed a fast path-wise coordinate descent algorithm for use in the time series setting. The Whittle approximation has also been studied in the Bayesian framework; see for example Tank et al. (2015).

In this work, we also estimate the inverse spectrum directly using a penalised Whittle likelihood. However, a key challenge in our point process framework lies in controlling the error for a preliminary multivariate estimate of the spectral density matrix (e.g. the periodogram), which is used as input to the Whittle likelihood. We establish asymptotic properties of a mean-corrected version of the periodogram under classical mixing conditions (Brillinger, 1972) in a multi-trial design setting relevant to our neuroscience application. To enable a truly high-dimensional analysis of our proposed estimators, we then study the finite sample properties of the periodogram for the multivariate point process under a weakly-mixing framework, as developed in Doukhan & Louhichi (1999); Doukhan & Neumann (2007). In terms of regularisation, we consider both a ridge type penalty (shrinking all elements of the inverse spectra) and a lasso type penalty. The latter imposes a sparsity assumption on the inverse spectral density matrix, easing interpretation of the spectrum, and enabling association of the point process with a form of graphical model. To the best of our knowledge, we believe this work is the first to consider regularisation of the Whittle likelihood in the context of multivariate point processes.

2. Preliminaries

2.1. The Tapered Fourier Transform for Point Processes

A classical non-parametric estimator of the spectral density matrix for a stationary point process is based on the periodogram, that is a (Hermitian) outer-product of the Fourier transform of the data (Bartlett, 1963). This estimate is often improved upon by applying a taper to the data prior to taking the Fourier transform (Percival et al., 1993). Here, we propose to utilise a variant of the multitaper periodogram (2) to estimate the spectral density matrix. However, as the periodogram is based on the Fourier transform, it is first useful to understand some properties of this object in relation to the (continuous time) point process setting.

As in regular time series analysis, different choices of taper functions may be useful for different tasks, each having their own concentration properties in the Fourier domain. For the purposes of our discussion, we will focus on the canonical choice of $k=1,\dots ,m_T$ non-overlapping tapers that act to segment the process $N(t)$ into $m_T$ intervals of equal length $n_T$, where $m_T$ is a monotonically increasing sequence in $T$. This aligns with the experimental motivation for our methodology, where we can attribute a single taper to each of the $m_T$ trials in our neuronal-spiking experiment. For simplicity, we choose $h_k(t/T)={\left(T/2\pi z_T\right)}^{1/2}$ where $\text{supp}\left(h_k\right)\cap \text{supp}\left(h_l\right)=\varnothing ,\forall k\ne l$ and $\left|\text{supp}\left(h_k\right)\right|=z_T$ for all $k=1,\dots ,m_T$, where $z_T≔n_T/T$. However, the framework is extensible to both other choices of taper function, and differing trial lengths.

Definition 1. The tapered Fourier Transform of the process $N_q(t)$ for $t\in (0,T]$ is defined as

$$d_{T;k;q}\left(\omega \right)≔{\int }_0^T{h}_k\left(t/T\right)\text{exp}\left(-i\omega t\right)d{N}_q\left(t\right).$$

An alternative view of the Fourier transform is given by the random set of events induced by the point process. That is, instead of the counting process representation, we can consider the events $E_{k,q}=\left\{t\mid d{N}_q(t)=1,t/T\in \text{supp}\left(h_k\right)\right\}$. In this case, we can write

$$d_{T;k;q}(\omega )=\sum _{t\in E_{k,q}}{h}_k(t/T)\text{exp}(-i\omega t).$$

As the rate of the point process is non-zero and positive, it is important to remove this mean prior to obtaining estimates of the spectrum (Jarvis & Mitra, 2001; Rajala et al., 2023). Therefore, we will work with the mean-corrected variates (1) for the remainder of the paper.

Definition 2. The mean-corrected tapered Fourier Transform is defined as

$${\bar{d}}_{T;k;q}\left(\omega \right)≔d_{T;k;q}(\omega )-d_{T;k;q}(0)\frac{H_k(T\omega )}{H_k(0)},$$ (1)

where $H_k(\omega )$ denotes the Fourier transform of $h_k(z),i.e.H_k(\omega )=\int h_k(z)\text{exp}(-i\omega z)dz$.

At this point, it is useful to introduce further conditions on $N(t)$, enabling further exploration and understanding of the asymptotic ($T\to \infty$) behaviour of the Fourier transform.

Assumption 1. Assume the $p$-dimensional point process $N(t)$ is strictly stationary, i.e. the $j^{\mathit{\text{th}}}$-order cumulant

$$C_{q_1,\dots ,q_j}\left(t_1,\dots ,t_j\right)d{t}_1\dots d{t}_j=\text{cum}\left\{d{N}_{q_1}\left(t_1\right),\dots ,d{N}_{q_j}\left(t_j\right)\right\},$$

is invariant to shifts in time. Let us define the reduced cumulant density $r_{q_1,\dots ,q_j}\left(u_1,\dots ,u_{j-1}\right)=C_{q_1,\dots ,q_j}\left(u_1,\dots ,u_{j-1},0\right)$, then we further assume all moments exist, and that

$${\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }\left|r_{q_1,\dots ,q_j}\left(u_1,\dots ,u_{j-1}\right)\right|\text{d}{u}_1\cdots \text{d}{u}_{j-1}<\infty ,$$

holds for any $q_1,\dots ,q_j=1,\dots ,p$ and all $j\ge 2$. Finally, we assume the covariance density satisfies

$${\int }_{-\infty }^{\infty }\left|u\right|\left|r_{q,r}(u)\right|\text{d}u<\infty ,$$

for all $q,r=1,\dots ,p$.

The above conditions amount to a form of mixing condition (Assumption 2.2 Brillinger, 1972) which ensures that the dependence of the point process decays at a sufficient rate to enable a central limit-theorem on the mean corrected Fourier transform.

2.2. The Multi-Trial Periodogram

We now use this classic result to motivate studying the multi-trial periodogram

$${\hat{S}}_{m_T}\left(\omega \right)≔\frac{1}{m_T}\sum _{k=1}^{m_T}{\bar{d}}_{T;k}(\omega ){\bar{d}}_{T;k}^H(\omega ).$$ (2)

Under Assumption 1, the estimator in (2) is an asymptotically, $p$ fixed, $T\to \infty$, consistent estimator for the spectrum. In the sequel we will be interested in the finite sample properties of (2). However, we first recall some properties of ${\hat{S}}_{m_T}(\omega )$ as $T\to \infty$. In the context of our neuronal-spiking experiment, this corresponds to a setting in which we have a fixed number of infinitely long trials, i.e., $m_T=m$ and $n_T\to \infty$ as $T\to \infty$.

