A robust approach for outlier imputation: Singular spectrum decomposition

This study introduces singular spectrum decomposition (SSD), a new adaptive method for decomposing nonlinear and nonstationary time series in narrow-banded components. The method takes its origin from singular spectrum analysis (SSA), a nonparametric spectral estimation method used for analysis and prediction of time series. Unlike SSA, SSD is a decomposition method in which the choice of fundamental parameters has been completely automated. This is achieved by focusing on the frequency content of the signal. In particular, this holds for the choice of the window length used to generate the trajectory matrix of the data and for the selection of its principal components for the reconstruction of a specific component series. Moreover, a new definition of the trajectory matrix with respect to the standard SSA allows the oscillatory content in the data to be enhanced and guarantees decrease of energy of the residual. Through the numerical examples and simulations, the SSD method is shown to be able to accurately retrieve different components concealed in the data, minimizing at the same time the generation of spurious components. Applications on time series from both the biological and the physical domain are also presented highlighting the capability of SSD to yield physically meaningful components.

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.

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 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) 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.

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

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.

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.

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.

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.

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).

<p>The North Atlantic Oscillation (NAO) index describes the hemispheric meridional oscillation of atmospheric masses above near Iceland and the subtropical Atlantic regions. In specific, NAO index indicates the differences in atmospheric pressure patterns between the two regions. Strong positive NAO index values relate to significant pressure differences, exposing US East and Northern Europe to warmer weather conditions, and Southern Europe to colder weather conditions. Negative NAO index values indicate weaker pressure differences, exposing US East and Northern Europe to cold weather, and Southern Europe to warm weather. A significant portion of the Atlantic sector climate is associated with NAO index and its variability. Historically, NAO index values were in a positive trend between 1970s and 1980s. Highest positive values were reported in the early 1990s. By that time, it was suggested that NAO index positive trends contributed significantly to the global warming signal. Most recently, research outputs from climate model predictions suggest that NAO index values will be more at the positive phase as a result of strong global warming signal. In a warmer climate the overall number of storms is predicted to decrease but storms will be more intense. However, more research is needed to understand variability trends in NAO index and to what extent they might be attributed to climate change impacts. Hence referred in here as NAO natural and non-natural, or else, climate change related variability trends. This study investigated the NAO variability trends by use of Singular Spectrum Analysis (SSA) and SSA based Entropy index. SSA is a statistical mechanics tool used to study the non-linear behavioral characteristics in complex geophysical, meteorological and climatic systems, monitored by time series data. The main objective of this analysis is to reveal the evolution of the NAO index dynamical system and convey information about the changing dynamics of the system. By use of SSA Entropy based index, the chaotic behavior of NAO index is studied. An SSA entropy based chaotic descriptor might entail information of the non-natural variability trend for NAO index values. Also the same descriptor might prove capable of defining a historical milestone of when this NAO variability trend started changing in an unpredictable- non natural- way owed to climate change forcing factors. NAO Index data are extracted from Climatic Research Unit, University of East Anglia from 1979-2018. SSA Phase space reconstruction by method of delays has been applied first to characterize the statistical and chaotic behavior of NAO patterns, by calculating variability and inconsistency descriptors. Phase space reconstruction allows analyzing time series data within the dynamics systems theory context. Following that, reconstructed attractor from the NAO observed time series allowed to build an approximation of the unknown observed states. Results revealed a highly variable and inconsistent behavior in NAO patterns over time. SSA Entropy based index investigation is currently underway to further understand the nature of inconsistency revealed in NAO patterns. Further research is expected to establish wind and wave storm patterns connections with NAO index patterns, through transfer dynamics concepts.</p>