High-dimensional Nonstationary Time Series Research Articles

Causal structure learning for high-dimensional non-stationary time series

Learning the causal structure of high-dimensional non-stationary time series can help in understanding the data generation mechanism, which is a crucial task in machine learning. However, current causal discovery methods for high-dimensional non-stationary time series face several challenges, including the inability to effectively capture non-stationarity, failure to ensure acyclicity of causal graphs, and reliance on subjective threshold definitions, leading to suboptimal algorithm performance. To address these challenges, we introduce a novel Causal Structure Learning model for High-dimensional Non-stationary Time Series (CSL-HNTS). Firstly, this model presents a graph neural network to model the non-stationarity of time series. Secondly, it introduces a novel Directed Acyclic Graph (DAG) sampling method that transforms the space of DAGs into a continuous space, enabling the search for causal graphs within this continuous space to ensure acyclicity. Finally, the model proposes an automatic threshold definition method, without prior knowledge, to convert the weighted adjacency matrix into the Boolean adjacency matrix of the causal graph, thereby avoiding time-consuming postprocessing steps. The proposed approach is validated using simulation datasets and two real datasets, and is benchmarked against current state-of-the-art methods and ablation experiments. The results demonstrate a significant improvement over existing methods, highlighting the efficacy of the proposed model.

Frequency Detection and Change Point Estimation for Time Series of Complex Oscillation

We consider detecting the evolutionary oscillatory pattern of a signal when it is contaminated by non-stationary noises with complexly time-varying data generating mechanism. A high-dimensional dense progressive periodogram test is proposed to accurately detect all oscillatory frequencies. A further phase-adjusted local change point detection algorithm is applied in the frequency domain to detect the locations at which the oscillatory pattern changes. Our method is shown to be able to detect all oscillatory frequencies and the corresponding change points within an accurate range with a prescribed probability asymptotically. A Gaussian approximation scheme and an overlapping-block multiplier bootstrap methodology for sums of complex-valued high dimensional non-stationary time series without variance lower bounds are established, which could be of independent interest. This study is motivated by oscillatory frequency estimation and change point detection problems encountered in physiological time series analysis. An application to spindle detection and estimation in electroencephalogram recorded during sleep is used to illustrate the usefulness of the proposed methodology.

Adaptive Bayesian Spectral Analysis of High-Dimensional Nonstationary Time Series

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 increasingly shrunk toward zero as the column index increases. Formulated in a fully Bayesian framework, the time series is adaptively partitioned into approximately stationary segments, where both the number and locations of partition points are assumed unknown. Stochastic approximation Monte Carlo techniques are used to accommodate the unknown number of segments, and a conditional Whittle likelihood-based Gibbs sampler is developed for efficient sampling within segments. By averaging over the distribution of partitions, the proposed method can approximate both abrupt and slowly varying changes in spectral matrices. Performance of the proposed model is evaluated by extensive simulations and demonstrated through the analysis of high-density electroencephalography. Supplementary materials for this article are available online.

Estimation of Dynamic Networks for High-Dimensional Nonstationary Time Series.

This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified on the basis of comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered on the basis of a kernelized time-varying constrained -minimization for inverse matrix estimation (CLIME) estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this two-step approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under a certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S&P 500 index between 2003 and 2008.

HAR Testing for Spurious Regression in Trend

The usual t test, the t test based on heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators, and the heteroskedasticity and autocorrelation robust (HAR) test are three statistics that are widely used in applied econometric work. The use of these significance tests in trend regression is of particular interest given the potential for spurious relationships in trend formulations. Following a longstanding tradition in the spurious regression literature, this paper investigates the asymptotic and finite sample properties of these test statistics in several spurious regression contexts, including regression of stochastic trends on time polynomials and regressions among independent random walks. Concordant with existing theory (Phillips 1986, 1998; Sun 2004, 2014b) the usual t test and HAC standardized test fail to control size as the sample size n → ∞ in these spurious formulations, whereas HAR tests converge to well-defined limit distributions in each case and therefore have the capacity to be consistent and control size. However, it is shown that when the number of trend regressors K → ∞ , all three statistics, including the HAR test, diverge and fail to control size as n → ∞ . These findings are relevant to high-dimensional nonstationary time series regressions where machine learning methods may be employed.

A Near Unit Root Test for High-Dimensional Nonstationary Time Series

This paper proposes a new near-unit root test for a class of high-dimensional nonstationary time series. A central limit theorem for the proposed test is proposed and then evaluated by an extensive simulation study.

High Dimensional Nonstationary Time Series Modelling with Generalized Dynamic Semiparametric Factor Model

(High dimensional) time series which reveal nonstationary and possibly periodic behavior occur frequently in many fields of science. In this article, we separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via functional factor analysis. We propose a two-step estimation procedure. At the first step, we detect the deterministic trends of the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under various situations extending current variable selection studies. At the second step, we obtain the detrended low dimensional stochastic process, but it also poses an important question: is it justified, from an inferential point of view, to base further statistical inference on the estimated stochastic time series? We show that the difference of the inference based on the estimated time series and true unobserved time series is asymptotically negligible, which finally allows one to study the dynamics of the whole high-dimensional system with a low dimensional representation together with the deterministic trend. We apply the method to our motivating empirical problems: studies of the dynamic behavior of temperatures (further used for pricing weather derivatives), implied volatilities and risk patterns and correlated brain activities (neuro-economics related) using fMRI data, where a panel version model is also presented.