Using Assumption 1 and Theorem 4.2 of Brillinger (1972), one can show that ${\left\{{\bar{d}}_{T;k}(\omega )\right\}}_{k=1}^m$ are asymptotically independent complex Gaussian, and thus the multi-trial periodogram is distributed as a Complex Wishart distribution, i.e., ${\hat{S}}_m(\omega )~m^{-1}{𝒲}_p^C(m,S(\omega ))$ with $m$ degrees of freedom and centrality matrix $S(\omega )$. As a further corollary of the asymptotic complex normality of ${\left\{{\bar{d}}_k(\omega )\right\}}_{k=1}^m$, we have (Goodman, 1963) that if $p<m$, then the estimated squared spectral coherence ${\hat{R}}_{qr}^2(\omega )={\left|{\hat{S}}_{qr}(\omega )\right|}^2/\left\{{\hat{S}}_{qq}(\omega ){\hat{S}}_{rr}(\omega )\right\}$ has the density function

$$f_{{\hat{R}}_{qr}^2}(x)=(m-1)\left(1-R_{qr}^2\right){\left(1-x^2\right)}^{(m-2)}{{}_{2}F}_1\left(m,m;1;R_{qr}^{2}x\right),$$ (3)

where ${{}_{2}F}_1\left({\alpha }_1,{\alpha }_2;{\beta }_1;z\right)$ denote the hypergeometric function with 2 and 1 parameters ${\alpha }_1,{\alpha }_2$ and ${\beta }_1$ and scalar argument $z$.

The above properties are well known; however, it is nowadays of interest to examine the statistical properties of our multi-trial periodogram in high-dimensional, large $p\ge m$ settings. In these situations, the Goodman distribution (3) will not apply, however, the errors of the periodogram may still be bounded. As an alternative to the classic asymptotic framework (Brillinger, 1972), we propose to study finite-sample deviation bounds, i.e. for finite $m$, that can control the behaviour (error) of the multi-trial periodogram. We begin by discussing the asymptotic setting, where $T\to \infty$, before stating a deviation bound which holds for $T<\infty$ and where we allow $m_T$ to be a monotonically increasing sequence in $T$. We later harness this result as a lemma when deriving bounds on our regularised estimators estimates of the inverse spectrum.

Let us define the Lipschitz modulus of a function $g$ as Lip $g={\text{sup}}_{x\ne y}|g(x)-g(y)|/\Vert x-y{\Vert }_1$, where $\Vert z{\Vert }_1=\sum _i\left|z_i\right|$, then the following weak dependence condition (Doukhan & Louhichi, 1999) is sufficient for the finite sample analysis which follows.

Assumption 2. The variates ${\left\{{\bar{d}}_{T;k}(\omega )\right\}}_{k=1}^{m_T}$ are so-called $\eta$-weakly dependent. That is, there exists a sequence $\{\eta (\tau ){\}}_{\tau \in \mathbb{N}}$ converging to 0, such that for any $u$-tuple $\left(s_1,\dots ,s_u\right)\in {\mathbb{Z}}^u$ and any $v$-tuple $\left(t_1,\dots ,t_v\right)\in {\mathbb{Z}}^v$ with $s_1\le \cdots \le s_u\le s_u+\tau =t_1\cdots \le t_v$ and arbitrary measurable functions $f:{\mathbb{C}}^{up}\to \mathbb{R},g:{\mathbb{C}}^{vp}\to \mathbb{R}$ with $\Vert f{\Vert }_{\infty }\le 1,\Vert g{\Vert }_{\infty }\le 1$, Lip $f<\infty$ and Lip $g<\infty$, such that

$$\begin{gathered} \left.\mid \text{cov}\left[f\left\{{\bar{d}}_{T;s_1}\left(\omega \right),\dots ,\right.\right.\left.{\bar{d}}_{T;s_u}(\omega )\right\},g\left\{{\bar{d}}_{T;t_1}(\omega ),\dots ,{\bar{d}}_{T;t_v}(\omega )\right\}\right]\mid \\ \le \left\{u(\text{Lip}f)\Vert g{\Vert }_{\infty }+v(\text{Lip}g)\Vert f{\Vert }_{\infty }\right\}\eta (\tau ), \end{gathered}$$

where $(u,v)\in {\mathbb{N}}^{*}\times {\mathbb{N}}^{*}$.

Remark 1. Assumption 2 controls dependence between the trial specific Fourier transforms via the covariance of transformation functions $f,g$, for a specific number of trials. Intuitively, this assumption is more restrictive than Assumption 1 which bounds the cumulant function over unbounded intervals. Related work on the notion of weak-dependence for point processes as an alternative to more traditional mixing properties can be found in Doukhan & Lang (2016).

In the sequel, we introduce a useful result for the periodogram (2) which provides a deviation bound, firstly in the asymptotic setting, and then in the setting where $m_T$ and $T$ are both finite. Prior to stating this bound, we highlight that the periodogram is asymptotically unbiased. However, for finite samples there can be a trade-off between bias and variance, based on the choice of taper function. The bias of the periodogram is given by the following result.

Lemma 1. Define $Z_T\left(\omega ,{\omega }^{\prime }\right)≔H_k\left(T\omega -T{\omega }^{\prime }\right)-H_k(T\omega )H_k^{*}\left(T{\omega }^{\prime }\right)/H_k(0)$ and let ${\tilde{S}}_{qr}(\omega )≔\mathbb{E}\left\{{\hat{S}}_{m_T;qr}(\omega )\right\}$, then

$${\tilde{S}}_{qr}\left(\omega \right)=\int {\left|Z_T\left(\omega ,{\omega }^{\prime }\right)\right|}^{2}S\left({\omega }^{\prime }\right)d{\omega }^{\prime }.$$ (4)

Let us track the bias using ${\delta }_{b;T}(\omega )≔{\text{max}}_{qr}\left|{\tilde{S}}_{qr}(\omega )-S_{qr}(\omega )\right|$. If $S_{qr}(\omega )$ is smooth and has uniformly bounded second derivatives $S_{qr}^{″}(\omega )\le M_2$, then choosing ${\omega }_{*}$ such that $H_k\left(T{\omega }_{*}\right)=0$, and assuming $f_{W_T(\omega )}(x)={\left|H_k(T(\omega -x))\right|}^2$ forms a valid probability density, we have

$${\delta }_{b;T}\left({\omega }_{*}\right)\le S^{\prime }\left({\omega }_{*}\right)\left(\mathbb{E}\left[W_T\left({\omega }_{*}\right)\right]-{\omega }_{*}\right)+\frac{M_2}{2}Var\left[W_T\left({\omega }_{*}\right)\right],$$ (5)

where $W_T(\omega )$ is a random variable with density $f_{W_T(\omega )}(x)$.

