Nonstationarity In Variance Research Articles

Nonstationary Changes in Annual Rainfall over Indonesia’s Maritime Continent

The investigation into changing rainfall patterns in the Indonesian Maritime Continent (IMC) involved testing for trends, step changes, and variance nonstationarity using the Mann–Kendall, Pettitt, and White tests, respectively. The analysis covered data from 106 meteorological stations over the period from 1981 to 2021, with all tests conducted at annual time scales to understand the evolving precipitation dynamics in the region. According to the findings of this study, the average annual rainfall in IMC is nonstationary. Rainfall has increased dramatically by 12.72 mm with a significant shift point in 1994. However, this characteristic is likely to vary if the analysis period is extended, reduced, or conducted in different time frames. The spatial analysis indicates that 35.8% of meteorological stations observed a notable increase in rainfall, while 28.3% experienced a significant shift, and 16% displayed considerable variation. Consequently, the study only partially identified the nonstationary nature of rainfall in the IMC. Moreover, the research highlights a substantial rise in rainfall in the central to eastern IMC region, whereas the central to western region predominantly shows a decrease. This nonstationary test helps avoid errors in climatic conditions and analytical methods, presenting recommendations for hydrological projects. Consequently, employing this nonstationary test helps prevent inaccuracies in understanding climatic conditions and analytical techniques, thereby offering valuable suggestions for hydrological projects.

Tracing Time-Varying Characteristics of Meteorological Drought through Nonstationary Joint Deficit Index

Abstract Standardized precipitation index (SPI) is one of the frequently used meteorological drought indices. However, the time-varying characteristics observed in the historical precipitation data questions the reliability of SPI and motivated the development of nonstationary SPI. To overcome some of the limitations in the existing nonstationary drought indices, a new framework for drought index is proposed, incorporating the temporal dynamics in the precipitation. The proposed drought index is developed by coupling the joint deficit index with the extended time sliding window–based nonstationary modeling (TSW-NSM). The proposed nonstationary joint deficit index (NJDI) detects the signature of nonstationarity in the distribution parameter and models both long-term (i.e., trend) and short-term (i.e., step-change) temporal dynamics of distribution parameters. The efficacy of NJDI is demonstrated by employing it to identify the meteorological drought-prone areas over India. The changes observed in the distribution parameter of rainfall series reveal an increasing number of dry days in recent decades all over India, except the northeast. Comparison of NJDI and stationary joint deficit index (JDI) reveals that JDI overestimates drought when frequent severe dry events are clustered and underestimates when these events are scattered, which indicates that the traditional index is biased toward the lowest magnitude of precipitation while classifying the drought. Moreover, NJDI could closely capture historical droughts and their spatial variations, thereby reflecting the temporal dynamics of rainfall series and the changes in the pattern of dry events over India. NJDI proves to be a potentially reliable index for drought monitoring in a nonstationary climate. Significance Statement Drought is one of the most severe natural disasters and is expected to intensify under a warming climate. There has been progress on developing newer methodologies to characterize drought severity under the changing climate. However, some limitations remain in capturing the temporal changes of the precipitation, especially the nonstationarity in variance (rapid increases in extreme precipitation versus the average precipitation). We propose an extension to the time sliding window approach to capture the nonstationarity in variance and mean while incorporating short-term (step-change) and long-term (trend) temporal dynamics. We apply the methodology to a subcontinent-sized heterogeneous country of India with gridded rainfall dataset (0.25° × 0.25°, 1901–2013) from the India Meteorological Department. We find that the majority of grids exhibit negative and positive trends in shape and scale parameters, respectively, which ultimately leads to an increase in the number of drier events, whereas a contradictory pattern is exhibited in the northwest Indian region (decrease in the number of dry days). The proposed index gives a potential framework for drought monitoring under the changing climate and can be extended to develop a multivariate nonstationary joint deficit index by incorporating other hydrological variables (e.g., soil moisture, diurnal temperature range, streamflow).

An improved non-stationary geostatistical method for three-dimensional interpolation of Benzo(a)pyrene at a contaminated site

