Search

We establish a theoretical framework, an algorithmic basis, and a computational workflow for the statistical analysis of multi-variate multi-dimensional random fields - sampled (possibly irregularly, with missing data) and finite (possibly bounded irregularly). Our research is practically motivated by geodetic and scientific problems of topography and gravity analysis in geophysics and planetary physics, but our solutions fulfill the more general need for sophisticated methods of inference that can be applied to massive remote-sensing data sets, and as such, our mathematical, statistical, and computational solutions transcend any particular application. The generic problem that we are addressing is: two (or more) spatial fields are observed, e.g., by passive or active sensing, and we desire a parsimonious statistical description of them, individually and in their relation to one another. We consider the fields to be realizations of a random process, parameterized as a Matern covariance structure, a very flexible description that includes, as special cases, many of the known models in popular use (e.g. exponential, autoregressive, von Karman, Gaussian, Whittle, ...) Our fundamental question is how to find estimates of the parameters of a Matern process, and the distribution of those estimates for uncertainty quantification. Our answer is, fundamentally: via maximum-likelihood estimation. We now provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can still be conducted in O(n log n) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and real world settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected and augmented by the procedures that we developed.

We consider that a network is an observation, and a collection of observed networks forms a sample. In this setting, we provide methods to test whether all observations in a network sample are drawn from a specified model. We achieve this by deriving the joint asymptotic properties of average subgraph counts as the number of observed networks increases but the number of nodes in each network remains finite. In doing so, we do not require that each observed network contains the same number of nodes, or is drawn from the same distribution. Our results yield joint confidence regions for subgraph counts, and therefore methods for testing whether the observations in a network sample are drawn from: a specified distribution, a specified model, or from the same model as another network sample. We present simulation experiments and an illustrative example on a sample of brain networks where we find that highly creative individuals' brains present significantly more short cycles than found in less creative people. for this article are available online.

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs., Comment: 15 pages, 2 figures

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 Fr\'{e}chet 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 Fr\'{e}chet 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., Comment: 33 pages, 8 figures

The ubiquity of sensing devices, the low cost of data storage, and the commoditization of computing have together led to a big data revolution. We discuss the implication of this revolution for statistics, focusing on how our discipline can best contribute to the emerging field of data science.

We propose a new class of univariate non-stationary 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 characterizing global ocean circulation and therefore also for decadal to century-scale climate understanding.

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., To appear in IEEE Transactions on Signal Processing

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalize the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(nlogn) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and realworld settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.

Algorithms are tools for decision-making. England's A-level results fiasco shows what happens when our tools are ill-suited or ill-designed and their workings poorly explained. By Sofia Olhede and Patrick J. Wolfe

Stochastic process exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm). In particular, the spectral slope at high frequencies is associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(N log N) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

This paper introduces a new modeling and inference framework for multivariate and anisotropic point processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields, and from them a family of anisotropic point processes. We give conditions that make the proposed models valid. We also propose a Palm likelihood-based inference method for this type of point process, circumventing issues of likelihood tractability. Finally we illustrate the utility of the proposed modeling framework by analyzing spatial ecological observations of plants and trees in the Barro Colorado Island data.

The Whittle likelihood is a widely used and computationally efficient pseudolikelihood. However, it is known to produce biased parameter estimates with finite sample sizes for large classes of models. We propose a method for debiasing Whittle estimates for second-order stationary stochastic processes. The debiased Whittle likelihood can be computed in the same O(n log n) operations as the standard Whittle approach. We demonstrate the superior performance of our method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the debiased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to those of the 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) for Gaussian processes and for certain classes of non-Gaussian or nonlinear processes. This is established under weaker assumptions than in the standard theory, and in particular the power spectral density is not required to be continuous in frequency. We describe how the method can be readily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.

Summary We propose a method for detecting significant interactions in very large multivariate spatial point patterns. This methodology thus develops high dimensional data understanding in the point process setting. The method is based on modelling the patterns by using a flexible Gibbs point process model to characterize point-to-point interactions at different spatial scales directly. By using the Gibbs framework significant interactions can also be captured at small scales. Subsequently, the Gibbs point process is fitted by using a pseudolikelihood approximation, and we select significant interactions automatically by using the group lasso penalty with this likelihood approximation. Thus we estimate the multivariate interactions stably even in this setting. We demonstrate the feasibility of the method with a simulation study and show its power by applying it to a large and complex rainforest plant population data set of 83 species.

