The time series with periodic behavior, such as the periodic autoregressive (PAR) models belonging to the class of the periodically correlated processes, are present in various real applications. In the literature, such processes were considered in different directions, especially with the Gaussian-distributed noise. However, in most of the applications, the assumption of the finite-variance distribution seems to be too simplified. Thus, one can consider the extensions of the classical PAR model where the non-Gaussian distribution is applied. In particular, the Gaussian distribution can be replaced by the infinite-variance distribution, e.g. by the $\alpha-$stable distribution. In this paper, we focus on the multidimensional $\alpha-$stable PAR time series models. For such models, we propose a new estimation method based on the Yule-Walker equations. However, since for the infinite-variance case the covariance does not exist, thus it is replaced by another measure, namely the covariation. In this paper we propose to apply two estimators of the covariation measure. The first one is based on moment representation (moment-based) while the second one - on the spectral measure representation (spectral-based). The validity of the new approaches are verified using the Monte Carlo simulations in different contexts, including the sample size and the index of stability of the noise. Moreover, we compare the moment-based covariation-based method with spectral-based covariation-based technique. Finally, the real data analysis is presented.
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Tracking the fundamental frequency (f0) of a monophonic instrumental performance is effectively a solved problem with several solutions achieving 99% accuracy. However, the related task of automatic music transcription requires a further processing step to segment an f0 contour into discrete notes. This sub-task of note segmentation is necessary to enable a range of applications including musicological analysis and symbolic music generation. Building on CREPE, a state-of-the-art monophonic pitch tracking solution based on a simple neural network, we propose a simple and effective method for post-processing CREPE's output to achieve monophonic note segmentation. The proposed method demonstrates state-of-the-art results on two challenging datasets of monophonic instrumental music. Our approach also gives a 97% reduction in the total number of parameters used when compared with other deep learning based methods.
Numerical methods for computing the solutions of Markov backward stochastic differential equations (BSDEs) driven by continuous-time Markov chains (CTMCs) are explored. The main contributions of this paper are as follows: (1) we observe that Euler-Maruyama temporal discretization methods for solving Markov BSDEs driven by CTMCs are equivalent to exponential integrators for solving the associated systems of ordinary differential equations (ODEs); (2) we introduce multi-stage Euler-Maruyama methods for effectively solving "stiff" Markov BSDEs driven by CTMCs; these BSDEs typically arise from the spatial discretization of Markov BSDEs driven by Brownian motion; (3) we propose a multilevel spatial discretization method on sparse grids that efficiently approximates high-dimensional Markov BSDEs driven by Brownian motion with a combination of multiple Markov BSDEs driven by CTMCs on grids with different resolutions. We also illustrate the effectiveness of the presented methods with a number of numerical experiments in which we treat nonlinear BSDEs arising from option pricing problems in finance.
It has been widely observed that exact or approximate MAP (mode-seeking) decoding from natural language generation (NLG) models consistently leads to degenerate outputs (Stahlberg and Byrne, 2019, Holtzman et al., 2019). This has generally been attributed to either a fundamental inadequacy of modes in models or weaknesses in language modeling. Contrastingly in this work, we emphasize that degenerate modes can even occur in the absence of any model error, due to contamination of the training data. Specifically, we show that mixing even a tiny amount of low-entropy noise with a population text distribution can cause the data distribution's mode to become degenerate, implying that any models trained on it will be as well. As the unconditional mode of NLG models will often be degenerate, we therefore propose to apply MAP decoding to the model's distribution conditional on avoiding specific degeneracies. Using exact-search, we empirically verify that the length-conditional modes of machine translation models and language models are indeed more fluent and topical than their unconditional modes. For the first time, we also share many examples of exact modal sequences from these models, and from several variants of the LLaMA-7B model. Notably, the modes of the LLaMA models are still degenerate, showing that improvements in modeling have not fixed this issue. Because of the cost of exact mode finding algorithms, we develop an approximate mode finding approach, ACBS, which finds sequences that are both high-likelihood and high-quality. We apply this approach to LLaMA-7B, a model which was not trained for instruction following, and find that we are able to elicit reasonable outputs without any finetuning.
This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.
