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When it came to Covid-19, timing was everything. This paper considers the spatiotemporal dynamics of the Covid-19 pandemic via a developed methodology of non-Euclidean spatially aware functional registration. In particular, the daily SARS-CoV-2 incidence in each of 380 local authorities in the UK from March to June 2020 is analysed to understand the phase variation of the waves when considered as curves. This is achieved by adapting a traditional registration method (that of local variation analysis) to account for the clear spatial dependencies in the data. This adapted methodology is shown via simulation studies to perform substantially better for the estimation of the registration functions than the non-spatial alternative. Moreover, it is found that the driving time between locations represents the spatial dependency in the Covid-19 data better than geographical distance. However, since driving time is non-Euclidean, the traditional spatial frameworks break down; to solve this, a methodology inspired by multidimensional scaling is developed to approximate the driving times by a Euclidean distance which enables the established theory to be applied. Finally, the resulting estimates of the registration/warping processes are analysed by taking functionals to understand the qualitatively observable earliness/lateness and sharpness/flatness of the Covid-19 waves quantitatively., Comment: 24 pages, 13 figures, 1 table

In this paper we present the framework of symmetry in nonparametric regression. This generalises the framework of covariate sparsity, where the regression function depends only on at most $s < d$ of the covariates, which is a special case of translation symmetry with linear orbits. In general this extends to other types of functions that capture lower dimensional behavior even when these structures are non-linear. We show both that known symmetries of regression functions can be exploited to give similarly faster rates, and that unknown symmetries with Lipschitz actions can be estimated sufficiently quickly to obtain the same rates. This is done by explicit constructions of partial symmetrisation operators that are then applied to usual estimators, and with a two step M-estimator of the maximal symmetry of the regression function. We also demonstrate the finite sample performance of these estimators on synthetic data., Comment: 29 Pages, 4 Figures, 2 Appendices

For healthcare datasets, it is often not possible to combine data samples from multiple sites due to ethical, privacy or logistical concerns. Federated learning allows for the utilisation of powerful machine learning algorithms without requiring the pooling of data. Healthcare data has many simultaneous challenges which require new methodologies to address, such as highly-siloed data, class imbalance, missing data, distribution shifts and non-standardised variables. Federated learning adds significant methodological complexity to conventional centralised machine learning, requiring distributed optimisation, communication between nodes, aggregation of models and redistribution of models. In this systematic review, we consider all papers on Scopus that were published between January 2015 and February 2023 and which describe new federated learning methodologies for addressing challenges with healthcare data. We performed a detailed review of the 89 papers which fulfilled these criteria. Significant systemic issues were identified throughout the literature which compromise the methodologies in many of the papers reviewed. We give detailed recommendations to help improve the quality of the methodology development for federated learning in healthcare., Comment: Supplementary table of extracted data at the end of the document

Survival analysis is an integral part of the statistical toolbox. However, while most domains of classical statistics have embraced deep learning, survival analysis only recently gained some minor attention from the deep learning community. This recent development is likely in part motivated by the COVID-19 pandemic. We aim to provide the tools needed to fully harness the potential of survival analysis in deep learning. On the one hand, we discuss how survival analysis connects to classification and regression. On the other hand, we provide technical tools. We provide a new loss function, evaluation metrics, and the first universal approximating network that provably produces survival curves without numeric integration. We show that the loss function and model outperform other approaches using a large numerical study.

Clinical data is often affected by clinically irrelevant factors such as discrepancies between measurement devices or differing processing methods between sites. In the field of machine learning (ML), these factors are known as domains and the distribution differences they cause in the data are known as domain shifts. ML models trained using data from one domain often perform poorly when applied to data from another domain, potentially leading to wrong predictions. As such, developing machine learning models that can generalise well across multiple domains is a challenging yet essential task in the successful application of ML in clinical practice. In this paper, we propose a novel disentangled autoencoder (Dis-AE) neural network architecture that can learn domain-invariant data representations for multi-label classification of medical measurements even when the data is influenced by multiple interacting domain shifts at once. The model utilises adversarial training to produce data representations from which the domain can no longer be determined. We evaluate the model's domain generalisation capabilities on synthetic datasets and full blood count (FBC) data from blood donors as well as primary and secondary care patients, showing that Dis-AE improves model generalisation on multiple domains simultaneously while preserving clinically relevant information., Comment: 17 pages main body, 5 figures, 18 pages of appendix

