This article introduces autocorrelograms for time series of point processes. Such time series usually arise when a longer temporal or spatio-temporal point process is sliced into smaller time units; for example, when an annual process is sliced into 365 daily replications. We assume the point processes follow a doubly-stochastic Poisson model with log-Gaussian intensity functions. The proposed autocorrelograms are computationally simple and based on binning. The asymptotic distribution of the autocorrelations is established. The ability of the method to detect the patterns of common autoregressive and moving-average time series models is shown by simulation. Two examples of application to temporal and spatial point-process time series are shown, pertaining bike demand in the Divvy bike-sharing system of Chicago and street theft in Chicago, respectively.
We propose a new autocorrelation measure for functional time series that we term spherical autocorrelation. It is based on measuring the average angle between lagged pairs of series after having been projected onto the unit sphere. This new measure enjoys several complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of sign or direction of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and are used to construct …
One form of big data are signals - time series of consecutive values. In physical experiments, billions of values can now be measured within a second. Signals of heart and brain in intensive care, as well as seismic waves, are measured with 100 up to 1000 Hz over hours, days or even years. A song of 3 minutes on CD comprises 16 million values. Music and seismic vibrations basically consist of harmonic oscillations for which classical tools like autocorrelation and spectrogram work well. This note presents similar tools for all kinds of rhythmic processes, with non-linear distortion, artefacts,…
Concurrent time series commonly arise in various applications, including when monitoring the environment such as in air quality measurement networks, weather stations, oceanographic buoys, or in paleo form such as lake sediments, tree rings, ice cores, or coral isotopes, with each monitoring or sampling site providing one of the time series. The goal in such applications is to extract a common time trend or signal in the observed data. Other examples where the goal is to extract a common time trend for multiple time series are in stock price time series, neurological time series, and quality c…
The classical regular and partial autocorrelation functions are powerful tools for stationary time series modelling and analysis. However, it is increasingly recognized that many time series are not stationary and the use of classical global autocorrelations can give misleading answers. This article introduces two estimators of the local partial autocorrelation function and establishes their asymptotic properties. The article then illustrates the use of these new estimators on both simulated and real time series. The examples clearly demonstrate the strong practical benefits of local estimator…
We begin with an interpretation of the L1-distance between two power spectral densities and then, following an analogous rationale, we develop a natural metric for quantifying distance between respective covariance matrices.
Fractional lower-order covariance-based measures for cyclostationary time series with heavy-tailed distributions: application to dependence testing and model order identification
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学纯科学与应用数学学院，雨果·斯坦豪斯中心）
We propose a specification test for conditional location--scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence -- as measured by the pre-asymptotic tail copula -- with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the …
Regression-based frameworks for streamflow regionalization are built around catchment attributes that traditionally originate from catchment hydrology, flood frequency analysis and their interplay. In this work, we deviated from this traditional path by formulating and extensively investigating the first regression-based streamflow regionalization frameworks that largely emerge from general-purpose time series features for data science and, more precisely, from a large variety of such features. We focused on 28 features that included (partial) autocorrelation, entropy, temporal variation, seas…
This paper develops a new time series clustering procedure allowing for heteroskedasticity, non-normality and model's non-linearity. At this aim, we follow a fuzzy approach. Specifically, considering a Dynamic Conditional Score (DCS) model, we propose to cluster time series according to their estimated conditional moments via the Autocorrelation-based fuzzy C-means (A-FCM) algorithm. The DCS parametric modelling is appealing because of its generality and computational feasibility. The usefulness of the proposed procedure is illustrated using an experiment with simulated data and several empiri…
The main goal of this work is to present a new model able to deal with potentially misreported continuous time series. The proposed model is able to handle the autocorrelation structure in continuous time series data, which might be partially or totally underreported or overreported. Its performance is illustrated through a comprehensive simulation study considering several autocorrelation structures and two real data applications on human papillomavirus incidence in Girona (Catalunya, Spain) and COVID-19 incidence in the Chinese region of Heilongjiang.
