Mouloud Goubi
Our aim in this work is to give explicit formula of the linear processes solution of autoregressive time series AR(2) with hint of generating functions theory by using the Horadam numbers and polynomials.
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This chapter presents some novel information theoretic results for the analysis of stationary time series in the frequency domain. In particular, the spectral distribution that corresponds to the most uncertain or unpredictable time series with some values of the autocovariance function fixed, is the generalized von Mises spectral distribution. It is thus a maximum entropy spectral distribution and the corresponding stationary time series is called the generalized von Mises time series. The generalized von Mises distribution is used in directional statistics for modelling planar directions tha…
In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …
Factorization of a spectral density with smooth eigenvalues of a multidimensional stationary time series
本文将一维光滑谱密度的经典结果推广到多维情形，通过显式因式分解谱密度来计算Wold表示的组成部分，从而为多维平稳时间序列的最佳线性预测提供理论基础。
The aim of this paper is to give a simpler, more usable sufficient condition to the regularity of generic weakly stationary time series. Also, this condition is used to show how regular processes satisfying these sufficient conditions can be approximated by a lower rank \emph{regular} process. The relevance of these issues is shown by the ever increasing presence of high-dimensional data in many fields lately, and because of this, low rank processes and low rank approximations are becoming more important. Moreover, regular processes are the ones which are completely influenced by random innova…
We adapt the classical definition of locally stationary processes in discrete-time to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient condi…
A model-free bootstrap procedure for a general class of stationary time series is introduced. The theoretical framework is established, showing asymptotic validity of bootstrap confidence intervals for many statistics of interest. In addition, asymptotic validity of one-step ahead bootstrap prediction intervals is also demonstrated. Finite-sample experiments are conducted to empirically confirm the performance of the new method, and to compare with popular methods such as the block bootstrap and the autoregressive (AR)-sieve bootstrap.
The Portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.
We consider a time series $X=\{X_k, k\in\mathbb{Z}\}$ with memory parameter $d\in\mathbb{R}$. This time series is either stationary or can be made stationary after differencing a finite number of times. We study the "Local Whittle Wavelet Estimator" of the memory parameter $d$. This is a wavelet-based semiparametric pseudo-likelihood maximum method estimator. The estimator may depend on a given finite range of scales or on a range which becomes infinite with the sample size. We show that the estimator is consistent and rate optimal if $X$ is a linear process and is asymptotically normal if $X$…
The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail - both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then the paper focusses on linear processes where a more general theory is possible. First a general definition for linear processes is given and time varying spectral densities are discu…
We propose generalized portmanteau-type test statistics in the frequency domain to test independence between two stationary time series. The test statistics are formed analogous to the one in Chen and Deo (2004, Econometric Theory 20, 382-416), who extended the applicability of portmanteau goodness-of-fit test to the long memory case. Under the null hypothesis of independence, the asymptotic standard normal distributions of the proposed statistics are derived under fairly mild conditions. In particular, each time series is allowed to possess short memory, long memory or anti-persistence. A sim…
Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation has been proposed in Kirch et al. who prove posterior consistency for spectral density estimation in combination with the Bernstein-Dirichlet process prior for Gaussian time series. In this paper, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem of Shalizi for dependent data and misspecified model…
We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observa…
A measure of primal importance for capturing the serial dependence of a stationary time series at extreme levels is provided by the limiting cluster size distribution. New estimators based on a blocks declustering scheme are proposed and analyzed both theoretically and by means of a large-scale simulation study. A sliding blocks version of the estimators is shown to outperform a disjoint blocks version. In contrast to some competitors from the literature, the estimators only depend on one unknown parameter to be chosen by the statistician.
We develop an estimator for the high-dimensional covariance matrix of a locally stationary process with a smoothly varying trend and use this statistic to derive consistent predictors in non-stationary time series. In contrast to the currently available methods for this problem the predictor developed here does not rely on fitting an autoregressive model and does not require a vanishing trend. The finite sample properties of the new methodology are illustrated by means of a simulation study and a financial indices study.
We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.
