Advances in Time Series Analysis and Forecasting

In the study of fatigue crack growth and detection modeling, modern prognosis and health management (PHM) typically utilizes damage precursors and signal processing in order to determine structural health. However, modern PHM assessments are also subject to various uncertainties due to the probability of detection (POD) of damage precursors and sensory readings, and due to various measurement errors that have been overlooked. A powerful non-destructive testing (NDT) method to collect data and information for fatigue damage assessment, including crack length measurement is the use of the acoustic emission (AE) signals detected during crack initiation and growth. Specifically, correlating features of the AE signals such as their waveform ring-count and amplitude with crack growth rate forms the basis for fatigue damage assessment. An extension of the traditional applications of AE in fatigue analysis has been performed by using AE features to estimate the crack length recognizing the Gaussian correlation between the actual crack length and a set of predefined crack shaping factors (CSFs). Beside the traditional physics-based empirical models, the Gaussian process regression (GPR) approach is used to model the true crack path and crack length as a function of the proposed CSFs. Considering the POD of the micro-cracks and the AE signals along with associated measurement errors, the properties of the distribution representing the true crack is obtained. Experimental fatigue crack and the corresponding AE signals are then used to make a Bayesian estimation of the parameters of the combined GPR, POD, and measurement error models. The results and examples support the usefulness of the proposed approach.

Anthropogenic pressures are increasing and induce more frequent and stronger disturbances on ecosystems especially on coral reefs which is one of the most diverse on Earth. Long-term data series are increasingly needed to understand and evaluate the consequences of such pressures on ecosystems. This 30-years monitoring program allowed a description of the ability of the coral reef of Tiahura (French Polynesia) to recover after two main coral cover declines, due to Acanthaster planci outbreaks. The study is divided in two distinct periods framing the drop of coral cover and analyze the reaction of two herbivorous family: Acanthuridae and Scarinae. First we compared the successive roles they played in the herbivorous community, then we evaluated the changes in species composition that occurred for both Acanthuridae and Scarinae between these two periods. The long-term study of this coral reef ecosystem provided a valuable study case of the resilience over 30 years.

Language is constantly changing, with words being created or disappearing over time. Moreover, the usage of different words tends to fluctuate due to influences from different fields, such as historical events, cultural movements or scientific discoveries. These changes are reflected in the written texts and thus, by tracking them, one can determine the moment when these texts were written. In this paper, we present an application based on time series analysis built on top of the Google Books N-gram corpus to determine the time stamp of different written texts. The application is using two heuristics: words’ fingerprinting, to find the time interval when they were most probable used, and words’ importance for the given text, to weight the influence of words’ fingerprinting for estimating the text time stamp. Combining these two heuristics allows time stamping of that text.

Identifying the moment of time when a book was published is an important problem that might help solving the problem of authorship identification and could also shed some light into identifying the realities of the human society during different periods of time. In this paper, we present an attempt to estimate the publication date of books based on the time series analysis of their content. The main assumption of this experiment is that the subject of a book is often specific to a time period. Therefore, it is likely to use topic modeling to learn a model that might be used to timestamp different books, given for training many books from similar periods of time. To validate the assumption, we built a corpus of 10 thousand books and used LDA to extract the topics from them. Then, we extracted the time series of particular terms from each topic using Google Books N-gram Corpus. By heuristically combining the words’ time series and the topics from a document, we have built that document’s time series. Finally, we applied peak detection algorithms to timestamp the document.

We study and compare the reference of the Spanish stock market IBEX 35 with other international indices from consolidated as well as emerging economies. We look for similarities and differences between the Spanish index and the markets chosen, from a self-affine perspective. For it we compute fractal parameters which provide an indication of the erraticity of the data. We perform inference statistical tests, in order to elucidate if the computed parameters are significantly different in the Spanish selective. Beginning from the daily closing values of the IBEX of more than one decade, we investigate the stability in mean and variance, and test the necessity of the transformation of the record in order to improve its normality or stabilize and minimize the deviation. We use appropriate statistical methodologies, as ARIMA and ARCH, to obtain good explicative models of the series considered, and estimate its parameters of interest.