We complement the theory developed in Preinerstorfer and Pötscher (2016) with further finite sample results on size and power of heteroskedasticity and autocorrelation robust tests. These allows us, in particular, to show that the sufficient conditions for the existence of size-controlling critical values recently obtained in Pötscher and Preinerstorfer (2018) are often also necessary. We furthermore apply the results obtained to tests for hypotheses on deterministic trends in stationary time series regressions, and find that many tests currently used are strongly size-distorted.
I examine the consequences of modelling contagious influence in a social network with incomplete edge information, namely in the situation where each individual may name a limited number of friends, so that extra outbound ties are censored. In particular, I consider a prototypical time series configuration where a property of the "ego" is affected in a causal fashion by the properties of their "alters" at a previous time point, both in the total number of alters as well as the deviation from a central value. This is considered with three potential methods for naming one's friends: a strict upp…
A new version of the partial autocorrelation plot and a new family of subset autoregressive models are introduced. A comprehensive approach to model identification, estimation and diagnostic checking is developed for these models. These models are better suited to efficient model building of high-order autoregressions with long time series. Several illustrative examples are given.
A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. In this paper, we elaborate on the Toeplitz autocorrelation matrix to the autocorrelation and cross-correlation of residuals and squared residuals block matrix. We derive a new portmanteau test statistic using the log of the determinant of the mth autocorrelations and cross-correlations block matrix. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-squared distributions and can be approximated by a gamma distribution. This test is applied to identify …
German Aerospace Center（德国航空航天中心）；Institute of Data Science（数据科学研究所）；Technische Universität Berlin（柏林工业大学）
The paper introduces a novel conditional independence (CI) based method for linear and nonlinear, lagged and contemporaneous causal discovery from observational time series in the causally sufficient case. Existing CI-based methods such as the PC algorithm and also common methods from other frameworks suffer from low recall and partially inflated false positives for strong autocorrelation which is an ubiquitous challenge in time series. The novel method, PCMCI$^+$, extends PCMCI [Runge et al., 2019b] to include discovery of contemporaneous links. PCMCI$^+$ improves the reliability of CI tests …
Count time series are widely encountered in practice. As with continuous valued data, many count series have seasonal properties. This paper uses a recent advance in stationary count time series to develop a general seasonal count time series modeling paradigm. The model permits any marginal distribution for the series and the most flexible autocorrelations possible, including those with negative dependence. Likelihood methods of inference can be conducted and covariates can be easily accommodated. The paper first develops the modeling methods, which entail a discrete transformation of a Gauss…
Time series observations are ubiquitous in astronomy, and are generated to distinguish between different types of supernovae, to detect and characterize extrasolar planets and to classify variable stars. These time series are usually modeled using a parametric and/or physical model that assumes independent and homoscedastic errors, but in many cases these assumptions are not accurate and there remains a temporal dependency structure on the errors. This can occur, for example, when the proposed model cannot explain all the variability of the data or when the parameters of the model are not prop…
A semi-analytic method is proposed for the generation of realizations of a multivariate process of a given linear correlation structure and marginal distribution. This is an extension of a similar method for univariate processes, transforming the autocorrelation of the non-Gaussian process to that of a Gaussian process based on a piece-wise linear marginal transform from non-Gaussian to Gaussian marginal. The extension to multivariate processes involves the derivation of the autocorrelation matrix from the marginal transforms, which determines the generating vector autoregressive process. The …
Analyzing ecological data often requires modeling the autocorrelation created by spatial and temporal processes. Many of the statistical methods used to account for autocorrelation can be viewed as regression models that include basis functions. Understanding the concept of basis functions enables ecologists to modify commonly used ecological models to account for autocorrelation, which can improve inference and predictive accuracy. Understanding the properties of basis functions is essential for evaluating the fit of spatial or time-series models, detecting a hidden form of multicollinearity,…
