Time Series

In multivariate time series, outlying data may be often observed that do not fit the common pattern. Occurrences of outliers are unpredictable events that may severely distort the analysis of the multivariate time series. For instance, model building, seasonality assessment, and forecasting may be seriously affected by undetected outliers. The structure dependence of the multivariate time series gives rise to the well-known smearing and masking phenomena that prevent using most outliers' identification techniques. It may be noticed, however, that a convenient way for representing multiple outliers consists of superimposing a deterministic disturbance to a gaussian multivariate time series. Then outliers may be modeled as nongaussian time series components. Independent component analysis is a recently developed tool that is likely to be able to extract possible outlier patterns. In practice, independent component analysis may be used to analyze multivariate observable time series and separate regular and outlying unobservable components. In the factor models framework too, it is shown that independent component analysis is a useful tool for detection of outliers in multivariate time series. Some algorithms that perform independent component analysis are compared. It has been found that all algorithms are effective in detecting various types of outliers, such as patches, level shifts, and isolated outliers, even at the beginning or the end of the stretch of observations. Also, there is no appreciable difference in the ability of different algorithms to display the outlying observations pattern.

The ability of the locomotor system to maintain continuous walking despite very small external or internal disturbances is called local dynamic stability (LDS). The importance of the LDS requires constantly working on different aspects of its assessment method which is based on the short-term largest Lyapunov exponent (LLE). A state space structure is a vital aspect of the LDS assessment because the algorithm of the LLE computation for experimental data requires a reconstruction of a state space trajectory. The gait kinematic data are usually one- or three-dimensional, which enables to construct a state space based on a uni- or multivariate time series. Furthermore, two variants of the short-term LLE are present in the literature which differ in length of a time span, over which the short-term LLE is computed. Both a state space structure and the consistency of the observations based on values of both short-term LLE variants were analyzed using time series representing the joint angles at ankle, knee, and hip joints. The short-term LLE was computed for individual joints in three state spaces constructed on the basis of either univariate or multivariate time series. Each state space revealed walkers' locally unstable behavior as well as its attenuation in the current stride. The corresponding conclusions made on the basis of both short-term LLE variants were consistent in ca. 59% of cases determined by a joint and a state space. Moreover, the authors present an algorithm for estimation of the embedding dimension in the case of a multivariate gait time series.

In finance, cluster analysis is a tool particularly useful for classifying stock market multivariate time series data related to daily returns, volatility daily stocks returns, commodity prices, volume trading, index, enhanced index tracking portfolio, and so on. In the literature, following different methodological approaches, several clustering methods have been proposed for clustering multivariate time series. In this paper by adopting a fuzzy approach and using the Partitioning Around Medoids strategy, we suggest to cluster multivariate financial time series by considering the dynamic time warping distance. In particular, we proposed a robust clustering method capable to neutralize the negative effects of possible outliers in the clustering process. The clustering method achieves its robustness by adopting a suitable trimming procedure to identify multivariate financial time series more distant from the bulk of data. The proposed clustering method is applied to the stocks composing the FTSE MIB index to identify common time patterns and possible outliers.

Multivariate networked time series data are ubiquitous in many applications, where multiple variables of interest are sequentially collected over time for each vertex as multivariate time series. Data of different vertices may have heterogeneous influence on each other through the network topology. These time series may usually exhibit multimodal marginal distributions, due to complex system variations. In this article, we propose a novel approach for such data modeling. In particular, we assume that each vertex has multiple latent states and exhibits state-switching behaviors according to a Markov process. The multivariate time series of each vertex depend on a defined latent effect variable influenced by both its own latent state and the latent effects of its neighbors through a heterogeneous network autoregressive model according to the network topology. Furthermore, the influence of some exogenous covariates on the time series can also be incorporated in the model. Some model properties are discussed, and a variational EM algorithm is proposed for model parameter estimation and state inference. Extensive synthetic experiments and a real-world case study demonstrate the effectiveness and applicability of the proposed model.