Sofia Olhede and Patrick Wolfe discuss the current state of data-driven automation and its implications for jobs

Lithospheric strength variations both influence and are influenced by many tectonic processes, including orogenesis and rifting cycles. The long, complex, and highly anisotropic histories of the continental lithosphere might lead to a natural expectation of widespread mechanical anisotropy. Anisotropy in the coherence between topography and gravity anomalies is indeed often observed, but whether it corresponds to an elastic thickness that is anisotropic remains in question. If coherence is used to estimate flexural strength of the lithosphere, the null-hypothesis of elastic isotropy can only be rejected when there is significant anisotropy in both the coherence and the elastic strengths derived from it, and if interference from anisotropy in the data themselves can be plausibly excluded. We consider coherence estimates made using multitaper and wavelet methods, from which estimates of effective elastic thickness are derived. We develop a series of statistical and geophysical tests for anisotropy, and specifically evaluate the potential for spurious results with synthetically generated data. Our primary case study, the North American continent, does not exhibit meaningful anisotropy in its mechanical strength. Similarly, a global reanalysis of continental gravity and topography using multitaper methods produces only scant evidence for lithospheric flexural anisotropy.

Changes in the spatial distribution of plankton populations are thought to have a profound effect on the oceanic ecosystem across all levels. This study aims to address how the spatial distribution of different plankton assemblages has changed over a multidecadal period. The multivariate structure on the CPR dataset is analysed using a technique called sparse principal component analysis. We identify functional groups of species and show that there have been changes in the ecoregions in the North East Atlantic over a multidecadal period. This technique is data-driven and can be used to identify biologically defined ecoregions based on dominant assemblages from the dataset without relying on prior knowledge. Whilst there is a change in ecoregions across time for both zooplankton and diatoms, the nature of the changes differs for the two assemblages. For zooplankton species there has been a shift in the ecoregions towards higher latitudes, implying that cold water zooplankton have moved further into arctic waters. For species that were previously restricted to the south of the region, these have been identified with increasing frequency further north. The change in spatial distribution for different species assemblages can be attributed to different factors. For example, the primary driver of zooplankton abundance across all spatial locations appears to be temperature. It is speculated that the observed northward movement of zooplankton species is a response to rising sea surface temperatures. The abundance of diatom species is instead highly correlated with the Atlantic Multidecadal Oscillation (AMO), with the spatial patterns becoming more clearly defined in its positive phase. For the diatom species, changes may be cyclic and so mean reverting to a certain extent, but for the zooplankton, continued changes can be expected if the current warming trend continues.

Summary 1. Species are seldom distributed at random across a community, but instead show spatial structure that is determined by environmental gradients and/or biotic interactions. Analysis of the spatial co-associations of species may therefore reveal information on the processes that helped to shape those patterns. 2. We propose a multivariate approach that uses the spatial co-associations between all pairs of species to find subcommunities of species whose distribution in the study area is positively correlated. Our method, which begins with the patterns of individuals, is particularly well-suited for communities with large numbers of species and gives rare species an equal weight. We propose a method to quantify a maximum number of subcommunities that are significantly more correlated than expected under a null model of species independence. 3. Using data on the distribution of tree and shrub species from a 50 ha forest plot on Barro Colorado Island (BCI), Panama, we show that our method can be used to construct biologically meaningful subcommunities that are linked to the spatial structure of the plant community. As an example, we construct spatial maps from the subcommunities that closely follow habitats based on environmental gradients (such as slope) as well as different biotic conditions (such as canopy gaps). 4. We discuss extensions and adaptations to our method that might be appropriate for other types of spatially referenced data and for other ecological communities. We make suggestions for other ways to interpret the subcommunities using phylogenetic relationships, biological traits and environmental variables as covariates and note that subcommunities that are hard to interpret may suggest groups of species and/or regions of the landscape that merit further attention.

As nations race for dominance in the field of artificial intelligence, Sofia Olhede and Patrick Wolfe consider the implications for statistics and statisticians

As automated decisions affect more and different areas of our lives, we are faced with ethical and legal questions that are likely to change the way we think about algorithms and the law. By Sofia Olhede and Patrick Wolfe

This paper introduces the novel class of modulated cyclostationary processes, a class of nonstationary 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. In a simulation study, we illustrate the performance of our estimation method on a model that realistically captures the features observed in the electroencephalogram (EEG) data. 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.

