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In clinical trials of longitudinal continuous outcomes, reference based imputation (RBI) has commonly been applied to handle missing outcome data in settings where the estimand incorporates the effects of intercurrent events, e.g. treatment discontinuation. RBI was originally developed in the multiple imputation framework, however recently conditional mean imputation (CMI) combined with the jackknife estimator of the standard error was proposed as a way to obtain deterministic treatment effect estimates and correct frequentist inference. For both multiple and CMI, a mixed model for repeated measures (MMRM) is often used for the imputation model, but this can be computationally intensive to fit to multiple data sets (e.g. the jackknife samples) and lead to convergence issues with complex MMRM models with many parameters. Therefore, a step-wise approach based on sequential linear regression (SLR) of the outcomes at each visit was developed for the imputation model in the multiple imputation framework, but similar developments in the CMI framework are lacking. In this article, we fill this gap in the literature by proposing a SLR approach to implement RBI in the CMI framework, and justify its validity using theoretical results and simulations. We also illustrate our proposal on a real data application.

In this paper, I present three closed-form approximations of the two-sample Pearson Bayes factor. The techniques rely on some classical asymptotic results about gamma functions. These approximations permit simple closed-form calculation of the Pearson Bayes factor in cases where only the summary statistics are available (i.e., the t-score and degrees of freedom).

In this paper, we investigate building a sequence to sequence architecture for motion to language translation and synchronization. The aim is to translate motion capture inputs into English natural-language descriptions, such that the descriptions are generated synchronously with the actions performed, enabling semantic segmentation as a byproduct, but without requiring synchronized training data. We propose a new recurrent formulation of local attention that is suited for synchronous/live text generation, as well as an improved motion encoder architecture better suited to smaller data and for synchronous generation. We evaluate both contributions in individual experiments, using the standard BLEU4 metric, as well as a simple semantic equivalence measure, on the KIT motion language dataset. In a follow-up experiment, we assess the quality of the synchronization of generated text in our proposed approaches through multiple evaluation metrics. We find that both contributions to the attention mechanism and the encoder architecture additively improve the quality of generated text (BLEU and semantic equivalence), but also of synchronization. Our code will be made available at \url{//github.com/rd20karim/M2T-Segmentation/tree/main}

In this paper we consider the numerical approximation of infinite horizon problems via the dynamic programming approach. The value function of the problem solves a Hamilton-Jacobi-Bellman (HJB) equation that is approximated by a fully discrete method. It is known that the numerical problem is difficult to handle by the so called curse of dimensionality. To mitigate this issue we apply a reduction of the order by means of a new proper orthogonal decomposition (POD) method based on time derivatives. We carry out the error analysis of the method using recently proved optimal bounds for the fully discrete approximations. Moreover, the use of snapshots based on time derivatives allow us to bound some terms of the error that could not be bounded in a standard POD approach. Some numerical experiments show the good performance of the method in practice.

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step graded mesh procedure based on an expansion of the vector field using orthonormal Jacobi polynomials. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

In this paper, we present a novel second-order generalised rotational discrete gradient scheme for numerically approximating the orthonormal frame gradient flow of biaxial nematic liquid crystals. This scheme relies on reformulating the original gradient flow system into an equivalent generalised "rotational" form. A second-order discrete gradient approximation of the energy variation is then devised such that it satisfies an energy difference relation. The proposed numerical scheme has two remarkable properties: (i) it strictly obeys the orthonormal property of the tensor field and (ii) it satisfies the energy dissipation law at the discrete level, regardless of the time step sizes. We provide ample numerical results to validate the accuracy, efficiency, unconditional stability and SO(3)-preserving property of this scheme. In addition, comparisons of the simulation results between the biaxial orthonormal frame gradient flow model and uniaxial Oseen-Frank gradient flow are made to demonstrate the ability of the former to characterize non-axisymmetric local anisotropy.

In this paper, we propose and analyze an efficient numerical method for the anisotropic phase field dendritic crystal growth model, which is challenging because we are facing the nonlinear coupling and anisotropic coefficient in the model. The proposed method is a two-step scheme. In the first step, an intermediate solution is computed by using BDF schemes of order up to three for both the phase-field and heat equations. In the second step the intermediate solution is stabilized by multiplying an auxiliary variable. The key of the second step is to stabilize the overall scheme while maintaining the convergence order of the stabilized solution. In order to overcome the difficulty caused by the gradient-dependent anisotropic coefficient and the nonlinear terms, some stabilization terms are added to the BDF schemes in the first step. The second step makes use of a generalized auxiliary variable approach with relaxation. The Fourier spectral method is applied for the spatial discretization. Our analysis shows that the proposed scheme is unconditionally stable and has accuracy in time up to third order. We also provide a sophisticated implementation showing that the computational complexity of our schemes is equivalent to solving two linear equations and some algebraic equations. To the best of our knowledge, this is the cheapest unconditionally stable schemes reported in the literature. Some numerical examples are given to verify the efficiency of the proposed method.

In this study, we present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptical equations. We did not impose a shape-regularity mesh condition for the analysis. Therefore, anisotropic meshes can be used. The main contributions of this study include providing new proof of the consistency term. This enabled us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart--Thomas and Morley finite element spaces. Our results show optimal convergence rates and imply that the modified Morley FEM may be effective for errors.

While analogies are a common way to evaluate word embeddings in NLP, it is also of interest to investigate whether or not analogical reasoning is a task in itself that can be learned. In this paper, we test several ways to learn basic analogical reasoning, specifically focusing on analogies that are more typical of what is used to evaluate analogical reasoning in humans than those in commonly used NLP benchmarks. Our experiments find that models are able to learn analogical reasoning, even with a small amount of data. We additionally compare our models to a dataset with a human baseline, and find that after training, models approach human performance.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.