We present a novel quasi-Bayesian method to weight multiple dynamical models by their skill at capturing both potentially non-linear trends and first-order autocorrelated variability of the underlying process, and to make weighted probabilistic projections. We validate the method using a suite of one-at-a-time cross-validation experiments involving Atlantic meridional overturning circulation (AMOC), its temperature-based index, as well as Korean summer mean maximum temperature. In these experiments the method tends to exhibit superior skill over a trend-only Bayesian model averaging weighting …
Testing restrictions on regression coefficients in linear models often requires correcting the conventional F-test for potential heteroskedasticity or autocorrelation amongst the disturbances, leading to so-called heteroskedasticity and autocorrelation robust test procedures. These procedures have been developed with the purpose of attenuating size distortions and power deficiencies present for the uncorrected F-test. We develop a general theory to establish positive as well as negative finite-sample results concerning the size and power properties of a large class of heteroskedasticity and au…
Multiple-group (Controlled) Interrupted Time Series Analysis with Higher-Order Autoregressive Errors: A Simulation Study Comparing Newey-West and Prais-Winsten Methods
University of California, San Francisco（加州大学旧金山分校）
Wind farms can be regarded as complex systems that are, on the one hand, coupled to the nonlinear, stochastic characteristics of weather and, on the other hand, strongly influenced by supervisory control mechanisms. One crucial problem in this context today is the predictability of wind energy as an intermittent renewable resource with additional non-stationary nature. In this context, we analyze the power time series measured in an offshore wind farm for a total period of one year with a time resolution of 10 min. Applying detrended fluctuation analysis, we characterize the autocorrelation of…
Hydroclimatic time series analysis focuses on a few feature types (e.g., autocorrelations, trends, extremes), which describe a small portion of the entire information content of the observations. Aiming to exploit a larger part of the available information and, thus, to deliver more reliable results (e.g., in hydroclimatic time series clustering contexts), here we approach hydroclimatic time series analysis differently, i.e., by performing massive feature extraction. In this respect, we develop a big data framework for hydroclimatic variable behaviour characterization. This framework relies on…
Given observations from a stationary time series, permutation tests allow one to construct exactly level $α$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null hypothesis of interest is that the underlying process is an uncorrelated sequence, permutation tests are not necessarily level $α$, nor are they approximately level $α$ in large samples. In addition, permutation tests may have large Type 3, or directional, errors, in which a two-sided test rejects the null hypothesis and concludes that the first-order autocorrel…
In many industrial manufacturing processes, the quality of products depends on the relation between two main ingredients or characteristics. Often, this calls for monitoring the ratio of two normal random variables with statistical process control (SPC) techniques. A large number of studies related to designing control charts monitoring this ratio have been published. However, these studies are based purely on the assumption of independent observations. In practice, autocorrelation between observations can exist and should be modeled to protect against the false alarm rate inflation when imple…
Motivated by electricity consumption metering, we extend existing nonnegative matrix factorization (NMF) algorithms to use linear measurements as observations, instead of matrix entries. The objective is to estimate multiple time series at a fine temporal scale from temporal aggregates measured on each individual series. Furthermore, our algorithm is extended to take into account individual autocorrelation to provide better estimation, using a recent convex relaxation of quadratically constrained quadratic program. Extensive experiments on synthetic and real-world electricity consumption datas…
Stationary and ergodic time series can be constructed using an s-vine decomposition based on sets of bivariate copula functions. The extension of such processes to infinite copula sequences is considered and shown to yield a rich class of models that generalizes Gaussian ARMA and ARFIMA processes to allow both non-Gaussian marginal behaviour and a non-Gaussian description of the serial partial dependence structure. Extensions of classical causal and invertible representations of linear processes to general s-vine processes are proposed and investigated. A practical and parsimonious method for …
In many statistical modeling frameworks, goodness-of-fit tests are typically administered to the estimated residuals. In the time series setting, whiteness of the residuals is assessed using the sample autocorrelation function. For many time series models, especially those used for financial time series, the key assumption on the residuals is that they are in fact independent and not just uncorrelated. In this paper, we apply the auto-distance covariance function (ADCV) to evaluate the serial dependence of the estimated residuals. Distance covariance can discriminate between dependence and ind…
