Sample Autocovariance Research Articles

Asymptotic normality of spectral means of Hilbert space valued random processes

A variety of statistics for functional time series allows for a representation as weighted average of corresponding periodogram operators over the frequency domain. We study consistency and asymptotic normality of such spectral mean estimators under mild assumptions. We show that weak convergence of these estimators can be deduced from the (joint) weak convergence of the sample autocovariance operators. The latter is established for a large class of weakly dependent functional time series, which admit expansions as Bernoulli shifts and the weak dependence is quantified by the condition of L4-m-approximability.

Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample autocovariance-based statistics.

Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.

Testing of two-dimensional Gaussian processes by sample cross-covariance function.

In this paper, we address the issue of testing two-dimensional Gaussian processes with a defined cross-dependency structure. Multivariate Gaussian processes are widely used in various applications; therefore, it is essential to identify the theoretical model that accurately describes the data. While it is relatively straightforward to do so in a one-dimensional case, analyzing multi-dimensional vectors requires considering the dependency between the components, which can significantly affect the efficiency of statistical methods. The testing methodology presented in this paper is based on the sample cross-covariance function and can be considered a natural generalization of the approach recently proposed for testing one-dimensional Gaussian processes based on the sample autocovariance function. We verify the efficiency of this procedure on three classes of two-dimensional Gaussian processes: Brownian motion, fractional Brownian motion, and two-dimensional autoregressive discrete-time process. The simulation results clearly demonstrate the effectiveness of the testing methodology, even for small sample sizes. The theoretical and simulation results are supported by analyzing two-dimensional real-time series that describe the main risk factors of a mining company, namely, copper price and exchange rates (USDPLN). We believe that the introduced methodology is intuitive and relatively simple to implement, and thus, it can be applied in many real-world scenarios where multi-dimensional data are examined.

On the asymptotic distribution of sample autocovariance differences of long-memory processes

On the asymptotic distribution of sample autocovariance differences of long-memory processes

A spectral approach to estimate the autocovariance function

This paper proposes two novel alternative estimators for the autocovariance function of a short-range dependent stationary process based on a spectral approach. One is based on the classical periodogram, and the other one uses the M-periodogram. The theoretical properties of both estimators are investigated for linear processes. Monte-Carlo experiments are conducted to compare these estimators to the standard sample autocovariance function in autoregressive moving average (ARMA) models with and without additive outliers. In the non-contaminated scenario, the empirical investigation shows that the three estimators display similar performance. However, in the contaminated case, the estimator based on the M-periodogram remains unaffected in the presence of additive outliers, while the two other estimators are corrupted.

Theory of evolutionary spectra for heteroskedasticity and autocorrelation robust inference in possibly misspecified and nonstationary models

The literature on heteroskedasticity and autocorrelation robust (HAR) inference is extensive but its usefulness relies on stationarity of the relevant process, say Vt, usually a function of the data and estimated model residuals. Yet, a large body of work shows widespread evidence of various forms of nonstationarity in the latter. Also, many testing problems are such that Vt is stationary under the null hypothesis but nonstationary under the alternative. In either case, the consequences are possible size distortions and, especially, a reduction in power which can be substantial (e.g., non-monotonic power), since all such estimates are based on weighted sums of the sample autocovariances of Vt, which are inflated. We propose HAR inference methods valid under a broad class of nonstationary processes, labeled Segmented Local Stationary, which possess a spectrum that varies both over frequencies and time. It is allowed to change either slowly and continuously and/or abruptly at some time points, thereby encompassing most nonstationary models used in applied work. We introduce a double kernel estimator (DK-HAC) that applies a smoothing over both lagged autocovariances and time. The optimal kernels and bandwidth sequences are derived under a mean-squared error criterion. The data-dependent bandwidths rely on the “plug-in” approach using approximating parametric models having time-varying parameters estimated with standard methods applied to local data. Our method yields tests with good size and power in finite-samples. In particular, the power gains are achieved without notable size distortions.