Gaussian processes (GPs) are a popular class of Bayesian nonparametric models, but its training can be computationally burdensome for massive training datasets. While there has been notable work on scaling up these models for big data, existing methods typically rely on a stationary GP assumption for approximation, and can thus perform poorly when the underlying response surface is non-stationary, i.e., it has some regions of rapid change and other regions with little change. Such non-stationarity is, however, ubiquitous in real-world problems, including our motivating application for surrogate modeling of computer experiments. We thus propose a new Product of Sparse GP (ProSpar-GP) method for scalable GP modeling with massive non-stationary data. The ProSpar-GP makes use of a carefully-constructed product-of-experts formulation of sparse GP experts, where different experts are placed within local regions of non-stationarity. These GP experts are fit via a novel variational inference approach, which capitalizes on mini-batching and GPU acceleration for efficient optimization of inducing points and length-scale parameters for each expert. We further show that the ProSpar-GP is Kolmogorov-consistent, in that its generative distribution defines a valid stochastic process over the prediction space; such a property provides essential stability for variational inference, particularly in the presence of non-stationarity. We then demonstrate the improved performance of the ProSpar-GP over the state-of-the-art, in a suite of numerical experiments and an application for surrogate modeling of a satellite drag simulator.
This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK methods of arbitrary order based on an essential set of trees only. Then, we explicitly provide the 36 order conditions required for sixth-order methods and present convergence results. In addition, we are able to solve the 36 stiff order conditions in both their weak and strong forms, resulting in two families of sixth-order parallel stages ExpRK schemes. Interestingly, while these new schemes require a high number of stages, they can be implemented efficiently similar to the cost of a 6-stage method. Numerical experiments are given to confirm the accuracy and efficiency of the new schemes.
It has been classically conjectured that the brain assigns probabilistic models to sequences of stimuli. An important issue associated with this conjecture is the identification of the classes of models used by the brain to perform this task. We address this issue by using a new clustering procedure for sets of electroencephalographic (EEG) data recorded from participants exposed to a sequence of auditory stimuli generated by a stochastic chain. This clustering procedure indicates that the brain uses renewal points in the stochastic sequence of auditory stimuli in order to build a model.
Over the recent past data-driven algorithms for solving stochastic optimal control problems in face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics the analysis has so far been restricted to the scalar case. In this paper we fill this gap by studying a multivariate singular control problem for reversible diffusions with controls of reflection type. Our contributions are threefold. We first explicitly determine the long-run average costs as a domain-dependent functional, showing that the control problem can be equivalently characterized as a shape optimization problem. For given diffusion dynamics, assuming the optimal domain to be strongly star-shaped, we then propose a gradient descent algorithm based on polytope approximations to numerically determine a cost-minimizing domain. Finally, we investigate data-driven solutions when the diffusion dynamics are unknown to the controller. Using techniques from nonparametric statistics for stochastic processes, we construct an optimal domain estimator, whose static regret is bounded by the minimax optimal estimation rate of the unreflected process' invariant density. In the most challenging situation, when the dynamics must be learned simultaneously to controlling the process, we develop an episodic learning algorithm to overcome the emerging exploration-exploitation dilemma and show that given the static regret as a baseline, the loss in its sublinear regret per time unit is of natural order compared to the one-dimensional case.
This study compares the performance of (1) fine-tuned models and (2) extremely large language models on the task of check-worthy claim detection. For the purpose of the comparison we composed a multilingual and multi-topical dataset comprising texts of various sources and styles. Building on this, we performed a benchmark analysis to determine the most general multilingual and multi-topical claim detector. We chose three state-of-the-art models in the check-worthy claim detection task and fine-tuned them. Furthermore, we selected three state-of-the-art extremely large language models without any fine-tuning. We made modifications to the models to adapt them for multilingual settings and through extensive experimentation and evaluation. We assessed the performance of all the models in terms of accuracy, recall, and F1-score in in-domain and cross-domain scenarios. Our results demonstrate that despite the technological progress in the area of natural language processing, the models fine-tuned for the task of check-worthy claim detection still outperform the zero-shot approaches in a cross-domain settings.
In this paper, we develop an arbitrary-order locking-free enriched Galerkin method for the linear elasticity problem using the stress-displacement formulation in both two and three dimensions. The method is based on the mixed discontinuous Galerkin method in [30], but with a different stress approximation space that enriches the arbitrary order continuous Galerkin space with some piecewise symmetric-matrix valued polynomials. We prove that the method is well-posed and provide a parameter-robust error estimate, which confirms the locking-free property of the EG method. We present some numerical examples in two and three dimensions to demonstrate the effectiveness of the proposed method.
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