Remark 2. In the discrete time setting, the bias properties of the periodogram depend on the choice of taper function $h(\cdot )$ (see for example Walden (2000) for detailed discussion). In our case, we find $\mathbb{E}\left[W_T(\omega )\right]=\omega$, and $Var\left[W_T(\omega )\right]=𝒪\left(T^{-2}\right)$ (see supplementary material for details). It is worth noting that the minimum allowable spacing between frequencies ${\omega }_{*}$ decreases as $1/T$. Consequently, while it is not possible to estimate the spectrum at all frequencies without bias, the bias diminishes as the observation interval $T$ increases. Therefore, asymptotically, the estimator becomes unbiased. The sidelobe behaviour around ${\omega }_{*}$ will depend on the form of the chosen taper. A generalised understanding of the multi-taper periodogram in the point process setting awaits future work. However, in our empirical experiments the bias appears to have minimal impact, and the main issue is the high variance of our periodogram estimates.

Proposition 1. Deviation Bound for the Multi-Trial Periodogram.

1. Let $N(t)$ satisfy Assumption 1 and let ${\hat{S}}_m(\omega )$ be the multi-trial periodogram estimator defined in (2) with $m_T=m$ and $n_T=T/m$, then

   $$\text{pr}\left\{\left|{\hat{S}}_{m;qr}(\omega )-S_{qr}(\omega )\right|\ge \delta \right\}\le 8\text{exp}\left\{\frac{-m{\delta }^2}{c_1{S}_{qq}(\omega )S_{rr}(\omega )}\right\},$$ (6)

   for some $c_1>0$ and all $\delta \in \left[0,12{\left\{S_{qq}(\omega )S_{rr}(\omega )\right\}}^{1/2}\right]$ as $T\to \infty$.
2. Let ${\hat{S}}_{m_T}(\omega )$ be the multi-trial periodogram estimator defined in (2) satisfying Assumption 2. Furthermore, assume that the bias is bounded uniformly by ${\delta }_b(\omega )\le (1-\alpha )\delta$ for some $\alpha \in (0,1),\delta >0$, then

   $$\text{pr}\left(\left|{\hat{S}}_{m_T;qr}(\omega )-S_{qr}(\omega )\right|\ge \delta \right)\le 8\text{exp}\left\{-\frac{m_T(\alpha \delta {)}^2}{c_2{\tilde{S}}_{qq}(\omega ){\tilde{S}}_{rr}(\omega )}\right\},$$ (7)

   for some $c_2>0$ and all $\delta \in \left[0,{\left\{{\tilde{S}}_{qq}(\omega ){\tilde{S}}_{rr}(\omega )\right\}}^{1/2}/\alpha \right]$.

Proof. The proof, which is based on results in Bickel & Levina (2008) and Doukhan & Neumann (2007), can be found in the Supplementary Material. □

To demonstrate how these properties hold in practice, and the challenges when $p\ge m_T$, we give an example for the empirical distribution of, and error incurred, by ${\hat{S}}_{m_T}(\omega )$. To illustrate the behaviour, we consider a simple homogeneous Poisson point process with rate $\lambda =1$ and let $T=1000$, and obtain 1000 Monte-Carlo replicates of the multi-trial spectrum with $m_T=10$ trials. The frequency $\omega =0.0628$ is considered without loss of generality since the process is Poisson. The results, depicted in Figure 1, confirm the appropriateness of the asymptotic distribution (3), and how the deviations grow as a function of $\text{log}(p)$. Empirical 95% confidence intervals are shown in red. To highlight the issue of instability of the periodogram as one approaches the high-dimensional setting, we plot the condition number of the multivariate spectrum, defined as the ratio ${\varphi }_{\text{max}}/{\varphi }_{\text{min}}$ of its largest to its smallest eigenvalue, in Fig 1c. We observe that the variability of the eigenvalues of ${\hat{S}}_{m_T}(\omega )$ is not well-controlled when $p\approx m_T$. Therefore, estimates of the inverse spectrum obtained by simply inverting ${\hat{S}}_{m_T}(\omega )$ would be highly inaccurate (Böhm & von Sachs, 2009). It is these scenarios, i.e., when $p\approx m_T$, and $p>m_T$ where the methods proposed in this paper will be of most utility.

3. Regularised Spectral Estimation

3.1. Methodology

We propose a novel methodology to estimate the inverse spectral density matrix for a high-dimensional multivariate point process at a specified frequency of interest. While estimates of a similar nature have undergone significant development for time series data (Jung et al., 2015; Nadkarni et al., 2016; Tugnait, 2022; Dallakyan et al., 2022; Deb et al., 2024) to the best of our knowledge, this is the first examination of related estimators in the point process framework.

Remark 3. Going forward, we will focus on regularised spectral estimators for high-dimensional settings. We will therefore work under Assumption 2 and let $m_T>1$ be a monotonically increasing sequence in $T$. The results presented here hold for all frequencies at which the bias can be bounded ${\delta }_b(\omega )\le (1-\alpha )\delta$ and consider the doubly asymptotic setting where both $p,m_T$ grow, and where it is possible that $p>m_T$. The results can be applied in the setting $\delta \asymp {\left\{\text{log}(p)/m_T\right\}}^{1/2}$ and when $T$ is finite.

Our estimator is motivated by the asymptotic complex normality of our multi-trial Fourier coefficients (Theorem 4.2 from Brillinger (1972)), where we appeal to the following pseudo (negative) log-likelihood (Whittle, 1953)

$$ℒ(\Theta (\omega ))≔-\text{log}\text{det}(\Theta (\omega ))+\text{tr}\left\{{\hat{S}}_{m_T}(\omega )\Theta (\omega )\right\},$$ (8)

where $\text{det}(\cdot )$ and $\text{tr}\{\cdot \}$ denote the determinant and trace of a matrix respectively.

