arXiv subjects

Sofia C. Olhede

Publications and source records attributed to Sofia C. Olhede.

At least 37 records · Page 2Linked to original sources

arXiv2022

Edge coherence in multiplex networks

This paper introduces a nonparametric framework for the setting where multiple networks are observed on the same set of nodes, also known as multiplex networks. Our objective is to provide a simple parameterization which explicitly captures linear dependence between the different layers of networks. For non-Euclidean observations, such as shapes and graphs, the notion of "linear" must be defined appropriately. Taking inspiration from the representation of stochastic processes and the analogy of the multivariate spectral representation of a stochastic process with joint exchangeability of Bernoulli arrays, we introduce the notion of edge coherence as a measure of linear dependence in the graph limit space. Edge coherence is defined for pairs of edges from any two network layers and is the key novel parameter. We illustrate the utility of our approach by eliciting simple models such as a correlated stochastic blockmodel and a correlated inhomogeneous graph limit model.

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arXiv2021

The Elliptical Ornstein-Uhlenbeck Process

We introduce the elliptical Ornstein-Uhlenbeck (OU) process, which is a generalisation of the well-known univariate OU process to bivariate time series. This process maps out elliptical stochastic oscillations over time in the complex plane, which are observed in many applications of coupled bivariate time series. The appeal of the model is that elliptical oscillations are generated using one simple first order stochastic differential equation (SDE), whereas alternative models require more complicated vectorised or higher order SDE representations. The second useful feature is that parameter estimation can be performed semi-parametrically in the frequency domain using the Whittle Likelihood. We determine properties of the model including the conditions for stationarity, and the geometrical structure of the elliptical oscillations. We demonstrate the utility of the model by measuring periodic and elliptical properties of Earth's polar motion.

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arXiv2020

Modeling Network Populations via Graph Distances

This article introduces a new class of models for multiple networks. The core idea is to parametrize a distribution on labelled graphs in terms of a Fréchet mean graph (which depends on a user-specified choice of metric or graph distance) and a parameter that controls the concentration of this distribution about its mean. Entropy is the natural parameter for such control, varying from a point mass concentrated on the Fréchet mean itself to a uniform distribution over all graphs on a given vertex set. We provide a hierarchical Bayesian approach for exploiting this construction, along with straightforward strategies for sampling from the resultant posterior distribution. We conclude by demonstrating the efficacy of our approach via simulation studies and two multiple-network data analysis examples: one drawn from systems biology and the other from neuroscience.

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arXiv2018

The De-Biased Whittle Likelihood

The Whittle likelihood is a widely used and computationally efficient pseudo-likelihood. However, it is known to produce biased parameter estimates for large classes of models. We propose a method for de-biasing Whittle estimates for second-order stationary stochastic processes. The de-biased Whittle likelihood can be computed in the same $\mathcal{O}(n\log n)$ operations as the standard approach. We demonstrate the superior performance of the method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the de-biased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of $n^{-1/2}$, under weaker assumptions than standard theory, where we do not require that the power spectral density is continuous in frequency. We describe how the method can be easily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.

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arXiv2018

Multivariate Geometric Anisotropic Cox Processes

This paper introduces a new modelling framework for multivariate anisotropic Cox processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields and construct a family of anisotropic point processes from it. We give conditions that make the models valid, and we provide additional understanding of valid point process dependence. We also propose a likelihood-based inference mechanism for this type of process. Finally we illustrate the utility of the proposed modelling framework by analysing spatial ecological observations of plants and trees in the Barro Colorado Island study.

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arXiv2017

Topology reveals universal features for network comparison

The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path provides a notion of scale, which is vitally important in characterizing dominant modes of system behavior. Here, by combining topology with scale, we prove the existence of universal features which reveal the dominant scales of any network. We use these features to compare several canonical network types in the context of a social media discussion which evolves through the sharing of rumors, leaks and other news. Our analysis enables for the first time a universal understanding of the balance between loops and tree-like structure across network scales, and an assessment of how this balance interacts with the spreading of information online. Crucially, our results allow networks to be quantified and compared in a purely model-free way that is theoretically sound, fully automated, and inherently scalable.