This review provides an overview of concepts relating to the communication of statistical and empirical evidence in times of crisis, with a special focus on COVID-19. In it, we consider topics relating both to the communication of numbers -- such as the role of format, context, comparisons, and visualization -- and the communication of evidence more broadly -- such as evidence quality, the influence of changes in available evidence, transparency, and repeated decision making. A central focus is on the communication of the inherent uncertainties in statistical analysis, especially in rapidly changing informational environments during crises. We present relevant literature on these topics and draw connections to the communication of statistics and empirical evidence during the COVID-19 pandemic and beyond. We finish by suggesting some considerations for those faced with communicating statistics and evidence in times of crisis., Comment: 33 pages; 4 figures

We present a method for estimating the maximal symmetry of a continuous regression function. Knowledge of such a symmetry can be used to significantly improve modelling by removing the modes of variation resulting from the symmetries. Symmetry estimation is carried out using hypothesis testing for invariance strategically over the subgroup lattice of a search group G acting on the feature space. We show that the estimation of the unique maximal invariant subgroup of G generalises useful tools from linear dimension reduction to a non linear context. We show that the estimation is consistent when the subgroup lattice chosen is finite, even when some of the subgroups themselves are infinite. We demonstrate the performance of this estimator in synthetic settings and apply the methods to two data sets: satellite measurements of the earth's magnetic field intensity; and the distribution of sunspots., Comment: 47 Pages, 16 figures

Krylov subspace methods are a powerful family of iterative solvers for linear systems of equations, which are commonly used for inverse problems due to their intrinsic regularization properties. Moreover, these methods are naturally suited to solve large-scale problems, as they only require matrix-vector products with the system matrix (and its adjoint) to compute approximate solutions, and they display a very fast convergence. Even if this class of methods has been widely researched and studied in the numerical linear algebra community, its use in applied medical physics and applied engineering is still very limited. e.g. in realistic large-scale Computed Tomography (CT) problems, and more specifically in Cone Beam CT (CBCT). This work attempts to breach this gap by providing a general framework for the most relevant Krylov subspace methods applied to 3D CT problems, including the most well-known Krylov solvers for non-square systems (CGLS, LSQR, LSMR), possibly in combination with Tikhonov regularization, and methods that incorporate total variation (TV) regularization. This is provided within an open source framework: the Tomographic Iterative GPU-based Reconstruction (TIGRE) toolbox, with the idea of promoting accessibility and reproducibility of the results for the algorithms presented. Finally, numerical results in synthetic and real-world 3D CT applications (medical CBCT and {\mu}-CT datasets) are provided to showcase and compare the different Krylov subspace methods presented in the paper, as well as their suitability for different kinds of problems., Comment: submitted

In this perspective, we argue that despite the democratization of powerful tools for data science and machine learning over the last decade, developing the code for a trustworthy and effective data science system (DSS) is getting harder. Perverse incentives and a lack of widespread software engineering (SE) skills are among many root causes we identify that naturally give rise to the current systemic crisis in reproducibility of DSSs. We analyze why SE and building large complex systems is, in general, hard. Based on these insights, we identify how SE addresses those difficulties and how we can apply and generalize SE methods to construct DSSs that are fit for purpose. We advocate two key development philosophies, namely that one should incrementally grow -- not biphasically plan and build -- DSSs, and one should always employ two types of feedback loops during development: one which tests the code's correctness and another that evaluates the code's efficacy.