Factorization and discrete-time representation of multivariate CARMA processes

In this paper we show that stationary and non-stationary multivariate continuous-time ARMA (MCARMA) processes have the representation as a sum of multivariate complex-valued Ornstein-Uhlenbeck processes under some mild assumptions. The proof benefits from properties of rational matrix polynomials. A conclusion is an alternative description of the autocovariance function of a stationary MCARMA process. Moreover, that representation is used to show that the discrete-time sampled MCARMA(p,q) process is a weak VARMA(p,p-1) process if second moments exist. That result complements the weak VARMA(p,p-1) representation derived in Chambers and Thornton (2012). In particular, it relates the right solvents of the autoregressive polynomial of the MCARMA process to the right solvents of the autoregressive polynomial of the VARMA process; in the one-dimensional case the right solvents are the zeros of the autoregressive polynomial. Finally, a factorization of the sample autocovariance function of the noise sequence is presented which is useful for statistical inference.

New estimation method for periodic autoregressive time series of order 1 with additive noise

The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.

Simultaneous inference for autocovariances based on autoregressive sieve bootstrap

In this article, maximum deviations of sample autocovariances and autocorrelations from their theoretical counterparts over an increasing set of lags are considered. The asymptotic distribution of such statistics for physically dependent stationary time series, which is of Gumbel type, only depends on second‐order properties of the underlying time series. Since the autoregressive sieve bootstrap is able to mimic the second‐order structure asymptotically correctly it is an obvious problem whether the autoregressive (AR) sieve bootstrap, which has been shown to work for a number of relevant statistics in time series analysis, asymptotically works for maximum deviations of autocovariances and autocorrelations as well. This article shows that the question can be answered positively. Moreover, potential applications including spectral density estimation and an investigation of finite sample properties of the AR‐sieve bootstrap proposal by simulation are given.

Large sample autocovariance matrices of linear processes with heavy tails

We provide asymptotic theory for certain functions of the sample autocovariance matrices of a high-dimensional time series with infinite fourth moment. The time series exhibits linear dependence across the coordinates and through time. Assuming that the dimension increases with the sample size, we provide theory for the eigenvectors of the sample autocovariance matrices and find explicit approximations of a simple structure, whose finite sample quality is illustrated for simulated data. We also obtain the limits of the normalized eigenvalues of functions of the sample autocovariance matrices in terms of cluster Poisson point processes. In turn, we derive the distributional limits of the largest eigenvalues and functionals acting on them. In our proofs, we use large deviation techniques for heavy-tailed processes, point process techniques motivated by extreme value theory, and related continuous mapping arguments.

Testing for unit roots based on sample autocovariances

Summary We propose a new unit-root test for a stationary null hypothesis $H_0$ against a unit-root alternative $H_1$. Our approach is nonparametric as $H_0$ assumes only that the process concerned is $I(0)$, without specifying any parametric forms. The new test is based on the fact that the sample autocovariance function converges to the finite population autocovariance function for an $I(0)$ process, but diverges to infinity for a process with unit roots. Therefore, the new test rejects $H_0$ for large values of the sample autocovariance function. To address the technical question of how large is large, we split the sample and establish an appropriate normal approximation for the null distribution of the test statistic. The substantial discriminative power of the new test statistic is due to the fact that it takes finite values under $H_0$ and diverges to infinity under $H_1$. This property allows one to truncate the critical values of the test so that it has asymptotic power 1; it also alleviates the loss of power due to the sample-splitting. The test is implemented in $\texttt{R}$.

Discriminating Gaussian processes via quadratic form statistics.

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein-Uhlenbeck (OU) process, which is stationary. We show that the considered statistics' ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process.

