Stationary Time Series Research Articles

Forecasting GDP Growth Rates of Bangladesh: An Empirical Study

The Gross Domestic Product (GDP) is the market value of all goods and services produced within the boundary of a nation in a year. This paper aims to apply time series tools and forecast GDP growth in the Bangladesh economy. Forecasting of time series is an important topic in macroeconomics. We collected the data from World Development Indicators (WDI) and it has been collected over a period of 37 years by WDI, World Bank. Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) tests were applied to investigate the stationary character of the data. Stata and R statistical software was used to build a class of Autoregressive Integrated Moving Average (ARIMA) and exponential smoothing methods to model the GDP growth. We applied several ARIMA (P, I, Q) models and employed the ARIMA (1,1,1) model as best for forecasting. This ARIMA (1,1,1) model was chosen based on the minimum values of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Also, we applied the Exponential Smoothing to forecast the GDP growth rate. In addition, among the Exponential Smoothing models, the triple exponential model better analyzed the data based on lowest Sum of Square Error (SSE) and Root Mean Square Error (RMSE). Using these models, the values of future GDP growth rates are forecasted. Statistical results show that Bangladesh’s GDP growth rate is an increasing trend that will continue rising in the future. This finding will help policymakers and academicians to formulate economic and business strategies more precisely . Keywords: Stationary time series, ARIMA, Time Series Forecasting, Exponential Smoothing, GDP growth rate, GDP growth in Bangladesh DOI : 10.7176/JESD/10-23-02 Publication date: December 31 st 2019

Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis

Nonparametric Bayesian inference has seen a rapid growth over the last decade but only few nonparametric Bayesian approaches to time series analysis have been developed. Most existing approaches use Whittle’s likelihood for Bayesian modelling of the spectral density as the main nonparametric characteristic of stationary time series. It is known that the loss of efficiency using Whittle’s likelihood can be substantial. On the other hand, parametric methods are more powerful than nonparametric methods if the observed time series is close to the considered model class but fail if the model is misspecified. Therefore, we suggest a nonparametric correction of a parametric likelihood that takes advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. We use a nonparametric Bernstein polynomial prior on the spectral density with weights induced by a Dirichlet process and prove posterior consistency for Gaussian stationary time series. Bayesian posterior computations are implemented via an MH-within-Gibbs sampler and the performance of the nonparametrically corrected likelihood for Gaussian time series is illustrated in a simulation study and in three astronomy applications, including estimating the spectral density of gravitational wave data from the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO).

Pricing analysis of stock index linked structured financial products of Commercial Banks

Financial products of commercial banks are an investment tool designed and issued independently by commercial banks. Nowadays, commercial banks have issued a variety of financial products, and the theoretical research on this aspect mainly focuses on marketing, the analysis of influencing factors of return rate and etc, but involving less pricing. Most of the research on pricing is limited to products with a simple structure. The case of this study is a structured financial product related to stock index. And besides an in-depth analysis of the structure characteristics and risk characteristics of the product, this study also calculates out the theoretical pricing of the product on the basis of using the stationary time series for conditional heteroscedastic model and the Monte Carlo Simulation Method (with the aid of Eviews software), and compares it with the actual pricing set by the Agricultural Bank of China, and then determines the rationality of the actual pricing. And suggestions are given accordingly.

Differences in the Dynamics of Radial Tree Increment in the Pine Looper Outbreak Area and in Undamaged Stands and Assessment of the Risk of Pests’ Attack

The time series of the radial increment of pine Pinus sylvestris L. in 1945–2015 were studied on the territory of the Krasnoturansky pine forest in the stands damaged during the outbreak of the pine looper Bupalus piniarius L. in 1974–1978 and in the stands not damaged by the pest. To assess the differences in the processes of radial increment of trees in damaged and undamaged stands before the outbreak, it is proposed to use the regulatory characteristics of time series, such as the values ​​of n delay (autocorrelation) and the amplitude s of fluctuations in time series. According to these indicators, one can numerically assess the inertia of the processes of regulation of radial increment. Since the regulatory characteristics can only be correctly calculated for stationary time series, the series of the first differences of the radial increment were used for the calculations. The trees in the stands that were attacked by insects were characterized either by the absence of delay in the time series, or by sufficiently large values ​​of the order of autocorrelation (n). Moreover, fluctuations in the current radial increment relative to the age trend were significantly greater in trees in damaged stands than in undamaged ones. The rate of growth recovery in trees damaged by insects depending on the regulatory characteristics of the time series of radial increment before the onset of damage was assessed. The obtained characteristics can be used to assess the risk of phyllophages’ attacks on stands and the rate of restoration of stands damaged by insects.

