Short Length Time Series Research Articles

Spectral Estimation of Electroencephalogram Signal using ARMAX Model and Particle Swarm Optimization

Electroencephalogram (EEG) signal is short-length time series data. Parametric and non-parametric methods are used to estimate the power spectral density (PSD) of the time series data. While the non-parametric approach of spectrum estimate of short-length data is unreliable, the parametric approach of spectrum estimate is widely suggested. In the presented work, a parametric autoregressive moving average with exogenous input (ARMAX) model was implemented to find the PSD of the EEG signal. The performance of the implemented ARMAX model was contrasted with other parametric methods like autoregressive AR) and autoregressive moving average (ARMA) and nonparametric methods like periodogram. Coefficients of ARMAX and (ARMA) were estimated using the particle swarm optimization (PSO) technique based on swarm intelligence. The PSO algorithm adapted with the system identification technique and statistics yielded highly satisfactory results in finding the coefficients of the ARMAX model. All the programming and visualization were performed in a MATLAB environment.

Spatial extreme model for rainfall depth: application to the estimation of IDF curves in the Basque country

Intensity-duration-frequency (IDF) curves are commonly used in engineering practice for the hydraulic design of flood protection infrastructures and flood risk management. IDF curves are simple functions between the rainfall intensity, the timescale at which the rainfall process is studied, and the return period. This work proposes and tests a new methodological framework for the spatial analysis of extreme rainfall depth at different timescales, taking advantage of two different precipitation datasets: local observational and gridded products. On the one hand, the proposed method overcomes or reduces known issues related to observational datasets (missing data and short temporal coverage, outliers, systematic biases and inhomogeneities, etc.). On the other hand, it allows incorporating appropriately terrain dependencies on the spatial distribution of the extreme precipitation regime. Finally, it allows to estimate the IDF curves at regional level overcoming the deficiencies of the classical regional approaches commonly used in practice. The method has been tested to compute IDF curves all over the Basque Country, contrasting results with respect to local analyses. Results show the method robustness against outliers, missing data, systematic biases and short length time series. Moreover, since generalized extreme value (GEV)-parameters from daily gridded dataset are used as covariates, the proposed approach allows coherent spatial interpolation/extrapolation of IDF curves properly accounting for the influence of orographic factors. In addition, due to the current coexistence of local observations and gridded datasets at regional (e.g. The Alps), national (e.g. Spain, France, etc.) or international scale (e.g. E-OBS for Europe or Daymet for the United States of America), the proposed methodology has a wide range of applicability in order to fulfill the known gaps of the observational datasets and reduce the uncertainty related to analysis and characterization of the extreme precipitation regime.

Statistical analysis of electricity supply deficits from renewable energy sources across an Alpine transect

While most of renewable integration studies rely on indicators that are integrated in time, little is known about the statistical properties of the electricity supply deficit periods. This study investigates the frequency and intensity of the electricity supply deficit using the Complete-Time-Analysis (CTA), an approach that allows assessing recurrence intervals of values over thresholds. Conversely to typical extreme analysis, CTA makes uses of the entire time series rather than only using values above the chosen thresholds. As such, it is suitable for short length time series and does not require the independence of the events above thresholds. Intensity-Duration-Frequency (IDF) curves are derived for the considered renewable sources and the role of the complementarity among sources and the use of electricity storage is investigated to reduce the occurrence of electricity supply deficit. The approach is illustrated for a region in North-Eastern Italy where solar photovoltaic (PV) and small run-of the river (RoR) hydropower are the main sources of electricity. Our results highlight that solar PV and small electricity storage capacities are effective to reduce significantly the occurrence of deficits, and RoR hydropower and large electricity storage is more effective to prevent long periods of deficit from happening.