Gaussian Approximation For Non-stationary Time Series with Optimal Rate and Explicit Construction
本文针对非平稳时间序列提出两种高斯逼近构造方法，获得最优速率并适用于广泛模型，进而用于变点检测和同时推断，并通过模拟与真实数据验证。
The extremal index $θ$, a number in the interval $[0,1]$, is known to be a measure of primal importance for analyzing the extremes of a stationary time series. New rank-based estimators for $θ$ are proposed which rely on the construction of approximate samples from the exponential distribution with parameter $θ$ that is then to be fitted via the method of moments. The new estimators are analyzed both theoretically as well as empirically through a large-scale simulation study. In specific scenarios, in particular for time series models with $θ\approx 1$, they are found to be superior to recent …
Forecasting the evolution of complex systems is one of the grand challenges of modern data science. The fundamental difficulty lies in understanding the structure of the observed stochastic process. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by an auto-regressive process of slowly diverging order. When linear prediction with ${\cal L}^2$ loss is concerned, the latter result facilitates a unified globally-optimal short-term forecasting theory for a w…
In this paper, we consider the estimation and inference of precision matrices of a rich class of locally stationary and nonlinear time series assuming that only one realization of the time series is observed. Using a Cholesky decomposition technique, we show that the precision matrices can be directly estimated via a series of least squares linear regressions with smoothly time-varying coefficients. The method of sieves is utilized for the estimation and is shown to be efficient and optimally adaptive in terms of estimation accuracy and computational complexity. We establish an asymptotic theo…
Inference for Non-Stationary Heavy Tailed Time Series
针对重尾非平稳时间序列，本文用时变线性过程的局部平稳逼近构造一致估计量，并以自加权方案在有无有限方差时均恢复渐近正态性。
A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a Glivenko--Cantelli-type convergence result. The results use conditions based on the metric entropy of the index class. In contrast to related earlier work, no Gaussian assumption is made. As applications, quasi-likelihood estimation, goodness-of-fit testing and inference under model misspecification are discussed. In an extended application, uniform rates of convergence are deriv…
Inference for Multiple Change-points in Piecewise Locally Stationary Time Series
本文提出了一种基于似然的方法，用于推断局部平稳时间序列中的多个转折点，区分跳跃和拐点，并建立了相应的渐近分布。
Kernel-based Joint Independence Tests for Multivariate Stationary and Non-stationary Time Series
本文扩展 dHSIC，提出适用于平稳与非平稳多元时间序列的基于核的联合独立性检验，并通过重采样在合成及真实数据中验证其高阶依赖发现能力。
ANOVA for High-dimensional Non-stationary Time Series
本文针对高维非平稳时间序列，提出考虑时间依赖性的方差分析检验统计量，并开发自助法算法进行假设检验，同时建立了相关理论结果并验证了有限样本性能。
The approximation of a stationary time-series by finite order autoregressive (AR) and moving averages (MA) is a problem that occurs in many applications. In this paper we study asymptotic behavior of the spectral density of finite order approximations of wide sense stationary time series. It is shown that when the on the spectral density is non-vanishing in $[-π,π]$ and the covariance is summable, the spectral density of the approximating autoregressive sequence converges at the origin. Under additional mild conditions on the coefficients of the Wold decomposition it is also shown that the spe…
Given a heterogeneous time-series sample, the objective is to find points in time (called change points) where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown stationary ergodic distributions. No modelling, independence or mixing assumptions are made. A novel, computationally efficient, nonparametric method is proposed, and is shown to be asymptotically consistent in this general framework. The theoretical results are complemented with experimental evaluations.
This paper deals with the comparison of several stationary processes with unequal sample sizes. We provide a detailed theoretical framework on the testing problem for equality of spectral densities in the bivariate case, after which the generalization of our approach to the m dimensional case and to other statistical applications (like testing for zero correlation or clustering of time series data with different length) is straightforward. We prove asymptotic normality of an appropriately standardized version of the test statistic both under the null and the alternative and investigate the fin…
A log-linear model for non-stationary time series of counts
针对具有强趋势的非平稳整数值时间序列，提出一种基于近尺度不变条件分布族的对数线性模型，并证明了该计数过程具有指数衰减系数的绝对正则性。
A portmanteau test for multivariate non-stationary functional time series with an increasing number of lags
针对多元局部平稳函数型时间序列，提出一种无需降维的portmanteau白噪声检验，基于递增滞后阶数的退化U统计量高斯逼近，通过bootstrap实现。
Auto-Regressive Approximations to Non-stationary Time Series, with Inference and Applications
本文证明一般非平稳非线性时间序列可由缓慢发散阶的白噪声驱动AR过程全局逼近，并提出系数推断的高维L²检验、乘子自助法和自适应稳定性检验，进而建立局部平稳序列的全局最优短期预测理论。
Bounds in Wasserstein Distance for Locally Stationary Functional Time Series
本文针对局部平稳函数型时间序列，在协变量处于半度量空间的设定下研究Nadaraya-Watson条件分布估计，并在小球概率与混合条件下给出其关于Wasserstein距离的收敛速率，同时通过模拟和真实数据验证了有限样本性能。
Asymptotic inference in a stationary quantum time series
本文研究了平稳量子高斯状态的统计模型，将其与经典高斯平稳时间序列类比，通过量子谱密度参数建立渐近等价关系，推导出几何回归模型，并探讨其与经典白噪声模型的联系。
Sequential Outlier Detection in Non-Stationary Time Series
提出一种基于极值理论控制错误概率的非平稳时间序列序贯异常值检测方法，并验证其渐近与有限样本性能。
On the partial autocorrelation function for locally stationary time series: characterization, estimation and inference