Everyday painful experiences are usually single events accompanied by tissue damage, and yet most experimental studies of cutaneous nociceptive processing in the brain use repeated laser, thermal, or electrical stimulations that do not damage the skin. In this study the nociceptive activity in the brain evoked by tissue-damaging skin lance was analyzed with electroencephalography (EEG) in 20 healthy adult volunteers (13 men and 7 women) aged 21–40 yr. Time-frequency analysis of the evoked activity revealed a distinct late event-related vertex potential (lance event-related potential, LERP) at 100–300 ms consisting of a phase-locked energy increase between 1 and 20 Hz (delta-beta bands). A pairwise comparison between lance and sham control stimulation also revealed a period of ultralate stronger desynchronization after lance in the delta band (1–5 Hz). Skin application of mustard oil before lancing, which sensitizes a subpopulation of nociceptors expressing the cation channel TRPA1, did not affect the ultralate desynchronization but reduced the phase-locked energy increase in delta and beta bands, suggesting a central interaction between different modalities of nociceptive inputs. Verbal descriptor screening of individual pain experience revealed that lance pain is predominantly due to Aδ fiber activation, but when individuals describe lances as C fiber mediated, an ultralate delta band event-related desynchronization occurs in the brain-evoked activity. We conclude that pain evoked by acute tissue damage is associated with distinct Aδ and C fiber-mediated patterns of synchronization and desynchronization of EEG oscillations in the brain.

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., Link to published version: http://ieeexplore.ieee.org/abstract/document/7539658/

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 $\gamma$. For integer values of $\gamma$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $\gamma=3$. 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.

The current spatial pattern of a population is the result of previous individual birth, death, and dispersal events. We present a simple model followed by a comparative analysis for a species-rich plant community to show how the current spatial aggregation of a population may hold information about recent population dynamics. Previous research has shown how locally restricted seed dispersal often leads to stronger aggregation in less abundant populations than it does in more abundant populations. In contrast, little is known about how changes in the local abundance of a species may affect the spatial distribution of individuals. If the level of aggregation within a species depends to some extent on the abundance of the species, then changes in abundance should lead to subsequent changes in aggregation. However, an overall change of spatial pattern relies on many individual birth and death events, and a surplus of deaths or births may have short-term effects on aggregation that are opposite to the long-term change predicted by the change in abundance. The change in aggregation may therefore lag behind the change in abundance, and consequently, the current aggregation may hold information about recent population dynamics. Using an individual-based simulation model with local dispersal and density-dependent competition, we show that, on average, recently growing populations should be more aggregated than shrinking populations of the same current local abundance. We tested this hypothesis using spatial data on individuals from a long-term tropical rain forest plot, and find support for this relationship in canopy trees, but not in understory and shrub species. On this basis we argue that current spatial aggregation is an important characteristic that contains information on recent changes in local abundance, and may be applied to taxonomic groups where dispersal is limited and within-species aggregation is observed.

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., 21 pages, 10 figures

Data is often described as the “new oil” that powers the modern information economy – but the comparison is flawed and oversimplified, write Sofia Olhede and Russell Rodrigues

Human–algorithm interactions are making the digital world more personal – and polarised. By Sofia Olhede and Russell Rodrigues

Sofia Olhede and Russell Rodrigues discuss recent efforts to ensure greater scrutiny of machine-generated decisions

This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.

This paper presents a computationally explicit formula of the chaotic representation property (CRP) for the powers of increments of a Levy process. The formula can be used to obtain the integrands of the CRP in terms of orthogonal compensated power jump processes and the CRP in terms of Poisson random measures. Simulation results demonstrate that the performance of the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of the powers of increments of the underlying Levy process using Taylor's expansion.

This paper proposes a novel multiscale estimator for the integrated volatility of an Ito process in the presence of market microstructure noise (observation error). The multiscale structure of the observed process is represented frequency by frequency, and the concept of the multiscale ratio is introduced to quantify the bias in the realized integrated volatility due to the observation error. The multiscale ratio is estimated from a single sample path, and a frequency-by-frequency bias correction procedure is proposed, which simultaneously reduces variance. We extend the method to include correlated observation errors and provide the implied time-domain form of the estimation procedure. The new method is implemented to estimate the integrated volatility for the Heston and other models, and the improved performance of our method over existing methods is illustrated by simulation studies.

This paper extends the 1-D analytic wavelet transform to the 2-D monogenic wavelet transform. The transformation requires care in its specification to ensure suitable transform coefficients are calculated, and it is constructed so that the wavelet transform may be considered as both local and monogenic. This is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform. Classes of monogenic wavelets are proposed with suitable local properties. It is shown that the monogenic wavelet annihilates anti-monogenic signals, that the monogenic wavelet transform is phase-shift covariant and that the transform magnitude is phase-shift invariant. A simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived. The monogenic wavelet ridges of local plane waves are given.

