Ricardo Ríos, Luis Rodríguez
In this paper we derive the consistency of the penalized likelihood method for the number state of the hidden Markov chain in autoregressive models with Markov regimen. Using a SAEM type algorithm to estimate the models parameters. We test the null hypothesis of hidden Markov Model against an autoregressive process with Markov regime.
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This report introduces a parsimonious structure for mixture of autoregressive models, where the weighting coefficients are determined through latent random variables as functions of all past observations. These variables follow a hidden Markov model. We modify EM and Baum-Welch algorithms to estimate the parameters of the model.
This paper introduces a new parsimonious structure for mixture of autoregressive models. the weighting coefficients are determined through latent random variables, following a hidden Markov model. We propose a dynamic programming algorithm for the application of forecasting. We also derive the limiting behavior of unconditional first moment of the process and an appropriate upper bound for the limiting value of the variance. This can be considered as long run behavior of the process. Finally we show convergence and stability of the second moment. Further, we illustrate the efficacy of the prop…
This work concerns estimation of linear autoregressive models with Markov-switching using expectation maximisation (E.M.) algorithm. Our method generalise the method introduced by Elliot for general hidden Markov models and avoid to use backward recursion.
We construct efficient robust truncated sequential estimators for the pointwise estimation problem in nonparametric autoregression models with smooth coefficients. For Gaussian models we propose an adaptive procedure based on the constructed sequential estimators. The minimax nonadaptive and adaptive convergence rates are established. It turns out that in this case these rates are the same as for regression models.
Bifurcating autoregressive processes, which can be seen as an adaptation of au-toregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions and we use nonparametric techniques. We investigate both nonasymp-totic and asymptotic behavior of the Nadaraya-Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by Tn, up to the depth n. Estimators achieve the classical rate…
This paper studies some temporal dependence properties and addresses the issue of parametric estimation for a class of state-dependent autoregressive models for nonlinear time series in which we assume a stochastic autoregressive coefficient depending on the first lagged value of the process itself. We call such a model state-dependent first-order autoregressive process, (SDAR). We introduce some assumptions under which this class of models is strictly stationary and uniformly ergodic and we establish consistency and asymptotic normality of the quasi-maximum likelihood estimator of the paramet…
This paper studies the threshold estimation of a TAR model when the underlying threshold parameter is a random variable. It is shown that the Bayesian estimator is consistent and its limit distribution is expressed in terms of a limit likelihood ratio. Furthermore, convergence of moments of the estimators is also established. The limit distribution can be computed via explicit simulations from which testing and inference for the threshold parameter can be conducted. The obtained results are illustrated with numerical simulations.
In this article we develop a tractable procedure for testing strict stationarity in a double autoregressive model and formulate the problem as testing if the top Lyapunov exponent is negative. Without strict stationarity assumption, we construct a consistent estimator of the associated top Lyapunov exponent and employ a random weighting approach for its variance estimation, which in turn are used in a t-type test. We also propose a GLAD estimation for parameters of interest, relaxing key assumptions on the commonly used QMLE. All estimators, except for the intercept, are shown to be consistent…
We investigate the estimation of parameters in the random coefficient autoregressive model. We consider a nonstationary RCA process and show that the innovation variance parameter cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for the remaining model parameters is proven so the unit root problem does not exist in the random coefficient autoregressive model.
Matrix-variate integer-valued autoregressive processes
本文针对社会学和经济学中矩阵值整数值时间序列的建模需求，提出矩阵值整数值自回归模型，通过左右矩阵薄化算子定义模型，研究其性质，开发投影与迭代最小二乘估计，并证明其优于现有模型。
Consider an autoregressive model with measurement error: we observe $Z_i=X_i+ε_i$, where $X_i$ is a stationary solution of the equation $X_i=f_{θ^0}(X_{i-1})+ξ_i$. The regression function $f_{θ^0}$ is known up to a finite dimensional parameter $θ^0$. The distributions of $X_0$ and $ξ_1$ are unknown whereas the distribution of $ε_1$ is completely known. We want to estimate the parameter $θ^0$ by using the observations $Z_0,..,Z_n$. We propose an estimation procedure based on a modified least square criterion involving a weight function $w$, to be suitably chosen. We give upper bounds for the ri…
Spatial unilateral autoregressive model $X_{k,\ell}=αX_{k-1,\ell}+βX_{k,\ell-1}+γX_{k-1,\ell-1}+ε_{k,\ell}$ is investigated in the unit root case, that is when the parameters are on the boundary of the domain of stability that forms a tetrahedron with vertices $(1,1,-1), \ (1,-1,1),\ (-1,1,1)$ and $(-1,-1,-1)$. It is shown that the limiting distribution of the least squares estimator of the parameters is normal and the rate of convergence is $n$ when the parameters are in the faces or on the edges of the tetrahedron, while on the vertices the rate is $n^{3/2}$.
