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In spatial blind source separation the observed multivariate random fields are assumed to be mixtures of latent spatially dependent random fields. The objective is to recover latent random fields by estimating the unmixing transformation. Currently, the algorithms for spatial blind source separation can only estimate linear unmixing transformations. Nonlinear blind source separation methods for spatial data are scarce. In this paper we extend an identifiable variational autoencoder that can estimate nonlinear unmixing transformations to spatially dependent data and demonstrate its performance for both stationary and nonstationary spatial data using simulations. In addition, we introduce scaled mean absolute Shapley additive explanations for interpreting the latent components through nonlinear mixing transformation. The spatial identifiable variational autoencoder is applied to a geochemical dataset to find the latent random fields, which are then interpreted by using the scaled mean absolute Shapley additive explanations. Finally, we illustrate how the proposed method can be used as a pre-processing method when making multivariate predictions.

In areal unit data with missing or suppressed data, it desirable to create models that are able to predict observations that are not available. Traditional statistical methods achieve this through Bayesian hierarchical models that can capture the unexplained residual spatial autocorrelation through conditional autoregressive (CAR) priors, such that they can make predictions at geographically related spatial locations. In contrast, typical machine learning approaches such as random forests ignore this residual autocorrelation, and instead base predictions on complex non-linear feature-target relationships. In this paper, we propose CAR-Forest, a novel spatial prediction algorithm that combines the best features of both approaches by fusing them together. By iteratively refitting a random forest combined with a Bayesian CAR model in one algorithm, CAR-Forest can incorporate flexible feature-target relationships while still accounting for the residual spatial autocorrelation. Our results, based on a Scottish housing price data set, show that CAR-Forest outperforms Bayesian CAR models, random forests, and the state-of-the-art hybrid approach, geographically weighted random forest, providing a state-of-the-art framework for small-area spatial prediction.

Multinomial prediction models (MPMs) have a range of potential applications across healthcare where the primary outcome of interest has multiple nominal or ordinal categories. However, the application of MPMs is scarce, which may be due to the added methodological complexities that they bring. This article provides a guide of how to develop, externally validate, and update MPMs. Using a previously developed and validated MPM for treatment outcomes in rheumatoid arthritis as an example, we outline guidance and recommendations for producing a clinical prediction model, using multinomial logistic regression. This article is intended to supplement existing general guidance on prediction model research. This guide is split into three parts: 1) Outcome definition and variable selection, 2) Model development, and 3) Model evaluation (including performance assessment, internal and external validation, and model recalibration). We outline how to evaluate and interpret the predictive performance of MPMs. R code is provided. We recommend the application of MPMs in clinical settings where the prediction of a nominal polytomous outcome is of interest. Future methodological research could focus on MPM-specific considerations for variable selection and sample size criteria for external validation.

Object data analysis is concerned with statistical methodology for datasets whose elements reside in an arbitrary, unspecified metric space. In this work we propose the object shape, a novel measure of shape/symmetry for object data. The object shape is easy to compute and interpret, owing to its intuitive interpretation as interpolation between two extreme forms of symmetry. As one major part of this work, we apply object shape in various metric spaces and show that it manages to unify several pre-existing, classical forms of symmetry. We also propose a new visualization tool called the peeling plot, which allows using the object shape for outlier detection and principal component analysis of object data.

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. Numerical experiments suggest that the presence of an interface seems to be of minor importance for the continuation of the solution beyond the data domain. On the other hand, certain convexity properties of the geometry are crucial as has already been observed for many other problems without interfaces.