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arXiv2017

Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals

There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations. These include procedures for parametrically modeling signal coherence, estimating model parameters using the Whittle likelihood, performing semi-parametric modeling, and choosing between classes of nested models using model choice. We also provide a new method of testing for impropriety in complex-valued signals, which tests for noncircular or anisotropic second-order statistical structure when the signal is represented in the complex plane. Finally, we demonstrate the usefulness of our methodology in capturing the anisotropic structure of signals observed from fluid dynamic simulations of turbulence.

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arXiv2017

A Widely Linear Complex Autoregressive Process of Order One

We propose a simple stochastic process for modeling improper or noncircular complex-valued signals. The process is a natural extension of a complex-valued autoregressive process, extended to include a widely linear autoregressive term. This process can then capture elliptical, as opposed to circular, stochastic oscillations in a bivariate signal. The process is order one and is more parsimonious than alternative stochastic modeling approaches in the literature. We provide conditions for stationarity, and derive the form of the covariance and relation sequence of this model. We describe how parameter estimation can be efficiently performed both in the time and frequency domain. We demonstrate the practical utility of the process in capturing elliptical oscillations that are naturally present in seismic signals.

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arXiv2017

Analysis of nonstationary modulated time series with applications to oceanographic flow measurements

We propose a new class of univariate nonstationary time series models, using the framework of modulated time series, which is appropriate for the analysis of rapidly-evolving time series as well as time series observations with missing data. We extend our techniques to a class of bivariate time series that are isotropic. Exact inference is often not computationally viable for time series analysis, and so we propose an estimation method based on the Whittle-likelihood, a commonly adopted pseudo-likelihood. Our inference procedure is shown to be consistent under standard assumptions, as well as having considerably lower computational cost than exact likelihood in general. We show the utility of this framework for the analysis of drifting instruments, an analysis that is key to characterising global ocean circulation and therefore also for decadal to century-scale climate understanding.

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arXiv2015

A Power Variance Test for Nonstationarity in Complex-Valued Signals

We propose a novel algorithm for testing the hypothesis of nonstationarity in complex-valued signals. The implementation uses both the bootstrap and the Fast Fourier Transform such that the algorithm can be efficiently implemented in O(NlogN) time, where N is the length of the observed signal. The test procedure examines the second-order structure and contrasts the observed power variance - i.e. the variability of the instantaneous variance over time - with the expected characteristics of stationary signals generated via the bootstrap method. Our algorithmic procedure is capable of learning different types of nonstationarity, such as jumps or strong sinusoidal components. We illustrate the utility of our test and algorithm through application to turbulent flow data from fluid dynamics.

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arXiv2015

Lagrangian Time Series Models for Ocean Surface Drifter Trajectories

This paper proposes stochastic models for the analysis of ocean surface trajectories obtained from freely-drifting satellite-tracked instruments. The proposed time series models are used to summarise large multivariate datasets and infer important physical parameters of inertial oscillations and other ocean processes. Nonstationary time series methods are employed to account for the spatiotemporal variability of each trajectory. Because the datasets are large, we construct computationally efficient methods through the use of frequency-domain modelling and estimation, with the data expressed as complex-valued time series. We detail how practical issues related to sampling and model misspecification may be addressed using semi-parametric techniques for time series, and we demonstrate the effectiveness of our stochastic models through application to both real-world data and to numerical model output.

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arXiv2014

Network histograms and universality of blockmodel approximation

In this article we introduce the network histogram: a statistical summary of network interactions, to be used as a tool for exploratory data analysis. A network histogram is obtained by fitting a stochastic blockmodel to a single observation of a network dataset. Blocks of edges play the role of histogram bins, and community sizes that of histogram bandwidths or bin sizes. Just as standard histograms allow for varying bandwidths, different blockmodel estimates can all be considered valid representations of an underlying probability model, subject to bandwidth constraints. Here we provide methods for automatic bandwidth selection, by which the network histogram approximates the generating mechanism that gives rise to exchangeable random graphs. This makes the blockmodel a universal network representation for unlabeled graphs. With this insight, we discuss the interpretation of network communities in light of the fact that many different community assignments can all give an equally valid representation of such a network. To demonstrate the fidelity-versus-interpretability tradeoff inherent in considering different numbers and sizes of communities, we analyze two publicly available networks - political weblogs and student friendships - and discuss how to interpret the network histogram when additional information related to node and edge labeling is present.