The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive latent process which forms a Markov chain in random environments. Contrarily to existing contributions in the field of Markov chains in random environments, the state space is not discrete and we do not use small set type assumptions or uniform contraction conditions for the random Markov kernels. Our assumptions are quite general and allows to deal with mode…
This paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison to the optimal tests for Markov-switching parameters of Carrasco…
Large sample statistical analysis of threshold autoregressive (TAR) models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter.
Parameter estimation for fractional autoregressive process with periodic structure
本文提出由分数高斯噪声驱动的周期分数自回归过程（PFAR）以建模降水蒸散发，结合GLSE和GPH方法构建初始估计并经One-Step程序优化，证明了估计量的一致性与渐近正态性。
A regularized vector autoregressive hidden semi-Markov model is developed to analyze multivariate financial time series with switching data generating regimes. Furthermore, an augmented EM algorithm is proposed for parameter estimation by embedding regularized estimators for the state-dependent covariance matrices and autoregression matrices in the M-step. The performance of the proposed regularized estimators is evaluated both in the simulation experiments and on the New York Stock Exchange financial portfolio data.
The two dominant approaches to neural text generation are fully autoregressive models, using serial beam search decoding, and non-autoregressive models, using parallel decoding with no output dependencies. This work proposes an autoregressive model with sub-linear parallel time generation. Noting that conditional random fields with bounded context can be decoded in parallel, we propose an efficient cascaded decoding approach for generating high-quality output. To parameterize this cascade, we introduce a Markov transformer, a variant of the popular fully autoregressive model that allows us to …
A new matrix-variate integer-valued autoregressive process with matrical negative binomial thinning
本文针对矩阵值整数时间序列的过度离散问题，提出带矩阵负二项稀疏的新型整数自回归模型，开发两种估计方法并验证其性能，该模型拟合效果优于两类对比模型。
The functional autoregressive model is a Markov model taylored for data of functional nature. It revealed fruitful when attempting to model samples of dependent random curves and has been widely studied along the past few years. This article aims at completing the theoretical study of the model by adressing the crucial issue of weak convergence for estimates from the model. The main difficulties stem from an underlying inverse problem as well as from dependence between the data. Traditional facts about weak convergence in non parametric models appear : the normalizing sequence is an o($sqrt{n}…
Motivated by a variety of applications, high-dimensional time series have become an active topic of research. In particular, several methods and finite-sample theories for individual stable autoregressive processes with known lag have become available very recently. We, instead, consider multiple stable autoregressive processes that share an unknown lag. We use information across the different processes to simultaneously select the lag and estimate the parameters. We prove that the estimated process is stable, and we establish rates for the forecasting error that can outmatch the known rate in…
We study ergodic properties of some Markov chains models in random environments when the random Markov kernels that define the dynamic satisfy some usual drift and small set conditions but with random coefficients. In particular, we adapt a standard coupling scheme used for getting geometric ergodic properties for homogeneous Markov chains to the random environment case and we prove the existence of a process of randomly invariant probability measures for such chains, in the spirit of the approach of Kifer for chains satisfying some Doeblin type conditions. We then deduce ergodic properties of…
Testing for regime switching when the regime switching probabilities are specified either as constants (`mixture models') or are governed by a finite-state Markov chain (`Markov switching models') are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (`observation-dependent regime switching'). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is hig…
In the last decades, due to the huge technological growth observed, it has become increasingly common that a collection of temporal data rapidly accumulates in vast amounts. This provides an opportunity for extracting valuable information through the estimation of increasingly precise models. But at the same time it imposes the challenge of continuously updating the models as new data become available. Currently available methods for addressing this problem, the so-called online learning methods, use current parameter estimations and novel data to update the estimators. These approaches avoid …