We propose a framework for jointly modeling the geometry and functionality in high dimensional functional surfaces. The proposed mixed effects model characterizes effects of subject-specific covariates and exogenous stimuli on functional surfaces while accounting for potential mutual-influence of their geometry and functionality. This is achieved through a computationally efficient estimation method that incorporates regularized estimation of the precision matrix of the random effects. We perform a thorough analysis of cerebral cortical surface structural MRI and task fMRI data from the Human Connectome Project and discover relationships between the geometric shapes of cortical surface and neuronal activation responding to task stimuli. Our findings highlight new modes of correspondence between cortical surface shape and functional activation relevant to emotion processing.

Functional depth is the functional data analysis technique that orders a functional data set. Unlike the case of data on the real line, defining this order is non-trivial, and particularly, with functional data, there are a number of properties that any depth should satisfy. We propose a new depth which both satisfies the properties required of a functional depth but also one which can be used in the case where there are a very large number of functional observations or in the case where the observations are functions of several continuous variables (such as images, for example). We give theoretical justification for our choice, and evaluate our proposed depth through simulation. We finally apply the proposed depth to the problem of yielding a completely non-parametric deconvolution of Positron Emission Tomography (PET) data for a very large number of curves across the image, as well as to the problem of finding a representative subject from a set of PET scans.

Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete, imputed, samples. The focus of the machine learning researcher is then to optimise the downstream classification performance. In this study, we highlight that it is imperative to consider the quality of the imputation. We demonstrate how the commonly used measures for assessing quality are flawed and propose a new class of discrepancy scores which focus on how well the method recreates the overall distribution of the data. To conclude, we highlight the compromised interpretability of classifier models trained using poorly imputed data., Comment: 17 pages, 10 figures, 30 supplementary pages

Invariant and equivariant models incorporate the symmetry of an object to be estimated (here non-parametric regression functions $f : \mathcal{X} \rightarrow \mathbb{R}$). These models perform better (with respect to $L^2$ loss) and are increasingly being used in practice, but encounter problems when the symmetry is falsely assumed. In this paper we present a framework for testing for $G$-equivariance for any semi-group $G$. This will give confidence to the use of such models when the symmetry is not known a priori. These tests are independent of the model and are computationally quick, so can be easily used before model fitting to test their validity., Comment: 15 Pages, 6 Figures

Functional Principal Component Analysis (FPCA) has become a widely-used dimension reduction tool for functional data analysis. When additional covariates are available, existing FPCA models integrate them either in the mean function or in both the mean function and the covariance function. However, methods of the first kind are not suitable for data that display second-order variation, while those of the second kind are time-consuming and make it difficult to perform subsequent statistical analyses on the dimension-reduced representations. To tackle these issues, we introduce an eigen-adjusted FPCA model that integrates covariates in the covariance function only through its eigenvalues. In particular, different structures on the covariate-specific eigenvalues -- corresponding to different practical problems -- are discussed to illustrate the model's flexibility as well as utility. To handle functional observations under different sampling schemes, we employ local linear smoothers to estimate the mean function and the pooled covariance function, and a weighted least square approach to estimate the covariate-specific eigenvalues. The convergence rates of the proposed estimators are further investigated under the different sampling schemes. In addition to simulation studies, the proposed model is applied to functional Magnetic Resonance Imaging scans, collected within the Human Connectome Project, for functional connectivity investigation., Comment: 31 pages, 4 figures

We present a statistical framework that jointly models brain shape and functional connectivity, which are two complex aspects of the brain that have been classically studied independently. We adopt a Riemannian modeling approach to account for the non-Euclidean geometry of the space of shapes and the space of connectivity that constrains trajectories of co-variation to be valid statistical estimates. In order to disentangle genetic sources of variability from those driven by unique environmental factors, we embed a functional random effects model in the Riemannian framework. We apply the proposed model to the Human Connectome Project dataset to explore spontaneous co-variation between brain shape and connectivity in young healthy individuals., Comment: 27 pages

In medical image analysis, constructing an atlas, i.e. a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterise and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the $L^\infty$ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e. more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which existence of minimisers, analysis of a numerical method of resolution, asymptotic results and a PCA analysis, as well as numerical simulations demonstrating the ability of the modeling to produce an atlas exhibiting sharp edges, high contrast and a consistent shape.