Identifying Multisubject Cortical Activation in Functional MRI: A Frequency Domain Approach

Functional magnetic resonance imaging (fMRI) has, since its de scription fifteen years ago, become the most common in-vivo neuroimaging technique. FMRI allows the identification of brain areas which are related to specific tasks, by statistical analysis of the BOLD (blood oxigenation level dependent) signal. Classically, the observed BOLD signal is compared to an expected haemodynamic response function (HRF) using a general linear model (GLM). However, the results of GLM rely on the HRF specification, which is usually determined in an ad hoc fashion. For periodic experimental designs, we propose a multisubject frequency domain brain mapping, which requires only the stimulation frequency, and consequently avoids subjective choices of HRF. We present some computational simulations, which demon strate a good performance of the proposed approach in short length time series. In addition, an application to real fMRI datasets is also presented.

Early detection of thermoacoustic instabilities in a cryogenic rocket thrust chamber using combustion noise features and machine learning.

We present a data-driven method for the early detection of thermoacoustic instabilities. Recurrence quantification analysis is used to calculate characteristic combustion features from short-length time series of dynamic pressure sensor data. Features like recurrence rate are used to train support vector machines to detect the onset of instability a few hundred milliseconds in advance. The performance of the proposed method is investigated on experimental data from a representative LOX/H 2 research thrust chamber. In most cases, the method is able to timely predict two types of thermoacoustic instabilities on test data not used for training. The results are compared with state-of-the-art early warning indicators.

Data-driven Detection and Early Prediction of Thermoacoustic Instability in a Multi-nozzle Combustor

ABSTRACT Thermoacoustic instability (TAI) is a critical issue in modern lean-burn gas-turbine combustors, which is induced by a strong coupling between the resonant combustor acoustics and fluctuations in the heat release rate. This instability may lead to high-amplitude pressure waves that generate undesirable noise levels as well as fatigue stresses in mechanical structures of the combustor. The intense pressure fluctuations due to TAI may also cause large flow perturbations and possibly flow reversal that may lead to flame oscillations, flame liftoff, and even flame blow-out. Hence, there is a strong need for exercising control actions in a timely fashion to mitigate the TAI phenomena. Anomaly detection is an essential prerequisite to the design of a good controller and such a detector must be able to reliably predict a forthcoming TAI. To detect and predict the onset of a TAI from an ensemble of pressure time series, this paper investigates three data-driven methods: Fast Fourier transform (FFT), symbolic time series analysis (STSA), and hidden Markov modeling (HMM). The main focus of the paper is to make a comparative evaluation of these three anomaly detection methods for classification of the current regime of operation into stable and unstable categories as well as for real-time identification of precursors to impending instabilities with short-length time series of measured variables (e.g., pressure oscillations). The results, generated on experimental data from a multi-nozzle combustor apparatus, have been compared to evaluate the performance of FFT, STSA, and HMM methods for TAI analysis.

Symbolic Time Series Analysis for Anomaly Detection in Measure-Invariant Ergodic Systems

Abstract This paper presents a novel framework of symbolic time series analysis (STSA) for anomaly detection in dynamical systems. The core concept is built upon a property of measure-preserving transformation (MPT) sequence, acting on a probability space with ergodic measure, that the eigenfunctions of these transformations would be time-invariant. As a result, unlike a standard STSA that is required to generate time-homogeneous Markov chains, the proposed MPT-based STSA is allowed to have time-inhomogeneous Markov chains, where the (possibly time-varying) state transition probability matrices have time-invariant eigenvectors. Such a time-invariance facilitates analysis of the dynamical system by using short-length time series of measurements. This is particularly important in applications, where the underlying dynamics and process anomalies need fast monitoring and control actions in order to mitigate any potential structural damage and/or to avoid catastrophic failures. The MPT-based STSA has been applied for low-delay detection of fatigue damage, which is a common source of failures in mechanical structures and which is known to have uncertain dynamical characteristics. The underlying algorithm has been validated with experimental data generated from a laboratory apparatus that uses ultrasonic sensors to detect fatigue damage in polycrystalline–alloy specimens. The performance of the proposed MPT-based STSA is evaluated by comparison with those of a standard STSA and a hidden Markov model (HMM) on the same experimental data. The results consistently show superior performance of the MPT-based STSA.

