Trends In Time Series Data Research Articles

Accurate Changing Point Detection for l1 Mean Filtering

It is often desirable to find the underlying trends in time series data. This is a well known signal processing problem that has many applications in areas such as financial data analysis, climatology, biological and medical sciences. Mean filtering finds a piece-wise constant trend in the data while trend filtering finds a piece-wise linear trend. When the signal is noisy, the main difficulty is finding the changing points in the data that mark the transition points when the mean or the trend changes. Previously proposed methods based on ${\ell _1}$ filtering suffer from the occurrence of false changing points in the estimate. This is known as the stair-case effect. The main contribution in this paper is incorporating a technique to remove these false changing points to a fast mean filtering algorithm, referred to as the taut-string method, resulting in an efficient procedure with accurate change point detection and thus the removal of the stair-case effect.

The distribution of Spearman’s rho trend statistic for persistent hydrologic data

Spearman’s rho, a distribution-free statistic, has been suggested in the literature for testing the significance of trend in time series data. Although the use of the test based on Spearman’s rho (also known as the Daniels test) is less widespread than that based on Kendall’s tau (the Mann-Kendall test), the two tests have been shown in the literature to be equivalent for time series with independent observations. The distribution of the Mann-Kendall trend statistic for persistent data has been previously addressed in the literature. In this paper, the distribution of Spearman’s rho as a trend test statistic for persistent data is studied. Following the same procedures used for Kendall’s tau in earlier work, an exact expression for the variance of Spearman’s rho for persistent data with multivariate Gaussian dependence is derived, and a method for calculating the exact full distribution of rho for small sample sizes is also outlined. Approximations for moderate and large sample sizes are also discussed. A case study of testing the significance of trends in a group of world river flow station data using both Kendall’s tau and Spearman’s rho is presented. Both the theoretical results and those of the case study confirm the equivalence of trend testing based on Spearman’s rho and Kendall’s tau for persistent hydrologic data.Editor Z. W. Kundzewicz; Associate editor S. Grimaldi

A Robust Test for OLS Trends in Daily Temperature Data

Trends in time series data estimated with OLS linear regression may be tested with a robust procedure that is less sensitive to influential observations and violations of regression assumptions. The test consists of comparing the average age of data tritiles. If the higher tritiles are newer a rising trend is indicated for the sample period. If the higher tritiles are older a declining trend is indicated. If neither of these conditions is met, no sustained trend in the sample period may be inferred from the data. Daily temperature data from selected USHCN and USCRN stations are used to demonstrate the utility of the proposed methodology.

5. Detection of Regularity Violations of Cyclic Processes in a Temperature Monitoring System Using Patterns Form

Periodicity mining is used for predicting trends in time series data. Discovering the rate at which the time series is periodic has always been an obstacle for fully automated periodicity mining. In this paper, a method for detecting the weather temperature series periodicity is proposed. The proposed method, based on DFT, effectively discovered the series periodicity and determined the periodic patterns and their repetition frequencies. Then, the series has been divided into equal time slots based on the pattern repetition frequency. A reference series has been constructed as repetitions for a template pattern, which was constructed from the patterns averages of the original temperature series. The reference series is very useful in temperature series analysis, as the patterns deviations, the future patterns predictions, and the anomalies detections. Experimental results show that the proposed method accurately discovers periodicity rates and periodic patterns.

Continuous treatments of estrogens through polymerization and regeneration of electrolytic cells

This study proposes a novel electrolytic method for simultaneous removal of trace estrogens and regeneration of electrolytic cells for long-term wastewater treatment. Continuous treatments of estrogens estrone (E1), 17β-estradiol (E2) and 17α-ethinyl estradiol (EE2) were theoretically and experimentally studied using an electrolytic reactor equipped with a multi-packed granular glassy carbon electrode reactor. Experimental results demonstrated that E1, E2 and EE2 were effectively removed through electro-polymerization on the granular glassy carbon (and Pt/Ti) anode counter. Polymer formed during continuous treatment was quickly decomposed and electrodes were regenerated completely by OH radicals produced through the reduction of ozone. Calculated overall energy consumptions were less than 10Wh/m3, demonstrating extremely low energy consumptions. In addition, a mathematical model developed based on the limiting mass transfer rate and post-regeneration could represent general trends in time series data observed in experiments.