CLT for largest eigenvalues and unit root testing for high-dimensional nonstationary time series

Let $\{Z_{ij}\}$ be independent and identically distributed (i.i.d.) random variables with $EZ_{ij}=0$, $E\vert Z_{ij}\vert^{2}=1$ and $E\vert Z_{ij}\vert^{4}<\infty$. Define linear processes $Y_{tj}=\sum_{k=0}^{\infty}b_{k}Z_{t-k,j}$ with $\sum_{i=0}^{\infty}\vert b_{i}\vert <\infty$. Consider a $p$-dimensional time series model of the form $\mathbf{x}_{t}=\boldsymbol{\Pi} \mathbf{x}_{t-1}+\Sigma^{1/2}\mathbf{y}_{t}, 1\leq t\leq T$ with $\mathbf{y}_{t}=(Y_{t1},\ldots,Y_{tp})'$ and $\Sigma^{1/2}$ be the square root of a symmetric positive definite matrix. Let $\mathbf{B}=(1/p)\mathbf{XX}^{*}$ with $\mathbf{X}=(\mathbf{x_{1}},\ldots,\mathbf{x_{T}})'$ and $X^{*}$ be the conjugate transpose. This paper establishes both the convergence in probability and the asymptotic joint distribution of the first $k$ largest eigenvalues of $\mathbf{B}$ when $\mathbf{x}_{t}$ is nonstationary. As an application, two new unit root tests for possible nonstationarity of high-dimensional time series are proposed and then studied both theoretically and numerically.

CLT for Largest Eigenvalues and Unit Root Tests for High-Dimensional Nonstationary Time Series

This paper proposes a new unit-root test for the case where a high-dimensional vector of nonstationary time series is considered. A new CLT is being established and studied both theoretically and numerically.

Multivariate time-series analysis and diffusion maps

Dimensionality reduction in multivariate time series analysis has broad applications, ranging from financial data analysis to biomedical research. However, high levels of ambient noise and various interferences result in nonstationary signals, which may lead to inefficient performance of conventional methods. In this paper, we propose a nonlinear dimensionality reduction framework using diffusion maps on a learned statistical manifold, which gives rise to the construction of a low-dimensional representation of the high-dimensional nonstationary time series. We show that diffusion maps, with affinity kernels based on the Kullback–Leibler divergence between the local statistics of samples, allow for efficient approximation of pairwise geodesic distances. To construct the statistical manifold, we estimate time-evolving parametric distributions by designing a family of Bayesian generative models. The proposed framework can be applied to problems in which the time-evolving distributions (of temporally localized data), rather than the samples themselves, are driven by a low-dimensional underlying process. We provide efficient parameter estimation and dimensionality reduction methodologies, and apply them to two applications: music analysis and epileptic-seizure prediction.

An approach for identifying and predicting economic recessions in real-time using time-frequency functional models

Identifying periods of recession and expansion is a challenging topic of ongoing interest with important economic and monetary policy implications. Given the current state of the global economy, significant attention has recently been devoted to identifying and forecasting economic recessions. Consequently, we introduce a novel class of Bayesian hierarchical probit models that take advantage of dimension-reduced time–frequency representations of various market indices. The approach we propose can be viewed as a Bayesian mixed frequency data regression model, as it relates high-frequency daily data observed over several quarters to a binary quarterly response indicating recession or expansion. More specifically, our model directly incorporates time–frequency representations of the entire high-dimensional non-stationary time series of daily log returns, over several quarters, as a regressor in a predictive model, while quantifying various sources of uncertainty. The necessary dimension reduction is achieved by treating the time–frequency representation (spectrogram) as an “image” and finding its empirical orthogonal functions. Subsequently, further dimension reduction is accomplished through the use of stochastic search variable selection. Overall, our dimension reduction approach provides an extremely powerful tool for feature extraction, yielding an interpretable image of features that predict recessions. The effectiveness of our model is demonstrated through out-of-sample identification (nowcasting) and multistep-ahead prediction (forecasting) of economic recessions. In fact, our results provide greater than 85% and 80% out-of-sample forecasting accuracy for recessions and expansions respectively, even three quarters ahead. Finally, we illustrate the utility and added value of including time–frequency information from the NASDAQ index when identifying and predicting recessions. Copyright © 2012 John Wiley & Sons, Ltd.

High Dimensional Nonstationary Time Series Modelling with Generalized Dynamic Semiparametric Factor Model

(High dimensional) time series which reveal nonstationary and possibly periodic behavior occur frequently in many scientic elds. In this article, we separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via functional factor analysis. We propose a two-step estimation procedure. At the rst step, we detect the deterministic trends of the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under various situations extending current variable selection studies. At the second step, we obtain the detrended low dimensional stochastic process. However an important issue arises: is it justied, from an inferential point of view, to base further statistical inference on the estimated stochastic time series? We show that the dierence between the inference based on the estimated time series and true unobserved time series is asymptotically negligible, which ultimately allows one to study the dynamics of the whole high-dimensional system with a low dimensional representation together with the deterministic trend. We apply the method to our motivating empirical problems: studies of the dynamic behavior of temperatures (further used for pricing weather derivatives), implied volatilities and risk patterns and correlated brain activities (neuro-economics related) using fMRI data, where a panel version model is also presented.

Determining the number of factors in a multivariate error correction-volatility factor model

In order to describe the comovements in both conditional mean and conditional variance of high dimensional nonstationary time series by dimension reduction, we introduce the conditional heteroscedasticity with factor structure to the error correction model. The new model is called the error correction volatility factor model. Some specification and estimation approaches are developed. In particular, the determination of the number of factors is discussed. Our setting is general in the sense that we impose neither i.i.d assumption on idiosyncratic components in the factor structure nor independence between factors and idiosyncratic errors. We illustrate the proposed approach with a Monte Carlo simulation and a real data example.