Cloud Changes in the Period of Global Warming: The Results of the International Satellite Project

The results of analysis of climatic series of global and regional cloudiness for 1983–2009. Data were obtained in the framework of the international satellite project ISCCP. The technology of statistical time series analysis including smoothing algorithm and wavelet analysis is described. Both methods are intended for the analysis of non-stationary series. The results of the analysis show that both global and regional cloudiness show a decrease of 2–6%. The greatest decrease is observed in the tropics and over the oceans. Over land, the decrease is minimal. The correlation coefficient between the global cloud series on the one hand and the global air and ocean surface temperature series on the other hand reaches values (–0.84) — (–0.86). The coefficient of determination that characterizes the accuracy of the regression for the prediction of global temperature changes based on data on changes in the lower cloud, in this case is 0.316.

Statistical inference of time-varying single-index coefficient models for locally stationary time series

Statistical inference of time-varying single-index coefficient models for locally stationary time series

Statistics of nonnumeric data in forty years (review)

Forty years ago, the statistics of nonnumeric data was singled out into an independent area of mathematical research methods. First the term «statistics of objects of nonnumeric nature» was used. The textbook on nonnumeric statistics is entitled «Nonnumeric Statistics». Statistics of nonnumeric data is one of the four main fields of applied statistics (along with the statistics of numbers, multivariate statistical analysis, statistics of time series and random processes). Statistics of nonnumeric data is divided into statistics in spaces of general nature and sections devoted to specific types of nonnumeric data (statistics of interval data, statistics of fuzzy sets, statistics of binary relations, etc.). Currently, statistics in spaces of general nature is the central part of applied statistics, and the statistics of nonnumeric data that includes it is the main area of applied statistics. This statement is confirmed by analysis of publications in the section «Mathematical Methods of Research» of our journal. This article is devoted to analysis of the main ideas of statistics of nonnumeric data against the background of the development of applied statistics on the basis of the new paradigm of the methods of mathematical research. Different types of nonnumeric data are considered along with the history of development of the statistics of nonnumeric data and statistical science. The basic ideas of statistics in spaces of general nature are analyzed: mean values, laws of large numbers, extreme statistical problems, nonparametric estimators of the probability distribution density, classification methods (diagnostics and cluster analysis), and statistics of the integral type. Some statistical methods for data analysis in specific nonnumeric spaces are briefly discussed: nonparametric statistics (in most cases, real distributions significantly differ from normal), statistics of fuzzy sets, expert estimation theory (Kemeny median as a sample average of expert orderings), etc. Some unsolved problems in statistics on nonnumeric data are also discussed.

Probabilistic Structure of Extreme Run Lengths in a Stationary Hydrologic Time Series

AbstractCurrent probabilistic methods for maximum and minimum run lengths in a hydrologic time series need further development. Using the first-order Markov chain model and an extreme-value theory ...

The effects of extremes and temporal scale on multifractal properties of river flow time series

AbstractFor accurate forecasting of extreme events in rivers, streamflow time series with sub‐daily temporal resolution (1–6 hour) are preferable, but discharge time series for long rivers are usually available at daily or monthly resolution. In this study, the scaling properties of hourly and daily streamflow time series were measured. As an innovation, the effects of extreme values on the multifractal behavior of these series were evaluated. Interestingly, both hourly and daily discharge records led to nearly identical scaling trends and identical crossover times. Daily and hourly discharge time series appeared to be non‐stationary when the timescale ranged from 75 to 366 days. Otherwise, the signals may be considered stationary time series. In addition, the results indicated that the extreme values strongly contribute to the multifractality of the series. The width of singularity spectra decreased considerably when the extreme events were removed from both hourly and daily discharge records.