Empirical anomaly measure for finite-variance processes

Anomalous diffusion phenomena are observed in many areas of interest. They manifest themselves in deviations from the laws of Brownian motion (BM), e.g. in the non-linear growth (mostly power-law) in time of the ensemble average mean squared displacement (MSD). When we analyze the real-life data in the context of anomalous diffusion, the primary problem is the proper identification of the type of the anomaly. In this paper, we introduce a new statistic, called empirical anomaly measure (EAM), that can be useful for this purpose. This statistic is the sum of the off-diagonal elements of the sample autocovariance matrix for the increments process. On the other hand, it can be represented as the convolution of the empirical autocovariance function with time lags. The idea of the EAM is intuitive. It measures dependence between the ensemble-averaged MSD of a given process from the ensemble-averaged MSD of the classical BM. Thus, it can be used to measure the distance between the anomalous diffusion process and normal diffusion. In this article, we prove the main probabilistic characteristics of the EAM statistic and construct the formal test for the recognition of the anomaly type. The advantage of the EAM is the fact that it can be applied to any data trajectories without the model specification. The only assumption is the stationarity of the increments process. The complementary summary of the paper constitutes of Monte Carlo simulations illustrating the effectiveness of the proposed test and properties of EAM for selected processes.

A Simple Asymptotically F-Distributed Portmanteau Test for Diagnostic Checking of Time Series Models With Uncorrelated Innovations

We propose a simple asymptotically F-distributed portmanteau test for diagnostically checking whether the innovations in a parametric time series model are uncorrelated while allowing them to exhibit higher-order dependence of unknown forms. A transform of sample residual autocovariances removing the influence of parameter estimation uncertainty makes the test simple. Further, by employing the orthonormal series variance estimator, a special sample autocovariances estimator that is asymptotically invariant to parameter estimation uncertainty, we show that the proposed test statistic is asymptotically F-distributed under fixed-smoothing asymptotics. The asymptotic F-theory accounts for the estimation error of the variance estimator that the asymptotic chi-squared theory ignores. Moreover, an extensive Monte Carlo study demonstrates that the F-test has more accurate finite sample size than existing tests with virtually no power loss. An application to S&P 500 returns illustrates the merits of the proposed methodology.

Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks.

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.

Estimating wold matrices and vector moving average processes

The Wold decomposition gives a moving average (MA) representation of a purely non‐deterministic stationary process. In this article, we derive estimates of the Wold matrices for a d‐dimensional process by using a Cholesky decomposition of a banded and tapered version of the sample autocovariance matrix, and we derive convergence rates for the estimation error of the (possibly infinite) sequence of Wold matrices. By analogy to lag‐window estimates of the spectral density, this method can be used to obtain finite vector MA models with an adaptive lag‐order. We additionally show how these results can be applied to impulse response analysis and to derive a bootstrap procedure. Our theoretical results are complemented by simulations which investigate the finite sample performance of the estimator.

On the behavior of large empirical autocovariance matrices between the past and the future

The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support [Formula: see text] is characterized. It is also established that the squared singular values are almost surely located in a neighborhood of [Formula: see text].

Testing equality of autocovariance operators for functional time series

We consider strictly stationary stochastic processes of Hilbert space‐valued random variables and focus on fully functional tests for the equality of the lag‐zero autocovariance operators of several independent functional time series. A moving block bootstrap (MBB)‐based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomessome common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert–Schmidt distance of the corresponding sample lag‐zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested MBB‐based testing procedure is illustrated through simulations and an application to a real‐life dataset is discussed.

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Suppose X is an N×n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1∕n and whose fourth moment is of order O(n−2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z≠0 is a complex number. First we consider the matrix XAX∗−z, and obtain asymptotic probability bounds for its smallest singular value when N and n diverge to infinity and N∕n→γ,0<γ<∞. Then we consider the special case where A=J=[1i−j=1modn] is a circulant matrix. Using the above result, we show that the limit spectral distribution of XJX∗ exists when N∕n→γ,0<γ<∞ and describe the limit explicitly. Assuming that X represents a ℂN-valued time series which is observed over a time window of length n, the matrix XJX∗ represents the one-step sample autocovariance matrix of this time series. A whiteness test against an MA correlation model for this time series is introduced based on the above limit result. Numerical simulations show the excellent performance of this test.

Limit theorems in the context of multivariate long-range dependence

This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.