Singular Spectrum Analysis Algorithm Research Articles

Ten daily rainfall forecasting using SSA algorithms and Seasonal ARIMA model to determine the beginning of the rainy season

Rainfall is one of the most important hydrological parameters. Based on the prediction of rainfall, several plans related to rainfall can be carried out, including predicting the start of the planting season based on information on the beginning of the rainy season and preparing cropping patterns. The purpose of this study is to predict the ten daily rainfall for 12 months from January to December of the following year and predict the start of the rainy season based on BMKG criteria. Modeling and forecasting is carried out on the ten daily regional rainfall in the catchment of Citarum-Majalaya water gauge using the Singular Spectrum Analysis (SSA) algorithm and the Seasonal ARIMA model. Forecasting model performance is measured using error values, namely mean absolute percentage error (MAPE). The results of these performance tests illustrate that the SSA algorithm and the Seasonal ARIMA model have fairly good accuracy for predicting rainfall for the next 6 months, where each has a MAPE value of 36.8 percent and 40.0 percent. Prediction of the beginning of the rainy season at the study location based on SSA forecasting results is most likely to occur on the 3rd of October, in accordance with BMKG predictions. whereas based on the results of the SARIMA forecasting model, the beginning of the rainy season is likely to occur on the 1st of November, and another possibility that the beginning of the rainy season will occur on the 2nd of October.

Statistical Analysis of Plasma Dynamics in Gyrokinetic Simulations of Stellarator Turbulence.

A geometrical method for assessing stochastic processes in plasma turbulence is investigated in this study. The thermodynamic length methodology allows using a Riemannian metric on the phase space; thus, distances between thermodynamic states can be computed. It constitutes a geometric methodology to understand stochastic processes involved in, e.g., order-disorder transitions, where a sudden increase in distance is expected. We consider gyrokinetic simulations of ion-temperature-gradient (ITG)-mode-driven turbulence in the core region of the stellarator W7-X with realistic quasi-isodynamic topologies. In gyrokinetic plasma turbulence simulations, avalanches, e.g., of heat and particles, are often found, and in this work, a novel method for detection is investigated. This new method combines the singular spectrum analysis algorithm with a hierarchical clustering method such that the time series is decomposed into two parts: useful physical information and noise. The informative component of the time series is used for the calculation of the Hurst exponent, the information length, and the dynamic time. Based on these measures, the physical properties of the time series are revealed.

An Improved Method of Heart Rate Extraction Algorithm Based on Photoplethysmography for Sports Bracelet

Heart rate measurement employing photoplethysmography (PPG) is a prevalent technique for wearable devices. However, the acquired PPG signal is often contaminated with motion artifacts, which need to be accurately removed. In cases where the PPG and accelerometer (ACC) spectra overlap at the actual heart rate, traditional discrete Fourier transform (DFT) algorithms fail to compute the heart rate accurately. This study proposed an enhanced heart rate extraction algorithm based on PPG to address the issue of PPG and ACC spectral overlap. The spectral overlap is assessed according to the morphological characteristics of both the PPG and ACC spectra. Upon detecting an overlap, the singular spectrum analysis (SSA) algorithm is employed to calculate the heart rate at the given time. The SSA algorithm effectively resolves the issue of spectral overlap by removing motion artifacts through the elimination of ACC-related time series in the PPG signal. Experimental results reveal that the accuracy of the proposed algorithm surpasses that of the traditional DFT method by 19.01%. The proposed method makes up for the deficiency posed by artifact and heart rate signal overlap in conventional algorithms and significantly improves heart rate extraction accuracy.

3-D Data Interpolation and Denoising by an Adaptive Weighting Rank-Reduction Method Using Multichannel Singular Spectrum Analysis Algorithm

Addressing insufficient and irregular sampling is a difficult challenge in seismic processing and imaging. Recently, rank reduction methods have become popular in seismic processing algorithms for simultaneous denoising and interpolating. These methods are based on rank reduction of the trajectory matrices using truncated singular value decomposition (TSVD). Estimation of the ranks of these trajectory matrices depends on the number of plane waves in the processing window; however, for the more complicated data, the rank reduction method may fail or give poor results. In this paper, we propose an adaptive weighted rank reduction (AWRR) method that selects the optimum rank in each window automatically. The method finds the maximum ratio of the energy between two singular values. The AWRR method selects a large rank for the highly curved complex events, which leads to remaining residual errors. To overcome the residual errors, a weighting operator on the selected singular values minimizes the effect of noise projection on the signal projection. We tested the efficiency of the proposed method by applying it to both synthetic and real seismic data.

