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Even though novel imaging techniques have been successful in studying brain structure and function, the measured biological signals are often contaminated by multiple sources of noise, arising due to e.g. head movements of the individual being scanned, limited spatial/temporal resolution, or other issues specific to each imaging technology. Data preprocessing (e.g. denoising) is therefore critical. Preprocessing pipelines have become increasingly complex over the years, but also more flexible, and this flexibility can have a significant impact on the final results and conclusions of a given study. This large parameter space is often referred to as multiverse analyses. Here, we provide conceptual and practical tools for statistical analyses that can aggregate multiple pipeline results along with a new sensitivity analysis testing for hypotheses across pipelines such as "no effect across all pipelines" or "at least one pipeline with no effect". The proposed framework is generic and can be applied to any multiverse scenario, but we illustrate its use based on positron emission tomography data.

This paper addresses the asymptotic performance of popular spatial regression estimators on the task of estimating the linear effect of an exposure on an outcome under "spatial confounding" -- the presence of an unmeasured spatially-structured variable influencing both the exposure and the outcome. The existing literature on spatial confounding is informal and inconsistent; this paper is an attempt to bring clarity through rigorous results on the asymptotic bias and consistency of estimators from popular spatial regression models. We consider two data generation processes: one where the confounder is a fixed function of space and one where it is a random function (i.e., a stochastic process on the spatial domain). We first show that the estimators from ordinary least squares (OLS) and restricted spatial regression are asymptotically biased under spatial confounding. We then prove a novel main result on the consistency of the generalized least squares (GLS) estimator using a Gaussian process (GP) covariance matrix in the presence of spatial confounding under in-fill (fixed domain) asymptotics. The result holds under very general conditions -- for any exposure with some non-spatial variation (noise), for any spatially continuous confounder, using any choice of Mat\'ern or square exponential Gaussian process covariance used to construct the GLS estimator, and without requiring Gaussianity of errors. Finally, we prove that spatial estimators from GLS, GP regression, and spline models that are consistent under confounding by a fixed function will also be consistent under confounding by a random function. We conclude that, contrary to much of the literature on spatial confounding, traditional spatial estimators are capable of estimating linear exposure effects under spatial confounding in the presence of some noise in the exposure. We support our theoretical arguments with simulation studies.

We show that confidence intervals for a variance component or proportion, with asymptotically correct uniform coverage probability, can be obtained by inverting certain test-statistics based on the score for the restricted likelihood. The results apply in settings where the variance or proportion is near or at the boundary of the parameter set. Simulations indicate the proposed test-statistics are approximately pivotal and lead to confidence intervals with near-nominal coverage even in small samples. We illustrate our methods' application in spatially-resolved transcriptomics where we compute approximately 15,000 confidence intervals, used for gene ranking, in less than 4 minutes. In the settings we consider, the proposed method is between two and 28,000 times faster than popular alternatives, depending on how many confidence intervals are computed.

Keeping the balance between electricity generation and consumption is becoming increasingly challenging and costly, mainly due to the rising share of renewables, electric vehicles and heat pumps and electrification of industrial processes. Accurate imbalance forecasts, along with reliable uncertainty estimations, enable transmission system operators (TSOs) to dispatch appropriate reserve volumes, reducing balancing costs. Further, market parties can use these probabilistic forecasts to design strategies that exploit asset flexibility to help balance the grid, generating revenue with known risks. Despite its importance, literature regarding system imbalance (SI) forecasting is limited. Further, existing methods do not focus on situations with high imbalance magnitude, which are crucial to forecast accurately for both TSOs and market parties. Hence, we propose an ensemble of C-VSNs, which are our adaptation of variable selection networks (VSNs). Each minute, our model predicts the imbalance of the current and upcoming two quarter-hours, along with uncertainty estimations on these forecasts. We evaluate our approach by forecasting the imbalance of Belgium, where high imbalance magnitude is defined as $|$SI$| > 500\,$MW (occurs 1.3% of the time in Belgium). For high imbalance magnitude situations, our model outperforms the state-of-the-art by 23.4% (in terms of continuous ranked probability score (CRPS), which evaluates probabilistic forecasts), while also attaining a 6.5% improvement in overall CRPS. Similar improvements are achieved in terms of root-mean-squared error. Additionally, we developed a fine-tuning methodology to effectively include new inputs with limited history in our model. This work was performed in collaboration with Elia (the Belgian TSO) to further improve their imbalance forecasts, demonstrating the relevance of our work.

Cross-validation is usually employed to evaluate the performance of a given statistical methodology. When such a methodology depends on a number of tuning parameters, cross-validation proves to be helpful to select the parameters that optimize the estimated performance. In this paper, however, a very different and nonstandard use of cross-validation is investigated. Instead of focusing on the cross-validated parameters, the main interest is switched to the estimated value of the error criterion at optimal performance. It is shown that this approach is able to provide consistent and efficient estimates of some density functionals, with the noteworthy feature that these estimates do not rely on the choice of any further tuning parameter, so that, in that sense, they can be considered to be purely empirical. Here, a base case of application of this new paradigm is developed in full detail, while many other possible extensions are hinted as well.

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

Random utility maximisation (RUM) models are one of the cornerstones of discrete choice modelling. However, specifying the utility function of RUM models is not straightforward and has a considerable impact on the resulting interpretable outcomes and welfare measures. In this paper, we propose a new discrete choice model based on artificial neural networks (ANNs) named "Alternative-Specific and Shared weights Neural Network (ASS-NN)", which provides a further balance between flexible utility approximation from the data and consistency with two assumptions: RUM theory and fungibility of money (i.e., "one euro is one euro"). Therefore, the ASS-NN can derive economically-consistent outcomes, such as marginal utilities or willingness to pay, without explicitly specifying the utility functional form. Using a Monte Carlo experiment and empirical data from the Swissmetro dataset, we show that ASS-NN outperforms (in terms of goodness of fit) conventional multinomial logit (MNL) models under different utility specifications. Furthermore, we show how the ASS-NN is used to derive marginal utilities and willingness to pay measures.

Any interactive protocol between a pair of parties can be reliably simulated in the presence of noise with a multiplicative overhead on the number of rounds (Schulman 1996). The reciprocal of the best (least) overhead is called the interactive capacity of the noisy channel. In this work, we present lower bounds on the interactive capacity of the binary erasure channel. Our lower bound improves the best known bound due to Ben-Yishai et al. 2021 by roughly a factor of 1.75. The improvement is due to a tighter analysis of the correctness of the simulation protocol using error pattern analysis. More precisely, instead of using the well-known technique of bounding the least number of erasures needed to make the simulation fail, we identify and bound the probability of specific erasure patterns causing simulation failure. We remark that error pattern analysis can be useful in solving other problems involving stochastic noise, such as bounding the interactive capacity of different channels.

We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $k\Delta_n$ for $k=0,\dots,n$, under the conditions $\Delta_n\to 0$ and $n\Delta_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection.

We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results.