Functional magnetic resonance imaging (fMRI) data provides information concerning activity in the brain and in particular the interactions between brain regions. Resting state fMRI data is widely used for inferring connectivities in the brain which are not due to external factors. As such analyzes strongly rely on stationarity, change point procedures can be applied in order to detect possible deviations from this crucial assumption. In this paper, we model fMRI data as functional time series and develop tools for the detection of deviations from covariance stationarity via change point alternatives. We propose a nonparametric procedure which is based on dimension reduction techniques. However, as the projection of the functional time series on a finite and rather low-dimensional subspace involves the risk of missing changes which are orthogonal to the projection space, we also consider two test statistics which take the full functional structure into account. The proposed methods are compared in a simulation study and applied to more than 100 resting state fMRI data sets., Comment: 39 pages, 11 figures

We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity., Comment: 40 pages

Given the recent controversies in some neuroimaging statistical methods, we compare the most frequently used functional Magnetic Resonance Imaging (fMRI) analysis packages: AFNI, FSL and SPM, with regard to temporal autocorrelation modeling. This process, sometimes known as pre-whitening, is conducted in virtually all task fMRI studies. We employ eleven datasets containing 980 scans corresponding to different fMRI protocols and subject populations. Though autocorrelation modeling in AFNI is not perfect, its performance is much higher than the performance of autocorrelation modeling in FSL and SPM. The residual autocorrelated noise in FSL and SPM leads to heavily confounded first level results, particularly for low-frequency experimental designs. Our results show superior performance of SPM's alternative pre-whitening: FAST, over SPM's default. The reliability of task fMRI studies would increase with more accurate autocorrelation modeling. Furthermore, reliability could increase if the packages provided diagnostic plots. This way the investigator would be aware of pre-whitening problems., Comment: compared to the third version, we investigated: (1) the impact of slice timing correction on pre-whitening and (2) the impact of pre-whitening on group results using the mixed effects model 3dMEMA

In Functional Data Analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called Functions on Surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries., Comment: 42 pages

Variational autoencoders (VAE) represent a popular, flexible form of deep generative model that can be stochastically fit to samples from a given random process using an information-theoretic variational bound on the true underlying distribution. Once so-obtained, the model can be putatively used to generate new samples from this distribution, or to provide a low-dimensional latent representation of existing samples. While quite effective in numerous application domains, certain important mechanisms which govern the behavior of the VAE are obfuscated by the intractable integrals and resulting stochastic approximations involved. Moreover, as a highly non-convex model, it remains unclear exactly how minima of the underlying energy relate to original design purposes. We attempt to better quantify these issues by analyzing a series of tractable special cases of increasing complexity. In doing so, we unveil interesting connections with more traditional dimensionality reduction models, as well as an intrinsic yet underappreciated propensity for robustly dismissing sparse outliers when estimating latent manifolds. With respect to the latter, we demonstrate that the VAE can be viewed as the natural evolution of recent robust PCA models, capable of learning nonlinear manifolds of unknown dimension obscured by gross corruptions.

Dialect variation is of considerable interest in linguistics and other social sciences. However, traditionally it has been studied using proxies (transcriptions) rather than acoustic recordings directly. We introduce novel statistical techniques to analyse geolocalised speech recordings and to explore the spatial variation of pronunciations continuously over the region of interest, as opposed to traditional isoglosses, which provide a discrete partition of the region. Data of this type require an explicit modeling of the variation in the mean and the covariance. Usual Euclidean metrics are not appropriate, and we therefore introduce the concept of $d$-covariance, which allows consistent estimation both in space and at individual locations. We then propose spatial smoothing for these objects which accounts for the possibly non convex geometry of the domain of interest. We apply the proposed method to data from the spoken part of the British National Corpus, deposited at the British Library, London, and we produce maps of the dialect variation over Great Britain. In addition, the methods allow for acoustic reconstruction across the domain of interest, allowing researchers to listen to the statistical analysis., Comment: 18 figures

