SINGULAR SPECTRUM DECOMPOSITION: A NEW METHOD FOR TIME SERIES DECOMPOSITION

Singular spectrum analysis (SSA) is a nonparametric method for separating time series data into a sum of small numbers of interpretable components (signal + noise). One of the steps of the SSA method, which is referenced to Embedding, is extremely sensitive to contamination of outliers which are often founded in time series analysis. To reduce the effect of outliers, SSA based on Singular Spectrum Decomposition (SSD) method is proposed. In this article, the ability of SSA based on SSD and basic SSA are compared in time series reconstruction in the presence of outliers. It is noteworthy that the matrix norm used in Basic SSA is the Frobenius norm or L 2-norm. There is a newer version of SSA that is based on L 1-norm and called L 1-SSA. It was confirmed that L 1-SSA is robust against outliers. In this regard, this research is also introduced a new version of SSD based on L 1-norm which is called L 1-SSD. A wide empirical study on both simulated and real data verifies the efficiency of basic SSA based on SSD and L 1-norm in reconstructing the time series where polluted by outliers.

Rotating machinery fault signals often consist of multiple components with time varying frequencies under variable speed conditions. Spectral overlap exists among these components, making it difficult to independently separate the features of the components. Singular spectrum decomposition (SSD), a singular spectrum analysis-based signal decomposition method, has shown its great potential in suppressing background noise and extracting fault-related components in complex background noise environments. However, SSD is a frequency domain decomposition method with equivalent filtering characteristics, and it is susceptible to the mode mixing when processing signals with spectral overlap. Moreover, the choice of a key parameter in the iteration decomposition process of SSD, the embedding dimension, is determined using an empirical formula, which might cause suboptimal decomposition outcomes. To address these issues, this paper proposes a generalized adaptive singular spectrum decomposition (GASSD) method, which combines generalized demodulation with improved embedding dimension selection for SSD. GASSD incorporates SSD into the framework of adaptive generalized demodulation to separate specific frequency domain features. Firstly, for an effective generalized demodulation analysis, a region block synchronous ridge extraction method is proposed to accurately estimate the instantaneous frequency ridges from the time-frequency plane, which helps construct proper demodulation phase functions. Secondly, to achieve optimal analysis of SSD, a Gini moderation decomposition index is designed to improve the construction of the trajectory matrix by determining an appropriate embedding dimension. Finally, the reliability of the proposed method is demonstrated by analyzing wind turbine generator bearing fault signals and rotor rubbing fault signals.

Singular spectrum analysis (SSA) is a nonparametric spectral estimation method that decomposes time series signals into interpretable components. With the rise of big time series, the demand for effective and scalable SSA techniques has become increasingly urgent. In this paper, we propose a novel multiway extension of SSA, called higher-order multivariate SSA (HO-MSSA), specifically designed for multivariate and multichannel time series signal analysis via tensor decomposition. HO-MSSA utilizes time-delay embedding and tensor singular value decomposition to transform multichannel time series signals into trajectory tensors, which are then decomposed into elementary components in the Fourier domain, rather than the time domain as in traditional SSA methods. These components are grouped into disjoint subsets using spectral clustering, enabling the reconstruction of the underlying source signals. Experimental results demonstrate that HO-MSSA outperforms state-of-the-art SSA methods in various biomedical applications, including electromyography (EMG), electrocardiography (ECG), and electroencephalogram (EEG) signals. • A novel tensor-based multivariate singular spectrum analysis is introduced. • A new time embedding technique embeds time series into a higher dimension. • Tensor SVD is used to factorize the trajectory tensor of time series. • Spectral clustering detects and groups the underlying time series components.

In the paper the algorithm of singular structural analysis and forecasting of a multidimensional series by the MSSA method is used. The program was developed, in which the steps of the method for the selection of singular decomposition components were implemented, the analysis and forecast of real time series was carried out. Singular spectrum analysis (SSA) is increasingly used in the study of time series. Unlike other methods of statistical research of time series, this method is used to study the structure, selection of individual components and forecast of both stationary and non-stationary time series. It does not require an analytical model of the series. In fact, this approach is based on the method of principal components. It is based on the transformation of a series into a matrix and its singular decomposition. After identifying the components of the singular decomposition, their grouping takes place, which leads to the decomposition of the original series into additive components, such as trend, oscillations (periodics), and noise. The SSA method allows you to continue the structure of the time series, thus building a forecast (continuation). An important direction of the development of the SSA method as a method of time series analysis is its generalization for the analysis of multidimensional time series. The method is known as MSSA (Multi-Channel SSA) or E-EOFs (Extended Empirical Orthogonal Functions). In this case, the expected result is the simultaneous decomposition of several series into interpreted components. However, a sufficiently complete theory for MSSA does not exist.