Hidden Markov Modeling-Based Decision-Making Using Short-Length Sensor Time Series

Real-time decision-making (e.g., monitoring and active control of dynamical systems) often requires feature extraction and pattern classification from short-length time series of sensor data. An example is thermoacoustic instabilities (TAI) in combustion systems, caused by spontaneous excitation of one or more natural modes of acoustic waves. The TAI are typically manifested by large-amplitude self-sustained pressure oscillations in time scales of milliseconds, which need to be mitigated by fast actuation of the control signals, requiring early detection of the forthcoming TAI. This issue is addressed in this technical brief by hidden Markov modeling (HMM) and symbolic time series analysis (STSA) for near-real-time recognition of anomalous patterns from short-length time series of sensor data. An STSA technique is first proposed, which utilizes a novel HMM-based partitioning method to symbolize the time series by using the Viterbi algorithm. Given the observed time series and a hidden Markov model, the algorithm generates a symbol string with maximum posterior probability. This symbol string is optimal in the sense of minimizing string error rates in the HMM framework. Then, an HMM likelihood-based detection algorithm is formulated and its performance is evaluated by comparison with the proposed STSA-based algorithm as a benchmark. The algorithms have been validated on a laboratory-scale experimental apparatus. The following conclusions are drawn from the experimental results: (1) superiority of the proposed STSA method over standard methods in STSA for capturing the dynamical behavior of the underlying process, based on short-length time series and (2) superiority of the proposed HMM likelihood-based algorithm over the proposed STSA method for different lengths of sensor time series.

Partial trend identification by change-point successive average methodology (SAM)

There are different trend identification methodologies in the literature, but almost all are for monotonic trend searches within a given hydro-meteorological time series partially at equal epochs or holistically over the whole record duration. The most important question is how to identify within the same time series successive trends of different durations and slopes. For this purpose, successive arithmetic average methodology (SAM) is proposed in this paper, first for a set of trends visual inspection and then their quantitative duration and slope calculations. This method provides to identify peak and valley change-points years, trends durations and slopes. The most significant benefit from SAM is that there is no need for preliminary assumptions and trends identifications are straight forward. The only limitation of the method is short length time series. Absolute relative percentage error control is suggested as ±10% for separation of successive partial trend differences. The application of SAM methodology is presented for more than seventy-year duration total annual rainfall records from seven geographic and climate regions in Turkey. It has been calculated that both increasing and decreasing trend durations have at the maximum 4.5-year duration. According to the calculations, except at one meteorology station almost all the trend slopes are negative, which may be an initial signal for climate change impact indicating the rainfall decreases in recent years over Turkey.

Early Detection of Thermoacoustic Instabilities Using Hidden Markov Models

ABSTRACTThis paper presents a dynamic data-driven method for early detection of thermoacoustic instabilities in combustors based on short-length time series of sensor data, where the objective is near-real-time monitoring and active control of pressure oscillations. The main idea is to use the available data at different regimes of the combustion process to train respective hidden-variable models using the concept of Hidden Markov Modeling (HMM) as a statistical learning tool; here, (short-length) time-series data of pressure oscillations are used to infer a Markov chain with unobserved (hidden) states. The proposed HMM-based method has been validated on experimental data collected from an electrically heated Rijke tube apparatus for predicting onset of thermoacoustic instabilities. The results have been compared with those of the current state-of-the-art measurement techniques for instability growth rate and associated computational complexity. The applicability of the proposed method has been demonstrated with respect to anomaly detection and regime identification with limited data requirements, making it a potential candidate for monitoring and active control of thermoacoustic instabilities in commercial-scale combustors.