Widespread use of information and communication technologies has caused that the decisions made in financial markets by investors are influenced by the use of techniques like fundamental analysis and technical analysis, and the methods used are from all branches of mathematical sciences. Recently the fractional Brownian motion has found its way to many applications. In this paper fractional Brownian motion is studied in connection with financial time series. We analyze open, high, low and close prices as a selfsimilar processes that are strongly correlated. We study their basic properties explained in Hurst exponent exponent, and we use them as a measure of predictability of time series.

Recurrence structures in univariate time series are challenging to detect. We propose a combination of symbolic and recurrence analysis in order to identify recurrence domains in the signal. This method allows to obtain a symbolic representation of the data. Recurrence analysis produces valid results for multidimensional data, however, in the case of univariate time series one should perform phase space reconstruction first. In this chapter, we propose a new method of phase space reconstruction based on the signal’s time-frequency representation and compare it to the delay embedding method. We argue that the proposed method outperforms the delay embedding reconstruction in the case of oscillatory signals. We also propose to use recurrence complexity as a quantitative feature of a signal. We evaluate our method on synthetic data and show its application to experimental EEG signals.

Climate dynamics were assessed using multifractal detrended fluctuation analysis (MF-DFA) for sites in Finland, Germany and Spain across a latitudinal transect. Meteorological time series were divided into the two subsets (1980–2001 and 2002–2010) and respective spectra of these subsets were compared to check whether changes in climate dynamics can be observed using MF-DFA. Additionally, corresponding shuffled and surrogate time series were investigated to evaluate the type of multifractality. All time series indicated underlying multifractal structures with considerable differences in dynamics and development between the studied locations. The source of multifractality of precipitation time series was two-fold, coming from the width of the probability function to a greater extent than for other time series. The multifractality of other analyzed meteorological series was mainly due to long-range correlations for small and large fluctuations. These results may be especially valuable for assessing the change of climate dynamics, as we found that larger changes in asymmetry and width parameters of multifractal spectra for divided datasets were observed for precipitation than for other time series. This suggests that precipitation is the most vulnerable meteorological quantity to change of climate dynamics.

In this paper, two traditional Autoregressive Moving Average models and two different Generalised Autoregressive Moving Average models are considered to forecast financial time series. These time series models are fitted to the financial time series data namely Dow Jones Utilities Index data set, Daily Closing Value of the Dow Jones Average and Daily Returns of the Dow Jones Utilities Average Index. Three different estimation methods such as Hannan-Rissanen Algorithm, Whittle’s Estimation and Maximum Likelihood Estimation are used to estimate the parameters of the models. Point forecasts have been done and the performance of all the models and the estimation methods are discussed.

In the current context of profound changes in the planning and operations of electrical systems, many Distribution System Operators (DSOs) are deploying Smart Meters at a large scale. The latter should participate in the effort of making the grid smarter through active management strategies such as storage or demand response. These considerations involve to model electrical quantities as locally as possible and on a sequential basis. This paper explores the possibility to model microscopic loads (individual loads) using Seasonal Auto-Regressive Moving Average (SARMA) time series based solely on Smart Meters data. A systematic definition of models for 18 customers has been applied using their consumption data. The main novelty is the qualitative analysis of complete SARMA models on different types of customers and an evaluation of their general performance in an LV network application. We find that residential loads are easily captured using a single SARMA model whereas other profiles of clients require segmentation due to strong additional seasonalities.

The multivariate relation between sample covariance matrices of errors and their residuals is an important tool in goodness-of-fit methods. This paper generalizes a widely used relation between sample covariance matrices of errors and their residuals proposed by Hosking (J Am Stat Assoc 75(371):602–608, 1980 [6]). Consequently, the asymptotic distribution of the residual correlation matrices is introduced. As an extension of Box and Pierce (J Am Stat Assoc 65(332):1509–1526, 1970 [11]), the asymptotic distribution recommends a graphical diagnostic method to select a proper VARMA(p, q) model. Several examples and simulations illustrate the findings.