Large-Scale Present-Day Plate Boundary Deformations in the Eastern Hemisphere Determined from VLBI Data: Implications for Plate Tectonics and Indian Ocean Growth

The dynamics of the planet Earth are manifestations of diverse plate tectonic processes which have been occurring since the Archean period of the Earth’s evolution and continue to deform the plate boundaries. Very long baseline interferometry (VLBI) is an efficient space geodetic method that enables precise measurement of plate motion and associated deformations. We analyze here VLBI measurements made during a period of approximately three decades at five locations on the Eastern hemisphere of the globe, which are geographically distributed over five continents (plates) around the Indian Ocean. Computed rate of change of baseline length show the deformation pattern and its rate at the boundaries between the major tectonic plates constituting the Eastern hemisphere of the Earth. The African (Nubian) and Antarctic plates are moving apart at 13.5 mm/year, which is mostly attributed to spreading of the South West Indian Ridge. Similarly, spreading of 59.0 mm/year is observed for the South East Indian Ridge that separates the Antarctic and Australian plates. Shortening at the rate of 3.9 mm/year is estimated across the subduction boundary between Africa (Nubia) and Eurasia. Similar convergence is evident between the Australian and Sunda blocks (of the Eurasian plate). The associated deformation of −54.8 mm/year seems to be chiefly accommodated along the Banda arc system, where the Australian plate is subducting under the Sunda block. VLBI sites within the Eurasian plate, Wettzell in Germany, and Seshan on the South China block, are moving apart at 3.6 mm/year. This relative motion between locations on the same plate is interpreted as a result of the deformation process along a large strike–slip fault, which is identified as the Western boundary of the South China block. Expansion of the Indian Ocean, at +91.5 m2/year, is also estimated from the rate of deformation estimated within the five baselines studied here. From the Hurst exponent values, which are indicators of the future trend of time series data, we predict deceleration of the various tectonic processes occurring at present.

Procedures for outlier detection in circular time series models

It is known that the occurrence of outliers in linear or non-linear time series models may have adverse effects on the modelling and statistical inference of the data. Consequently, extensive research has been conducted on developing outlier detection procedures so that outliers may be properly managed. However, no work has been done on the problem of outliers in circular time series data. This problem is the focus of this paper. The main objective is to develop novel numerical and graphical procedures for detecting these outliers in circular time series data.A number of circular time series models have been proposed including the circular autoregressive model. We extend the iterative outlier detection procedure which has been successfully used in linear time series models to the circular autoregressive model. The proposed procedure shows a good performance when investigated via simulation for the circular autoregressive model of order one. At the same time, several statistical techniques have been used to detect the change of preferred trend in time series data using SLIME and CUSUM plots. While the methods fail to indicate directly the outliers in circular time series data, we use the ideas employed to develop three novel graphical procedures for identifying the outliers. For illustration, we apply the procedures to a particular set of wind direction data. An agreement between the results of the graphical and iterative detection procedures is observed. These procedures could be very useful in improving the modelling and inferential processes for circular time series data.

Trend analysis of rotor-to-stator impact-rub based on smooth support vector regression

Traditional methods of mathematical modelling usually get complicated non-linear models for rub-impact fault. Besides, time-frequency domain analysis is very difficult to solve the problem because of its own limitation. Based on analysis need of degree and trend of rub-impact fault, support vector regression (SVR) arithmetic is imported, which adopts method of time series analysis. Smooth support vector regression (SSVR) using smoothing method in SVR suggests a best tradeoff between complexity of model and learning ability to establish linear model which is able to reflect the implicit mechanism of series data. Model of Jeffcott rotor was established to simulate actual rotor rub-impact fault, and SSVR was applied to analyse trend of time series data of rub-impact fault. The results indicate that SSVR is obviously superior to neural network, and is effective to analyse trend of rotor-to-stator impact-rub fault, which has guiding significance for using and maintenance of rotating machine.