Advanced LIGO and Advanced Virgo ground-based interferometers are instruments capable to detect gravitational wave signals exploiting advanced laser interferometry techniques. The underlying data analysis task consists in identifying specific patterns in noisy timeseries, but it is made extremely complex by the incredibly small amplitude of the target signals. In this scenario, the development of effective gravitational wave detection algorithms is crucial. We propose a novel layered framework for real-time detection of gravitational waves inspired by speech processing techniques and, in the present implementation, based on a state-of-the-art machine learning approach involving a hybridization of genetic programming and neural networks. The key aspects of the newly proposed framework are: the well structured, layered approach, and the low computational complexity. The paper describes the basic concepts of the framework and the derivation of the first three layers. Even if the layers are based on models derived using a machine learning approach, the proposed layered structure has a universal nature. Compared to more complex approaches, such as convolutional neural networks, which comprise a parameter set of several tens of MB and were tested exclusively for fixed length data samples, our framework has lower accuracy (e.g., it identifies 45% of low signal-to-noise-ration gravitational wave signals, against 65% of the state-of-the-art, at a false alarm probability of $10^{-2}$), but has a much lower computational complexity and a higher degree of modularity. Furthermore, the exploitation of short-term features makes the results of the new framework virtually independent against time-position of gravitational wave signals, simplifying its future exploitation in real-time multi-layer pipelines for gravitational-wave detection with new generation interferometers.

Neural networks are powerful tools in various applications, and quantifying their uncertainty is crucial for reliable decision-making. In the deep learning field, the uncertainties are usually categorized into aleatoric (data) and epistemic (model) uncertainty. In this paper, we point out that the existing popular variance attenuation method highly overestimates aleatoric uncertainty. To address this issue, we propose a new estimation method by actively de-noising the observed data \footnote{Source code available at \url{//github.com/wz16/DVA}.}. By conducting a broad range of experiments, we demonstrate that our proposed approach provides a much closer approximation to the actual data uncertainty than the standard method.

We study hypothesis testing under communication constraints, where each sample is quantized before being revealed to a statistician. Without communication constraints, it is well known that the sample complexity of simple binary hypothesis testing is characterized by the Hellinger distance between the distributions. We show that the sample complexity of simple binary hypothesis testing under communication constraints is at most a logarithmic factor larger than in the unconstrained setting and this bound is tight. We develop a polynomial-time algorithm that achieves the aforementioned sample complexity. Our framework extends to robust hypothesis testing, where the distributions are corrupted in the total variation distance. Our proofs rely on a new reverse data processing inequality and a reverse Markov inequality, which may be of independent interest. For simple $M$-ary hypothesis testing, the sample complexity in the absence of communication constraints has a logarithmic dependence on $M$. We show that communication constraints can cause an exponential blow-up leading to $\Omega(M)$ sample complexity even for adaptive algorithms.

Spatial data can come in a variety of different forms, but two of the most common generating models for such observations are random fields and point processes. Whilst it is known that spectral analysis can unify these two different data forms, specific methodology for the related estimation is yet to be developed. In this paper, we solve this problem by extending multitaper estimation, to estimate the spectral density matrix function for multivariate spatial data, where processes can be any combination of either point processes or random fields. We discuss finite sample and asymptotic theory for the proposed estimators, as well as specific details on the implementation, including how to perform estimation on non-rectangular domains and the correct implementation of multitapering for processes sampled in different ways, e.g. continuously vs on a regular grid.

Inferring causation from time series data is of scientific interest in different disciplines, particularly in neural connectomics. While different approaches exist in the literature with parametric modeling assumptions, we focus on a non-parametric model for time series satisfying a Markovian structural causal model with stationary distribution and without concurrent effects. We show that the model structure can be used to its advantage to obtain an elegant algorithm for causal inference from time series based on conditional dependence tests, coined Causal Inference in Time Series (CITS) algorithm. We describe Pearson's partial correlation and Hilbert-Schmidt criterion as candidates for such conditional dependence tests that can be used in CITS for the Gaussian and non-Gaussian settings, respectively. We prove the mathematical guarantee of the CITS algorithm in recovering the true causal graph, under standard mixing conditions on the underlying time series. We also conduct a comparative evaluation of performance of CITS with other existing methodologies in simulated datasets. We then describe the utlity of the methodology in neural connectomics -- in inferring causal functional connectivity from time series of neural activity, and demonstrate its application to a real neurobiological dataset of electro-physiological recordings from the mouse visual cortex recorded by Neuropixel probes.