Two different mixed first order AR(1) time series models are investigated when the marginal distribution is a two-parameter Beta \(\mathrm{B}_2(p,q)\). The asymptotics of Laplace transform for marginal distribution for large values of the argument shows a way to define novel mixed time-series models which marginals we call asymptotic Beta. The new model’s innovation sequences distributions are obtained using Laplace transform approximation techniques. Finally, the case of generalized functional Beta \(\mathrm{B}_2(G)\) distribution’s use is discussed as a new parent distribution. The chapter ends with an exhaustive references list.

The problem of predicting noisy time series, realization of processes of the type ARIMA (Auto Regressive Integrated Moving Average), is addressed in the framework of digital signal processing in conjunction with an iterative forecast procedure. Other than Gaussian random noise, deterministic shocks either superimposed to the signal at hand or embedded in the ARIMA excitation sequence, are considered. Standard ARIMA forecasting performances are enhanced by pre-filtering the observed time series according to a digital filter of the type Butterworth, whose cut-off frequency, iteratively determined, is the minimizer of a suitable loss function. An empirical study, involving computer generated time series with different noise levels, as well as real-life ones (macroeconomic and tourism data), will also be presented.

The problem of 1/f noise was identified by physicists about a century ago, while the puzzle posed by Hurst’s eponymous effect, originally identified by statisticians, hydrologists and time series analysts, is over 60 years old. Because these communities so often frame the problems in Fourier spectral language, the most famous solutions have tended to be the stationary ergodic long range dependent (LRD) models such as Mandelbrot’s fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal processes (FRP), I present the first results of my research into the history of Mandelbrot’s very little known work on the FRP in 1963–67. I discuss the differences between the Hurst effect, 1/f noise and LRD, concepts which are often treated as equivalent, and finally speculate about how the lack of awareness of his FRP papers in the physics and statistics communities may have affected the development of complexity science.

Outlier detection for time series data is a fundamental issue in time series analysis. In this work we develop statistical methods in order to detect outliers in time series of counts. More specifically we are interesting on detection of an Innovation Outlier (IO). Models for time series count data were originally proposed by Zeger (Biometrika 75(4):621–629, 1988) [28] and have subsequently generalized into GARMA family. The Maximum Likelihood Estimators of the parameters are discussed and the procedure of detecting an outlier is described. Finally, the proposed method is applied to a real data set.

The aim of this paper is to develop stochastic methods for detection whether a change in panel data occurred at some unknown time or not. Panel data of our interest consist of a moderate or relatively large number of panels, while the panels contain a small number of observations. Testing procedures to detect a possible common change in means of the panels are established. To this end, we consider several competing ratio type test statistics and derive their asymptotic distributions under the no change null hypothesis. Moreover, we prove the consistency of the tests under the alternative. The main advantage of the proposed approaches is that the variance of the observations neither has to be known nor estimated. The results are illustrated through a simulation study. An application of the procedure to actuarial data is presented.

Approximately 25% of the world population drinking water depends on karst aquifers. Nevertheless, due to their poor filtration properties, karst aquifers are very sensitive to pollutant transport and specifically to turbidity. As physical processes involved in solid transport (advection, diffusion, deposit…) are complicated and badly known in underground conditions, a black-box modelling approach using neural networks is promising. Despite the well-known ability of universal approximation of multilayer perceptron, it appears difficult to efficiently take into account hydrological conditions of the basin. Indeed these conditions depend both on the initial state of the basin (schematically wet or dry), and on the intensity of rainfalls. To this end, an original architecture has been proposed in previous works to take into account phenomenon at large temporal scale (moisture state), coupled with small temporal scale variations (rainfall). This architecture, called hereafter as “two-branches” multilayer perceptron is compared with the classical two layers perceptron for both kinds of modelling: recurrent and non-recurrent. Applied in this way to the Yport pumping well (Normandie, France) with 12 h lag time, it appears that both models proved crucial information: amplitude and synchronization are better with “two-branches” feed forward model when thresholds surpassing prediction is better using classical feed forward perceptron.

