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Effect modification occurs when the impact of the treatment on an outcome varies based on the levels of other covariates known as effect modifiers. Modeling of these effect differences is important for etiological goals and for purposes of optimizing treatment. Structural nested mean models (SNMMs) are useful causal models for estimating the potentially heterogeneous effect of a time-varying exposure on the mean of an outcome in the presence of time-varying confounding. A data-driven approach for selecting the effect modifiers of an exposure may be necessary if these effect modifiers are a priori unknown and need to be identified. Although variable selection techniques are available in the context of estimating conditional average treatment effects using marginal structural models, or in the context of estimating optimal dynamic treatment regimens, all of these methods consider an outcome measured at a single point in time. In the context of an SNMM for repeated outcomes, we propose a doubly robust penalized G-estimator for the causal effect of a time-varying exposure with a simultaneous selection of effect modifiers and use this estimator to analyze the effect modification in a study of hemodiafiltration. We prove the oracle property of our estimator, and conduct a simulation study for evaluation of its performance in finite samples and for verification of its double-robustness property. Our work is motivated by and applied to the study of hemodiafiltration for treating patients with end-stage renal disease at the Centre Hospitalier de l'Universit\'e de Montr\'eal. We apply the proposed method to investigate the effect heterogeneity of dialysis facility on the repeated session-specific hemodiafiltration outcomes.

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.

In this study, we tackle a modern research challenge within the field of perceptual brain decoding, which revolves around synthesizing images from EEG signals using an adversarial deep learning framework. The specific objective is to recreate images belonging to various object categories by leveraging EEG recordings obtained while subjects view those images. To achieve this, we employ a Transformer-encoder based EEG encoder to produce EEG encodings, which serve as inputs to the generator component of the GAN network. Alongside the adversarial loss, we also incorporate perceptual loss to enhance the quality of the generated images.

In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.

Under a generalised estimating equation analysis approach, approximate design theory is used to determine Bayesian D-optimal designs. For two examples, considering simple exchangeable and exponential decay correlation structures, we compare the efficiency of identified optimal designs to balanced stepped-wedge designs and corresponding stepped-wedge designs determined by optimising using a normal approximation approach. The dependence of the Bayesian D-optimal designs on the assumed correlation structure is explored; for the considered settings, smaller decay in the correlation between outcomes across time periods, along with larger values of the intra-cluster correlation, leads to designs closer to a balanced design being optimal. Unlike for normal data, it is shown that the optimal design need not be centro-symmetric in the binary outcome case. The efficiency of the Bayesian D-optimal design relative to a balanced design can be large, but situations are demonstrated in which the advantages are small. Similarly, the optimal design from a normal approximation approach is often not much less efficient than the Bayesian D-optimal design. Bayesian D-optimal designs can be readily identified for stepped-wedge cluster randomised trials with binary outcome data. In certain circumstances, principally ones with strong time period effects, they will indicate that a design unlikely to have been identified by previous methods may be substantially more efficient. However, they require a larger number of assumptions than existing optimal designs, and in many situations existing theory under a normal approximation will provide an easier means of identifying an efficient design for binary outcome data.

In this paper, we aim to perform sensitivity analysis of set-valued models and, in particular, to quantify the impact of uncertain inputs on feasible sets, which are key elements in solving a robust optimization problem under constraints. While most sensitivity analysis methods deal with scalar outputs, this paper introduces a novel approach for performing sensitivity analysis with set-valued outputs. Our innovative methodology is designed for excursion sets, but is versatile enough to be applied to set-valued simulators, including those found in viability fields, or when working with maps like pollutant concentration maps or flood zone maps. We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a kernel designed for set-valued outputs. After proposing a probabilistic framework for random sets, a first contribution is the proof that this kernel is characteristic, an essential property in a kernel-based sensitivity analysis context. To measure the contribution of each input, we then propose to use HSIC-ANOVA indices. With these indices, we can identify which inputs should be neglected (screening) and we can rank the others according to their influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally, we test the proposed method on three test cases of excursion sets.

In this study, we consider the application of orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in limited-aperture inverse scattering problem. We first apply the OSM with single source and show that the indicator function with single source can be expressed by the Bessel function of order zero of the first kind, infinite series of Bessel function of nonzero integer order of the first kind, range of signal receiver, and the location of emitter. Based on this result, we explain that the objects can be identified through the OSM with single source but the identification is significantly influenced by the location of source and applied frequency. For a successful improvement, we then consider the OSM with multiple sources. Based on the identified structure of the OSM with single source, we design an indicator function of the OSM with multiple sources and show that it can be expressed by the square of the Bessel function of order zero of the first kind an infinite series of the square of Bessel function of nonzero integer order of the first kind. Based on the theoretical results, we explain that the objects can be identified uniquely through the designed OSM. Several numerical experiments with experimental data provided by the Institute Fresnel demonstrate the pros and cons of the OSM with single source and how the designed OSM with multiple sources behave.

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpi\'{n}ski triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.

Based on interactions between individuals and others and references to social norms, this study reveals the impact of heterogeneity in time preference on wealth distribution and inequality. We present a novel approach that connects the interactions between microeconomic agents that generate heterogeneity to the dynamic equations for capital and consumption in macroeconomic models. Using this approach, we estimate the impact of changes in the discount rate due to microeconomic interactions on capital, consumption and utility and the degree of inequality. The results show that intercomparisons with others regarding consumption significantly affect capital, i.e. wealth inequality. Furthermore, the impact on utility is never small and social norms can reduce this impact. Our supporting evidence shows that the quantitative results of inequality calculations correspond to survey data from cohort and cross-cultural studies. This study's micro-macro connection approach can be deployed to connect microeconomic interactions, such as exchange, interest and debt, redistribution, mutual aid and time preference, to dynamic macroeconomic models.

We present an information-theoretic lower bound for the problem of parameter estimation with time-uniform coverage guarantees. Via a new a reduction to sequential testing, we obtain stronger lower bounds that capture the hardness of the time-uniform setting. In the case of location model estimation, logistic regression, and exponential family models, our $\Omega(\sqrt{n^{-1}\log \log n})$ lower bound is sharp to within constant factors in typical settings.