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal., Comment: 15 pages, 6 Postscript figures

Complex random variables arise naturally in many settings and their properties are of general interest. Past work on complex variables has mainly focused on their second-order structure, as well as that of their conjugates, whereas the main purpose of this correspondence is to clarify the concept of a density function for a complex random variable, and to discuss its properties. Two different functions play the role that the density of a real univariate random variable holds. Only one of these two functions can be correctly interpreted as a density, but both functions clarify the nature of a complex variable. The role played by the complex conjugate of the variable in this formulation is clarified, and the complex scalar nature of Z is discussed. As the properties of complex random variables are most naturally specified in terms of the complex quantities directly, and given in terms of the distribution of the complex variables rather than formulated in terms of the real and imaginary parts, ensuring that an interpretable complex formulation exists is important

The idea of forming a complex-valued (analytic) signal from a real-valued one by creating an imaginary part equal to the Hilbert transform of the real part is well known in exploration geophysics for seismic character mapping via instantaneous attributes. However in this paper we consider the denoising of bivariate signals (time series) where the two real-valued components become the real and imaginary parts of a single complex-valued signal, and concentrate on the case where the two real-valued components are 'in quadrature' and also the complex signal is analytic. The Hilbert transform is applied to the noisy complex-valued signal to produce a new analytic noisy complex-valued signal with a useful noise structure. Numerical calculations show that our proposed 'complex analytic denoising' is superior to two other approaches for (i) a synthetic signal which is both in quadrature and analytic, and (ii) phase estimation for a Rayleigh wave signal which is close to analytic.

SUMMARY We introduce so-called analytic stationary wavelet transform thresholding where, using the discrete Hilbert transform, we create a complex-valued 'analytic' vector from which an amplitude vector is defined. Thresholding of a real-valued wavelet coefficient at some transform level is carried out according to the corresponding value in this amplitude vector; relevant statistical results follow from properties of the discrete Hilbert transform. Analytic stationary wavelet transform thresholding is found to produce consistently a reduced mean squared error compared to using standard stationary wavelet transform, or 'cycle spinning', thresholding. For signals with extensive oscillations at some transform levels, this improvement is very marked. Furthermore we show that our thresholding test is invariant to phase shifts in the data, whereas, if complex wavelet filters are being used, the filters must be analytic or anti-analytic at each level of the wavelet transform.

Seasonally persistent models were first introduced by Andel (1986) and Gray et al. (1989) to extend autoregressive moving-average and fractionally differenced models and to encompass long-memory quasi-periodic behaviour. These models are, for certain ranges of parameters, stationary, and we prove here that the behaviour of the periodogram and other tapered estimators cannot be simply extended from the work of Kunsch (1986) and Hurvich & Beltrao (1993) on long memory induced by a pole at the origin. We demonstrate that potentially large both positive and negative bias can be found from the same value of the long-memory parameter, and that the new distribution can be easily written down in the case of Gaussian processes. We also consider using both the cosine taper and the sine taper. The extended least squares estimator is also considered in this context.

Johnstone, Kerkyacharian, Picard and Raimondo Johnstone, Kerkyacharian, Picard and Raimondo are interested in the inverse problem of estimating f where f has been convolved with g and then contaminated with white noise. This popular problem has been tackled by a wide variety of procedures and wavelet methods have recently generated considerable interest. Donoho’s (1995) seminal wavelet–vaguelette paper introduced the notion that wavelets would be a good choice for the representation of f since real life objects, such as images, are more likely to be efficiently represented using wavelets when compared with, for example, Fourier representations. Johnstone and his colleagues have moved the field on significantly. In particular, their procedure is more direct than wavelet–vaguelette or Abramovich and Silverman’s (1998) vaguelette–wavelet method; it can handle boxcar blur theoretically and practically, they have rates of convergence forp =2 (p defines the type of loss) and the paper innovates through use of the new maxiset approach. For me, the most appealing of these innovations is that of enabling the treatment of boxcar blur which is one of the most common types of inverse problem. However, is it really, really, the case that for rational a nothing can be done? Formula (4) compels us to say no, nothing can, but naively it still feels wrong. Formula (19) is the popular ‘signal-plus-noise’ model but here it is a little different from what normally appears in the literature because the quantities are complex-valued random variables. More specifically, the zl are zero-mean Gaussian variables which are complex valued and satisfy E.zlzk/= δlk. One question is why threshold the βk and not the yl directly? The covariance of the βk is given by cov.βk, βl/=n ∑ m ΨkmΨ l m