Intense industrial activities and complex hydrogeological conditions at contaminated sites make accurate three-dimensional (3D) mapping challenging. The cause is the non-stationarity in the variance of soil pollutants in geographical space (G-space), making the stationary hypothesis required by the Kriging method unsatisfactory. To handle the variance non-stationarity, a Variance-Octree-Kriging (VOK) method was proposed. VOK is a spatial deformation method that constructs a stationary deformation space (D-space) by stretching and shrinking the G-spaces with low and high spatial correlation, respectively. VOK method consists of 3D stratification in G-space, space scaling and transformation, and ordinary Kriging (OK) in D-space. 3D stratification uses variance octree (VOT) to generate a set of anchor points in the G-space. The spatial scaling and transformation use the virtual force algorithm (VFA) and thin-plate spline to evenly distribute the anchor points and obtain the D-space, where the OK is implemented. The method was applied to predict the distribution of soil Benzo(a)pyrene (BaP) at a contaminated site in North China Plain. The results show that the interpolation accuracy of VOK was 9% higher than that of OK. The VOK method also changed the spatial structure from anisotropic to isotropic. The root mean squared error (RMSE) of fill, silt and clay layers decreased by 4.67%, 11.39%, and 20.46%, respectively. This method is applicable to the 3D interpolation of pollutants at contaminated sites, with the advantages of high interpolation accuracy and the ability to handle the non-stationarity in variance.

Modelling Economic Time Series in the Presence of Variance Non-Stationarity: A Practical Approach

Although non-stationarity in the level of a time series is always tested (and there is a variety of tests for this purpose), non-stationarity in the variance is sometimes neglected in applied research. In this work, the consequences of neglecting variance non-stationarity in economic time series, and the conceptual difference between variance non-stationarity and conditional variance are discussed. An ad hoc method for testing and correcting for variance non-stationarity is suggested. It is shown that the presence of variance non-stationarity leads to misspecified univariate ARIMA models and correcting for it, the number of model parameters is vastly reduced. The implications of the tests for the hypothesis of weak form market efficiency (WFME) are discussed. More specifically it is argued that the usual autocorrelation tests are inappropriate when based on the differences of asset prices. Finally, it is shown how the analysis of outliers is affected by the presence of variance non-stationarity.

Does non-stationarity of extreme precipitation exist in the Poyang Lake Basin of China?

Study regionPoyang Lake Basin, China. Study focusThis study aimed to investigate whether there are non-stationary characteristics of extreme precipitation in the Poyang Lake Basin (PLB) of China, and the trends of non-stationary characteristics from 1959 to 2019. The spatio-temporal variations of extreme precipitation were analysed from three fundamental aspects: duration, frequency, and intensity, based on the prewhitening Mann-Kendall (PWMK) test. Non-stationary variations and the risk of extreme precipitation were investigated based on the generalized additive models for location, scale, and shape (GAMLSS). New hydrological insights for the region(1) the intensity and frequency of extreme precipitation increased significantly, whereas there was a significant decrease in extreme precipitation duration in the PLB. (2) The duration of extreme precipitation showed significant non-stationary characteristics in the western PLB. At the Nanchang site, 83.3 % of the extreme precipitation intensity indices showed non-stationary characteristics. The RX1day (maximum 1-day precipitation amount) and RX5day (maximum 5-day precipitation amount) increased significantly for different return periods under non-stationary conditions in the northwestern PLB. (3) The risk of extreme precipitation can be captured using the GAMLSS. The stationary method underestimated the extreme precipitation intensity (e.g., RX1day) compared to the GAMLSS for longer return periods in the PLB. More attention should be paid to the increase and fluctuation of the return period of extreme precipitation caused by the mean non-stationarity and variance non-stationarity.

Semiparametric outlier detection in nonstationary times series: Case study for atmospheric pollution in Brno, Czech Republic

Large environmental datasets usually include outliers which can have significant effects on further analysis and modelling. There exist various outlier detection methods that depend on the distribution of the analysed variable. However quite often the distribution of environmental variables can not be estimated. This paper presents an approach for identification of outliers in environmental time series which does not impose restrictions on the distribution of observed variables. The suggested algorithm combines kernel smoothing and extreme value estimation techniques for stochastic processes within considerations of nonstationary expected value of the process. The nonstationarity in variance is evaded by change point analysis which precedes the proposed algorithm. Possible outliers are identified as observations with rare occurrence and, in correspondence to extreme value methodology, the confidence limits for high values of observed variables are constructed. The proposed methodology can be especially convenient for cases where validation of the data has to be carried out manually, since it significantly reduces the number of implausible observations. For a case study, the technique is applied for outlier detection in time series of hourly PM10 concentrations in Brno, Czech Republic. The methodology is derived on solid theoretical results and seems to perform well for the series of PM10. However its flexibility makes it generally applicable not only to series of atmospheric pollutants. On the other hand, the choice of return level turns out to be crucial in sensitivity to the outliers. This issue should be left to the practitioners to decide with respect to specific application conditions.