Harmonic Elimination and Magnetic Resonance Sounding Signal Extraction Based on Matching Pursuit Algorithm

Magnetic resonance sounding (MRS) is a non-invasive, direct, and quantitative geophysical method for detecting groundwater, and has been widely used in groundwater survey, water resource assessment, and disaster water source forecasting. However, the MRS signal is weak (nV level) and highly susceptible to environmental noise, such as random noise and power-line harmonics, resulting in reduced quality of received data. Achieving reliable extraction of MRS signals under strong noise is difficult. To solve this problem, we propose a matching pursuit algorithm based on sparse decomposition theory for data noise suppression and MRS signal extraction. In accordance with the characteristics of the signal and noise, an oscillating atomic library is constructed as a sparse dictionary to realize signal sparse decomposition. A two-step denoising strategy is proposed to reconstruct the power-line harmonics and then extract the MRS signal. We simulated synthetic data with different signal-to-noise ratios (SNRs), relaxation times, and Larmor frequencies. Our results show that the proposed algorithm can effectively remove power-line harmonics and reduce random noise. SNR is significantly improved by up to 35.6 dB after denoising. The effectiveness and superiority of the proposed algorithm are further verified by the measured data and through comparison with the singular spectrum analysis algorithm and harmonic modeling cancellation algorithm.

Research on De-Noising Method of Grounded Electrical Source Airborne Transient Electromagnetic Data Based on Singular Spectrum Analysis

The grounded electrical source airborne transient electromagnetic (GREATEM) system is widely used in groundwater resources detection, geothermal resource detection, geological structure detection, and other fields due to its wide detection range, high detection efficiency, and high resolution. The field data received by the GREATEM system is easily affected by various noises, such as instrument system noise, power frequency noise, sferics noise, and other noise, which reduce the data signal-to-noise ratio (SNR) and affects the data interpretation accuracy. This paper proposes a singular spectrum analysis (SSA) for the GREATEM data de-noising in response to this problem. First, we calculate the electromagnetic response of a uniform half-space using a GREATEM system with an electrical source to verify the effectiveness of the SSA algorithm for GREATEM data de-noising. To determine the appropriate parameters for SSA, we propose a particle swarm optimization algorithm to choose the window length. Later, SSA is used to decompose a synthetic quasi-two-dimensional earth model of GREATEM data. After SSA, the SNR of the reconstructed signal increased by 36 dB, and the RMSE does not exceed 4.9 × 10−6, which verifies the feasibility of the SSA for de-noising GREATEM data. Finally, through field measurement data processing, the effectiveness of the method is further confirmed.

Earth orientation parameters prediction based on the hybrid SSA + LS + SVM model

The high-precision prediction of Earth orientation parameters (EOPs) is essential for astro-geodynamics, high-precision space navigation and positioning, on-board autonomous orbits determination and deep space exploration. However, the prediction accuracy of existing models is much lower than the estimation accuracy of geodetic technical solutions, which affects certain high-precision real-time users. To improve the prediction accuracy of EOP in short- and long-term period, we propose a hybrid model by combining the singular spectrum analysis (SSA), least squares (LSs) and support vector machine (SVM) in the study. Through SSA algorithm, the deterministic time-varying signal of EOP time series can be more precisely and reasonably detected and modeled. Based on the optimization theory, we reconstruct the EOP sequences using SSA and establish the LS extrapolation model based on the reconstructed series. Then, the residuals from SSA reconstruction and those from the LS model, are used for SVM training and prediction. The results of two-year prediction experiments based on the EOP 14 C04 series show that the proposed hybrid model has significant improvements in polar motion (PM) and length of day (LOD) for different prediction intervals (1–360 d) compared with the LS + autoregression (AR) model. The prediction error for x-component of polar motion (PMX) is reduced by 40.2%, 31.0% and 51.4% while that for y-component of polar motion (PMY) is 22.1%, 23.3% and 55.6% for prediction period of 30, 90 and 180 d respectively. For LOD, the maximum prediction improvement can reach to 53.8% during the predicted 360 d. In addition, the proposed method has better accuracy in mid- and long-term PM(x, y) predictions compared to the Bulletin A, with a 360 d prediction error of 27.273 and 21.741 mas.

