Detecting seismic response signals using singular spectrum analysis

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

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.

Singular Spectrum Analysis (SSA) is an effective method of decomposition of time series for separating key components in data, such as trends, seasonality, and noise. This study aims to analyze the role of Hankel matrix in the SSA embedding process and how window length (L) selection can affect the effectiveness of component separation in data time series. In this study, the data used includes public data that can be influenced by seasonal factors and unexpected events, such as natural disasters or regulatory changes. The research process begins with the data preprocessing stage, followed by the embedding stage to form a matrix used in decomposition with Singular Value Decomposition (SVD). To evaluate the similarity of separate components, w-correlation is used. The results show that the selection of optimal window lengths, in the range of N/4 < L < N/2 is very important to maintain a balance between temporal information and matrix dimensions. With the right window selection, the embedding process in SSA can be more effective in separating the trending, seasonal, and noise components in the data pattern. By understanding the structure of the Hankel matrix and selecting the right parameters, the embedding process in SSA can be more effective in separating the components of the time series and preserving temporal information.

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.

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.

Singular spectrum analysis and forecasting of hydrological time series

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 (SSA) has recently been applied to various geodetic time series studies. As a data-adaptive method, SSA is capable of extracting signals with non-constant phase and amplitudes. Although SSA is a competent method in the presence of white noise, the contribution of colored noise, having semi-periodic behavior, degrades its performance. Parts of colored noise can be absorbed in the SSA eigenmodes, which specifies signals and hence resulting in spurious modulation or losing significant signals. Signals and colored noise are thus to be discriminated in the signal identification procedure. Monte Carlo SSA (MCSSA) in its original formulation, providing a significance test against the AR(1) noise null hypothesis, can be misinterpreted when other colored noise structures contribute to the series. We propose an algorithm for MCSSA that is not limited to the AR(1) noise hypothesis. It estimates the noise model parameters using LS-VCE and generates the surrogate data using the Cholesky decomposition. The algorithm is adapted to GPS position time series where the underlying noise is a combination of white noise and flicker noise. GPS position time series, postulated real situation, are first simulated to include annual and semiannual signals plus white and flicker noise. The results indicate that MCSSA can extract the annual and semiannual signals with 2.11 and 1.25 mm amplitudes (the global mean values) from 20-year-long time series, with 95% confidence level, if flicker noise is less than 17 and 13 $$ {\text{mm/year}}^{1/4} $$ , respectively. The longer the time series or the stronger the signals are, the higher these thresholds will be. This conclusion is also verified when applying MCSSA to the up component of GPS position time series of 347 JPL stations.

This paper presents the technique of Singular Spectrum Analysis (SSA) and its application for electric load forecasting purposes. SSA is a relatively new non-parametric data-driven technique developed to model non-linear and/or non-stationary, noisy time series. SSA is able to decompose the original time series into the sum of independent components, which represent the trend, oscillatory behavior (periodic or quasi- periodic components) and noise. One of the main advantages of SSA compared to other non-parametric approaches is that only two parameters are required to model the time series under analysis. An example of application is given, with regards to forecasting the monthly electric load demand in a Venezuelan region served by wind power generators. In this case, careful demand estimation is required since the wind generation output could be highly variable and additional conventional generation or transmission links would be required to satisfy the load demand. A comparison with other classical time-series approaches (like exponential smoothing and Auto Regressive Integrated Moving Average models) is presented. The results show that SSA is a powerful approach to model time series, capable of identifying sub-time series (trends along with seasonal periodic components).

This paper proposes a novel approach for improving the identifiability of structural damage by using higher order responses and singular spectrum analysis (SSA)-based phase space topology (PST) technique. SSA is employed to decompose the structural dynamic acceleration response into several low-dimensional components. Then, the change of PST (CPST) reconstructed from the decomposed component with higher order responses of structures under intact and damaged states is utilized to identify the structural condition changes. Numerical studies on a planar truss structure are conducted to investigate the accuracy and reliability of the proposed approach. The dynamic acceleration responses of the truss structure corresponding to intact and several damage scenarios are utilized for signal decomposition by using SSA and damage detection with CPST. The effects of white noise, pink noise, and modeling uncertainties on the accuracy of damage detection are investigated. The proposed method is also validated through experimental studies on a Tee-section prestressed concrete beam model. Vibration acceleration responses under hammer impact loads are measured under intact and damaged states, which are used for the damage detection with the proposed approach. Damage patterns identified by using the proposed approach under different loading levels agree well with the observed crack distributions. Both numerical and experimental results demonstrate that using the higher order response components decomposed by SSA and the proposed method is sensitive to damage and reliable for structural damage detection.