Design impact and significance of non-stationarity of variance in extreme rainfall

Abstract. Stationarity in hydro-meteorological records is often investigated through an assessment of the mean value of the tested parameter. This is arguably insufficient for capturing fully the non-stationarity signal, and parameter variance is an equally important indicator. This study applied the Mann-Kendall linear and Mann-Whitney-Wilcoxon step change trend detection techniques to investigate the changes in the mean and variance of annual maximum daily rainfalls at eight stations in Dublin, Ireland, where long and high quality daily rainfall records were available. The eight stations are located in a geographically similar and spatially compact region (< 950 km2) and their rainfalls were shown to be well correlated. Results indicate that while significant positive step changes were observed in mean annual maximum daily rainfalls (1961 and 1997) at only two of the eight stations, a significant and consistent shift in the variance was observed at all eight stations during the 1980's. This period saw a widely noted positive shift in the winter North Atlantic Oscillation that greatly influences rainfall patterns in Northern Europe. Design estimates were obtained from a frequency analysis of annual maximum daily rainfalls (AM series) using the Generalised Extreme Value distribution, identified through application of the Modified Anderson Darling Goodness of Fit criterion. To evaluate the impact of the observed non-stationarity in variance on rainfall design estimates, two sets of depth-frequency relationships at each station for return periods from 5 to 100-years were constructed. The first was constructed with bootstrapped confidence intervals based on the full AM series assuming stationarity and the second was based on a partial AM series commencing in the year that followed the observed shift in variance. Confidence intervals distinguish climate signals from natural variability. Increases in design daily rainfall estimates obtained from the depth-frequency relationship developed from the truncated AM series, as opposed to those using the full series, ranged from 5 to 16% for the 5-year event and from 20 to 41% for the 100-year event. Results indicate that the observed trends exceed the envelopes of natural climate variability and suggest that the non-stationarity in variance is associated with a climate change signal. Results also illustrate the importance of considering trends in higher order moments (e.g. variance) of hydro-meteorological variables in assessing non-stationarity influences.

Maximization of Nonlinear Autocorrelation for Blind Source Separation of Non-stationary Complex Signals

Blind source separation of complex-valued signals has been a vital issue especially in the field of digital communication signal processing. This paper proposes a novel method based on nonlinear autocorrelation to solve the problem. Relying on the temporal structure with nonlinear autocorrelation of the signals, the method has a potential capability of extracting non-stationary complex sources with Gaussian or non-Gaussian distribution. Most traditional methods would fail in separating this kind of sources. We also analyze the stability conditions of the method in theory. Numerical simulations on artificial complex Gaussian data and orthogonal frequency division multiplexing sources corroborate the validity and efficiency of the proposed method. Moreover, with respect to classical methods, including cumulant-based approach using the non-stationarity of variance and complexity pursuit, our method offers equally good results with lower computational cost and better robustness. Finally, experiments for the separation of real communication signals illustrate that our method has good prospects in real-world applications.

Testing Second-Order Dynamics for Autoregressive Processes in Presence of Time-Varying Variance

This article considers the volatility modeling for autoregressive univariate time series. A benchmark approach is the stationary autoregressive conditional heteroscedasticity (ARCH) model of Engle. Motivated by real data evidence, processes with nonconstant unconditional variance and ARCH effects have been recently introduced. We take into account this type of nonstationarity in variance and propose simple testing procedures for ARCH effects. Adaptive McLeod and Li’s portmanteau and ARCH-LM tests for checking the presence of such second-order dynamics are provided. The standard versions of these tests, commonly used by practitioners, suppose constant unconditional variance. The failure of these standard tests with time-varying unconditional variance is highlighted. The theoretical results are illustrated by means of simulated and real data. Supplementary materials for this article are available online.

A comparison of time series similarity measures for classification and change detection of ecosystem dynamics

Time series of remote sensing imagery or derived vegetation indices and biophysical products have been shown particularly useful to characterize land ecosystem dynamics. Various methods have been developed based on temporal trajectory analysis to characterize, classify and detect changes in ecosystem dynamics. Although time series similarity measures play an important role in these methods, a quantitative comparison of the similarity measures is lacking. The objective of this study was to provide an overview and quantitative comparison of the similarity measures in function of varying time series and ecosystem characteristics, such as amplitude, timing and noise effects. For this purpose, the performance was evaluated for the commonly used similarity measures (D), ranging from Manhattan ( D Man ), Euclidean ( D E ) and Mahalanobis ( D Mah ) distance measures, to correlation ( D CC ), Principal Component Analysis (PCA; D PCA ) and Fourier based ( D FFT , D ξ , D Fk ) similarities. The quantitative comparison consists of a series of Monte-Carlo simulations based on subsets of global MODIS Normalized Difference Vegetation index (NDVI) and Enhanced Vegetation Index (EVI) and Leaf Area Index (LAI) data. Results of the simulations reveal four main groups of time series similarity measures with different sensitivities: (i) D Man , D E , D PCA , D Fk quantify the difference in time series values, (ii) D Mah accounts for temporal correlation and non-stationarity of variance, (iii) D CC measures the temporal correlation, and (iv) the Fourier based D FFT and D ξ show their specific sensitivity based on the selected Fourier components. The difference measures show relatively the highest sensitivity to amplitude effects, whereas the correlation based measures are highly sensitive to variations in timing and noise. The Fourier based measures, finally, depend highly on the signal to noise ratio and the balance between amplitude and phase dominance. The heterogeneity in sensitivity of each D stresses the importance of (i) understanding the time series characteristics before applying any classification of change detection approach and (ii) defining the variability one wants to identify/account for. This requires an understanding of the ecosystem dynamics and time series characteristics related to the baseline, amplitude, timing, noise and variability of the ecosystem time series. This is also illustrated in the quantitative comparison, where the different sensitivities of D for the NDVI, EVI, and LAI data relate specifically to the temporal characteristics of each data set. Additionally, the effect of noise and intra- and interclass variability is demonstrated in a case study based on land cover classification.