本文针对短程依赖局部平稳时间序列，刻画了局部PACF的性质，提出基于筛法OLS的估计与检验方法，配套乘子自助法和R包，经模拟与实证验证有效。
Entropy estimation, due in part to its connection with mutual information, has seen considerable use in the study of time series data including causality detection and information flow. In many cases, the entropy is estimated using $k$-nearest neighbor (Kozachenko-Leonenko) based methods. However, analytic results on this estimator are limited to independent data. In the article, we show rigorous bounds on the rate of decay of the bias in the number of samples, $N$, assuming they are drawn from a stationary process which satisfies a suitable mixing condition. Numerical examples are presented w…
Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert-Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the co…
Equivariant online predictions of non-stationary time series
本研究在决策理论框架下，证明随机游走动态线性模型在非平稳时间序列预测中具有极小极大最优性，并扩展至多预测密度合成，为模型比较提供理论基线。
Stationary subspace analysis (SSA) searches for linear combinations of the components of nonstationary vector time series that are stationary. These linear combinations and their number defne an associated stationary subspace and its dimension. SSA is studied here for zero mean nonstationary covariance processes. We characterize stationary subspaces and their dimensions in terms of eigenvalues and eigenvectors of certain symmetric matrices. This characterization is then used to derive formal statistical tests for estimating dimensions of stationary subspaces. Eigenstructure-based techniques ar…
The problem of time-series clustering is considered in the case where each data-point is a sample generated by a piecewise stationary ergodic process. Stationary processes are perhaps the most general class of processes considered in non-parametric statistics and allow for arbitrary long-range dependence between variables. Piecewise stationary processes studied here for the first time in the context of clustering, relax the last remaining assumption in this model: stationarity. A natural formulation is proposed for this problem and a notion of consistency is introduced which requires the sampl…
One of the latest self-supervised learning (SSL) methods, VICReg, showed a great performance both in the linear evaluation and the fine-tuning evaluation. However, VICReg is proposed in computer vision and it learns by pulling representations of random crops of an image while maintaining the representation space by the variance and covariance loss. However, VICReg would be ineffective on non-stationary time series where different parts/crops of input should be differently encoded to consider the non-stationarity. Another recent SSL proposal, Temporal Neighborhood Coding (TNC) is effective for …
Accurate forecasting of the U.K. gross value added (GVA) is fundamental for measuring the growth of the U.K. economy. A common nonstationarity in GVA data, such as the ABML series, is its increase in variance over time due to inflation. Transformed or inflation-adjusted series can still be challenging for classical stationarity-assuming forecasters. We adopt a different approach that works directly with the GVA series by advancing recent forecasting methods for locally stationary time series. Our approach results in more accurate and reliable forecasts, and continues to work well even when the…
DynaConF: Dynamic Forecasting of Non-Stationary Time Series
本文提出DynaConF方法，通过解耦平稳条件分布建模与非平稳动态建模，结合贝叶斯动态模型和分解输出空间的深度条件分布模型，提升对非平稳多元时间序列的预测适应能力。
Methods of estimation and forecasting for stationary models are well known in classical time series analysis. However, stationarity is an idealization which, in practice, can at best hold as an approximation, but for many time series may be an unrealistic assumption. We define a class of locally stationary processes which can lead to more accurate uncertainty quantification over making an invalid assumption of stationarity. This class of processes assumes the model parameters to be time-varying and parameterizes them in terms of a transformation of basis functions that ensures that the process…
The Model-free Prediction Principle of Politis (2015) has been successfully applied to general regression problems, as well as problems involving stationary time series. However, with long time series, e.g. annual temperature measurements spanning over 100 years or daily financial returns spanning several years, it may be unrealistic to assume stationarity throughout the span of the dataset. In the paper at hand, we show how Model-free Prediction can be applied to handle time series that are only locally stationary, i.e., they can be assumed to be as stationary only over short time-windows. Su…
We develop an anomaly-detection method when systematic anomalies, possibly statistically very similar to genuine inputs, are affecting control systems at the input and/or output stages. The method allows anomaly-free inputs (i.e., those before contamination) to originate from a wide class of random sequences, thus opening up possibilities for diverse applications. To illustrate how the method works on data, and how to interpret its results and make decisions, we analyze several actual time series, which are originally non-stationary but in the process of analysis are converted into stationary.…
Optimal Short-Term Forecast for Locally Stationary Functional Time Series
本文研究非平稳函数型时间序列的短期线性预测，提出统一的函数型自回归逼近与双重筛扩张方法，证明其渐近最优性，并通过电信流量数据验证实用性。
Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimat…
