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Using notions from linear algebraic graph theory, this article provides a unified perspective on some autocorrelation measures in different fields. They are as follows: (a) Orcutt’s first serial correlation coefficient, (b) Anderson’s first circular serial correlation coefficient, (c) Moran’s , and (d) Moran’s . The first two are autocorrelation measures for one-dimensional data equally spaced, such as time series data, and the last two are for spatial data. We prove that (a)–(c) are a kind of (d). For example, we show that (d) such that its spatial weight matrix equals the adjacency matrix of a path graph is the same as (a). The perspective is beneficial because studying the properties of (d) leads to studying the properties of (a)–(c) at the same time. For example, the bounds of (a)–(c) can be found from the bounds of (d).

The dependence between pairs of time series is commonly quantified by Pearson's correlation. However, if the time series are themselves dependent (i.e. exhibit temporal autocorrelation), the effective degrees of freedom (EDF) are reduced, the standard error of the sample correlation coefficient is biased, and Fisher's transformation fails to stabilise the variance. Since fMRI time series are notoriously autocorrelated, the issue of biased standard errors – before or after Fisher's transformation – becomes vital in individual-level analysis of resting-state functional connectivity (rsFC) and must be addressed anytime a standardised Z-score is computed. We find that the severity of autocorrelation is highly dependent on spatial characteristics of brain regions, such as the size of regions of interest and the spatial location of those regions. We further show that the available EDF estimators make restrictive assumptions that are not supported by the data, resulting in biased rsFC inferences that lead to distorted topological descriptions of the connectome on the individual level. We propose a practical “xDF” method that accounts not only for distinct autocorrelation in each time series, but instantaneous and lagged cross-correlation. We find the xDF correction varies substantially over node pairs, indicating the limitations of global EDF corrections used previously. In addition to extensive synthetic and real data validations, we investigate the impact of this correction on rsFC measures in data from the Young Adult Human Connectome Project, showing that accounting for autocorrelation dramatically changes fundamental graph theoretical measures relative to no correction. •Autocorrelation is a problem for sample correlation, breaking the variance-stabilising property of Fisher's transformation.•We show that fMRI autocorrelation varies systematically with region of interest size, and is heterogeneous over subjects.•Existing adjustment methods are themselves biased when true correlation is non-zero due to a confounding effect.•Our “xDF” method provides accurate Z-scores based on either of Pearson's or Fisher's transformed correlations.•Resting state fMRI autocorrelation considerably alters the graph theoretical description of human connectome.

This paper considers generalized least squares (GLS) estimation for linear panel data models. By estimating the large error covariance matrix consistently, the proposed feasible GLS estimator is more efficient than the ordinary least squares in the presence of heteroskedasticity, serial and cross-sectional correlations. The covariance matrix used for the feasible GLS is estimated via the banding and thresholding method. We establish the limiting distribution of the proposed estimator. A Monte Carlo study is considered. The proposed method is applied to an empirical application.

Commonly used tests to assess evidence for the absence of autocorrelation in a univariate time series or serial cross-correlation between time series rely on procedures whose validity holds for i.i.d. data. When the series are not i.i.d., the size of correlogram and cumulative Ljung–Box tests can be significantly distorted. This paper adapts standard correlogram and portmanteau tests to accommodate hidden dependence and nonstationarities involving heteroskedasticity, thereby uncoupling these tests from limiting assumptions that reduce their applicability in empirical work. To enhance the Ljung–Box test for non-i.i.d. data, a new cumulative test is introduced. Asymptotic size of these tests is unaffected by hidden dependence and heteroskedasticity in the series. Related extensions are provided for testing cross-correlation at various lags in bivariate time series. Tests for the i.i.d. property of a time series are also developed. An extensive Monte Carlo study confirms good performance in both size and power for the new tests. Applications to real data reveal that standard tests frequently produce spurious evidence of serial correlation.

The total quantity and energy delivered through a gas grid is calculated using simple formulæ that sum the increments measured at regular time intervals. These calculations are described in international standards (e.g., ISO 15112 and EN 1776) and guidelines (e.g., OIML R140). These guidelines recommend that the associated measurement uncertainty is evaluated assuming the measurement results to be mutually independent. This assumption leads to the underestimation of the measurement uncertainty. To address the growing concern among transmission and distribution system operators, the underlying assumptions of these uncertainty evaluations are revisited and reworked to be more adequate. The dependence of measurement results coming from, e.g., the same flow meter and gas chromatograph will be assessed for correlations, as well as other effects, such as the effect of the chosen mathematical approximation of the totalisation integral and fluctuations in the flow rate and gas quality. In this paper, an outline is given for improvements that can be implemented in the measurement models to render them more responsive to the error structure of the measurement data, temporal effects in these data, and the fluctuations in the gas quality and gas quantity. By impact assessment using a simple scenario involving the injection of (renewable) hydrogen into a natural gas grid, it is shown that these improvements lead to a substantive difference. This preliminary work demonstrates that correlations occur both in the instrumental measurement uncertainty and due to temporal effects in the gas grid. To obtain a fit-for-purpose uncertainty budget for custody transfer and grid balancing, it is key to enhance the current models and standards accordingly.