Testing for causality in variance under nonstationarity in variance

Van Dijk, Osborn and Sensier [van Dijk, D., Osborn, D.R., Sensier, M., 2005, Testing for causality in variance in the presence of breaks. Economics Letters 89, 193–199.] have recently shown, through Monte Carlo simulation, that causality-in-variance tests may suffer severe finite sample size distortions in the presence of neglected structural breaks. In this paper we provide the theoretical foundation to characterize such departures. The asymptotic theory allows to depict a more complete and general analysis.

The importance of variance stationarity in economic time series modelling. A practical approach

Although non-stationarity in the level of a time series is always tested (and there is a variety of tests for this purpose), non-stationarity in the variance is sometimes neglected in applied research. In this work, the consequences of neglecting variance non-stationarity in financial time series, and the conceptual difference between variance non-stationarity and conditional variance are discussed. An ad hoc method for testing and correcting for variance non-stationarity is suggested. It is shown that the presence of variance non-stationarity leads to misspecified univariate ARIMA models and correcting for it, the number of model parameters is vastly reduced. Implications for the tests of the hypothesis of weak form market efficiency (WFME) are discussed. More specifically it is argued that the usual autocorrelation tests are inappropriate when based on the differences of asset prices. Finally, it is shown how the analysis of outliers is affected by the presence of variance non-stationarity.

Blind source separation by nonstationarity of variance: a cumulant-based approach

Blind separation of source signals usually relies either on the nonGaussianity of the signals or on their linear autocorrelations. A third approach was introduced by Matsuoka et al. (1995), who showed that source separation can be performed by using the nonstationarity of the signals, in particular the nonstationarity of their variances. In this paper, we show how to interpret the nonstationarity due to a smoothly changing variance in terms of higher order cross-cumulants. This is based on the time-correlation of the squares (energies) of the signals and leads to a simple optimization criterion. Using this criterion, we construct a fixed-point algorithm that is computationally very efficient.

On regression-based tests for persistence in logarithmic volatility models

Building on the work of Pantula (1986), this paper discusses how the hypothesis of conditional variance nonstationarity in the logarithmic family of generalized autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility processes may be tested using regression-based tests. The latter are easy to implement, have well-defined large-sample distributions, and are less sensitive to structural changes than tests based on the quasimaximum likelihood estimator.

Space-time analysis of water status in a volcanic Vesuvian soil

Many studies have shown that the observations of soil hydraulic properties should not be considered purely random, given that they possess a structure which may be described by means of stochastic processes. The techniques used for analyzing such a structure have essentially been based either on the theory of regionalized variables or, to a lesser extent, on the analysis of historical series. This work attempts to combine the two approaches mentioned above by means of a study of parameters h and θ, which characterize soil water status, in the space-time domain. The data of the analyses were recorded in the open field during a controlled drainage process, evaporation being prevented, along a 50 m transect in a volcanic Vesuvian soil. The statistical models used may be traced back to a generalization of the classical ARMA models, commonly used in time series analyses, so as to take the following into account; (a) spatial anisotropy in the phenomena studied; (b) simultaneous physical relations between h and θ; (c) space-time interrelations which characterize these parameters. In particular, in the work h and θ are shown to present: (i) marked non-stationarity on mean attributable to the time dimension; (ii) a strong non-stationarity in variance only for parameter h and in the time dimension; (iii) a different space-time structure which cannot be influenced by possible crops; (iv) strong anisotropy just for the space dimension; (v) simultaneous linear relations confined to lesser time values.

Predicting currency return volatility

Standard deviations and elasticity of variance coefficients implied in currency option prices are employed to predict subsequent currency return volatility. Despite documented variance non-stationarity, strong predictive relationships are found. The predictive accuracies of standard deviations and elasticity coefficients are statistically indistinguishable. Finally, predictive accuracy is invariant to the weighting techniques employed.