This article extends the autoregressive count time series model class by allowing for a model with regimes, that is, some of the parameters in the model depend on the state of an unobserved Markov chain. We develop a quasi-maximum likelihood estimator by adapting the extended Hamilton-Grey algorithm for the Poisson log-linear autoregressive model, and we perform a simulation study to check the finite sample behaviour of the estimator. The motivation for the model comes from the study of corporate defaults, in particular the study of default clustering. We provide evidence that time series of c…
In this work we consider the problem of anomaly detection in heterogeneous, multivariate, variable-length time series datasets. Our focus is on the aviation safety domain, where data objects are flights and time series are sensor readings and pilot switches. In this context the goal is to detect anomalous flight segments, due to mechanical, environmental, or human factors in order to identifying operationally significant events and provide insights into the flight operations and highlight otherwise unavailable potential safety risks and precursors to accidents. For this purpose, we propose a f…
Autoregressive Processes on Riemannian Manifolds
提出黎曼自回归模型（R-AR(1)），将经典离散时间随机过程推广到流形值数据，通过Fréchet均值和自回归参数控制平稳性与遍历性，并建立依赖样本的强相合估计理论。
DAMNETS: A Deep Autoregressive Model for Generating Markovian Network Time Series
本文提出DAMNETS，一种可扩展的深度自回归生成模型，用于生成马尔可夫网络时间序列，并在真实与合成数据集上全面超越现有方法。
In this article, we propose the fractional lower order covariance method (FLOC) for estimating the parameters of vector autoregressive process (VAR) of order $p$, $p\geq 1$ with symmetric stable noise. Further, we show the efficiency, accuracy, and simplicity of our methods through Monte-Carlo simulation.
Quasi-Score Matching Estimation for Spatial Autoregressive Model with Random Weights Matrix and Regressors
针对SAR模型QMLE在大规模数据下计算不可扩展且经典假设不切实际的问题，提出拟得分匹配估计法，降低计算复杂度并建立随机权重矩阵与回归量下的渐近性质，为新理论框架提供支撑。
Empirical study of periodic autoregressive models with additive noise -- estimation and testing
针对受加性噪声污染的有限方差PAR时间序列，本文提出四种基于Yule-Walker方程和自协方差函数的估计方法，并构建模型检验程序，通过仿真和空气质量数据验证其有效性。
The autoregressive (AR) models are used to represent the time-varying random process in which output depends linearly on previous terms and a stochastic term (the innovation). In the classical version, the AR models are based on normal distribution. However, this distribution does not allow describing data with outliers and asymmetric behavior. In this paper, we study the AR models with normal inverse Gaussian (NIG) innovations. The NIG distribution belongs to the class of semi heavy-tailed distributions with wide range of shapes and thus allows for describing real-life data with possible jump…
We consider the estimation of the transition matrix in the high-dimensional time-varying vector autoregression (TV-VAR) models. Our model builds on a general class of locally stationary VAR processes that evolve smoothly in time. We propose a hybridized kernel smoothing and $\ell^1$-regularized method to directly estimate the sequence of time-varying transition matrices. Under the sparsity assumption on the transition matrix, we establish the rate of convergence of the proposed estimator and show that the convergence rate depends on the smoothness of the locally stationary VAR processes only t…
Latent autoregressive models are useful time series models for the analysis of infectious disease data. Evaluation of the likelihood function of latent autoregressive models is intractable and its approximation through simulation-based methods appears as a standard practice. Although simulation methods may make the inferential problem feasible, they are often computationally intensive and the quality of the numerical approximation may be difficult to assess. We consider instead a weighted pairwise likelihood approach and explore several computational and methodological aspects including estima…
Estimation of High-dimensional Nonlinear Vector Autoregressive Models
本文提出了一种高维非线性向量自回归模型估计方法，通过基函数展开构建模型，并推导了收敛速率和模型选择一致性，同时展示了在基因表达时间序列数据中的应用优势。