Motivated by increasing trends of relating brain images to a clinical outcome of interest, we propose a functional domain selection (FuDoS) method that effectively selects subregions of the brain associated with the outcome. View each individual's brain as a 3D functional object, the statistical aim is to distinguish the region where a regression coefficient $\beta(t)=0$ from $\beta(t)\neq0$, where $t$ denotes spatial location. FuDoS is composed of two stages of estimation. We first segment the brain into several small parts based on the correlation structure. Then, potential subsets are built using the obtained segments and their predictive performance are evaluated to select the best subset, augmented by a stability selection criterion. We conduct extensive simulations both for 1D and 3D functional data, and evaluate its effectiveness in selecting the true subregion. We also investigate predictive ability of the selected stable regions. To find the brain regions related to cognitive ability, FuDoS is applied to the ADNI's PET data. Due to the induced sparseness, the results naturally provide more interpretable information about the relations between the regions and the outcome. Moreover, the selected regions from our analysis show high associations with the expected anatomical brain areas known to have memory-related functions.

We propose a novel approach to the analysis of covariance operators making use of concentration inequalities. First, non-asymptotic confidence sets are constructed for such operators. Then, subsequent applications including a k sample test for equality of covariance, a functional data classifier, and an expectation-maximization style clustering algorithm are derived and tested on both simulated and phoneme data., Comment: 15 pages, 2 figures, 6 tables

Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional manifold. For this purpose a regularization approach is adopted, introducing a smoothing penalty coherent with the geodesic distance over the manifold. The model introduced can be applied to any manifold topology, can naturally handle missing data and functional samples evaluated in different grids of points. We approach the discretization task by means of finite element analysis and propose an efficient iterative algorithm for its resolution. We compare the performances of the proposed algorithm with other approaches classically adopted in literature. We finally apply the proposed method to resting state functional magnetic resonance imaging data from the Human Connectome Project, where the method shows substantial differential variations between brain regions that were not apparent with other approaches., Comment: 33 pages

We provide a complete description of possible covariance matrices consistent with a Gaussian latent tree model for any tree. We then present techniques for utilising these constraints to assess whether observed data is compatible with that Gaussian latent tree model. Our method does not require us first to fit such a tree. We demonstrate the usefulness of the inverse-Wishart distribution for performing preliminary assessments of tree-compatibility using semialgebraic constraints. Using results from Drton et al. (2008) we then provide the appropriate moments required for test statistics for assessing adherence to these equality constraints. These are shown to be effective even for small sample sizes and can be easily adjusted to test either the entire model or only certain macrostructures hypothesized within the tree. We illustrate our exploratory tetrad analysis using a linguistic application and our confirmatory tetrad analysis using a biological application., Comment: 15 pages

The historical and geographical spread from older to more modern languages has long been studied by examining textual changes and in terms of changes in phonetic transcriptions. However, it is more difficult to analyze language change from an acoustic point of view, although this is usually the dominant mode of transmission. We propose a novel analysis approach for acoustic phonetic data, where the aim will be to statistically model the acoustic properties of spoken words. We explore phonetic variation and change using a time-frequency representation, namely the log-spectrograms of speech recordings. We identify time and frequency covariance functions as a feature of the language; in contrast, mean spectrograms depend mostly on the particular word that has been uttered. We build models for the mean and covariances (taking into account the restrictions placed on the statistical analysis of such objects) and use these to define a phonetic transformation that models how an individual speaker would sound in a different language, allowing the exploration of phonetic differences between languages. Finally, we map back these transformations to the domain of sound recordings, allowing us to listen to the output of the statistical analysis. The proposed approach is demonstrated using recordings of the words corresponding to the numbers from "one" to "ten" as pronounced by speakers from five different Romance languages.