If $m_T>p$, the negative log-likelihood achieves its minimum for $\hat{\Theta }(\omega )={\hat{S}}_{m_T}(\omega {)}^{-1}$. However, as discussed in Section 2.2, in the high-dimensional setting where $p>m_T$, the inverse of the multi-trial periodogram is not uniquely defined. To overcome this, it is common to add a penalty to $ℒ(\omega )$ in order to constrain the maximum likelihood estimate $\hat{\Theta }(\omega )$ to a feasible region.

Definition 3. Let $\lambda (\omega )>0$, the regularised spectral estimator for $\Theta (\omega )$ is defined as the M-estimator

$$\hat{\Theta }(\omega )≔\text{arg}\underset{\Theta (\omega )\in C}{\text{min}}[ℒ\{\Theta (\omega )\}+\lambda (\omega )P\{\Theta (\omega )\}],$$ (9)

with constraint set $C≔\left\{\Theta (\omega )\in {\mathbb{C}}^{p\times p}:\Theta (\omega )\succ 0\right\}$. We consider two distinct penalties as below, each possessing their unique advantages

$$P_1\{\Theta (\omega )\}=\Vert \Theta (\omega ){\Vert }_1≔\sum _{qr}\left|{\Theta }_{qr}(\omega )\right|=\sum _{qr}{\left[{\text{Re}}^2\left\{{\Theta }_{qr}(\omega )\right\}+{\text{Im}}^2\left\{{\Theta }_{qr}(\omega )\right\}\right]}^{1/2}$$ (10)

$$P_2\{\Theta (\omega )\}=\text{tr}\{\Theta (\omega )\}.$$ (11)

We refer to the regularised spectral estimator with penalties (10) and (11) as the lasso and ridge estimators, respectively.

The penalty in (10) encourages sparsity in the inverse spectrum by jointly penalising both the real and imaginary parts of the entries in $\Theta (\omega )$. This can be seen an example of the group-lasso penalty (Huang & Zhang, 2010). In cases where the inverse spectrum is known to be sparse, it is beneficial to use the lasso-type penalty. By contrast, the penalty in (11) proportionally shrinks elements of $\Theta (\omega )$ (without inducing sparsity) and can therefore be viewed as a ridge type estimator. Similar estimators for the estimation of high-dimensional covariance matrices have been discussed in Friedman (1989) and Warton (2008).

An advantage of the ridge estimator is that solutions to (9) can be found analytically. Specifically, $\hat{\Theta }(\omega )={\left\{{\hat{S}}_{m_T}(\omega )+\lambda (\omega )\mathbb{I}\right\}}^{-1}$, where ${\hat{S}}_{m_T}(\omega )$ is an estimate of the spectrum and $\mathbb{I}$ is the identity matrix. When the lasso penalty in (10) is used, the convex optimisation problem (9) can be solved numerically using the alternating direction method of multipliers algorithm, or via the complex graphical lasso (Deb et al., 2024). In this paper, we adopt the former approach due to its simplicity. This approach has been used throughout the time series literature to solve optimisation problems similar to those considered (Jung et al., 2015; Nadkarni et al., 2016; Dallakyan et al., 2022). A full description of the algorithm is given in the Supplementary Material.

3.2. Error Bounds for the Regularised-Spectral Estimator

In this section, we establish asymptotic properties of the lasso and ridge estimators, providing rates in Frobenius norm. We begin by specifying the following assumptions about the true $\Theta (\omega )$ and $S(\omega )$.

• A1. Let the set $E\{\Theta (\omega )\}≔\left\{(q,r)\mid {\Theta }_{qr}(\omega )\ne 0,q\ne r\right\}$. Then card $\left[E\right\{\Theta (\omega )\left\}\right]\le s$.
• A2. $0<\underset{\_}{k}\le {\varphi }_{\text{min}}(S(\omega ))\le \bar{k}<\infty$.

The second assumption provides a lower bound on the eigenvalues of the spectral density matrix, and thus ensures positive definiteness of $S(\omega )$. This guarantees the existence of $\Theta (\omega )$.

Proposition 2. Let ${\hat{\Theta }}_R(\omega )$ be the regularised spectral estimator defined in (9) with penalty (11), and regularisation parameter $\lambda (\omega )=c_1{\left(p\tau \text{log}p/m_T\right)}^{1/2}$ for any $\tau >2$ and constant $c_1>0$. Then, under A2, there exists a constant $c_2>0$ such that using sufficient trials,

$$m_T>\tau \text{log}p\text{max}\left[\alpha {\left\{{\tilde{S}}_{qq}(\omega ){\tilde{S}}_{rr}(\omega )\right\}}^{-1},p^2{\underset{\_}{k}}^{-2}{c}_2^2\right],$$

we have, with probability at least $1-8/p^{\tau -2}$,

$${‖{\hat{\Theta }}_R(\omega )-\Theta (\omega )‖}_F\le \frac{c_2{\tau }^{1/2}}{{\underset{\_}{k}}^2}{\left(\frac{p^2\text{log}p}{m_T}\right)}^{1/2}.$$

Proof. The proof is given in the Supplementary Material. □

The above result is based on lower-bounding the eigenvalues of the estimated spectrum, alongside bounding the deviations of the periodogram based on Proposition 1. This leads to the stated bound, holding for $m_T=\Omega \left(p^2\text{log}p\right)$ which ensures that ${‖{\hat{\Theta }}_R(\omega )‖}_2$ is bounded. We note that the bound is not particularly tight, however, it is feasible that tighter bounds can be obtained if the true spectrum maintained a specific structure, e.g., if it were approximately low rank then it may be possible to extend the results of Bunea & Xiao (2015) to the spectral setting. As an alternative, we now present a result on the lasso spectral estimator, which can make use of the structure within the edge set $E\{\Theta (\omega )\}$ by selecting certain entries of the inverse spectral density matrix to be exactly zero.