This paper will examine the productivity of the public sectors in the US across the states. Because there is heterogeneity across states in terms of public services provided that could impact its productivity. In fact, there could be a convergence among the states. The services provided by the public sectors have come under increased scrutiny with the ongoing process of reform in recent years. The public sector unlike the private sector or in the absence of contestable markets, and the information and incentives provided by these markets, performance information, particularly measures of comparative performance, have been used to gauge the productivity of the public service sector. This paper will examine the productivity of the public sector across states throughout the United States. The research methodology marries exploratory (i.e. Kohonen clustering) and empirical techniques (panel model) via the Cobb-Douglas production function. Given that there is a homogeneity across states in terms of the use of a standard currency, it will be easy to identify the nature of the convergence process in the public sectors by states throughout the United States.

Time series data is abundant in many areas of practical life such as medical and health related applications, biometric or process industry, financial or economical analysis etc. The categorization of multivariate time series (MTS) data poses problem due to its dynamical nature and conventional machine learning algorithms for static data become unsuitable for time series data processing. For classification or clustering, a similarity measure to assess similarity between two MTS data is needed. Though various similarity measures have been developed so far, dynamic time warping (DTW) and its variants have been found to be the most popular. An approach of time series classification with a similarity measure (Cross Translational Error CTE) based on multidimensional delay vector (MDV) representation of time series has been proposed previously. In this work another new similarity measure (Dynamic Translational Error DTE), an improved version of CTE, and its two variants are proposed and the performance study of DTE(1) and DTE(2) in comparison to several other currently available similarity measures have been done using 43 publicly available bench mark data sets with simulation experiments. It has been found that the new measures produce the best recognition accuracy in larger number of data sets compared to other measures.

In this study, a fuzzy time series modeling method that utilizes a class of customized and flexible parametric membership functions in the fuzzy rule consequents is introduced. The novelty of the proposed methodology lies in the flexibility of this membership function, which we call the composite kappa membership function, and its curve may take various shapes, such as a symmetric or asymmetric bell, triangular, or quasi trapezoid. In our approach, the fuzzy c-means clustering algorithm is used for fuzzification and for the establishment of fuzzy rule antecedents and a heuristic is introduced for identifying the quasi optimal number of clusters to be formed. The proposed technique does not require any preliminary parameter setting, hence it is easy-to-use in practice. In a real-life example, the modeling capability of the proposed method was compared to those of Winters’ method, the Autoregressive Integrated Moving Average technique and Adaptive Neuro-Fuzzy Inference System. Based on the empirical results, the proposed method may be viewed as a viable time series modeling technique.

Consider a time series of signal measurements, x(t), where x has two or more components. This paper shows how to perform nonlinear blind source separation; i.e., how to determine if these signals are equal to linear or nonlinear mixtures of the state variables of two or more statistically independent subsystems. First, the local distributions of measurement velocities are processed in order to derive vectors at each point in x-space. If the data are separable, each of these vectors must be directed along a subspace of \(x \text{-space }\) that is traversed by varying the state variable of one subsystem, while all other subsystems are kept constant. Because of this property, these vectors can be used to construct a small set of mappings, which must contain the “unmixing” function, if it exists. Therefore, nonlinear blind source separation can be performed by examining the separability of the data after they have been transformed by each of these mappings. The method is analytic, constructive, and model-independent. It is illustrated by blindly recovering the separate utterances of two speakers from nonlinear combinations of their audio waveforms.

This paper addresses the problem of constructing nonlinear relationships in complex time-dependent data. We present an approach for learning nonlinear mappings that combines convex optimization for the model order selection problem followed by non-convex optimization for model refinement. This approach exploits the linear system that arises with radial basis function approximations. The first phase of the learning employs the Dantzig-Selector convex optimization problem to determine the number and candidate locations of the RBFs. At this preliminary stage maintaining the supervised learning relationships is not part of the objective function but acts as a constraint in the optimization problem. The model refinement phase is a non-convex optimization problem the goal of which is to optimize the shape and location parameters of the skew RBFs. We demonstrate the algorithm on on the Mackey-Glass chaotic time-series where we explore time-delay embedding models in both three and four dimensions. We observe that the initial centers obtained by the Dantzig-Selector provide favorable initial conditions for the non-convex refinement problem.