The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional {data analysis} settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full non-separable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset., Comment: 47 pages, 10 figures, 4 tables

Positron Emission Tomography (PET) is an imaging technique which can be used to investigate chemical changes in human biological processes such as cancer development or neurochemical reactions. Most dynamic PET scans are currently analyzed based on the assumption that linear first order kinetics can be used to adequately describe the system under observation. However, there has recently been strong evidence that this is not the case. In order to provide an analysis of PET data which is free from this compartmental assumption, we propose a nonparametric deconvolution and analysis model for dynamic PET data based on functional principal component analysis. This yields flexibility in the possible deconvolved functions while still performing well when a linear compartmental model setup is the true data generating mechanism. As the deconvolution needs to be performed on only a relative small number of basis functions rather than voxel by voxel in the entire 3-D volume, the methodology is both robust to typical brain imaging noise levels while also being computationally efficient. The new methodology is investigated through simulations in both 1-D functions and 2-D images and also applied to a neuroimaging study whose goal is the quantification of opioid receptor concentration in the brain., Comment: 33 pages, 10 figures

Prior to adjustment, accounting conditions between national accounts data sets are frequently violated. Benchmarking is the procedure used by economic agencies to make such data sets consistent. It typically involves adjusting a high frequency time series (e.g. quarterly data) so it becomes consistent with a lower frequency version (e.g. annual data). Various methods have been developed to approach this problem of inconsistency between data sets. This paper introduces a new statistical procedure; namely wavelet benchmarking. Wavelet properties allow high and low frequency processes to be jointly analysed and we show that benchmarking can be formulated and approached succinctly in the wavelet domain. Furthermore the time and frequency localisation properties of wavelets are ideal for handling more complicated benchmarking problems. The versatility of the procedure is demonstrated using simulation studies where we provide evidence showing it substantially outperforms currently used methods. Finally, we apply this novel method of wavelet benchmarking to official Office of National Statistics (ONS) data., Comment: 33 pages, 6 figures

Evolutionary models of languages are usually considered to take the form of trees. With the development of so-called tree constraints the plausibility of the tree model assumptions can be addressed by checking whether the moments of observed variables lie within regions consistent with trees. In our linguistic application, the data set comprises acoustic samples (audio recordings) from speakers of five Romance languages or dialects. We wish to assess these functional data for compatibility with a hereditary tree model at the language level. A novel combination of canonical function analysis (CFA) with a separable covariance structure provides a method for generating a representative basis for the data. This resulting basis is formed of components which emphasize language differences whilst maintaining the integrity of the observational language-groupings. A previously unexploited Gaussian tree constraint is then applied to component-by-component projections of the data to investigate adherence to an evolutionary tree. The results indicate that while a tree model is unlikely to be suitable for modeling all aspects of the acoustic linguistic data, certain features of the spoken Romance languages highlighted by the separable-CFA basis may indeed be suitably modeled as a tree., Comment: 48 pages

While there is considerable work on change point analysis in univariate time series, more and more data being collected comes from high dimensional multivariate settings. This paper introduces the asymptotic concept of high dimensional efficiency which quantifies the detection power of different statistics in such situations. While being related to classic asymptotic relative efficiency, it is different in that it provides the rate at which the change can get smaller with dimension while still being detectable. This also allows for comparisons of different methods with different null asymptotics as is for example the case in high-dimensional change point settings. Based on this new concept we investigate change point detection procedures using projections and develop asymptotic theory for how full panel (multivariate) tests compare with both oracle and random projections. Furthermore, for each given projection we can quantify a cone such that the corresponding projection statistic yields better power behavior if the true change direction is within this cone. The effect of misspecification of the covariance on the power of the tests is investigated, because in many high dimensional situations estimation of the full dependency (covariance) between the multivariate observations in the panel is often either computationally or even theoretically infeasible. It turns out that the projection statistic is much more robust in this respect in terms of size and somewhat more robust in terms of power. The theoretic quantification by the theory is accompanied by simulation results which confirm the theoretic (asymptotic) findings for surprisingly small samples. This shows in particular that the concept of high dimensional efficiency is indeed suitable to describe small sample power, and this is demonstrated in a multivariate example of market index data., Comment: 37 pages, 6 figures

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved sub-problems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed Divide-and-Conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-Conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging sub-problems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem.