We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an M-estimation efficiency bound, equal to that found in work using an adaptive LAM/LAN approach. A major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies. This is relevant also when the innovation density is parameterized, and we provide a result for that case also. Our semiparametric estimates employ nonparametric series estimation, which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models; our work thus also contributes to methods and theory for nonfractional time series models, such as autoregressive moving averages. A Monte Carlo study of finite sample performance of the semiparametric estimates is included.
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…
We propose NonStGM, a general nonparametric graphical modeling framework for studying dynamic associations among the components of a nonstationary multivariate time series. It builds on the framework of Gaussian Graphical Models (GGM) and stationary time series Gaussian Graphical model (StGM), and complements existing works on parametric graphical models based on change point vector autoregressions (VAR). Analogous to StGM, the proposed framework captures conditional noncorrelations (both intertemporal and contemporaneous) in the form of an undirected graph. In addition, to describe the more n…
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…
For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance …
We review the advancement of nonstationary time series analysis from the perspective of Cowles Commission structural equation approach. We argue that despite the rich repertoire nonstationary time series analysis provides to analyze how do variables respond dynamically to shocks through the decomposition of a dynamic system into long-run and short-run relations, nonstationarity does not invalid the classical concerns of structural equation modeling -- identification and simultaneity bias. The same rank condition for identification holds for stationary and nonstationary data and some sort of in…
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…
We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quan…
Understanding and predicting environmental phenomena often requires the construction of spatio-temporal statistical models, which are typically Gaussian processes. A common assumption made on Gaussian processes is that of covariance stationarity, which is unrealistic in many geophysical applications. In this article, we introduce a deep-learning-inspired approach to construct descriptive nonstationary spatio-temporal models by modeling stationary processes on warped spatio-temporal domains. The warping functions we use are constructed using several simple injective warping units which, when co…
In this paper, we propose a fast, well-performing, and consistent method for segmenting a piecewise-stationary, linear time series with an unknown number of breakpoints. The time series model we use is the nonparametric Locally Stationary Wavelet model, in which a complete description of the piecewise-stationary second-order structure is provided by wavelet periodograms computed at multiple scales and locations. The initial stage of our method is a new binary segmentation procedure, with a theoretically justified and rapidly computable test criterion that detects breakpoints in wavelet periodo…
How does soil pollution affect a plant's circadian clock? Are there any differences between how the clock reacts when exposed to different concentrations of elements of the periodic table? If so, can we characterise these differences? We approach these questions by analysing and modelling circadian plant data, where the levels of expression of a luciferase reporter gene were measured at regular intervals over a number of days after exposure to different concentrations of lithium. A key aspect of circadian data analysis is to determine whether a time series (derived from experimental data) is `…
CREST, ENSAE, Université Paris Saclay（巴黎萨克雷大学国立统计与经济行政学院 CREST）；AGM UMR8088 University Paris-Seine（巴黎塞纳大学 AGM UMR8088）；CIMFAV, Universidad de Valparaiso（瓦尔帕莱索大学 CIMFAV）；CAM, Tianjin University（天津大学 CAM）
Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behavior is non-stationary. In this paper, we extend the basic tools of Dedecker and Fan (2015) to nonstationary Mark…
Tests for structural breaks in time series should ideally be sensitive to breaks in the parameter of interest, while being robust to nuisance changes. Statistical analysis thus needs to allow for some form of nonstationarity under the null hypothesis of no change. In this paper, estimators for integrated parameters of locally stationary time series are constructed and a corresponding functional central limit theorem is established, enabling change-point inference for a broad class of parameters under mild assumptions. The proposed framework covers all parameters which may be expressed as nonli…
In this chapter we present an overview of the main ideas and methods in the fractional integration and cointegration literature. We do not attempt to give a complete survey of this enormous literature, but rather a more introductory treatment suitable for a researcher or graduate student wishing to learn about this exciting field of research. With this aim, we have surely overlooked many relevant references for which we apologize in advance. Knowledge of standard time series methods, and in particular methods related to nonstationary time series, at the level of a standard graduate course or a…
Argo floats measure seawater temperature and salinity in the upper 2,000 m of the global ocean. Statistical analysis of the resulting spatio-temporal dataset is challenging due to its nonstationary structure and large size. We propose mapping these data using locally stationary Gaussian process regression where covariance parameter estimation and spatio-temporal prediction are carried out in a moving-window fashion. This yields computationally tractable nonstationary anomaly fields without the need to explicitly model the nonstationary covariance structure. We also investigate Student-$t$ dist…
Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments i…
We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification …
A common class of methods for analyzing of multivariate time series, stationary and nonstationary, decomposes the observed series into latent sources. Methods such as principal compoment analysis (PCA), independent component analysis (ICA) and Stationary Subspace Analysis (SSA) assume the observed multivariate process is generated by latent sources that are stationary or nonstationary. We develop a method that tracks changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. We study complexity through the latent sources …
Within the neurosciences, to observe variability across time in the dynamics of an underlying brain process is neither new nor unexpected. Wavelets are essential in analyzing brain signals because, even within a single trial, brain signals exhibit nonstationary behaviour. However, neurological signals generated within an experiment may also potentially exhibit evolution across trials (replicates). As neurologists consider localised spectra of brain signals to be most informative, here we develop a novel wavelet-based tool capable to formally represent process nonstationarities across both time…
This work addresses the problem of missing data in time-series analysis focusing on (a) estimation of model parameters in the presence of missing data and (b) reconstruction of missing data. Standard approaches used to solve these problems like the maximum likelihood estimation or the Bayesian inference rely on a priori assumptions like the Gaussian or stationary behavior of missing data and might lead to biased results where these assumptions are unfulfilled. In order to go beyond, we extend the Finite Element Methodology (FEM) for Vector Auto-Regressive models with eXogenous factors and boun…
This article introduces a nonparametric approach to multivariate time-varying power spectrum analysis. The procedure adaptively partitions a time series into an unknown number of approximately stationary segments, where some spectral components may remain unchanged across segments, allowing components to evolve differently over time. Local spectra within segments are fit through Whittle likelihood based penalized spline models of modified Cholesky components, which provide flexible nonparametric estimates that preserve positive definite structures of spectral matrices. The approach is formulat…
This article introduces a nonparametric approach to spectral analysis of a high-dimensional multivariate nonstationary time series. The procedure is based on a novel frequency-domain factor model that provides a flexible yet parsimonious representation of spectral matrices from a large number of simultaneously observed time series. Real and imaginary parts of the factor loading matrices are modeled independently using a prior that is formulated from the tensor product of penalized splines and multiplicative gamma process shrinkage priors, allowing for infinitely many factors with loadings incr…
University of California, Davis（加州大学戴维斯分校）；Princeton University（普林斯顿大学）
Bandit algorithms have been predominantly analyzed in the convex setting with function-value based stationary regret as the performance measure. In this paper, motivated by online reinforcement learning problems, we propose and analyze bandit algorithms for both general and structured nonconvex problems with nonstationary (or dynamic) regret as the performance measure, in both stochastic and non-stochastic settings. First, for general nonconvex functions, we consider nonstationary versions of first-order and second-order stationary solutions as a regret measure, motivated by similar performanc…
In this paper the class of ARCH$(\infty)$ models is generalized to the nonstationary class of ARCH$(\infty)$ models with time-varying coefficients. For fixed time points, a stationary approximation is given leading to the notation ``locally stationary ARCH$(\infty)$ process.'' The asymptotic properties of weighted quasi-likelihood estimators of time-varying ARCH$(p)$ processes ($p<\infty$) are studied, including asymptotic normality. In particular, the extra bias due to nonstationarity of the process is investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in terms of st…
We build and study a data-driven procedure for the estimation of the stationary density f of an additive fractional SDE. To this end, we also prove some new concentrations bounds for discrete observations of such dynamics in stationary regime.
The goal of this paper is to extend independent subspace analysis (ISA) to the case of (i) nonparametric, not strictly stationary source dynamics and (ii) unknown source component dimensions. We make use of functional autoregressive (fAR) processes to model the temporal evolution of the hidden sources. An extension of the ISA separation principle--which states that the ISA problem can be solved by traditional independent component analysis (ICA) and clustering of the ICA elements--is derived for the solution of the defined fAR independent process analysis task (fAR-IPA): applying fAR identific…