Frequency-Weighted Singular Spectrum Analysis for Time Series

Singular spectrum analysis (SSA) is a nonparametric spectral decomposition of a time series. A time series is exactly separated into arbitrary number of additive subsequences with the singular value decomposition. Previously, we have shown that SSA algorithm can equivalently be formulated as an optimality condition for the generation of adaptive filters. In this paper, based on our optimal-filter viewpoint, we show that the spectral weight factor can naturally be introduced into the SSA algorithm. With this extension, we can selectively focus on the specific frequency domain of the time series, and then the detailed study of the time series with complicated spectral structure becomes possible.

Singular Spectrum Analysis and Circulant Maximum Variance Frames

Singular spectrum analysis (SSA) is a powerful method that is frequently used in dynamical systems theory and time series analysis. However, the algorithm itself is only partially understood. In this paper, we tackle the problem of a thorough interpretation of the complete basic SSA algorithm. We point out the relationship between SSA and Fourier analysis (FA), linear filtering (LF), principal component analysis (PCA), and power spectrum density estimation (PSDE). The main contribution is a reformulation and comprehensive interpretation of SSA by means of a novel class of data-adaptive circulant frames. Based on this class of frames, we introduce a formalism for time series decomposition and reconstruction. This mathematical framework generalizes principle component analysis and SSA. We show that SSA, PCA, FA, LF, and PSDE can be found from special cases of the proposed frame structures. This way we are able to provide a formal description of SSA and its relationship to the aforementioned methods. Whilst previous approaches to the interpretation of SSA relied on approximations, we provide exact mathematical descriptions of the relation between the above-mentioned techniques.

Sea state from ocean video with singular spectrum analysis and extended Kalman filter

A method for estimating key parameters of ocean waves (the dominant frequency and the significant wave height) from uncalibrated monoscopic video is proposed, based on temporal variation of the wave field, specifically time series of pixel intensities. The methodology tracks the principal component of the movement of water in the video, which we propose is associated with the dominant frequency of the ocean. To accomplish this, the singular spectrum analysis algorithm and the extended Kalman filter are used. Then, the shape of an empirical spectrum is used in order to translate the dominant frequency output into a significant wave height estimation.

Singular Spectrum Analysis for Source Separation in Drone-Based Audio Recording

The usage of drones is increasingly spreading into new fields of application, ranging from agriculture to security. One of these new applications is sound recording in areas of difficult access. The challenge that arises when using drones for this purpose is that the sound of the recorded sources must be separated from the noise produced by the drone. The intensity of the noise emitted by the drone depends on several factors such as engine power, propeller rotation speed, or propeller type. Noise reduction is thus one of the greatest challenges for the next generations of unmanned aerial vehicles (UAVs) and unmanned aerial systems (UAS). Even though some advances have been made on that matter, drones still produce a considerable noise. In this article, we approach the problem of removing drone noise from single-channel audio recordings using blind source separation (BSS) techniques, and in particular, the singular spectrum analysis algorithm (SSA). Furthermore, we propose an optimization of this algorithm with a spatial complexity of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (nt), which is significantly lower than the naive implementation which has a spatial complexity of O(tk <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) (where n is the number of sounds to be recovered, t is the signal length and k is the window size). The best value for each parameter (window length and number of components used to reconstruct the source) is selected by testing a wide range of values on different noise-sound ratios. Our system can greatly reduce the noise produced by the drone on said recordings. On average, after the recording has been processed by our method, the noise is reduced by 1.41 decibels.