A new multivariate stochastic volatility estimation procedure for financial time series is proposed. A Wishart autoregressive process is considered for the volatility precision covariance matrix, for the estimation of which a two step procedure is adopted. The first step is the conditional inference on the autoregressive parameters and the second step is the unconditional inference, based on a Newton-Raphson iterative algorithm. The proposed methodology, which is mostly Bayesian, is suitable for medium dimensional data and it bridges the gap between closed-form estimation and simulation-based …
Regularized Estimation of High-Dimensional Matrix-Variate Autoregressive Models
本文针对高维矩阵变量时间序列，提出了带状和LASSO稀疏两种正则化估计方法以降低双线性自回归模型的参数维度，推导了渐近性质，并通过模拟与真实数据验证了其优越的预测性能。
Moving Aggregate Modified Autoregressive Copula-Based Time Series Models (MAGMAR-Copulas)
本文提出MAGMAR-皮克罗模型，通过引入移动聚合部分改进传统皮克罗时间序列模型，以处理非马尔可夫过程，通过D-vine框架推导分布特性，并在美通胀数据上验证模型性能。
Many nonlinear time series models have been proposed in the last decades. Among them, the models with regime switchings provide a class of versatile and interpretable models which have received a particular attention in the literature. In this paper, we consider a large family of such models which generalize the well known Markov-switching AutoRegressive (MS-AR) by allowing non-homogeneous switching and encompass Threshold AutoRegressive (TAR) models. We prove various theoretical results related to the stability of these models and the asymptotic properties of the Maximum Likelihood Estimates …
Fast and Interpretable Autoregressive Estimation with Neural Network Backpropagation
本文提出利用神经网络反向传播进行自回归估计，通过嵌入自回归结构到前馈网络中，实现参数估计的同时保持可解释性，实验表明该方法在计算效率和稳定性上优于传统方法。
High-dimensional time series data exist in numerous areas such as finance, genomics, healthcare, and neuroscience. An unavoidable aspect of all such datasets is missing data, and dealing with this issue has been an important focus in statistics, control, and machine learning. In this work, we consider a high-dimensional estimation problem where a dynamical system, governed by a stable vector autoregressive model, is randomly and only partially observed at each time point. Our task amounts to estimating the transition matrix, which is assumed to be sparse. In such a scenario, where covariates a…
Estimation and Autoregressive Modeling of Mittag-Leffler Distributions with Application in Finance
本文提出利用经验拉普拉斯变换方法估计Mittag-Leffler分布参数，并建立自回归模型用于高频交易数据分析。
The Vector AutoRegressive Moving Average (VARMA) model is fundamental to the theory of multivariate time series; however, identifiability issues have led practitioners to abandon it in favor of the simpler but more restrictive Vector AutoRegressive (VAR) model. We narrow this gap with a new optimization-based approach to VARMA identification built upon the principle of parsimony. Among all equivalent data-generating models, we use convex optimization to seek the parameterization that is "simplest" in a certain sense. A user-specified strongly convex penalty is used to measure model simplicity,…
Risk Estimate under a Time-Varying Autoregressive Model for Data-Driven Reproduction Number Estimation
本文提出广义时变自回归模型估计COVID-19再生数，通过推广Stein无偏风险估计构建数据驱动预言机选择超参数，提出每周缩放Poisson模型提升鲁棒性，并在合成与真实数据上验证有效性。
Estimation of Poisson Autoregressive Model for Multiple Time Series
本文将泊松自回归（PAR）模型视为加性模型，采用三次平滑样条与极大似然估计的混合方法进行参数估计，并将其推广至多重计数时间序列，发现该方法在计数值较大时优于状态空间方法且对结构变化具有稳健性。
Tail-robust estimation of factor-adjusted vector autoregressive models for high-dimensional time series
本文研究了通过因子调整向量自回归模型对高维厚尾时间序列数据建模问题，结合少量因子捕捉变量共动性和稀疏VAR模型捕捉剩余互联性，并通过数据截断和两阶段估计法处理厚尾问题，推导出估计率并验证其在模拟数据和宏观经济指标预测中的有效性。
We consider the problem of estimating the parameters of a linear univariate autoregressive model with sub-Gaussian innovations from a limited sequence of consecutive observations. Assuming that the parameters are compressible, we analyze the performance of the $\ell_1$-regularized least squares as well as a greedy estimator of the parameters and characterize the sampling trade-offs required for stable recovery in the non-asymptotic regime. In particular, we show that for a fixed sparsity level, stable recovery of AR parameters is possible when the number of samples scale sub-linearly with the …
High-dimensional time series data appear in many scientific areas in the current data-rich environment. Analysis of such data poses new challenges to data analysts because of not only the complicated dynamic dependence between the series, but also the existence of aberrant observations, such as missing values, contaminated observations, and heavy-tailed distributions. For high-dimensional vector autoregressive (VAR) models, we introduce a unified estimation procedure that is robust to model misspecification, heavy-tailed noise contamination, and conditional heteroscedasticity. The proposed met…
An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.
High-dimensional vector autoregressive (VAR) models are important tools for the analysis of multivariate time series. This paper focuses on high-dimensional time series and on the different regularized estimation procedures proposed for fitting sparse VAR models to such time series. Attention is paid to the different sparsity assumptions imposed on the VAR parameters and how these sparsity assumptions are related to the particular consistency properties of the estimators established. A sparsity scheme for high-dimensional VAR models is proposed which is found to be more appropriate for the tim…
Network estimation from multi-variate point process or time series data is a problem of fundamental importance. Prior work has focused on parametric approaches that require a known parametric model, which makes estimation procedures less robust to model mis-specification, non-linearities and heterogeneities. In this paper, we develop a semi-parametric approach based on the monotone single-index multi-variate autoregressive model (SIMAM) which addresses these challenges. We provide theoretical guarantees for dependent data and an alternating projected gradient descent algorithm. Significantly w…
Causal inference in multivariate time series is challenging due to the fact that the sampling rate may not be as fast as the timescale of the causal interactions. In this context, we can view our observed series as a subsampled version of the desired series. Furthermore, due to technological and other limitations, series may be observed at different sampling rates, representing a mixed frequency setting. To determine instantaneous and lagged effects between time series at the true causal scale, we take a model-based approach based on structural vector autoregressive (SVAR) models. In this cont…
Nonstationary time series data exist in various scientific disciplines, including environmental science, biology, signal processing, econometrics, among others. Many Bayesian models have been developed to handle nonstationary time series. The time-varying vector autoregressive (TV-VAR) model is a well-established model for multivariate nonstationary time series. Nevertheless, in most cases, the large number of parameters presented by the model results in a high computational burden, ultimately limiting its usage. This paper proposes a computationally efficient multivariate Bayesian Circular La…
Nonlinear autoregressive models are very useful for modeling many natural processes, however, the size of the class of these models is large. Functional-coefficient autoregressive models (FCAR) are useful structures for reducing the size of the class of these models. Although this structure reduces the class of nonlinear models, it is broad enough to include some common time series models as specific cases. A recent development in estimating nonlinear time series data is the spline backfitted kernel (SBK) method. This method combines the computational speed of splines with the asymptotic prope…
This paper introduces an alternative approach to sampling from autoregressive models. Autoregressive models are typically sampled sequentially, according to the transition dynamics defined by the model. Instead, we propose a sampling procedure that initializes a sequence with white noise and follows a Markov chain defined by Langevin dynamics on the global log-likelihood of the sequence. This approach parallelizes the sampling process and generalizes to conditional sampling. Using an autoregressive model as a Bayesian prior, we can steer the output of a generative model using a conditional lik…
Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of ARCH and GARCH processes. More recently multivariate variants of these processes have been in the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Due to the need for estimation of many parameters, however, these models may not be suitable for modeling now prevalent high-frequency volatility data. One potentially useful way to bypass these issues is to take a functional app…
Dynamic Estimation of Learning Rates Using a Non-Linear Autoregressive Model
针对优化器学习率与动量的动态调节问题，提出融入动量的自适应Nlar模型及三种学习率估计器，并通过理论证明和多数据集与强化学习实验验证其稳健快速收敛。
Signed Matrix Thinning and Projection Estimation for Integer-Valued Autoregressive Models
本文针对MINAR模型无法处理负值的缺陷，提出Z-MINAR模型，通过符号矩阵稀疏化算子和扩展泊松分布实现正负整数处理，经模拟和实证验证其估计精度等性能优于基准模型。
Simultaneous Estimation and Group Identification for Network Vector Autoregressive Model with Heterogeneous Nodes
本文针对网络节点异质性动态模式，提出带潜在群体结构的网络向量自回归模型，通过最小二乘目标函数同时估计模型参数与节点成员归属，并允许群体数量过度指定以提升稳健性，同时给出群体数量一致识别的数据驱动准则。
A Random Graph-based Autoregressive Model for Networked Time Series
本文提出一种基于随机图的联网时间序列自回归模型，通过引入灵活的潜在变量刻画同质性，并结合一阶差分法与邻居平滑技术实现参数估计，无需指定网络生成机制，理论证明与实验均验证了其有效性。
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