A statistical approach to bioclimatic trend detection in the airborne pollen records of Catalonia (NE Spain)

Airborne pollen records are a suitable indicator for the study of climate change. The present work focuses on the role of annual pollen indices for the detection of bioclimatic trends through the analysis of the aerobiological spectra of 11 taxa of great biogeographical relevance in Catalonia over an 18-year period (1994-2011), by means of different parametric and non-parametric statistical methods. Among others, two non-parametric rank-based statistical tests were performed for detecting monotonic trends in time series data of the selected airborne pollen types and we have observed that they have similar power in detecting trends. Except for those cases in which the pollen data can be well-modeled by a normal distribution, it is better to apply non-parametric statistical methods to aerobiological studies. Our results provide a reliable representation of the pollen trends in the region and suggest that greater pollen quantities are being liberated to the atmosphere in the last years, specially by Mediterranean taxa such as Pinus, Total Quercus and Evergreen Quercus, although the trends may differ geographically. Longer aerobiological monitoring periods are required to corroborate these results and survey the increasing levels of certain pollen types that could exert an impact in terms of public health.

$\ell_1$ Trend Filtering

The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick-Prescott (H-P) filtering, a widely used method for trend estimation. The proposed $\ell_1$ trend filtering method substitutes a sum of absolute values (i.e., $\ell_1$ norm) for the sum of squares used in H-P filtering to penalize variations in the estimated trend. The $\ell_1$ trend filtering method produces trend estimates that are piecewise linear, and therefore it is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interior-point methods, $\ell_1$ trend filtering can be carried out with not much more effort than H-P filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties and give some illustrative examples. We show how the method is related to $\ell_1$ regularization-based methods in sparse signal recovery and feature selection, and we list some extensions of the basic method.

De-Trending Time Series Data for Variability Surveys

AbstractWe present an algorithm for the removal of trends in time series data. The trends could be caused by various systematic and random noise sources such as cloud passages, change of airmass or CCD noise. In order to determine the trends, we select template stars based on a hierarchical clustering algorithm. The hierarchy tree is constructed using the similarity matrix of light curves of stars whose elements are the Pearson correlation values. A new bottom-up merging algorithm is developed to extract clusters of template stars that are highly correlated among themselves, and may thus be used to identify the trends. We then use the multiple linear regression method to de-trend all individual light curves based on these determined trends. Experimental results with simulated light curves which contain artificial trends and events are presented. We also applied our algorithm to TAOS (Taiwan-American Occultation Survey) wide field data observed with a 0.5m f/1.9 telescope equipped with 2k by 2k CCD. With our approach, we successfully removed trends and increased signal to noise in TAOS light curves.

Trend Time–Series Modeling and Forecasting With Neural Networks

Despite its great importance, there has been no general consensus on how to model the trends in time-series data. Compared to traditional approaches, neural networks (NNs) have shown some promise in time-series forecasting. This paper investigates how to best model trend time series using NNs. Four different strategies (raw data, raw data with time index, detrending, and differencing) are used to model various trend patterns (linear, nonlinear, deterministic, stochastic, and breaking trend). We find that with NNs differencing often gives meritorious results regardless of the underlying data generating processes (DGPs). This finding is also confirmed by the real gross national product (GNP) series.

Fast Normalization-Transformed Subsequence Matching in Time-Series Databases

Normalization transform is known to be very useful for finding the overall trend of time-series data since it enables finding sequences with similar fluctuation patterns. Previous subsequence matching methods with normalization transform, however, would incur index overhead both in storage space and in update maintenance since they should build multiple indexes for supporting query sequences of arbitrary length. To solve this problem, we adopt a single-index approach in the normalization-transformed subsequence matching that supports query sequences of arbitrary length. For the single-index approach, we first provide the notion of inclusion-normalization transform by generalizing the original definition of normalization transform. To normalize a window, the inclusion-normalization transform uses the mean and the standard deviation of a subsequence that includes the window while the original transform uses those of the window itself. Next, we formally prove the correctness of the proposed normalization-transformed subsequence matching method that uses the inclusion-normalization transform. We then propose subsequence matching and index-building algorithms to implement the proposed method. Experimental results for real stock data show that our method improves performance by up to 2.5 ~ 2.8 times compared with the previous method.