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arXiv2013

Nonparametric graphon estimation

We propose a nonparametric framework for the analysis of networks, based on a natural limit object termed a graphon. We prove consistency of graphon estimation under general conditions, giving rates which include the important practical setting of sparse networks. Our results cover dense and sparse stochastic blockmodels with a growing number of classes, under model misspecification. We use profile likelihood methods, and connect our results to approximation theory, nonparametric function estimation, and the theory of graph limits.

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arXiv2013

Degree-based network models

We derive the sampling properties of random networks based on weights whose pairwise products parameterize independent Bernoulli trials. This enables an understanding of many degree-based network models, in which the structure of realized networks is governed by properties of their degree sequences. We provide exact results and large-sample approximations for power-law networks and other more general forms. This enables us to quantify sampling variability both within and across network populations, and to characterize the limiting extremes of variation achievable through such models. Our results highlight that variation explained through expected degree structure need not be attributed to more complicated generative mechanisms.

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arXiv2013

Maximum-likelihood estimation of lithospheric flexural rigidity, initial-loading fraction, and load correlation, under isotropy

Topography and gravity are geophysical fields whose joint statistical structure derives from interface-loading processes modulated by the underlying mechanics of isostatic and flexural compensation in the shallow lithosphere. Under this dual statistical-mechanistic viewpoint an estimation problem can be formulated where the knowns are topography and gravity and the principal unknown the elastic flexural rigidity of the lithosphere. In the guise of an equivalent "effective elastic thickness", this important, geographically varying, structural parameter has been the subject of many interpretative studies, but precisely how well it is known or how best it can be found from the data, abundant nonetheless, has remained contentious and unresolved throughout the last few decades of dedicated study. The popular methods whereby admittance or coherence, both spectral measures of the relation between gravity and topography, are inverted for the flexural rigidity, have revealed themselves to have insufficient power to independently constrain both it and the additional unknown initial-loading fraction and load-correlation fac- tors, respectively. Solving this extremely ill-posed inversion problem leads to non-uniqueness and is further complicated by practical considerations such as the choice of regularizing data tapers to render the analysis sufficiently selective both in the spatial and spectral domains. Here, we rewrite the problem in a form amenable to maximum-likelihood estimation theory, which we show yields unbiased, minimum-variance estimates of flexural rigidity, initial-loading frac- tion and load correlation, each of those separably resolved with little a posteriori correlation between their estimates. We are also able to separately characterize the isotropic spectral shape of the initial loading processes.

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arXiv2012

Covariance of Replicated Modulated Cyclical Time Series

This paper introduces the novel class of modulated cyclostationary processes, a class of non-stationary processes exhibiting frequency coupling, and proposes a method of their estimation from repeated trials. Cyclostationary processes also exhibit frequency correlation but have Loeve spectra whose support lies only on parallel lines in the dual-frequency plane. Such extremely sparse structure does not adequately represent many biological processes. Thus, we propose a model that, in the time domain, modulates the covariance of cyclostationary processes and consequently broadens their frequency support in the dual-frequency plane. The spectra and the cross-coherence of the proposed modulated cyclostationary process are first estimated using multitaper methods. A shrinkage procedure is then applied to each trial-specific estimate to reduce the estimation risk. Multiple trials of each series are observed. When combining information across trials, we carefully take into account the bias that may be introduced by phase misalignment and the fact that the Loeve spectra and cross-coherence across replicates may only be "similar" - but not necessarily identical - across replicates. The application of the inference methods developed for the modulated cyclostationary model to EEG data also demonstrates that the proposed model captures statistically significant cross-frequency interactions, that ought to be further examined by neuroscientists.

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arXiv2012

Order statistics of observed network degrees

This article discusses the properties of extremes of degree sequences calculated from network data. We introduce the notion of a normalized degree, in order to permit a comparison of degree sequences between networks with differing numbers of nodes. We model each normalized degree as a bounded continuous random variable, and determine the properties of the ordered k-maxima and minima of the normalized network degrees when they comprise a random sample from a Beta distribution. In this setting, their means and variances take a simplified form given by their ordering, and we discuss the relation of these quantities to other prescribed decays such as power laws. We verify the derived properties from simulated sets of normalized degrees, and discuss possible extensions to more flexible classes of distributions.

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arXiv2012

Generalized Morse Wavelets as a Superfamily of Analytic Wavelets

The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed bandwidth may be modified by varying a second parameter, called $γ$. For integer values of $γ$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $γ=3$. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use.

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