We propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. As an important prerequisite for applications in statistics, we prove absolute regularity of the count process with exponentially decaying coefficients.
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ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
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The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.
It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type calculus (ETTC). ETTC is a calculus of specific typed terms, which represent tuples of strings, more precisely bipartite graphs decorated with strings. Types are derived from linear logic formulas, and rules correspond to concrete operations on these string-labeled graphs, so that they can be conveniently visualized. This provides the above mentioned fragment of MLL1 that is relevant for language modeling not only with some alternative syntax and intuitive geometric representation, but also with an intrinsic deductive system, which has been absent. In this work we consider a non-trivial notationally enriched variation of the previously introduced {\bf ETTC}, which allows more concise and transparent computations. We present both a cut-free sequent calculus and a natural deduction formalism.
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.
A Milstein-type method is proposed for some highly non-linear non-autonomous time-changed stochastic differential equations (SDEs). The spatial variables in the coefficients of the time-changed SDEs satisfy the super-linear growth condition and the temporal variables obey some H\"older's continuity condition. The strong convergence in the finite time is studied and the convergence order is obtained.
We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multi-dimensional Ornstein-Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high-dimensional optimal control of SDEs and training stochastic point process models of limit order book events.
The block maxima method is a classical and widely applied statistical method for time series extremes. It has recently been found that respective estimators whose asymptotics are driven by empirical means can be improved by using sliding rather than disjoint block maxima. Similar results are derived for general non-degenerate U-statistics of arbitrary order, in the multivariate time series case. Details are worked out for selected examples: the empirical variance, the probability weighted moment estimator and Kendall's tau statistic. The results are also extended to the case where the underlying sample is piecewise stationary. The finite-sample properties are illustrated by a Monte Carlo simulation study.
We present new Dirichlet-Neumann and Neumann-Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet-Neumann and Neumann-Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
We propose a computationally and statistically efficient procedure for segmenting univariate data under piecewise linearity. The proposed moving sum (MOSUM) methodology detects multiple change points where the underlying signal undergoes discontinuous jumps and/or slope changes. Theoretically, it controls the family-wise error rate at a given significance level asymptotically and achieves consistency in multiple change point detection, as well as matching the minimax optimal rate of estimation when the signal is piecewise linear and continuous, all under weak assumptions permitting serial dependence and heavy-tailedness. Computationally, the complexity of the MOSUM procedure is $O(n)$ which, combined with its good performance on simulated datasets, making it highly attractive in comparison with the existing methods. We further demonstrate its good performance on a real data example on rolling element-bearing prognostics.
In many scientific applications the aim is to infer a function which is smooth in some areas, but rough or even discontinuous in other areas of its domain. Such spatially inhomogeneous functions can be modelled in Besov spaces with suitable integrability parameters. In this work we study adaptive Bayesian inference over Besov spaces, in the white noise model from the point of view of rates of contraction, using $p$-exponential priors, which range between Laplace and Gaussian and possess regularity and scaling hyper-parameters. To achieve adaptation, we employ empirical and hierarchical Bayes approaches for tuning these hyper-parameters. Our results show that, while it is known that Gaussian priors can attain the minimax rate only in Besov spaces of spatially homogeneous functions, Laplace priors attain the minimax or nearly the minimax rate in both Besov spaces of spatially homogeneous functions and Besov spaces permitting spatial inhomogeneities.
Neural dynamical systems with stable attractor structures, such as point attractors and continuous attractors, are hypothesized to underlie meaningful temporal behavior that requires working memory. However, working memory may not support useful learning signals necessary to adapt to changes in the temporal structure of the environment. We show that in addition to the continuous attractors that are widely implicated, periodic and quasi-periodic attractors can also support learning arbitrarily long temporal relationships. Unlike the continuous attractors that suffer from the fine-tuning problem, the less explored quasi-periodic attractors are uniquely qualified for learning to produce temporally structured behavior. Our theory has broad implications for the design of artificial learning systems and makes predictions about observable signatures of biological neural dynamics that can support temporal dependence learning and working memory. Based on our theory, we developed a new initialization scheme for artificial recurrent neural networks that outperforms standard methods for tasks that require learning temporal dynamics. Moreover, we propose a robust recurrent memory mechanism for integrating and maintaining head direction without a ring attractor.
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