Assessment of the length of rainfall time series for rainwater harvesting in buildings

[1] A recent article by Rosenlof and Reid [2008] (hereinafter referred to as RR08) examines the relationship in the tropics between long-term cooling in the stratosphere and warming of the sea surface, with special emphasis in the western Pacific. In the stratosphere, radiosonde observations seem to show that most of the cooling took place since the mid-1990s, apparently coincident to a considerable extent with an increase in the rate of warming of sea surface temperatures (SSTs). A link was proposed by RR08 whereby the increase in SSTs leads to enhanced convection, which is responsible, dynamically, for cooling in the stratosphere. [2] The stratospheric analyses of RR08 rely considerably on tropical radiosonde observations, particularly those from several island stations in the western tropical Pacific. There is extensive evidence that the long-term variation in worldwide radiosonde temperatures is affected critically by inhomogeneities introduced through changes in instruments and measurement practices [e.g., Gaffen, 1994; Gaffen et al., 2000; Lanzante et al . , 2003a, 2003b; Climate Change Science Program (CCSP), 2006], a fact which is acknowledged by RR08. However, they minimize the importance of any such inhomogeneities in the six stations located in the ‘‘warm pool’’ region which they employ [RR08, paragraph 10]: ‘‘However, for our analysis, and in particular for changes noted post-1990, these discontinuities are not sufficient to preclude the use of these stations.’’ The purpose of this comment is to demonstrate that in fact such discontinuities are present, post-1990, and critically affect the evaluation of stratospheric temperatures made by RR08. [3] In their Figures 2, 3, and 5, RR08 display temperature time series for several levels in the lower stratosphere for the four stations located in the heart of the western Pacific warm pool. They note that while most of the earlier part of the records show little long-term trend, a pronounced cooling is seen for about the last 10 years of the record ( 1995–2005). Furthermore, they note an abrupt rise in temperature in early 1999 that lasts for about 2 years. As further evidence to bolster their findings, they note [RR08, paragraph 12] the similarity of the time series from the different stations: ‘‘. . . the overall similarity of the time series in Figure 2 shows that the assumption of homogeneity over the warm pool region is reasonable.’’ Unfortunately, there is considerable evidence to dispute the veracity of these claims. [4] The apparent substantial cooling during the final 10 years of the record can be explained largely by a change in radiosonde instrument type, specifically the transition from Viz to Vaisala, hereafter referred to as the VVT. Abrupt cooling upon introduction of Vaisala radiosondes was noted by Parker et al. [1997] for stations in Australia and New Zealand in the mid to late 1980s, and for U.S. controlled tropical Pacific stations beginning in 1995 [Stendel et al., 2000]. More generally, there has been a systematic worldwide change to Vaisala radiosondes since the 1980s with an expected artificial cooling, especially in the stratosphere [Lanzante et al., 2003a; Christy et al., 2003; CCSP, 2006] (see also Integrated Global Radiosonde Archive (IGRA) station history metadata available at http:// www1.ncdc.noaa.gov/pub/data/igra/igra-metadata.txt). [5] The IGRA metadata indicate that the VVT occurs in November or December 1995 for five of the six stations shown in Figure 1 of RR08 (Lihue, Truk, Koror, Yap, and Pago Pago). The literature cited above suggests that the major part of the drop in stratospheric temperature after 1995 (as seen quite clearly in Figure 5 of RR08) is almost certainly artificial. To further illustrate this point, Figure 1a shows several time series of stratospheric temperatures at Koror. The thick black curve is based on the IGRA data [Durre et al., 2006], which were used by RR08. Note that the IGRA data, although of very high quality (i.e., they have been carefully checked for consistency in time and in the vertical, random errors and outliers, as well as duplicate soundings, etc.), have not been adjusted to account for artificial inhomogeneities. A number of teams have produced adjusted radiosonde temperature time series, each using a different homogenization method. Also shown in Figure 1a are corresponding adjusted time series (HadAT2) [Thorne et al., 2005], RATPAC [Free et al., 2005], IUK [Sherwood et al., 2008], RAOBCORE 1.4 [Haimberger et al., 2008], and RICH [Haimberger et al., 2008] produced by these teams. In comparing the IGRA with the various adjusted time series in Figure 1a it is seen that the adjusted versions all show much less long-term cooling, as indicated by the accompanying trend lines. A closer look at Figure 1a reveals that most of the difference is attributable to the transition across 1995. The adjusted time series shows a JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, D12104, doi:10.1029/2008JD010542, 2009 Click Here for Full Article