Structural breaks in panel data: Large number of panels and short length time series

ABSTRACTThe detection of (structural) breaks or the so called change point problem has drawn increasing attention from the theoretical, applied economic and financial fields. Much of the existing research concentrates on the detection of change points and asymptotic properties of their estimators in panels when N, the number of panels, as well as T, the number of observations in each panel are large. In this paper we pursue a different approach, i.e., we consider the asymptotic properties when N→∞ while keeping T fixed. This situation is typically related to large (firm-level) data containing financial information about an immense number of firms/stocks across a limited number of years/quarters/months. We propose a general approach for testing for break(s) in this setup. In particular, we obtain the asymptotic behavior of test statistics. We also propose a wild bootstrap procedure that could be used to generate the critical values of the test statistics. The theoretical approach is supplemented by numerous simulations and by an empirical illustration. We demonstrate that the testing procedure works well in the framework of the four factors CAPM model. In particular, we estimate the breaks in the monthly returns of US mutual funds during the period January 2006 to February 2010 which covers the subprime crises.

The ordinal Kolmogorov-Sinai entropy: A generalized approximation

We introduce the multi-dimensional ordinal arrays complexity as a generalized approximation of the ordinal Komogorov-Sinai entropy. The ordinal arrays entropy (OAE) is defined as the Shannon entropy of a series of m-ordinal patterns encoded symbols, while the ordinal arrays complexity (OAC) is defined as the differential of the OAE with respect to m. We theoretically establish that the OAC provides a better estimate of the complexity measure for short length time series. Simulations were carried out using discrete maps, and confirm the efficiency of the OAC as complexity measure from a small data set even in a noisy environment.

The coupling analysis between stock market indices based on permutation measures

Many information-theoretic methods have been proposed for analyzing the coupling dependence between time series. And it is significant to quantify the correlation relationship between financial sequences since the financial market is a complex evolved dynamic system. Recently, we developed a new permutation-based entropy, called cross-permutation entropy (CPE), to detect the coupling structures between two synchronous time series. In this paper, we extend the CPE method to weighted cross-permutation entropy (WCPE), to address some of CPE’s limitations, mainly its inability to differentiate between distinct patterns of a certain motif and the sensitivity of patterns close to the noise floor. It shows more stable and reliable results than CPE does when applied it to spiky data and AR(1) processes. Besides, we adapt the CPE method to infer the complexity of short-length time series by freely changing the time delay, and test it with Gaussian random series and random walks. The modified method shows the advantages in reducing deviations of entropy estimation compared with the conventional one. Finally, the weighted cross-permutation entropy of eight important stock indices from the world financial markets is investigated, and some useful and interesting empirical results are obtained.

Modeling Metabolism and Disease in Bioarcheology.

We examine two important measures that can be made in bioarcheology on the remains of human and vertebrate animals. These remains consist of bone, teeth, or hair; each shows growth increments and each can be assayed for isotope ratios and other chemicals in equal intervals along the direction of growth. In each case, the central data is a time series of measurements. The first important measures are spectral estimates in spectral analyses and linear system analyses; we emphasize calculation of periodicities and growth rates as well as the comparison of power in bands. A low frequency band relates to the autonomic nervous system (ANS) control of metabolism and thus provides information about the life history of the individual of archeological interest. Turning to nonlinear system analysis, we discuss the calculation of SM Pinus' approximate entropy (ApEn) for short or moderate length time series. Like the concept that regular heart R-R interval data may indicate lack of health, low values of ApEn may indicate disrupted metabolism in individuals of archeological interest and even that a tipping point in deteriorating metabolism may have been reached just before death. This adds to the list of causes of death that can be determined from minimal data.

Measuring the self-similarity exponent in Lévy stable processes of financial time series