Proposition 3. Let ${\hat{\Theta }}_G(\omega )$ be the regularised spectral estimator defined in (9) with penalty (10) and regularisation parameter $\lambda (\omega )=c{\left(\tau \text{log}p/m_T\right)}^{1/2}$ for any $\tau >2$ and some constant $c>0$. Then, under A1–2 and with sufficient trials,

$$m_T>\tau \text{log}p\text{max}\left[\alpha {\left\{{\tilde{S}}_{qq}(\omega ){\tilde{S}}_{rr}(\omega )\right\}}^{-1},64c^2{\underset{\_}{k}}^{-2}(p+s)\right],$$

we have, with probability at least $1-8/p^{\tau -2}$,

$${‖{\hat{\Theta }}_G(\omega )-\Theta (\omega )‖}_F\le \frac{8c{\tau }^{1/2}}{{\underset{\_}{k}}^2}{\left\{\frac{(p+s)\text{log}p}{m_T}\right\}}^{1/2}.$$ (12)

Proof. The proof is given in the Supplementary Material. □

The stated bound for the lasso estimator (12) is similar to that in Rothman et al. (2008) for sparse estimates of the inverse covariance matrix. The result for the lasso estimator sharpens the rate by order $p^{1/2}$ compared with the ridge. The order $(p+s{)}^{1/2}$ term in the bound originates as we need to estimate both the diagonal and off-diagonal structure in the inverse spectral density. In comparison to Proposition 2, this bound will hold as long as $m=\Omega ((p+s)\text{log}p)$, whilst this allows a relatively high-dimensional setting we still need sufficient samples to ensure ${‖{\hat{\Theta }}_G(\omega )‖}_2$ is bounded. In the following section we further sharpen these bounds by introducing a more restrictive incoherence condition on the spectrum.

3.3. Consistent Estimation of Partial Coherence Graphs

The lasso regularised spectral estimator in (9) can be used to construct a frequency specific partial coherence graph $𝒢(\omega )=(V,E)$ which can be used to visualise conditional relationships between dimensions of the multivariate point process (Dahlhaus, 2000). In this graph, the nodes $V=\{1,\dots ,p\}$ represent the $q=1,\dots ,p$ processes, $N_q(t)$, and the edge set is characterised by the squared partial coherence. If the squared partial spectral coherence ${\rho }_{qr}^2(\omega )=0$ for all frequencies $\omega$, then the edge between nodes $q$ and $r$ would be excluded from the corresponding conditional independence graph (Eichler et al., 2003).

Whilst we have shown consistency of the lasso estimator in (12), it remains to demonstrate that, with high probability, this estimate can correctly identify the zero pattern of the matrix $\Theta (\omega )$. In this section, we state a result on the model selection consistency of the lasso estimator, which builds on the work presented in Ravikumar et al. (2011) and Deb et al. (2024). Before stating the result, we first set out some quantities required in our analysis.

Recall from Assumption A1 that the set of off-diagonal entries in $\Theta (\omega )$ used to define the edge set of the graph is denoted by $E\{\Theta (\omega )\}$, we define the maximum degree of the graph associated with these edges as $d=\text{max}\text{deg}\left\{E\right\{\Theta (\omega )\}$, or equivalently the maximum number of non-zeros on any row of $\Theta (\omega )$. We denote the supported elements of the inverse spectral density by the set $ℳ(\omega )≔E(\Theta (\omega ))\cup \{(1,1),\dots ,(p,p)\}$. Furthermore, we use ${ℳ}^{\perp }(\omega )$ to denote the complement of $ℳ(\omega )$, corresponding to all pairs $(q,r)$ for which ${\Theta }_{qr}=0$. For convenience, we adopt the shorthand $ℳ$ and ${ℳ}^{\perp }$ respectively, but these sets may still remain frequency specific.

Let the Hessian of the Whittle likelihood at frequency $\omega$ be given by $\Gamma (\omega )=S(\omega )\otimes S(\omega )\in {\mathbb{C}}^{p^2\times p^2}$. We define the term

$${\kappa }_S≔|\Vert S(\omega ){\Vert |}_{\infty }=\underset{q=1,\dots ,p}{\text{max}}\sum _{r=1}^p\left|S_{qr}(\omega )\right|,$$ (13)

corresponding to the ${\ell }_{\infty }$ norm of the true spectral density matrix. Similarly, we consider the subset of the Hessian relevant to the true model subset ${\Gamma }_{ℳℳ}(\omega )$ and let

$${\kappa }_{\Gamma }=|\Vert {\left\{{\Gamma }_{ℳℳ}\left(\omega \right)\right\}}^{-1}{\Vert |}_{\infty }.$$ (14)

The two quantities (13) and (14) play an important role in describing the behaviour of the lasso estimator, and are used in the final result.

We now introduce a frequency specific incoherence condition, that holds across all frequencies, which is required for the model selection consistency of the lasso estimator.

• A3. Let $\Gamma ≔\Gamma (\omega )$. Then for frequency $\omega \in \mathbb{R}$ there exists some $\alpha \in (0,1]$ such that

$$\underset{e\in {ℳ}^{\perp }}{\text{max}}‖{\Gamma }_{eℳ}{\left({\Gamma }_{ℳℳ}\right)}^{-1}‖\le (1-\alpha ).$$ (15)

The intuition behind this assumption is that it limits the influence of the non-edge terms, indexed by ${ℳ}^{\perp }$, on the edge terms, indexed by $ℳ$; see Saegusa & Shojaie (2016) for further details.

The next result establishes an elementwise deviation bound for the lasso estimator which can be used to establish consistent estimation of the resulting partial coherence graphs.