Detecting Cointegrating Relations in Non-stationary Matrix-Valued Time Series
本文提出矩阵误差修正模型，允许行列分别协整，用信息准则选秩，经模拟与宏观应用验证估计可靠。
Testing for nonlinearity is one of the most important preprocessing steps in nonlinear time series analysis. Typically, this is done by means of the linear surrogate data methods. But it is a known fact that the validity of the results heavily depends on the stationarity of the time series. Since most physiological signals are non-stationary, it is easy to falsely detect nonlinearity using the linear surrogate data methods. In this document, we propose a methodology to extend the procedure for generating constrained surrogate time series in order to assess nonlinearity in non-stationary data. …
Structural Classification of Locally Stationary Time Series Based on Second-order Characteristics
本文提出一种基于自回归近似、集成聚合和距离阈值的局部平稳时间序列分类方法，理论上渐近零误分类率，并在模拟和真实脑电数据上优于多种现有方法。
With developing of computation tools in the last years, data analysis methods to find insightful information are becoming more common among industries and researchers. This paper is the first part of the times series analysis of New England electricity price and demand to find anomaly in the data. In this paper time-series stationary criteria to prepare data for further times-series related analysis is investigated. Three main analysis are conducted in this paper, including moving average, moving standard deviation and augmented Dickey-Fuller test. The data used in this paper is New England bi…
Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stoch…
Monitoring economic conditions and financial stability with an early warning system serves as a prevention mechanism for unexpected economic events. In this paper, we investigate the statistical performance of sequential break-point detectors for stationary time series regression models with extensive simulation experiments. We employ an online sequential scheme for monitoring economic indicators from the European as well as the American financial markets that span the period during the 2008 financial crisis. Our results show that the performance of these tests applied to stationary time serie…
Missing data in non-stationary multivariate time series from digital studies in Psychiatry
针对精神病学数字研究中非平稳多变量时间序列的缺失数据难题，提出MCEM-SSM算法，经多场景模拟验证优势后应用于精神疾病患者研究，探究数字社交与消极情绪的关联。
Time series with long-term structure arise in a variety of contexts and capturing this temporal structure is a critical challenge in time series analysis for both inference and forecasting settings. Traditionally, state space models have been successful in providing uncertainty estimates of trajectories in the latent space. More recently, deep learning, attention-based approaches have achieved state of the art performance for sequence modeling, though often require large amounts of data and parameters to do so. We propose Stanza, a nonlinear, non-stationary state space model as an intermediate…
In this note we show that the locally stationary wavelet process can be decomposed into a sum of signals, each of which following a moving average process with time-varying parameters. We then show that such moving average processes are equivalent to state space models with stochastic design components. Using a simple simulation step, we propose a heuristic method of estimating the above state space models and then we apply the methodology to foreign exchange rates data.
This paper proposes some novel one-sided omnibus tests for independence between two multivariate stationary time series. These new tests apply the Hilbert-Schmidt independence criterion (HSIC) to test the independence between the innovations of both time series. Under regular conditions, the limiting null distributions of our HSIC-based tests are established. Next, our HSIC-based tests are shown to be consistent. Moreover, a residual bootstrap method is used to obtain the critical values for our HSIC-based tests, and its validity is justified. Compared with the existing cross-correlation-based…
We study causality between bivariate curve time series using the Granger causality generalized measures of correlation. With this measure, we can investigate which curve time series Granger-causes the other; in turn, it helps determine the predictability of any two curve time series. Illustrated by a climatology example, we find that the sea surface temperature Granger-causes the sea-level atmospheric pressure. Motivated by a portfolio management application in finance, we single out those stocks that lead or lag behind Dow-Jones industrial averages. Given a close relationship between S&P 500 …
To capture the slowly time-varying spectral content of real-world time-series, a common paradigm is to partition the data into approximately stationary intervals and perform inference in the time-frequency domain. However, this approach lacks a corresponding nonstationary time-domain generative model for the entire data and thus, time-domain inference occurs in each interval separately. This results in distortion/discontinuity around interval boundaries and can consequently lead to erroneous inferences based on any quantities derived from the posterior, such as the phase. To address these shor…
State space model multiple imputation for missing data in non-stationary multivariate time series with application in digital Psychiatry
针对非平稳多元时间序列缺失数据问题，提出基于状态空间模型的多重插补方法SSMmp及其高效变体SSMimpute，通过理论与模拟验证其优势，并应用于双相情感障碍患者智能手机数据中社交网络规模与负面情绪关联的研究。
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