Chapter 5 - Time-series Analysis Methods

The ice core time series from Vostok Station in Antarctica and the North Greenland Ice Core Project have seasonal variation corresponding to the Milankovitch cycles. After removing these cycles, and interpolating to equal time intervals, stationary time series models are fitted. The series show clear directionality and this feature is modelled by either non-Gaussian errors or non-linear time series models. Threshold autoregressive models are fitted by penalized least squares and compared with non-threshold autoregressive models. Since both ice core time series are reasonably modelled as first order autoregressive series with parameters close to one, directionality will arise from non-symmetric error distributions. However, two regime threshold autoregressive models, of order one and two for Greenland and Vostok, respectively, give an improved match to the observed directionality and a reduced sum of squared residuals. Realizations from the threshold autoregressive models are noticeably different from the non-threshold models. Since the non-threshold models are a restricted case of the threshold models, and the threshold models are a better fit to the observed time series, threshold models should provide more realistic realizations. References C. Chatfield. The Analysis of Time Series: An Introduction . CRC Press, 2004. https://www.crcpress.com/The-Analysis-of-Time-Series-An-Introduction-Sixth-Edition/Chatfield/p/book/9781584883173 A. J. Lawrance. Directionality and reversibility in time series. Int. Stat. Rev. 59(1):67–79, 1991. doi:10.2307/1403575 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Threshold autoregressive models for directional time series. In I. Rojas and H. Pomares (Eds.), Time Series Analysis and Forecasting Selected Contributions from the ITISE Conference (ITISE 2015) . pp. 13–25, 2016. doi:10.1007/978-3-319-28725-6 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Modelling directionality in stationary geophysical time series. International work-conference on Time Series (ITISE 2015) . http://www.researchgate.net/publication/281835075 M. M. Mansor, D. A. Green and A. V. Metcalfe. Modelling and simulation of directional financial time series. Proceedings of the 21st International Congress on Modelling and Simulation (MODSIM 2015) , pp. 1022–1028, 2015. http://www.mssanz.org.au/modsim2015/E4/mansor.pdf M. M. Mansor, D. A. Green and A. V. Metcalfe. Directionality and volatility in electroencephalogram time series. Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS 2016), AIP Conf. Proc. 1739:020080, 2016. doi:10.1063/1.4952560 North Greenland Ice Core Project members. High-resolution record of Northern Hemisphere climate extending into the last interglacial period. Nature , 431:147–151, 2004. doi:10.1038/nature02805 J. R. Petit, J. Jouzel, D. Raynaud, N. I. Barkov, J.-M. Barnola, I. Basile, M. Bender, J. Chappellaz, M. Davis, G. Delaygue, M. Delmotte, V. M. Kotlyakov, M. Legrand, V. Y. Lipenkov, C. Lorius, L. Pepin, C. Ritz, E. Saltzman and M. Stievenard. Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature , 399:429–436, 1999. doi:10.1038/20859 S. Soubeyrand, C. E. Morris and E. K. Bigg. Analysis of fragmented time directionality in time series to elucidate feedbacks in climate data. Environ. Modell. Softw. 61:78–86, 2014. doi:10.1016/j.envsoft.2014.07.003