Proposition 4. Consistent Estimation of Partial Coherence Graphs. Consider a distribution satisfying the incoherence assumption (15) with parameter $\alpha \in (0,1]$. Let ${\hat{\Theta }}_G(\omega )$ be the regularised spectral estimator defined by (9) with penalty (10) and regularisation parameter $\lambda (\omega )=(8c/\alpha ){\left(\text{log}{p}^{\tau }/m_T\right)}^{1/2}$ for any $\tau >2$. Then, there exists a constant $c>0$ such that under assumptions A1–A3 with $p>3$ and sufficient trials,

$$m_T>\tau \text{log}p\text{max}\left[\alpha {\left\{{\tilde{S}}_{qq}(\omega ){\tilde{S}}_{rr}(\omega )\right\}}^{-1},{\left\{6c(1+8/\alpha {)}^{2}d{\kappa }_S^3{\kappa }_{\Gamma }^2\right\}}^2,{\left\{6c(1+8/\alpha )d{\kappa }_S^2{\kappa }_{\Gamma }\text{max}\left(1,{\kappa }_S{\kappa }_{\Gamma }\right)\right\}}^2\right],$$

with probability greater than $1-8/p^{\tau -2}$, we have:

1. The estimate ${\hat{\Theta }}_G(\omega )$ satisfies the element-wise ${\ell }_{\infty }$-bound:

   $${‖{\hat{\Theta }}_G(\omega )-\Theta (\omega )‖}_{\infty }\le 2c{\kappa }_{\Gamma }\left(1+\frac{8}{\alpha }\right){\left(\frac{\tau \text{log}p}{m_T}\right)}^{\frac{1}{2}}.$$ (16)

2. The estimate ${\hat{\Theta }}_G(\omega )$ specifies an edge set $E\left({\hat{\Theta }}_G(\omega )\right)$ that is a subset of the true edge set $E(\Theta (\omega ))$, and includes all edges $(q,r)$ with $\left|{\Theta }_{qr}(\omega )\right|>2c{\kappa }_{\Gamma }\left(1+{\alpha }^{-1}8\right){\left(\tau \text{log}p/m_T\right)}^{1/2}$.

Proof. The proof, which, similar to Deb et al. (2024), adopts the primal dual witness approach (Ravikumar et al., 2011), is given in the Supplementary Material.

Part ($b$) of Proposition 4 states that the estimated edge set is a subset of the true edge set. More specifically, it asserts that $E\left({\hat{\Theta }}_G(\omega )\right)$ correctly excludes all non-edges and only includes edges that are ‘large’ relative to decay of the error $c(\tau \text{log}p/m{)}^{1/2}$.

Sign consistency can also be achieved by placing a lower bound on the entries in the inverse spectral density matrix. In this case, not only will the estimator have the same edge set as $\Theta (\omega )$, but it will also be able to recover the correct signs on these edges. Mathematically, this translates to the event

$$𝒦(\hat{\Theta }(\omega );\Theta (\omega ))≔\left\{\text{sign}\left({\hat{\Theta }}_{qr}(\omega )\right)=\text{sign}\left({\Theta }_{qr}(\omega )\right)\forall (q,r)\in ℳ(\omega )\right\}.$$

Using this notation, and with a sufficiently modified bound on the number of trials, the estimator ${\hat{\Theta }}_G(\omega )$ is model selection consistent with high probability as $p\to \infty$,

$$\text{pr}(𝒦(\hat{\Theta }(\omega );\Theta (\omega )))\ge 1-8/p^{\tau -2}\to 1.$$

The sign consistency can also be achieved without requiring Assumption A3 and instead using a thresholded or adaptive lasso penalty (Bühlmann & van de Geer, 2011).

4. Synthetic Experiments

4.1. Simulation Setting

We use synthetic data to compare the performance of the lasso and ridge estimators to existing statistical approaches where applicable. The goal of the study is two-fold: (1) to assess the accuracy of the recovered inverse spectrum and (2) to investigate the conditional relationships of high-dimensional multivariate point processes. Throughout, we use simulated data from different parameterisations of the multivariate Hawkes process. This particular point process has been used extensively to model neuronal interactions in the brain, and as such will serve as an illustrative example in our discussion (see for example Pernice et al., 2011; Chen et al., 2017; Lambert et al., 2018; Wang & Shojaie, 2021).

In the simulation study, we consider three different parameterisations of stationary multivariate Hawkes processes. Motivated by applications in neuroscience, we consider processes that are governed by exponential excitation functions. More specifically, the conditional intensity function of the process can be written as $\Lambda =(\mathbb{I}-G(0){)}^{-1}v$, where $v$ is a vector of background intensities, $\mathbb{I}$ is the identity matrix and $G$ is the Fourier transform of the excitation function defined elementwise in the exponential case as $G_{qr}(\omega )={\int }_{-\infty }^{\infty }\text{exp}(-i\omega \tau ){\alpha }_{qr}\text{exp}\left(-{\beta }_{qr}\tau \right)d\tau ={\alpha }_{qr}/\left({\beta }_{qr}+i\omega \right)$. The spectral density matrix of a stationary multivariate Hawkes process is subsequently given as $S(\omega )=1/2\pi \{\mathbb{I}-G(\omega ){\}}^{-1}D{\left\{\mathbb{I}-G^{\top }(-\omega )\right\}}^{-1}$, where $D=\text{diag}(\Lambda )$. Further details regarding the multivariate Hawkes process used in this section can be found in the Supplementary Material.

We consider multivariate Hawkes processes of dimension $p=12,48\text{and}96$ and simulate $m=10$ and $m=50$ independent trials of synthetic data using Ogata’s method (Ogata, 1981), via the hawkes¹ package in R. $T=200$ seconds were generated for each process.

Multiple scenarios are considered in this study. In the first instance, we simulate from a $p$-dimensional Hawkes process whose dependence structure is captured by a block diagonal excitation matrix $\alpha$. Each block consists of a 3 × 3 matrix ${\alpha }^{*}$ designed to ensure both self- and cross-excitation between the dimensions in a particular block. Thus, dependence between dimensions of the process is null between the blocks and non-null within each block. We consider two models of this nature: model (a) in which there are interactions between two of the processes within a block, and an additional independent process and model (b) where all three processes in a particular block interact with each other. The excitation matrix for model (a) is low-rank compared to model (b) which is full rank. In the final setting (c) we simulate from a $p$-dimensional Hawkes process whose dependence structure is driven by an arbitrarily sparse excitation matrix $\alpha$, which is also low-rank. All parameters were chosen in such a way to ensure both stability and stationarity of the multivariate process. For further details on the specification of the Hawkes process used in this study, see the Supplementary Material.