UNIT ROOT AND COINTEGRATION TESTING: GUEST EDITORS' INTRODUCTION

By pointing out the spurious regression problem, Granger and Newbold (1974) have shown the importance of stochastic trends in time series data in the context of linear regression models. At the time, removing trends by differencing was already common practice in univariate time series modeling as part of the Box–Jenkins approach (Box and Jenkins, 1976). These new developments, however, emphasized the importance of autoregressive (AR) unit roots and motivated the development of statistical procedures for their detection. Dickey and Fuller (1979) and Fuller (1976) were pioneers in developing tests for unit roots that became widely used. The foundation of asymptotic theory for regressions involving stochastic trends was led by Phillips (1986, 1987) with the introduction of the functional limit theory, weak convergence methods, convergence to stochastic integrals, nonparametric unit root testing, and continuous record asymptotics. Phillips and Durlauf (1986) extended some of these results to the multivariate setting by presenting the multivariate invariance principles and the asymptotic theory of multivariate nonstationary and cointegrating regressions. These contributions provided the asymptotic tools that have served as the basis for most of the limit results derived in the context of unit root nonstationarity, and they have stimulated extensive subsequent research.We are grateful to Peter Phillips for proposing a special issue of Econometric Theory for papers from our conference Unit Root and Cointegration Testing. We thank all participants of the conference who contributed their papers to this special issue. We are also very grateful to those colleagues who agreed to serve as referees for the papers. They not only provided generous help through comments but also respected our rather tight deadlines. We are grateful to all those who helped us to complete the special issue unusually fast. To protect their identities, we do not list them here. They will be included in the general list of referees for Econometric Theory. Our conference would not have been possible without the generous financial support of a number of sponsors. We are grateful to the Bank of Portugal, to the Portuguese Science Foundation (FCT), to the Luso-American Foundation for Development (FLAD), and to the Journal of Applied Econometrics for their financial support.

Periodicity detection in time series databases

Periodicity mining is used for predicting trends in time series data. Discovering the rate at which the time series is periodic has always been an obstacle for fully automated periodicity mining. Existing periodicity mining algorithms assume that the periodicity, rate (or simply the period) is user-specified. This assumption is a considerable limitation, especially in time series data where the period is not known a priori. In this paper, we address the problem of detecting the periodicity rate of a time series database. Two types of periodicities are defined, and a scalable, computationally efficient algorithm is proposed for each type. The algorithms perform in O(n log n) time for a time series of length n. Moreover, the proposed algorithms are extended in order to discover the periodic patterns of unknown periods at the same time without affecting the time complexity. Experimental results show that the proposed algorithms are highly accurate with respect to the discovered periodicity rates and periodic patterns. Real-data experiments demonstrate the practicality of the discovered periodic patterns.

A New Challenge for Applying Time Series Metrics Data to Software Quality Estimation

In typical software development, a software reliability growth model (SRGM) is applied in each testing activity to determine the time to finish the testing. However, there are some cases in which the SRGM does not work correctly. That is, the SRGM sometimes mistakes quality for poor quality products. In order to tackle this problem, we focussed on the trend of time series data of software defects among successive testing phases and tried to estimate software quality using the trend. First, we investigate the characteristics of the time series data on the detected faults by observing the change of the number of detected faults. Using the rank correlation coefficient, the data are classified into four kinds of trends. Next, with the intention of estimating software quality, we investigate the relationship between the trends of the time series data and software quality. Here, software quality is defined by the number of faults detected during six months after shipment. Finally, we find a relationship between the trends and metrics data collected in the software design phase. Using logistic regression, we statistically show that two review metrics in the design and coding phase can determine the trend.