Singular spectrum analysis (SSA) is relatively new method for analysis of non-stationary time series. The weakness of SSA is lack of analytical model representation of time series, e.g., as a sum of simple functions, which could be clearer and easier for interpretation than a large number of components in form of time series. In this paper we propose to use variative modeling, based on joint use of SSA and method of modeleteka, for obtaining of analytical model of time series, providing necessary level of adequacy, compactness and interpretability. First, time series are decomposed into components using SSA, significant components are selected using formal indicators (e.g. variance contributed by component, etc.). Second, each significant component is identified according to the purpose of identification with simple and interpretable model from preformed modeleteka. The result is final model of time series in additive or additive-multiplicative form. Applicability of the method is shown on synthetic data and time series of daily changes of water turbidity in the river in the city of Chelyabinsk in 2005.

Singular spectrum analysis and forecasting of hydrological time series

Singular spectrum analysis and Fisher–Shannon analysis of spring flow time series: An application to Anjar Spring, Lebanon

Singular spectrum analysis (SSA) is relatively new method for analysis of non-stationary time series. In this paper we propose to use variative modeling, based on joint use of SSA and method of modeleteka (models warehouse), for obtaining of analytical model of time series that combines adequacy, compactness, and interpretability. First, time series are decomposed into components using SSA; significant components are selected using formal indicators. Second, each significant component is identified according to the purpose of identification with simple and interpretable model from preformed modeleteka. The result is final model of time series in additive or additive-multiplicative form. Applicability of the method is illustrated on synthetic data.

Non-stationary time series plays a prominent role in the analysis of performance time series of many real-world systems. Recently, fuzzy time series models have been extended to forecast non-stationary time series. Over different phases of time, performance time series may show drastic changes. Therefore, a non-stationary time series is partitioned according to different phases of time. These phases may be taken as weeks, months, or years. Over different phases of time, the universe of discourse, knowledge base, and rule base may vary. A common constraint in the modelling of time-variant fuzzy time series is their incapability to address the phase change of time in the model and accordingly incorporate the necessary changes in the universe of discourse, knowledge base, and rule base over different time phases. To address this issue, a Phase-cum-Time Variant Fuzzy Time Series Model (PTVFTS) is presented in this paper. The PTVFTS model is developed so that it will address the problem of phase change as well as the time variations within each phase simultaneously. The concept of the model rebuilding process is applied to handle changes over each phase and the modified parameter adaptation technique is used for the time variations within each phase. The developed model is applied to the daily closing price time series for the years 2017, 2018, 2019, 2020, 2021, and 2022 separately of stock market indices, NASDAQ, S&P 500, Dow Jones, and TAIEX. The comparison of the developed model is made with the time-variant fuzzy time series model known as the non-stationary fuzzy time series model (NSFTS), Dynamic Evolving Neural-Fuzzy Inference System (DENFIS) model, Long Short-Term Memory (LSTM) model, and the classical Auto-Regressive Integrated Moving Average (ARIMA) model. The efficiency of the developed PTVFTS model is tested using forecasting metrics and statistical tests. The comparison shows that the developed model is more efficient in forecasting phase-cum-time variant non-stationary time series.

Accurate clustering of time series is a challenging problem for data arising from areas such as financial markets, biomedical studies, and environmental sciences, especially when some, or all, of the series exhibit nonlinearity and nonstationarity. When a subset of the series exhibits nonlinear characteristics, frequency domain clustering methods based on higher-order spectral properties, such as the bispectra or trispectra are useful. While these methods address nonlinearity, they rely on the assumption of series stationarity. We propose the Bispectral Smooth Localized Complex EXponential (BSLEX) approach for clustering nonlinear and nonstationary time series. BSLEX is an extension of the SLEX approach for linear, nonstationary series, and overcomes the challenges of both nonlinearity and nonstationarity through smooth partitions of the nonstationary time series into stationary subsets in a dyadic fashion. The performance of the BSLEX approach is illustrated via simulation where several nonstationary or nonlinear time series are clustered, as well as via accurate clustering of the records of 16 seismic events, eight of which are earthquakes and eight are explosions. We illustrate the utility of the approach by clustering S&P 100 financial returns.

Singular Spectrum Analysis (SSA) is a novel non-parametric technique based on principle of multivariate statistics. The original time series is decomposed into a number of additive time series, each of which can be easily identified as being part of the modulated signals, or as being part of the random noise. It shows that the embedding dimension m of a dynamical time series can be conducted by the Singular Value Decomposition (SVD) experiments. This SVD scheme was used to detect low-dimensionally dynamic signals and the residuals. It provides trend extraction involves a decomposition of a time series into low-frequency trends and high-frequency variability. In this study, first, SSA is used to decompose the nonlinear seismic responses of reinforced concrete frames and to elucidate permanent deformation. Then, damage feature extraction is conducted using the high-frequency variability of SSA to identify the occurrence of damage. Comparison the results with the Holder exponent and the Level-1 detail of the discrete wavelet component to detect the damage occurrence was made. Finally, using SSA to estimate the permanent deformation using recorded acceleration data was also discussed. In this study, four reinforced concrete frame test data collected in response to various degrees of seismic excitation are used to demonstrate the application of SSA in damage detection.