4.2. Parameter Tuning and Performance Metrics

The ridge and lasso estimators require careful specification of a tuning parameter $\lambda$ (where we have dropped the dependence on $\omega$ for ease of notation). To reduce the computational complexity of our experiments, a group of independently simulated samples are used to select sensible values of the regularisation parameter for use in our simulation study. Going forward, we will refer to this independent set of samples as the ‘tuning set’. More specifically, estimates of $\Theta (\omega )$ are obtained over a fixed grid of $\lambda$ values, and an optimal value of $\lambda$ is selected according to an appropriate performance metric. Often, one may consider a measure of accuracy (i.e. precision of the recovered inverse spectrum), or edge selection (i.e. estimation of the correct sparsity pattern). For the former, we use the notion of mean squared error (MSE) defined in (18) and for the latter, we consider the F₁ score defined as F₁ = TP/{TP + 1/2(FP + FN)} where TP is the number of correctly identified edges in the estimate graph, while FP and FN are the number of false positives and false negatives respectively. In the absence of a ground truth comparison, we recommend the use of the Extended Bayesian information criterion (eBIC) for tuning parameter selection, which is defined in Foygel & Drton (2010) as

$${\text{eBIC}}_{\gamma }\left({\hat{\Theta }}_{\lambda }(\omega )\right)=2{\ell }_m\left({\hat{\Theta }}_{\lambda }(\omega )\right)+\left|E\right|\text{log}(m)+4\gamma \left|E\right|\text{log}(p),$$ (17)

where $E$ is the edge set of a candidate graph, ${\ell }_m(\hat{\Theta }(\omega ))$ denotes the maximised log-likelihood of the associated model and $\gamma$ is a tuning parameter. For each sample in the tuning set, an optimal choice of $\lambda$ is chosen either by maximising the F₁ score or minimising the MSE and eBIC. These values are then averaged across the number of samples in the tuning set to arrive at a final optimal parameter ${\lambda }^{*}$.

We estimate the inverse spectral density matrix across $N=100$ Monte Carlo samples. Then, we quantify the accuracy of our estimation using the mean squared error (MSE) defined as

$$\text{MSE}\left\{{\hat{\Theta }}^{(n)}(\omega ),\Theta (\omega )\right\}=\frac{2}{Np(p-1)}\sum _{n=1}^N\sum _{q<r}{\left({\hat{\Theta }}_{qr}^{(n)}-{\Theta }_{qr}\right)}^2$$ (18)

where $N$ denotes the number of Monte Carlo samples in our simulation study. We also report results for the recovery of the sparsity pattern for the lasso estimate, and calculate the area under receiver operating characteristic (AUROC) curve. Since the ridge and periodogram estimators do not produce sparse results, we only report AUROC scores for the lasso estimate. Numerical results averaged across 100 repetitions are presented in Table 1.

Table 1:

Simulation results over 100 replications for estimating the inverse spectral density matrix, with mean squared error (MSE) and AUROC. All results are recorded at a particular frequency $\omega =0.0628$ and are in the form of mean (standard error). Standard errors of < 10⁻² are omitted for brevity. Hyphenated entries (−) denote that the multi-trial periodogram matrix could not be inverted.

		Mean Squared Error - ISDM					AUROC
$\mathit{p}$	$\mathit{m}$	IP	Ridge	Lasso1	Lasso2	Lasso3	Lasso1	Lasso2	Lasso3
Model (a)
12	10	-	3.67 (0.04)	2.50 (0.05)	5.26 (0.01)	5.20 (0.02)	0.61	0.90 (0.01)	0.91
	50	5.14	1.36 (0.02)	0.86 (0.01)	4.85 (0.04)	2.97 (0.02)	0.55	0.99	0.90
48	10	-	1.18	0.88 (0.01)	1.27	1.26	0.78	0.85	0.86
	50	38522.09	0.81 (0.01)	0.34	1.19	0.95	0.67	1.00	0.99
96	10	-	0.61	0.48	0.63	0.63	0.85	0.84	0.84
	50	-	0.51	0.21	0.59	0.48	0.74	1.00	1.00
Model (b)
12	10	-	0.34 (0.03)	0.18	0.87	0.44 (0.01)	0.61	0.90	0.56
	50	0.94	0.18 (0.02)	0.07	0.81	0.09	0.59	0.99	0.64
48	10	-	0.14	0.07	0.21	0.21	0.73	0.79	0.74
	50	6411.92	0.08	0.02	0.20	0.14	0.65	0.99	0.90
96	10	-	0.09	0.04	0.10	0.10	0.81	0.74	0.72
	50	-	0.04	0.01	0.10	0.09	0.69	0.99	0.99
Model (c)
12	10	-	3.35 (0.01)	2.24 (0.05)	4.50	3.15 (0.04)	0.75	0.95	0.88
	50	26.07	2.33 (0.01)	1.56 (0.02)	4.28	1.87 (0.02)	0.69	0.94	0.84
48	10	-	0.93	0.69 (0.01)	1.09	1.03	0.86	0.93	0.93
	50	214203.50	0.77	0.30	1.08	0.71	0.82	0.95	0.92
96	10	-	0.48	0.42	0.54	0.52	0.88	0.88	0.94
	50	-	0.42	0.17	0.54	0.49	0.84	0.95	0.94

4.3. Simulation Results

Table 1 reports simulation results for Models (a)-(c), respectively. Numerical results are averaged across 100 Monte Carlo samples, and each entry of the table is of the form mean (standard error). We omit standard errors of ≤ 10⁻² for brevity. Lasso₁, lasso₂ and lasso₃ refer to the lasso estimator tuned using the MSE, F₁ score and eBIC, respectively.

The results across all simulation settings reveal that the multi-trial periodogram matrix cannot be inverted in settings where $p>m_T$, thus motivating the need for regularisation or shrinkage.

In terms of MSE, the ridge and lasso estimators are comparable across all experimental settings, though generally the lasso estimator yields (slightly) more favourable results. In particular, lasso₁ outperforms all estimators in every scenario, while the ridge estimator is most competitive with the lasso₃ estimator in settings (a) and (b). This highlights a potential drawback of the ridge estimator in that it is biased towards a scaled identity matrix, of which the sparsity pattern of models (a) and (b) somewhat resemble (Fiecas et al., 2019). It also suggests the use of the lasso estimator when the sparsity pattern of the inverse spectrum is unspecified rather than containing some sort of banded structure. Notably, the periodogram estimate is somewhat competitive in terms of MSE in scenarios where $p=12$ and $m=50$. This highlights the competence of the multi-trial estimate in scenarios where $p<m$, while also emphasising the importance for appropriate regularisation in high-dimensional settings.