INVESTIGATION ON SPATIAL REPRESENTATIVENESS OF POINT MEASUREMENTS

The spatial representativeness of turbulent fluxes on the basis of point measurements has been investigated through a theoretical approach and numerical experiments using Large Eddy Simulation (LES) Model. These investigations have been done for the daytime atmospheric boundary layers heated uniformly over the flat surface with no synoptic vertical motion under several geostrophic winds from 1 to 4 (ms-1). The following results were obtained;(1) The spatial representativeness of turbulent fluxes based on the time-series analysis are shown to be problematic: Their horizontally averaged value has a negative imbalance irrespective of the density of measurements points, and their spatial variability increases, in accordance with averaging time, or under lower horizontal wind.(2) The imbalance intensities are closely related to turbulent organized structure (TOS). The points over the updraft regions are likely to have positive imbalance due to highly activated thermal plumes. The temporal and spatial change of TOS pattern cause low-frequency trends in time-series data, which can account for the large horizontal scatter of flux estimates.(3) High-pass filter excluding the temperature trend is theoretically useful to improve the representativeness of point measurements, although the filter choice actually needs a great care not to damage the turbulent components.

Power of the Mann–Kendall and Spearman's rho tests for detecting monotonic trends in hydrological series

In many hydrological studies, two non-parametric rank-based statistical tests, namely the Mann–Kendall test and Spearman's rho test are used for detecting monotonic trends in time series data. However, the power of these tests has not been well documented. This study investigates the power of the tests by Monte Carlo simulation. Simulation results indicate that their power depends on the pre-assigned significance level, magnitude of trend, sample size, and the amount of variation within a time series. That is, the bigger the absolute magnitude of trend, the more powerful are the tests; as the sample size increases, the tests become more powerful; and as the amount of variation increases within a time series, the power of the tests decrease. When a trend is present, the power is also dependent on the distribution type and skewness of the time series. The simulation results also demonstrate that these two tests have similar power in detecting a trend, to the point of being indistinguishable in practice. The two tests are implemented to assess the significance of trends in annual maximum daily streamflow data of 20 pristine basins in Ontario, Canada. Results indicate that the P-values computed by these different tests are almost identical. By the binomial distribution, the field significant downward trend was assessed at the significance level of 0.05. Results indicate that a higher number of sites show evidence of decreasing trends than one might expect due to chance alone.

Improved Tests for Trend in Time Series Data

The difficult problem of testing for linear trend in the presence of correlated residuals is addressed. Because the residuals are correlated, tests for trend based on the classical least-squares regression techniques are inappropriate. Whenever the residuals are even moderately correlated, the procedures in the literature that are intended to adjust for the correlation in the residuals tend to have the problem that the observed significance levels are higher than nominal levels for small to moderate realization lengths. Although it is well known that this tendency exists, the extent of the problem is not widely appreciated. We introduce a bootstrap-based procedure to test for trend in this setting. This procedure is better adapted to controlling the significance levels and is studied via simulation results. The procedure is then applied to the problem of testing for trend in global atmospheric temperature data and in data from models for ocean acoustic travel time along a path.

Global Warming and the Problem of Testing for Trend in Time Series Data

Abstract In recent years a number of statistical tests have been proposed for testing the hypothesis that global warming is occurring. The standard approach is to examine one or two of the more prominent global temperature datasets by letting Yt = a + bt + Et, where Yt represents the temperature at time t and Et represents error from the trend line, and to test the hypothesis that b = 0. Several authors have applied these tests for trend to determine whether or not a significant long-term or deterministic trend exists, and have generally concluded that there is a significant deterministic trend in the data. However, we show that certain autoregressive-moving average (ARMA) models may also be very reasonable models for these data due to the random trends present in their realizations. In this paper, we provide simulation evidence to show that the tests for trend detect a deterministic trend in a relatively high percentage of realizations from a wide range of ARMA models, including those obtained for the te...