Extended D-TomoSAR Displacement Monitoring for Nanjing (China) City Built Structure Using High-Resolution TerraSAR/TanDEM-X and Cosmo SkyMed SAR Data

The availability of high-resolution spaceborne synthetic aperture radar (SAR) data coupled with the ongoing refinement of tomographic SAR (TomoSAR) technology has made use of radar data feasible for preventive monitoring and assessment of built structures. In this study, we first applied extended differential TomoSAR (D-TomoSAR) to a set of 26 scenes of TerraSAR/TanDEM-X (TSX/TDX) (2013–2015) and 32 scenes of Cosmo-SkyMed (CSK) (2015–2017) images to estimate motions of skyscrapers, bridges and historical monuments in Nanjing City, China. The calculation and isolation of unknown parameters in the D-TomoSAR model, including linear velocity, thermal dynamics and structural heights, helped to estimate millimetric statistics of motion time series. Then, aforementioned two SAR datasets were tentatively tested using amplitude dispersion and phase stability indicators, highlighting the performance and sensitivity of X-band SAR in structural displacement monitoring. Experimental results demonstrated that motion indexes, e.g., heterogeneities of thermal amplitudes and spatiotemporal displacements, were useful to evaluate the conditions of built structures being monitored, in particular when their structural topology were visible owing to the enhanced density of persistent scatterer (PS) measurements. This study implies the value of high-resolution D-TomoSAR tools in the preventive monitoring and health diagnosis of built structures elsewhere over the world.

Walsh Fourier Transform of Locally Stationary Time Series

A new time‐frequency model and a method to classify time series data are proposed in this article. By viewing the observed signals as realizations of locally dyadic stationary (LDS) processes, a LDS model can be used to provide a time‐frequency decomposition of the signals, under which the evolutionary Walsh spectrum and related statistics can be defined and estimated. The classification procedure is as follows. First choose a training data set that comprises two groups of time series with a known group. Then compute the time frequency feature (the energy) using the training data set, and use a best tree method to maximize the discrepancy of this feature between the two groups. Finally, choose the testing data set with the unknown group as validation data, and use a discriminant statistic to classify the validation data to one of the groups. The classification method is illustrated via an electroencephalographic dataset and the Ericsson B transaction time dataset. The proposed classification method performs better for integer‐valued time series in terms of classification error rates in both simulations and real‐life applications.

A stochastic Functional Model based method for random vibration based robust fault detection under variable non–measurable operating conditions with application to railway vehicle suspensions

The problem of random vibration based robust fault detection under variable and non–measurable Environmental and Operating Conditions (EOCs) is considered, and a novel stochastic Functional Model (FM) based method is postulated. It is a data–driven method, of the Statistical Time Series (STS) type, and aims at overcoming the well known drawbacks of available methods by achieving high detection performance while eliminating their drawbacks, such as the need for measurable EOCs, for measurement of a high number of vibration signals for proper training, for subjective judgement in selecting method parameters, and for high dimensional non–convex optimization procedures. The method is based on representing the system dynamics, under any set of EOCs, in a proper feature space, within which the healthy dynamics are represented by a proper healthy subspace constructed via a Functional Model. Fault detection is then based upon determining, at a certain risk level, whether or not the current dynamics resides within the healthy subspace. The method's assessment is achieved via simulation results with a case study pertaining to fault detection in a railway vehicle suspension under variable payload, with high detection performance, clearly exceeding that of an alternative Principal Component Analysis (PCA) based method.

Multiseasonal grapevine water consumption – Drivers and forecasting

The interactions between temperature, relative humidity, radiation, wind speed and their effect on plant transpiration in the context of water consumption for irrigation purposes have been studied for over a century. Leaf area has also been established as an important factor affecting water consumption. We analyzed a multivariable time series composed of both meteorological and vegetative variables with a daily temporal resolution for the growing seasons of 2013–2016 for Vitis vinfera ‘Cabernet Sauvignon’ vineyards in the mountainous region in Israel. Time-series analysis of this data was used to characterize seasonal patterns affecting water consumption (ETc) of vines and to quantify interrelations between meteorological and vegetative factors affecting vine water consumption. Moreover, we applied a machine learning regression model to determine the relative influence of meteorological and vegetative factors on ETc during four growing seasons. Finally, we developed an ensemble model for temporally forecasting vine ETc for an additional season using a training dataset of multiple variables. Our findings show that decomposing the time-series dataset uncovered a wider variety of underlying temporal patterns, and enabled quantification of seasonal and daily relationships. Leaf area had a substantial impact on ETc and was found to have a relative influence ranging between 62 and 86% for the different growing seasons. Mean temperature was ranked second followed by minor effects of relative humidity, solar radiation and wind speed that were interchangeably ordered. The ensemble model produced reliable results, with cross validation coefficients ~ 0.9. Incorporating leaf area measurements into the regression model improved both the performance of the model and the training data correlation. Using time-series statistics to explore meteorological and vegetative temporal characteristics, patterns, interrelations and relative effect on evapotranspiration may facilitate the understanding of water consumption processes and assist in generating more effective and skillful irrigation models.

Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach

Many Bayesian nonparametric approaches to multivariate time series rely on Whittle’s Likelihood, involving the second order structure of a stationary time series by means of its spectral density matrix. In this work, we model the spectral density matrix by means of random measures that are constructed in such a way that positive definiteness is ensured. This is in line with existing approaches for the univariate case, where the normalized spectral density is modeled similar to a probability density, e.g. with a Dirichlet process mixture of Beta densities. We present a related approach for multivariate time series, with matrix-valued mixture weights induced by a Hermitian positive definite Gamma process. The latter has not been considered in the literature, allows to include prior knowledge and possesses a series representation that will be used in MCMC methods. We establish posterior consistency and contraction rates and small sample performance of the proposed procedure is shown in a simulation study and for real data.

Time scale of resilience loss: Implications for managing critical transitions in water quality.

Regime shifts involving critical transitions are a type of rapid ecological change that are difficult to predict, but may be preceded by decreases in resilience. Time series statistics like lag-1 autocorrelation may be useful for anticipating resilience declines; however, more study is needed to determine whether the dynamics of autocorrelation depend on the resolution of the time series being analyzed, i.e., whether they are time-scale dependent. Here, we examined timeseries simulated from a lake eutrophication model and gathered from field measurements. The field study involved collecting high frequency chlorophyll fluorescence data from an unmanipulated reference lake and a second lake undergoing experimental fertilization to induce a critical transition in the form of an algal bloom. As part of the experiment, the fertilization was halted in response to detected early warnings of the algal bloom identified by increased autocorrelation. We tested these datasets for time-scale dependence in the dynamics of lag-1 autocorrelation and found that in both the simulation and field experiment, the dynamics of autocorrelation were similar across time scales. In the simulated time series, autocorrelation increased exponentially approaching algal bloom development, and in the field experiment, the difference in autocorrelation between the manipulated and reference lakes increased sharply. These results suggest that, as an early warning indicator, autocorrelation may be robust to the time scale of the analysis. Given that a time scale can be shortened by increasing sampling frequency, or lengthened by aggregating data during analysis, these results have important implications for management as they demonstrate the potential for detecting early warning signals over a wide range of monitoring frequencies and without requiring analysts to make situation-specific decisions regarding aggregation. Such an outcome provides promise that data collection procedures, especially by automated sensors, may be used to monitor and manage ecosystem resilience without the need for strict attention to time scale.

Statistical arbitrage under the efficient market hypothesis

When a financial derivative can be traded consecutively and its terminal payoffs can be adjusted into a stationary time series, there might be a statistical arbitrage opportunity even under the efficient market hypothesis. In particular, we show the examples of selling put options of the three major ETFs (Exchange Traded Funds) in the U.S. market.

Edge Exclusion Tests for Graphical Model Selection: Complex Gaussian Vectors and Time Series

We consider the problem of inferring the conditional independence graph (CIG) of a stationary multivariate Gaussian time series. A $p$ -variate Gaussian time series graphical model associated with an undirected graph with $p$ vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph. In some existing methods, partial coherence has been used as a test statistic for graphical model selection. To test inclusion/exclusion of a given edge in the graph, the test is applied at distinct frequencies, requiring multiple tests and leading to a loss in test power. The nonparametric method of Matsuda uses the Kullback-Leibler divergence measure to define a test statistic which does not need multiple testing to test between two competing models. In this paper we propose a generalized likelihood ratio test (GLRT) based edge exclusion test statistic that also does not need multiple testing. It is computationally significantly faster than the method of Matsuda, and simulations show that we achieve comparable power levels. Our proposed approach is based on a novel formulation of a GLRT based edge exclusion test for $p$ -variate complex Gaussian graphical models (CGGMs); this result is also of independent interest. The computational complexity of the proposed statistic is $\mathcal {O}(p^3)$ compared to $\mathcal {O}(p^5)$ for the existing result. We also apply our time series graphical model selection method to a foreign exchange data set consisting of monthly trends of foreign exchange rates of 10 countries.