In this study, a soft computational method for modeling and forecasting bathtub-shaped failure rate curves of consumer electronic goods is introduced. The empirical failure rate time series are modeled by a flexible function the parameters of which have geometric interpretations, and so the model parameters grab the characteristics of bathtub-shaped failure rate curves. The so-called typical standardized failure rate curve models, which are derived from the model functions through standardization and fuzzy clustering processes, are applied to predict failure rate curves of consumer electronics in a method that combines analytic curve fitting and soft computing techniques. The forecasting capability of the introduced method was tested on real-life data. Based on the empirical results from practical applications, the introduced method may be viewed as a novel, alternative reliability prediction technique.

Most big data analytics research is scattered across multiple disciplines such as applied statistics, machine learning, language technology or databases. Little attention has been paid to aligning big data solutions with end-user’s mental models for conducting exploratory and predictive data analysis. We are particularly interested in the way domain experts perform big data analysis by applying statistics to big data with a focus on statistical learning. In this paper we compare and contrast the different views about data between the fields of statistics and computer science. We review popular analysis techniques and tools within a defined analytics stack. We then propose a model-driven architecture that uses semantic and event processing technologies to achieve a separation of concerns between expressing the mathematical model and the computational requirements. The paper also describes an implementation of the proposed architecture with a case study in funds management.

We investigate various problems encountered when forecasting wind speeds for a network of measurements stations using outputs of numerical weather prediction (NWP) model as one of the predictors in a statistical forecasting model. First, it is interesting to analyze prediction error properties for different station types (professional and amateur). Secondly, the statistical model can be viewed as a calibration of the original NWP model. Hence, careful semi-parametric smoothing of NWP input can discover various weak points of the NWP, and at the same time, it improves forecasting performance. It turns out that useful information is contained not only in the latest prediction available. It is beneficial to combine different horizon NWP predictions to one target time. GARCH sub-model for the residuals then shows complicated structure usable for short-term forecasts.

Time series eye-tracking data, consisting of a sequence of fixations and saccades is a rich source of information for research in the area of cognitive neuroscience. With advanced eye-tracking equipments, many aspects of human perception and cognition are now analyzed from fixations and saccades data. Reading is a complex cognitive process involving a coordination of eye movements on the text and its comprehension. Reading necessitates both a vocabulary sufficient to cover the words in the text, as well as the ability to comprehend the syntax and composition of complex sentences therein. For rapid reading additional factors are involved, like a better peripheral vision. The motivation of this work is to pinpoint lacunae in reading, from her/his eye-tracking data while reading—whether the person lacks in vocabulary, or can not comprehend complex sentences, or needs scanning the text letter by letter which makes the reading very slow. Once the problem for an individual is identified, suggestions to improve reading ability could be made. We also investigated whether there is any basic difference how a native language (L1) and a second language (L2) are read? Is there any difference while reading a text in phonetic script and in logosyllabic script? Eye tracking data was collected while subjects were asked to read texts in their native language (L1) as well as in their second language (L2). Time series data of horizontal axis position and vertical axis position of the location where the fovea is focused, were collected. Simple features were extracted for analysis. For experiments with second language (in this work it is English) subjects belonged to 3 groups: expert, medium proficiency and poor in English. We proposed a formula to evaluate the reading ability, and compared scores with what they obtained in standardized English language test like TOEFL or TOEIC. We also find the correlation of a person’s ability of peripheral vision (by Schultz’s test) and reading speed. The final goal of this work is to build a platform for e-learning of foreign language, while eye-tracking data is analyzed in real-time and appropriate suggestions extended.

In this paper we approach the problem of predicting the start and the end dates for the pollen season of grasses (family Poaceae) and plantains (family Plantago) in the city of Madrid. A classification-based approach is introduced to forecast the main pollination season, and the proposed method is applied to a range of parameters such as the threshold level, which defines the pollen season, and several forecasting horizons. Different computational intelligence approaches are tested including Random Forests, Logistic Regression and Support Vector Machines. The model allows to predict risk exposures for patients and thus anticipate the activation of preventive measures for clinical institutions.