The behavior of the power function of autocorrelation tests such as the Durbin-Watson test in time series regressions or the Cliff-Ord test in spatial regression models has been intensively studied in the literature. When the correlation becomes strong, Krämer (1985) (for the Durbin-Watson test) and Krämer (2005) (for the Cliff-Ord test) have shown that the power can be very low, in fact can converge to zero, under certain circumstances. Motivated by these results, Martellosio (2010) set out to build a general theory that would explain these findings. Unfortunately, Martellosio (2010) does not…
We study goodness-of-fit testing for non-causal autoregressive time series with non-Gaussian stable noise. To model time series exhibiting sharp spikes or occasional bursts of outlying observations, the exponent of the non-Gaussian stable variables is assumed to be less than two. Under such conditions, the innovation variables have no finite second moment. We proved that the sample autocorrelation functions of the trimmed residuals are asymptotically normal. Nonparametric tests are also investigated. The rank correlations of the residuals or the squared residuals are shown to be asymptotically…
FreDF: Learning to Forecast in the Frequency Domain
Department of Control Science and Engineering, Zhejiang University（浙江大学控制科学与工程学院）；School of Automation, Central South University（中南大学自动化学院）；Department of Computer Science and Engineering, Shanghai Jiao Tong University（上海交通大学计算机科学与工程学院）；Trust and Safety Team, TikTok Sydney, ByteDance Inc.（字节跳动悉尼信任与安全团队）；Center for Data Science, Peking University（北京大学数据科学中心）；Generative AI Lab, College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院生成式人工智能实验室）
Traditional white noise testing, for example the Ljung-Box test, studies only the autocorrelation function (ACF). Time series can be heteroscedastic and therefore not i.i.d. but still white noise (that is, with zero ACF). An example of heteroscedasticity is financial time series: times of high variance (financial crises) can alternate with times of low variance (calm times). Here, absolute values of time series terms are not white noise. We could test for white noise separately for original and absolute values, for example using Ljung-Box tests for both. In this article, we create an omnibus t…
In this paper we introduce a new model named CARMA(p,q)-Hawkes process as the Hawkes model with exponential kernel implies a strictly decreasing behaviour of the autocorrelation function and empirically evidences reject the monotonicity assumption on the autocorrelation function. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process and specifically is able to reproduce more realistic dependence structures. We also study the conditions of stationarity and positivity for the intensity and the strong mixing property for…
There has been an increasing interest from the scientific community in using likelihood-free inference (LFI) to determine which parameters of a given simulator model could best describe a set of experimental data. Despite exciting recent results and a wide range of possible applications, an important bottleneck of LFI when applied to time series data is the necessity of defining a set of summary features, often hand-tailored based on domain knowledge. In this work, we present a data-driven strategy for automatically learning summary features from univariate time series and apply it to signals …
We propose a novel estimator of the autocorrelation function in presence of missing observations. We establish the consistency, the asymptotic normality, and we derive deviation bounds for various classes of weakly dependent stationary time series, including causal or non causal models. In addition, we introduce a modified version periodogram defined from these autocorrelation estimators and derive asymptotic distribution of linear functionals of this estimator.
Estimation of autocorrelations and spectral densities is of fundamental importance in many fields of science, from identifying pulsar signals in astronomy to measuring heart beats in medicine. In circumstances where one is interested in specific autocorrelation functions that do not fit into any simple families of models, such as auto-regressive moving average (ARMA), estimating model parameters is generally approached in one of two ways: by fitting the model autocorrelation function to a non-parameteric autocorrelation estimate via regression analysis or by fitting the model autocorrelation f…
The problem of test of fit for Vector AutoRegressive (VAR) processes with unconditionally heteroscedastic errors is studied. The volatility structure is deterministic but time-varying and allows for changes that are commonly observed in economic or financial multivariate series. Our analysis is based on the residual autocovariances and autocorrelations obtained from Ordinary Least Squares (OLS), Generalized Least Squares (GLS)and Adaptive Least Squares (ALS) estimation of the autoregressive parameters. The ALS approach is the GLS approach adapted to the unknown time-varying volatility that is …