Optical performance monitoring (OPM) becomes an inviting topic in high speed optical communication networks. In this paper, a novel technique of OPM based on a new elaborated computation approach of singular spectrum analysis (SSA) for time series prediction is presented. Indeed, various optical impairments among chromatic dispersion (CD), polarization mode dispersion (PMD) and amplified spontaneous emission (ASE) noise are a major factors limiting quality of transmission data in the systems with data rates lager than 40 Gbit/s. This technique proposed an independent and simultaneous multi-impairments monitoring, where we used SSA of time series analysis and forecasting. It has proven their usefulness in the temporal analysis of short and noisy time series in several fields, that it is based on the singular value decomposition (SVD). Also, advanced optical modulation formats (100 Gbit/s non-return-to zero dual-polarization quadrature phase shift keying (NRZ-DP-QPSK) and 160 Gbit/s DP-16 quadrature amplitude modulation (DP-16QAM)) offering high spectral efficiencies have been successfully employed by analyzing their asynchronously sampled amplitude. The simulated results proved that our method is efficient on CD, first-order PMD, Q-factor and OSNR monitoring, which enabled large monitoring ranges, the CD in the range of 170-1700 ps/nm.Km and 170-1110 ps/nm.Km for 100 Gbit/s NRZ-DP-QPSK and 160 Gbit/s DP-16QAM respectively, and also the DGD up to 20 ps is monitored. We could accurately monitor the OSNR in the range of 10-40 dB with monitoring error remains less than 1 dB in the presence of large accumulated CD.

This study proposes a model‐free approach for day‐ahead electricity price forecasting. The proposed approached is based on the singular spectrum analysis (SSA) technique. The SSA is a relatively new and powerful technique in time series analysis and forecasting thanks to its well‐known capabilities in extracting the main structure of the broad classes of the time series. In this study, it is shown that SSA can be employed to decompose the original electricity price series into trend, periodic and noisy components. The main part of the price series, that is, the trend and harmonic components, is reconstructed by removing the noise component from the original series. The reconstructed price series is then used for forecasting the day‐ahead electricity prices. The proposed approach is evaluated by analysing and forecasting of the day‐ahead electricity prices in the Australian and Spanish electricity markets. The forecasting results confirm the superiority of the SSA approach compared with some of the recently published forecasting techniques.

Rivers act as the primary harmonizer of the critical zone, reflecting the confluence of processes that regulate nutrient cycling, water quality, and energy flow within ecosystems. Understanding water quality dynamics and their relationship with hydrological and biogeochemical processes warrants the study of river chemistry at a detailed temporal resolution, but this is often hindered by logistical challenges in the acquisition of such data. Recent efforts in automated data acquisition, such as the RiverLab “lab-on-a-field” setup (Floury et al. 2017) have equipped us with a toolbox to peek into riverine systems at unprecedentedly fine temporal scales. This assemblage consists of a primary circuit that continuously samples water from the middle of a stream, and measures various physico-chemical parameters using probes. A portion of this water is also routed through a filtration system towards two ion chromatographs housed in the setup, allowing the measurement of dissolved ions. This system thus enables sub-hourly measurement of major ions and near-continuous monitoring of pH, Temperature, Conductivity, etc. However, despite these advances in data acquisition, technical challenges remain. The datasets often contain gaps due to practical limitations like sensor drift and malfunction, power outages, etc. These induce gaps in the multivariate time series datasets, and can impede the reliable application of statistical techniques, introduce bias in model outputs and obscure periodicities. Collectively, these pose a risk of misconstruing the true water quality patterns. To address these challenges and for an efficient understanding of high-frequency stream chemistry, here we present a novel two-pronged approach: Reconstruction and forecasting of stream chemistry using machine learning (ML) techniques; and Understanding trends and seasonalities in stream chemistry using Singular Spectrum Analysis (SSA). Reconstruction and forecasting of stream chemistry using machine learning (ML) techniques; and Understanding trends and seasonalities in stream chemistry using Singular Spectrum Analysis (SSA). We used one year of high-frequency stream chemistry measurements from the RiverLab at Orgeval Critical Zone Observatory (Loumagne and Tallec 2013), a constituent site of the OZCAR network (Gaillardet et al. 2018) of French Critical Zone Observatories. By incorporating the temporal structure and physico-chemical parameters (Discharge, Temperature, Conductivity, pH) as inputs, multiple ML-based models were developed and applied to predict the dissolved major ions in the stream. Preliminary results hold considerable promise, with the mean absolute percentage error (MAPE) on Cl- prediction remaining below 5% for forecasts extending up to one year. This underscores the potential of ML-assisted approaches in complementing cost-effective and accessible water quality monitoring, owing to the lower operating costs and near-continuous data output provided by the probes. Following this, we discuss the applications of Singular Spectrum Analysis (SSA) on the reconstructed RiverLab data. SSA is a non-parametric technique that combines aspects of classical time series analysis and multivariate statistics (e.g. Principal Component Analysis) to decompose the time series into trends, seasonalities and residual components without relying on the assumption of stationarity (Vautard and Ghil 1989). This study marks one of the first applications of SSA in hydrogeochemistry, and is proposed as a novel tool for exploring concentration-discharge (C-Q) relationships, water quality forecasting, anomaly detection and interpretation of periodic signals in streams.

A robust time series prediction method based on empirical mode decomposition and high-order fuzzy cognitive maps