The AUROC results highlight that the lasso₂ estimator generally yields more favourable results compared to lasso₁ in terms of sparse recovery of the spectrum; however, this comes at the expense of an increase in MSE. This confirms the previously observed challenges of selecting the tuning parameter of lasso penalty to provide satisfactory performances in model selection and parameter estimation (Meinshausen & Bühlmann, 2006). However, the lasso₃ estimator provides favourable results both in terms of MSE and sparse recovery of the spectrum, and thus we recommend its use in practical settings.

Overall, the lasso estimator is preferable when the goal is to learn the sparsity pattern of the inverse spectrum. However, the ridge estimator yields competitive results in terms of the accuracy of entries in the recovered estimate. Furthermore, the results presented here highlight that the proposed methodology improves upon existing estimators, such as the multi-trial periodogram, which break down in high-dimensional settings.

5. Neuronal Synchronicity and Partial Coherence Networks

We illustrate how our proposed methodology can be used to infer a network of interactions between a population of neurons, using the spike train data from Bolding & Franks (2018). In this experiment, an optogenetic approach was used to directly stimulate the olfactory bulb (OB) region of the mouse brain. In particular, 1 second light pulses were presented above the OB region at increasing intensity levels from $0\text{to}50mW/m{m}^2$ over a total of $m=10$ experimental trials.

The dataset consists of spike times for multiple mice, collected at 8 intensities of laser stimuli. As an illustrative example, we will consider data collected at three stimuli ($0,10\text{and}50mW/m{m}^2$) from a single mouse with $p=26$ neurons measured in the OB region.

For each of the considered levels of stimuli, data is collected during a ‘laser on’ and ‘laser off’ condition. For the laser on condition, we have 1 second worth of data, compared to a total of 3 seconds for the laser off condition. We use our lasso estimator to visualise functional connectivity for each condition, under the differing levels of laser stimuli. Since a laser stimulus of intensity $0mw/m{m}^2$ is equivalent to the laser off condition, we use this setting as a plausibility check for the proposed methodology.

Initial exploratory analysis showed that each of the 26 point processes are driven by lower frequencies in the spectrum, which in neuroscience are named as frequencies in the delta, (0, 4) Hz, and theta, (4, 8) Hz, bands (Buzsaki & Draguhn, 2004). Consequently, we will apply our method to these lower frequency bands. The overall aim of our analysis is to use the regularised estimator of the partial coherence matrix to determine if and how estimates of neuronal connectivity differ in response to the changing stimuli.

To estimate connectivity in a particular frequency band, we consider an adaptation of the lasso estimator whereby we use both a trial and frequency smoothed periodogram estimate as input to optimisation problem (9). Specifically, we define the trial-frequency smoothed estimator as

$$\hat{S}\left({\omega }_{*}\right)=\frac{1}{\text{card}(F)}\sum _{f\in F}\frac{1}{m}\sum _{k=1}^{m_T}{\bar{d}}_k\left({\omega }_f\right){\bar{d}}_k^H\left({\omega }_f\right),$$

where the frequencies $f$ lie in a specified range. This modified version of the lasso estimator is tuned using the eBIC in (17) with $\gamma =0.5$, as suggested in Foygel & Drton (2010). The degrees of freedom are determined by the number of non-zero elements in our estimate $\hat{\Theta }(\cdot )$.

Figure 2 shows the estimated partial coherence graphs obtained for the delta band under the laser on and laser off conditions. That is, we present estimates of neuronal interactions for frequencies in the range (0, 4) Hz that are specific to each experimental condition. The nodes in the graphs represent neurons and edges indicate non-zero values of partial coherence estimated using the lasso estimator. Each plot is annotated with the average number of spikes that are observed under each experimental condition. We highlight that for the laser on condition, the number of spikes observed increases with the increasing level of laser stimuli.

Fig. 2:

Estimated networks of neural interactions on the delta band using the spike train data from Bolding & Franks (2018). Each column shows the estimated networks for the laser off (top) and laser on (bottom) condition for laser stimuli (a) $0mw/m{m}^2$, (b) $10mw/m{m}^2$ and (c) $50mw/m{m}^2$. Common edges between the on and off conditions for each level of intensity are shown in red, and edges unique to the laser on condition are shown in black. All other edges are shown in grey.

In general, the network structure changes in response to the external stimuli. Less edges are detected when the laser is applied (40 under $10mW/m{m}^2$ and 29 under $50mW/m{m}^2$ for the laser on setting versus 40 under $10mW/m{m}^2$ and 50 under $50mW/m{m}^2$ for the laser off setting). For the $0mW/m{m}^2$ setting, 56 edges are detected in the ‘laser off’ condition versus 70 for the ‘laser on’ condition. We hypothesise that this difference in network structure is a consequence of the difference in the observed number of points. Overall, the results presented here align with the findings in Bolding & Franks (2018) where it is highlighted by neuroscientists that the OB response is sensitive to the application of external stimuli.

6. Discussion

We have proposed novel statistical methodology for the estimation of high-dimensional inverse spectral density matrices in the point process setting. The main advantage of our proposal is the ability to estimate the inverse spectrum when the dimension of the multivariate process is large, i.e. when $p>m_T$, unlike existing methods which break down in high-dimensional settings. Importantly, our lasso estimator yields sparse estimates of the inverse spectrum, easing interpretation of the resulting partial coherence networks. Moreover, under the stated incoherence condition, this estimator can correctly identify the sparse structure of the inverse spectrum. In general, we would recommend the lasso-based estimator due to this interpretability; however, in situations where the sparsity assumption may not hold, we demonstrate that a ridge-based regulariser can also provide estimates of the spectral density in high-dimensional scenarios.

There are many possible and interesting avenues for future work. Whilst we have demon-strated consistent estimation of the inverse spectrum, it remains to provide a way in which to quantify uncertainty around the resulting partial coherence graphs. Existing work on statistical inference has been discussed for high-dimensional inverse covariance estimation in Janková & van de Geer (2015). Furthermore, extending the methodology to non-stationary settings is also of interest, where one could easily extend the localised estimation methods provided in Roueff & Sachs (2019), which may find utility both in neuroscience, but also in high-frequency econometric applications. Moreover, one may consider further extensions beyond the current assumption of non-overlapping trials; for example, it would be relatively straightforward to apply our regularisation methods (and analysis) in the more general multi-tapering framework of Walden (2000); Cohen & Gibberd (2021).

Supplementary Material

The Supplementary Material includes additional results and proofs. The code associated with this paper is available at https://github.com/pinkney19/RSE