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In recent years, there has been considerable interest in estimating conditional independence graphs in high dimensions. Most previous work assumed that the variables are multivariate Gaussian or that the conditional means of the variables are linearly related. Unfortunately, if these assumptions are violated, the resulting conditional independence estimates can be inaccurate.We propose a semiparametric method, graph estimation with joint additive models, which allows the conditional means of the features to take an arbitrary additive form. We present an efficient algorithm for computation of our estimator, and prove that it is consistent. We extend our method to estimation of directed graphs with known causal ordering. Using simulated data, we show that our method performs better than existing methods when there are nonlinear relationships among the features, and is comparable to methods that assume multivariate normality when the conditional means are linear. We illustrate our method on a cell signalling dataset.

In most data assimilation and numerical weather prediction systems, the Gaussian assumption is prevalent for the behavior of the random variables/errors that are involved. At the Cooperative Institute for Research in the Atmosphere theory has been developed for different forms of variational data assimilation schemes that enables the Gaussian assumption to be relaxed. For certain variable types, a lognormally distributed random variable can be combined in a mixed Gaussian‐lognormal distribution to better capture the interactions of the errors of different distributions. However, assuming that a distribution can change in time, then developing techniques to know when to switch between different versions of the data assimilation schemes becomes very important. By dynamically changing the formulation of the data assimilation system we are able to assimilate observations in a way that reflects the flow‐dependent variability of their distribution. In this paper, we present results with a machine learning technique (the support vector machine) to switch between data assimilation methods based on the detection of a change in the Lorenz 1963 model's z component's probability distribution. Given the machine learning technique's detection/prediction of a change in the distribution, then either a Gaussian or a mixed Gaussian‐lognormal 3DVar based cost function is used to minimize the errors in this period of time. It is shown that switching from a Gaussian 3DVar to a lognormal 3DVar at lognormally distributed parts of the attractor improves the data assimilation analysis error compared to using one distribution type for the entire attractor. Key Points Machine learning can be used to detect non‐Gaussian distributions Given the machine learning detection, we can switch between optimal data assimilation schemes Data assimilation can be improved using machine learning techniques

Independent component analysis is a probabilistic method for learning a linear transform of a random vector. The goal is to find components that are maximally independent and non-Gaussian (non-normal). Its fundamental difference to classical multi-variate statistical methods is in the assumption of non-Gaussianity, which enables the identification of original, underlying components, in contrast to classical methods. The basic theory of independent component analysis was mainly developed in the 1990s and summarized, for example, in our monograph in 2001. Here, we provide an overview of some recent developments in the theory since the year 2000. The main topics are: analysis of causal relations, testing independent components, analysing multiple datasets (three-way data), modelling dependencies between the components and improved methods for estimating the basic model.

Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here, we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement, which we confirm by Bayesian phase estimation with sub–shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems.

We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

We analyse the transport of diffusive particles that switch between mobile and immobile states with finite rates. We focus on the effect of advection on the density functions and mean squared displacements (MSDs). At relevant intermediate time scales we find strong anomalous diffusion with cubic scaling in time of the MSD for high Péclet numbers. The cubic scaling exists for short and long mean residence times in the immobile state τ i m . For long τ i m the plateau in the MSD at intermediate times, previously found in the absence of advection, also exists for high Péclet numbers. Initially immobile tracers are subject to the newly observed regime of advection induced subdiffusion for short immobilisations and high Péclet numbers. In the long-time limit the effective advection velocity is reduced compared to advection in the mobile phase. In contrast, the MSD is enhanced by advection. We explore physical mechanisms behind the emerging non-Gaussian density functions and the features of the MSD.

Argo floats measure seawater temperature and salinity in the upper 2000 m of the global ocean. Statistical analysis of the resulting spatio-temporal dataset is challenging owing to its non-stationary structure and large size. We propose mapping these data using locally stationary Gaussian process regression where covariance parameter estimation and spatio-temporal prediction are carried out in a moving-window fashion. This yields computationally tractable non-stationary anomaly fields without the need to explicitly model the non-stationary covariance structure. We also investigate Student t -distributed fine-scale variation as a means to account for non-Gaussian heavy tails in ocean temperature data. Cross-validation studies comparing the proposed approach with the existing state of the art demonstrate clear improvements in point predictions and show that accounting for the non-stationarity and non-Gaussianity is crucial for obtaining well-calibrated uncertainties. This approach also provides data-driven local estimates of the spatial and temporal dependence scales for the global ocean, which are of scientific interest in their own right.

We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors (p) is large compared to the sample size (n); in fact, p can be much larger than n, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the Gaussian multiplier (or wild) bootstrap procedure. Here too, p can be large or even much larger than n. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our Gaussian approximations and the multiplier bootstrap can be used for modern highdimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain nonasymptotic bounds on approximation errors.

The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.

A standard method to obtain information on a quantum state is to measure marginal distributions along many different axes in phase space, which forms a basis of quantum-state tomography. We theoretically propose and experimentally demonstrate a general framework to manifest nonclassicality by observing a single marginal distribution only, which provides a unique insight into nonclassicality and a practical applicability to various quantum systems. Our approach maps the 1D marginal distribution into a factorized 2D distribution by multiplying the measured distribution or the vacuum-state distribution along an orthogonal axis. The resulting fictitious Wigner function becomes unphysical only for a nonclassical state; thus the negativity of the corresponding density operator provides evidence of nonclassicality. Furthermore, the negativity measured this way yields a lower bound for entanglement potential—a measure of entanglement generated using a nonclassical state with a beam-splitter setting that is a prototypical model to produce continuous-variable (CV) entangled states. Our approach detects both Gaussian and non-Gaussian nonclassical states in a reliable and efficient manner. Remarkably, it works regardless of measurement axis for all non-Gaussian states in finite-dimensional Fock space of any size, also extending to infinite-dimensional states of experimental relevance for CV quantum informatics. We experimentally illustrate the power of our criterion for motional states of a trapped ion, confirming their nonclassicality in a measurement-axis–independent manner. We also address an extension of our approach combined with phase-shift operations, which leads to a stronger test of nonclassicality, that is, detection of genuine non-Gaussianity under a CV measurement.