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Linear structural vector autoregressive models can be identified statistically without imposing restrictions on the model if the shocks are mutually independent and at most one of them is Gaussian. We show that this result extends to structural threshold and smooth transition vector autoregressive models incorporating a time-varying impact matrix defined as a weighted sum of the impact matrices of the regimes. Our empirical application studies the effects of the climate policy uncertainty shock on the U.S. macroeconomy. In a structural logistic smooth transition vector autoregressive model consisting of two regimes, we find that a positive climate policy uncertainty shock decreases production in times of low economic policy uncertainty but slightly increases it in times of high economic policy uncertainty. The introduced methods are implemented to the accompanying R package sstvars.

Reachability and other path-based measures on temporal graphs can be used to understand spread of infection, information, and people in modelled systems. Due to delays and errors in reporting, temporal graphs derived from data are unlikely to perfectly reflect reality, especially with respect to the precise times at which edges appear. To reflect this uncertainty, we consider a model in which some number $\zeta$ of edge appearances may have their timestamps perturbed by $\pm\delta$ for some $\delta$. Within this model, we investigate temporal reachability and consider the problem of determining the maximum number of vertices any vertex can reach under these perturbations. We show that this problem is intractable in general but is efficiently solvable when $\zeta$ is sufficiently large. We also give algorithms which solve this problem in several restricted settings. We complement this with some contrasting results concerning the complexity of related temporal eccentricity problems under perturbation.

We consider a special type of fast reaction-diffusion systems in which the coefficients of the reaction terms of the two substances are much larger than those of the diffusion terms while the diffusive motion to the substrate is negligible. Specifically speaking, the rate constants of the reaction terms are $O(1/\epsilon)$ while the diffusion coefficients are $O(1)$ where the parameter $\epsilon$ is small. When the rate constants of the reaction terms become highly large, i.e. $\epsilon$ tends to 0, the singular limit behavior of such a fast reaction-diffusion system is inscribed by the Stefan problem with latent heat, which brings great challenges in numerical simulations. In this paper, we adopt a semi-implicit scheme, which is first-order accurate in time and can accurately approximate the interface propagation even when the reaction becomes extremely fast, that is to say, the parameter $\epsilon$ is sufficiently small. The scheme satisfies the positivity, bound preserving properties and has $L^2$ stability and the linearized stability results of the system. For better performance on numerical simulations, we then construct a semi-implicit Runge-Kutta scheme which is second-order accurate in time. Numerous numerical tests are carried out to demonstrate the properties, such as the order of accuracy, positivity and bound preserving, the capturing of the sharp interface with various $\epsilon$ and to simulate the dynamics of the substances and the substrate, and to explore the heat transfer process, such as solid melting or liquid solidification in two dimensions.

Count time series data are frequently analyzed by modeling their conditional means and the conditional variance is often considered to be a deterministic function of the corresponding conditional mean and is not typically modeled independently. We propose a semiparametric mean and variance joint model, called random rounded count-valued generalized autoregressive conditional heteroskedastic (RRC-GARCH) model, to address this limitation. The RRC-GARCH model and its variations allow for the joint modeling of both the conditional mean and variance and offer a flexible framework for capturing various mean-variance structures (MVSs). One main feature of this model is its ability to accommodate negative values for regression coefficients and autocorrelation functions. The autocorrelation structure of the RRC-GARCH model using the proposed Laplace link functions with nonnegative regression coefficients is the same as that of an autoregressive moving-average (ARMA) process. For the new model, the stationarity and ergodicity are established and the consistency and asymptotic normality of the conditional least squares estimator are proved. Model selection criteria are proposed to evaluate the RRC-GARCH models. The performance of the RRC-GARCH model is assessed through analyses of both simulated and real data sets. The results indicate that the model can effectively capture the MVS of count time series data and generate accurate forecast means and variances.

High-dimensional, higher-order tensor data are gaining prominence in a variety of fields, including but not limited to computer vision and network analysis. Tensor factor models, induced from noisy versions of tensor decompositions or factorizations, are natural potent instruments to study a collection of tensor-variate objects that may be dependent or independent. However, it is still in the early stage of developing statistical inferential theories for the estimation of various low-rank structures, which are customary to play the role of signals of tensor factor models. In this paper, we attempt to ``decode" the estimation of a higher-order tensor factor model by leveraging tensor matricization. Specifically, we recast it into mode-wise traditional high-dimensional vector/fiber factor models, enabling the deployment of conventional principal components analysis (PCA) for estimation. Demonstrated by the Tucker tensor factor model (TuTFaM), which is induced from the noisy version of the widely-used Tucker decomposition, we summarize that estimations on signal components are essentially mode-wise PCA techniques, and the involvement of projection and iteration will enhance the signal-to-noise ratio to various extent. We establish the inferential theory of the proposed estimators, conduct rich simulation experiments, and illustrate how the proposed estimations can work in tensor reconstruction, and clustering for independent video and dependent economic datasets, respectively.

Two sequential estimators are proposed for the odds p/(1-p) and log odds log(p/(1-p)) respectively, using independent Bernoulli random variables with parameter p as inputs. The estimators are unbiased, and guarantee that the variance of the estimation error divided by the true value of the odds, or the variance of the estimation error of the log odds, are less than a target value for any p in (0,1). The estimators are close to optimal in the sense of Wolfowitz's bound.

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities.

The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels. This paper defines and proves a set of joint covariance properties under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, which make it possible to characterize how different types of image transformations interact with each other and the associated spatio-temporal receptive field responses. In this regard, we also extend the notion of scale-normalized derivatives to affine-normalized derivatives, to be able to obtain true affine-covariant properties of spatial derivatives, that are computed based on spatial smoothing with affine Gaussian kernels. The derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. As a side effect, the presented proof for the joint covariance property over the integrated combination of the different geometric image transformations also provides specific proofs for the individual transformation properties, which have not previously been fully reported in the literature. The paper also presents an in-depth theoretical analysis of geometric interpretations of the derived covariance properties, as well as outlines a number of biological interpretations of these results.

Twin nodes in a static network capture the idea of being substitutes for each other for maintaining paths of the same length anywhere in the network. In dynamic networks, we model twin nodes over a time-bounded interval, noted $(\Delta,d)$-twins, as follows. A periodic undirected time-varying graph $\mathcal G=(G_t)_{t\in\mathbb N}$ of period $p$ is an infinite sequence of static graphs where $G_t=G_{t+p}$ for every $t\in\mathbb N$. For $\Delta$ and $d$ two integers, two distinct nodes $u$ and $v$ in $\mathcal G$ are $(\Delta,d)$-twins if, starting at some instant, the outside neighbourhoods of $u$ and $v$ has non-empty intersection and differ by at most $d$ elements for $\Delta$ consecutive instants. In particular when $d=0$, $u$ and $v$ can act during the $\Delta$ instants as substitutes for each other in order to maintain journeys of the same length in time-varying graph $\mathcal G$. We propose a distributed deterministic algorithm enabling each node to enumerate its $(\Delta,d)$-twins in $2p$ rounds, using messages of size $O(\delta_\mathcal G\log n)$, where $n$ is the total number of nodes and $\delta_\mathcal G$ is the maximum degree of the graphs $G_t$'s. Moreover, using randomized techniques borrowed from distributed hash function sampling, we reduce the message size down to $O(\log n)$ w.h.p.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.