Rotating devices such as gears often generate fault vibration signals during eration. Under the interference of strong external noise, it is often difficult to extract these weak fault feature. To this end, a method based on Complementary Ensemble Empirical Mode Decomposition (CEEMD) and Singular Spectrum Analysis (SSA) is proposed to reduce the noise of the gear signals. Firstly, original vibration signals are decomposed into a series of intrinsic mode functions (IMFs) by CEEMD, and the reconstructed signals are obtained after the some IMFs are removed according to evaluation coefficients. Then, the reconstructed signals are further denoised using SSA. After singular value decomposition, grouping and reconstruction, the denoised time series are obtained. The actual gear vibration signals from the QPZZ-II rotating machinery fault simulation rig are used to verify the effectiveness of the proposed method. The results of comparing with the different denoising methods show that the presented method can effectively depress the strong noise, and extract weak fault feature.

This paper proposes a heuristic singular spectrum analysis (SSA) approach to extract signals from suspended sediment concentration (SSC) time series contaminated by multiplicative noise, in which multiplicative noise is converted to approximate additive noise by multiplying with the signal estimate of the time series. Therefore both the signal and noise components need to be recursively estimated. Since the converted additive noise is heterogeneous, a weight factor is introduced according to the variance of additive noise. The proposed heuristic SSA approach is employed to process the SSC series in San Francisco Bay compared to the traditional SSA and homomorphic log-transformation SSA approach. By using our heuristic SSA approach, the first 10 principal components derived can capture 96.49% of the total variance with the fitting error of 6.17 mg/L, better than those derived by traditional SSA approach and homomorphic log-transformation SSA approach that catch 88.97% and 87.35% of the total variance with the fitting errors of 14.47 mg/L and 15.03 mg/L, respectively. Therefore, our heuristic SSA approach can extract more signals than traditional SSA and homomorphic log-transformation SSA approach. Furthermore, the results from the simulation cases show that all the mean root mean squared errors and mean absolute errors derived by our heuristic SSA are smaller than the traditional and homomorphic log-transformation SSA, which indicate that the extracted signals by heuristic SSA approach are much closer to the real signals than those by the other two approaches. Therefore it can be conclude that our heuristic SSA approach performs better in extracting signals from SSC time series contaminated with multiplicative noise.

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.

Background: Rice plays a central role in Indonesia's food security and inflation dynamics, making accurate price forecasts crucial for effective planning policies. In East Kalimantan, weekly rice prices represent a significant risk factor that needs to be closely monitored to anticipate potential risks to household welfare and regional inflation. Objective: To predict rice prices in East Kalimantan over the next eight periods using the SSA method. Methods: Singular Spectrum Analysis (SSA) to predict weekly rice prices. The SSA procedure includes embedding (window length L = 48), singular value decomposition, eigentriple clustering (trend: 1–2; seasonality: 3–35; disturbance: 36–48), diagonal averaging, and R-forecasting using the Linear Recursive Formula (LRF). Results: MAPE of 0.084% (in-sample) and 3.299% (out-sample). Estimates indicate that rice prices will remain in the range of IDR 15,500–16,500/kg during August–October 2025. Conclusion: The SSA is effective in capturing basic price components and producing reliable forecasts to support food reserve maintenance, distribution strategies, and inflation risk mitigation.

In ocean acoustic fields, extracting the normal-mode interference spectrum (NMIS) from the received sound intensity spectrum (SIS) plays an important role in waveguide-invariant estimation and underwater source ranging. However, the received SIS often has a low signal-to-noise ratio (SNR) owing to ocean ambient noise and the limitations of the received equipment. This can lead to significant performance degradation for the traditional methods of extracting NMIS at low SNR conditions. To address this issue, a new deep neural network model called SSANet is proposed to obtain NMIS based on unrolling the traditional singular spectrum analysis (SSA) algorithm. First, the steps of embedding and singular value decomposition (SVD) in SSA is achieved by the convolutional network. Second, the grouping step of the SSA is simulated using the matrix multiply weight layer, ReLU layer, point multiply weight layer and matrix multiply weight layer. Third, the diagonal averaging step was implemented using a fully connected network. Simulation results in canonical ocean waveguide environments demonstrate that SSANet outperforms other traditional methods such as Fourier transform (FT), multiple signal classification (MUSIC), and SSA in terms of root mean square error, mean absolute error, and extraction performance.