An Adaptive Intrusion Detection Scheme for Cloud Computing

To provide dynamic resource management, live virtual machine migration is used to move a virtual machine from one host to another. However, virtual machine migration poses challenges to cloud intrusion detection systems because movement of VMs from one host to another makes it difficult to create a consistent normal profile for anomaly detection. Hence, there is a need to provide an adaptive anomaly detection system capable of adapting to changes that occur in the cloud data during VM migration. To achieve this, the authors proposed a scheme for adaptive IDS for Cloud computing. The proposed adaptive scheme is comprised of four components: an ant colony optimization-based feature selection component, a statistical time series change point detection component, adaptive classification, and model update component, and a detection component. The proposed adaptive scheme was evaluated using simulated datasets collected from vSphere and performance comparison shows improved performance over existing techniques.

Forecasting methods in engineering

Forecasting in engineering is one of the most important topics when it comes to optimisation, which is related to energy savings, material savings, increasing efficiency, and appropriate and correct decisions at the level of a company, institution, city, or region. Moreover, forecasting is indirectly related to cost savings and the sustainable development of society and environment. In the energy industry (electricity, natural gas, heat load), there are requirements to balance the supply and demand. Markets are very dynamic, and for this reason forecasting is more challenging. Forecasting errors are usually penalized drastically. However, a well-developed forecasting approach represents a competitive advantage, and so a company may significantly reduce expenditure and increase profits. Many publications on forecasting have been published during recent years. Long-term forecasting methods offer many opportunities for strategic planning and optimal scheduling, whereas a short-term forecasting approach would help attain optimal daily operations and the maximum utilisation of the company’s resources. Although different forecasting techniques can be used, the major conclusions are that exponential smoothing methods are the simplest and the least expensive. Distinguished by their simplicity, their forecasts are comparable to forecasts of more complex statistical time series models. In this paper, the forecasting performance of Additive, Multiplicative, and Extended Holt-Winters methods were analysed. We also analyse whether the data format influences the choice of the forecasting method: is the most accurate method for monthly data also the best method for quarterly data?

Processing time-series of randomly forced self-oscillators: The example of beer bottle whistling

We present a model-based, output-only parameter identification method for self-sustained oscillators forced by dynamic noise, which we illustrate experimentally with a simple aeroacoustic setup: a turbulent jet impinging a beer bottle and producing a distinct whistling tone in a finite range of jet angles and jet velocities. Given a low-order model of the system, the identification is based on analyzing stationary time series of a single key observable: the acoustic pressure fluctuations inside the bottle. Noting that this observable exhibits all the characteristics of weakly non-linear self-oscillations, we choose as a minimal model the classic Van der Pol (VdP) oscillator: a linear acoustic oscillator (the bottle) subject to stochastic forcing and non-linear deterministic forcing (the turbulent jet). Although very simple, the VdP oscillator driven by random noise proves to be a sufficient phenomenological description of the aeroacoustic limit cycle for the purpose of model-based linear growth rate identification. We derive the associated stochastic amplitude equation, which allows us to describe, on both sides of the bifurcation, the stochastic fluctuations of the acoustic amplitude resulting from a competition between (i) linear growth rate induced by the coherent unsteady vortex force, (ii) random forcing induced by the turbulence and (iii) non-linear saturation of the coherent flapping motion of the jet. Finally, we use the associated adjoint Fokker-Planck equation to reveal both the deterministic potential well and the stochastic forcing intensity that govern these random fluctuations. Additional experiments with an external control validate the identification results. We also observe that different operating conditions can yield similar statistics but different dynamics, showing that the identification method can disentangle deterministic and stochastic effects in systems forced stochastically. This output-only parameter identification approach can be used in a wide range of phenomena exhibiting stationary self-sustained oscillations in the presence of random forcing.