Multivariate time series (MTS) arise due to multisensor data collection in manufacturing. These data are complex in the sense that attributes have a varying scale, volitivity, continuity, and so on, and interattribute dependence also appears, which can mask the inherent information about system health status. Conventional machine learning-based process monitoring techniques are inefficient in predicting faults with MTS—their detection capability heavily relies on the input features, yet the classification power of MTS attributes is weakened by the complexity of multisensor data. Effective feature extraction is, therefore, necessary to facilitate fault prediction with MTS. This study proposes a fault prediction framework for MTS based on bag-of-words (BOW) feature extraction, statistical feature selection, and classification analysis. BOW models are for the first time adopted in a manufacturing context. Their superior capability in information preservation, local pattern recognition, and temporal effect accommodation has overcome the major limitations in current manufacturing practices with MTS. A comprehensive case study demonstrates the desirable performance of this framework on two MTS data sets from paper manufacturing and automotive manufacturing, as well as its superiority over conventional machine learning-based fault prediction. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Process monitoring in a multisensor environment has been a vital interest in manufacturing. The difficulty lies in the lack of detection power in many conventional techniques, e.g., control chart and logistic regression. A critical reason for such failure is the neglect of time effect in MTS—patterns associated with system faults tend to stretch a period, but a conventional control chart or classifier inspects each time stamp separately. Fault detection based on time series sequences is, therefore, essential. However, how to effectively extract features from MTS becomes a challenge. The framework proposed in this study adopts BOW models, specifically symbolic aggregate approximation (SAX), to extract features from MTS sequences, thus substantially improves the detection power against local patterns over time. Many manufacturing multisensor data are subject to such local patterns that point to the root cause of fault, so the proposed framework has a wide application in manufacturing.

Updating Landsat time series of surface-reflectance composites and forest change products with new observations

Many types of financial time series, most notably market returns, have been found to exhibit long-range memory as well as dramatic day-to-day swings that cannot be adequately represented by light-tailed distributions such as the normal distribution. In particular, this means that for such time series, the usual variance parameter (i.e., the sum of covariances at all time lags) is not defined because the covariance function does not converge to zero fast enough as the time lag increases. Moreover in such time series, often the tails of the marginal density converge to zero so slowly that higher-order marginal moments such as skewness and kurtosis fail to exist. Therefore conventional methods for analyzing simulation-generated time series cannot generally be applied to high-fidelity simulations of financial markets.Building on earlier work in fractal geometry and fractal time series, Mandelbrot et al. proposed the multifractal model of asset returns (MMAR) as an alternative to the ARCH models for analyzing time series exhibiting volatility clustering, long-range dependence, and heavy-tailed returns (Mandelbrot et al. 1997). They defined the multifractal spectrum as the renormalized probability density function of the Holder exponents observed in the time series; and they used the multifractal spectrum to measure the ability of MMAR to match the statistical properties of real data. In 2002 Kantelhardt et al. formulated multifractal detrended fluctuation analysis (MF-DFA), an algorithm for extracting the multifractal spectrum from a time series (Kantelhardt et al. 2002).Many economists have recently adopted agent-based simulation for modeling financial markets. Fads based on new products, shocks related to world events, scandals involving company leaders, or outright criminal activity can drastically change the processes and relationships governing a market. Already there is evidence that agent-based models yield more accurate approximations to the true or observed behavior in financial markets. Agent-based models exhibit emergent behaviors that have been linked to non-Gaussian interaction metrics and singularities in the time series they generate (Chan 2011). Farmer et al. constructed a agent-based model that is capable of exhibiting many of the statistical properties observed in real financial data simply due to the rules governing a double auction order book (Farmer et al. 2005).During the 20th century, economists relied heavily on Brownian motion to construct theoretical models of finance. The advantages of using Brownian motion stem from its foundation in the Gaussian distribution; and with advances in stochastic calculus and differential equations, it became a staple in models of empirical finance. The well-known Black-Scholes model for options pricing is a prime example of this application. Multifractal analysis has the ability to illuminate the underlying difference between a monofractal like Brownian motion and a considerably more diverse construction like the multifractal binomial measure. Thus, combining multifractal analysis with an agent-based model has the potential to provide insight into the underlying processes and behaviors of a dynamic economic system that is constantly changing and evolving.To analyze market-return time series exhibiting volatility clustering, long-range dependence, or heavy-tailed marginals, we exploit multifractal analysis and agent-based simulation. We develop a robust, automated software tool for extracting the multifractal spectrum of a time series based on MF-DFA. Guidelines are given for setting MF-DFA's parameters in practice. The software is tested on simulated data with closed-form monofractal and multifractal spectra as well as on observed data, and the results are analyzed. We also present a prototype agent-based financial market model based on the zero-intelligence model of Farmer et al. and analyze its output using MF-DFA. The figure below compares the multifractal spectrum of the output from the prototype model with that of the multifractal binomial measure. The ultimate objective is to expand this model to study the effects of microlevel agent behaviors on the macrolevel time series output as analyzed by MF-DFA. Finally we explore the potential for validating agent-based models using MF-DFA and thus being able to tune these models to the multifractal spectrum of empirical data.