Geometric method-based procedures, which will be called GM algorithms herein, were introduced in [M.A. Sánchez Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543-5551], to efficiently calculate the self-similarity exponent of a time series. In that paper, the authors showed empirically that these algorithms, based on a geometrical approach, are more accurate than the classical algorithms, especially with short length time series. The authors checked that GM algorithms are good when working with (fractional) Brownian motions. Moreover, in [J.E. Trinidad Segovia, M. Fernández-Martínez, M.A. Sánchez-Granero, A note on geometric method-based procedures to calculate the Hurst exponent, Phys. A 391 (2012) 2209-2214], a mathematical background for the validity of such procedures to estimate the self-similarity index of any random process with stationary and self-affine increments was provided. In particular, they proved theoretically that GM algorithms are also valid to explore long-memory in (fractional) Lévy stable motions.In this paper, we prove empirically by Monte Carlo simulation that GM algorithms are able to calculate accurately the self-similarity index in Lévy stable motions and find empirical evidence that they are more precise than the absolute value exponent (denoted by AVE onwards) and the multifractal detrended fluctuation analysis (MF-DFA) algorithms, especially with a short length time series. We also compare them with the generalized Hurst exponent (GHE) algorithm and conclude that both GM2 and GHE algorithms are the most accurate to study financial series. In addition to that, we provide empirical evidence, based on the accuracy of GM algorithms to estimate the self-similarity index in Lévy motions, that the evolution of the stocks of some international market indices, such as U.S. Small Cap and Nasdaq100, cannot be modelized by means of a Brownian motion.

Error Covariance Matrix Estimation of Noisy and Dynamically Coupled Time Series

We estimate the covariance matrix of the errors in several dynamically coupled time series corrupted by measurement errors. We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. The estimation of the covariance matrix of the errors is made using a noise reduction algorithm that efficiently exploits the information contained jointly in the dynamically coupled noisy time series. The method is particularly powerful for short length time series with high uncertainties.

A hypothesis test using bias-adjusted AR estimators for classifying time series in small samples

A new test of hypothesis for classifying stationary time series based on the bias-adjusted estimators of the fitted autoregressive model is proposed. It is shown theoretically that the proposed test has desirable properties. Simulation results show that when time series are short, the size and power estimates of the proposed test are reasonably good, and thus this test is reliable in discriminating between short-length time series. As the length of the time series increases, the performance of the proposed test improves, but the benefit of bias-adjustment reduces. The proposed hypothesis test is applied to two real data sets: the annual real GDP per capita of six European countries, and quarterly real GDP per capita of five European countries. The application results demonstrate that the proposed test displays reasonably good performance in classifying relatively short time series.

A note on geometric method-based procedures to calculate the Hurst exponent

Geometric method-based procedures, which we will call GM algorithms hereafter, were introduced in M.A. Sánchez-Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543–5551, to calculate the Hurst exponent of a time series. The authors proved that GM algorithms, based on a geometrical approach, are more accurate than classical algorithms, especially with short length time series. The main contribution of this paper is to provide a mathematical background for the validity of these two algorithms to calculate the Hurst exponent H of random processes with stationary and self-affine increments. In particular, we show that these procedures are valid not only for exploring long memory in classical processes such as (fractional) Brownian motions, but also for estimating the Hurst exponent of (fractional) Lévy stable motions.

Noise reduction by recycling dynamically coupled time series

We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties.

Bivariate nonlinear prediction to quantify the strength of complex dynamical interactions in short-term cardiovascular variability

A nonlinear prediction method for investigating the dynamic interdependence between short length time series is presented. The method is a generalization to bivariate prediction of the univariate approach based on nearest neighbor local linear approximation. Given the input and output series x and y, the relationship between a pattern of samples of x and a synchronous sample of y was approximated with a linear polynomial whose coefficients were estimated from an equation system including the nearest neighbor patterns in x and the corresponding samples in y. To avoid overfitting and waste of data, the training and testing stages of the prediction were designed through a specific out-of-sample cross validation. The robustness of the method was assessed on short realizations of simulated processes interacting either linearly or nonlinearly. The predictor was then used to characterize the dynamical interaction between the short-term spontaneous fluctuations of heart period (RR interval) and systolic arterial pressure (SAP) in healthy young subjects. In the supine position, the predictability of RR given SAP was low and influenced by nonlinear dynamics. After head-up tilt the predictability increased significantly and was mostly due to linear dynamics. These findings were related to the larger involvement of the baroreflex regulation from SAP to RR in upright than in supine humans, and to the simplification of the RR-SAP coupling occurring with the tilt-induced alteration of the neural regulation of the cardiovascular rhythms.