Functional magnetic resonance imaging (fMRI) functional connectivity between brain regions is often computed using parcellations defined by functional or structural atlases. Typically, some kind of voxel averaging is performed to obtain a single temporal correlation estimate per region pair. However, several estimators can be defined for this task, with various assumptions and degrees of robustness to local noise, global noise, and region size. In this paper, we systematically present and study the properties of 9 different functional connectivity estimators taking into account the spatial structure of fMRI data, based on a simple fMRI data spatial model. These include 3 existing estimators and 6 novel estimators. We demonstrate the empirical properties of the estimators using synthetic, animal, and human data, in terms of graph structure, repeatability and reproducibility, discriminability, dependence on region size, as well as local and global noise robustness. We prove analytically the link between regional intra-correlation and inter-region correlation, and show that the choice of estimator has a strong influence on inter-correlation values. Some estimators, including the commonly used correlation of averages (ca), are positively biased, and have more dependence to region size and intra-correlation than robust alternatives, resulting in spatially-dependent bias. We define the new local correlation of averages estimator with better theoretical guarantees, lower bias, significantly lower dependence on region size (Spearman correlation 0.40 vs 0.55, paired t-test T=27.2, p=1.1e−47), at negligible cost to discriminative power, compared to the ca estimator. The difference in connectivity pattern between the estimators is not distributed uniformly throughout the brain, but rather shows a clear ventral-dorsal gradient, suggesting that region size and intra-correlation plays an important role in shaping functional networks defined using the ca estimator, and leading to non-trivial differences in their connectivity structure. We provide an open source R package and equivalent Python implementation to facilitate the use of the new estimators, together with preprocessed rat time-series. •Most atlas-based functional connectivity estimates use the correlation of averages.•Six novel estimators are derived using a spatio-temporal signal and noise model•Synthetic, rat and Human Connectome Project data are used to validate estimators•Lower dependency on region size, intra-correlation shown analytically and empirically•Improved reproducibility on graph metrics using freely available codes, datasets

It is well-known that successive residuals may be correlated with each other, and serial correlation usually result in an inefficient estimate in time series analysis. In this paper, we investigate the serial correlation test of parametric regression models where the response is missing at random. Three test statistics based on the empirical likelihood method are proposed to test serial correlation. It is proved that three proposed empirical likelihood ratios admit limiting chi-square distribution under the null hypothesis of no serial correlation. The proposed test statistics are simple to calculate and convenient to use, and they can test not only zero first-order serial correlation, but also the higher-order serial correlation. A simulation study and a real data analysis are conducted to evaluate the finite sample performance of our proposed test methods.

In this paper, we first study the ARMA model and its derivative models, smooth the non-stationary time series by order difference, and estimate the coefficients of the ARMA model by the method of moment estimation. Then, hypothesis testing is conducted based on serial correlation of the relationship between corporate business administration and organizational effectiveness. In the process of hypothesis testing, the empirical likelihood method is used to test the serial correlation of the residual terms under weak hypotheses, which is then extended to apply to the test case where the residual terms contain finite and infinite variances. Finally, based on the model, we assessed the level of business administration and organizational effectiveness of Enterprise H, explored the correlation between the two, and analyzed the linkage effect between business administration and organizational effectiveness. The results show that there is a significant positive correlation between the dimensions of business administration and organizational effectiveness of enterprises, r=0.754-0.864, P<0.01. The correlation coefficients between the two fluctuate between 0.4 and 0.8 for a long period of time, with significant time-varying characteristics.

Environmental epidemiological studies of the health effects of air pollution frequently utilize the generalized additive model (GAM) as the standard statistical methodology, considering the ambient air pollutants as explanatory covariates. Although exposure to air pollutants is multi-dimensional, the majority of these studies consider only a single pollutant as a covariate in the GAM model. This model restriction may be because the pollutant variables do not only have serial dependence but also interdependence between themselves. In an attempt to convey a more realistic model, we propose here the hybrid generalized additive model-principal component analysis-vector auto-regressive (GAM-PCA-VAR) model, which is a combination of PCA and GAMs along with a VAR process. The PCA is used to eliminate the multicollinearity between the pollutants whereas the VAR model is used to handle the serial correlation of the data to produce white noise processes as covariates in the GAM. Some theoretical and simulation results of the methodology proposed are discussed, with special attention to the effect of time correlation of the covariates on the PCA and, consequently, on the estimates of the parameters in the GAM and on the relative risk, which is a commonly used statistical quantity to measure the effect of the covariates, especially the pollutants, on population health. As a main motivation to the methodology, a real data set is analysed with the aim of quantifying the association between respiratory disease and air pollution concentrations, especially particulate matter PM₁₀, sulphur dioxide, nitrogen dioxide, carbon monoxide and ozone. The empirical results show that the GAM-PCA-VAR model can remove the auto-correlations from the principal components. In addition, this method produces estimates of the relative risk, for each pollutant, which are not affected by the serial correlation in the data. This, in general, leads to more pronounced values of the estimated risk compared with the standard GAM model, indicating, for this study, an increase of almost 5.4% in the risk of PM₁₀, which is one of the most important pollutants which is usually associated with adverse effects on human health.

Autoregressive exogenous, hereafter ARX, models are widely adopted in time series-related domains as they can be regarded as the combination of an autoregressive process and a predictive regression. Within a more complex structure, extant diagnostic checking methods face difficulties in remaining validity in many conditions existing in real applications, such as heteroscedasticity and error correlations exhibited between the ARX model itself and its exogenous processes. For these reasons, we propose a new serial correlation test method based on the profile empirical likelihood. Simulation results, as well as two real data examples, show that our method has a good performance in all mentioned conditions.