Causality or driving in electrophysiological signal analysis

[EN] In this thesis it is introduced a method to filter GNSS height time series from vertical displacements derived from load variations in the Earth's surface, obtained from the monthly geopotential coefficients from the GRACE mission. These load variations are originated mainly by the seasonal hydrological variations as it can be confirmed from the hydrological GLDAS model. This filtering method is based on spectral analysis techniques that allow the manipulation of time series in both time and frequency domains.\n\nTo test this method data from 66 IGS Core Network reference stations have been used. The origin of the GNSS data are the weekly solutions provided by the First Data Reprocessing Campaign carried out by IGS. Monthly level-2 data from GRACE Mission provided by GFZ, with a time-span of more than 11 years, have been used to generate hydrological load and equivalent water thickness monthly time series at every reference station from which GNSS data are available. In addition, soil water content time series have also been created from monthly GLDAS data.\n\nWith the periodograms of the four time series datasets a signal with frequency close to 1 cycle per year is detected in the GNSS height, load, equivalent water thickness and GLDAS time series. As this signal has the same frequency in the equivalent water thickness and GLDAS time series it is confirmed that its origin is the hydrological cycle. Therefore the variations observed in the GNSS height and load time series can be explained, at least part of it, by these hydrological variations.\n\nThe spectral filtering technique introduced in this thesis is based on the detection and isolation of the annual signal in the load time series from GRACE data. This process is done in the frequency domain by using the Discrete Fourier Transform (DFT) with the Fast Fourier Transform (FFT) algorithm. Using the linearity property of the DFT, the isolated annual signal from the load time series is subtracted from the DFT of the GNSS height time series in the frequency domain. The new time series in the frequency domain is the GNSS height without the annual signal generated by the hydrological variations in the GNSS reference station area.\n\nAfter testing the spectral filter with times series dataset from 66 reference stations, it is demonstrated that for stations in areas with important hydrological activity the GNSS height time series reduce their WRMS by 18 % on average, with some special cases where WRMS reduction is higher than 25 %. Other parameters like GNSS height vs. GRACE load correlation before and after the filtering process show a reduction of the correlation from 53.9 % to 15.0 % In areas where the hydrological variations are not clearly detected by GNSS and GRACE, like islands, coasts or deserts, there is no WRMS reduction, and other parameters do not show any improvement in the GNSS height time series after the filtering process.

Rare earth elements price forecasting by means of transgenic time series developed with ARIMA models

The article will present the methodology for holistic time series analysis, based on time series transformation model REFII (REFII is an acronym for Raise-Equal-Fall and the model version is II or 2) Patel et al. (Mining motifs in massive time series databases, 2002) [1], Perng and Parker (SQL/LPP: a time series extension of SQL based on limited patience patterns, 1999) [2], Popivanov and Miller (Similarity search over time series data using wavelets, 2002) [3]. The main purpose of REFII model is to automate time series analysis through a unique transformation model of time series. The advantage of this approach to a time series analysis is the linkage of different methods for time series analysis linking traditional data mining tools in time series, and constructing new algorithms for analyzing time series. REFII model is not a closed system, which means that there is a finite set of methods. This is primarily a model used for transformation of values of time series, which prepares data used by different sets of methods based on the same model of transformation in the domain of problem space. REFII model gives a new approach in time series analysis based on a unique model of transformation, which is a base for all kind of time series analyses. In combination with elements of other methods, such as self-organizing maps or frequent-pattern trees, REFII models can make new hybrid methods for efficient time temporal data mining. Similar principle of hybridization could be used as a tool for time series temporal pattern recognition. The article describes real case study illustrating practical application of described methodology.