Physiological Tremor Filtering Without Phase Distortion for Robotic Microsurgery

All existing physiological tremor filtering algorithms, developed for robotic microsurgery, use nonlinear phase prefilters to isolate the tremor signal. Such filters cause phase distortion to the filtered tremor signal and limit the filtering accuracy. We revisited this long-standing problem to enable filtering of the physiological tremor without any phase distortion. We developed a combined estimation–prediction paradigm that offers zero-phase type filtering. The estimation is achieved with the mathematically modified recursive singular spectrum analysis algorithm, and the prediction is delivered with the standard extreme learning machine. In addition, to limit the computational cost, we developed two moving window versions of this structure, which are appropriate for real-time implementation. The proposed paradigm preserved the natural phase of the filtered tremor. It achieved the key performance index of error limitation below <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10\mu \text{m}$ </tex-math></inline-formula> , yielding the estimation accuracy larger than 70%, at a time delay of 36 ms only. Both moving window versions of the proposed approach restricted the computational cost considerably while offering the same performance. It is the first time that the effective estimation of the physiological tremor is achieved, without any prefiltering and phase distortion. This proposed method is feasible for real-time implantation. Clinical translation of the proposed paradigm can significantly enhance the outcome in hand-held surgical robotics. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The imprecision caused by physiological hand tremor in microsurgeries has motivated researchers to innovate an efficient tremor compensating technique that can improve surgical performance. Yet, all the existing tremor filtering algorithms, implemented in hand-held surgical instruments, use nonlinear phase prefilters to separate the tremor signal. The inherent phase distortion caused by such prefilters restricts the filtering performance significantly and renders the existing methods inadequate for hand-held robotic surgery. Motivated by this, we proposed a novel estimator-predictor-based framework, by adopting the modified recursive singular spectrum analysis estimator and the extreme learning machine predictor. The proposed framework filters the tremor signal accurately, without distorting it, but at a small fixed lag. In a set of rigorous testing performed by emulating real-time processing, the proposed algorithm showed higher performance compared with the state-of-the-art algorithms. This validates not only its suitability for real-time implantation but also its potential to improve surgical performance, which has been limited by the distorted filtering. Nonetheless, we have presented a proof-of-principle framework for distortion-free filtering, but its full implementation in a real surgical instrument, such as Micron or ITrem, requires a substantial amount of experimental testing and verification. It can be also applicable in a wide range of areas, including health-care, digital manufacturing, smart automation and control, and various other robotic technologies where efficient filtering of advanced sensor data is highly desirable. In the future, we will develop the multidimensional model of the proposed framework to enable filtering of tremor in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xyz</i> -axes simultaneously.

Non-Negative Time Series Reconstruction via Singular Spectrum Analysis: A Case Study of Precipitation Dynamics in Nigeria

A proper understanding and analysis of the processes involved in seasonal precipitation variability and dynamics is essential to provide reliable information about climate change and how it can affect matters of critical importance such as water availability and agricultural productivity in urban cities. Precipitation data, as many other time series data present only non-negative observations, are is not constrained by standard time series methods. In this paper, we propose a modified singular spectrum analysis (SSA) algorithm for decomposition and reconstruction of time series with non-negative values. Our approach uses a non-negative matrix factorization (NMF) instead of the singular value decomposition in the SSA algorithm. The new algorithm is compared with the classic SSA algorithm by considering a simulation study and observed data of monthly precipitation of four major cities in Nigeria (Lagos, Kano, Ibadan and Kaduna). Although in terms of mean stared errors both methods give similar results, the percentage of negative fitted values for reconstructions with the classical SSA algorithm reached more than [Formula: see text] in our real data application, which is inappropriate for non-negative time series. The proposed adaptation of the SSA algorithm for non-negative time series data provides an important development with applications in many fields where time series data has non-negative constraints.

A Template Addition Method for Eigentriple Rearrangement in Singular Spectrum Analysis for Processing Biopotential Signals With Extremely Lower SNRs

Biopotential signals captured by wearable devices are generally corrupted by different kinds of noises. Singular Spectrum Analysis (SSA) algorithm can be utilized for processing such signals by reconstructing the signal from the sum of a small number of independent and interpretable components. However, in case of extremely lower signal-to-ratio (SNR) signals, the components to be extracted are often located at the back of the eigentriple list. By analyzing the filtering characteristics of the SSA algorithm, a target component advancing method based on template addition is proposed. The template is constructed based on the frequency features of the processed signal. By splicing the template at the starting point of the processed signal, the energy of the components to be extracted is enhanced and therefore the eigentriples of interest can be effectively advanced to the front of the list. This method is applied to denoising and feature extraction of human physiologic signals with lower SNRs, such as non-contact Electrocardiogram (ECG), Ballistocardiogram (BCG) and Electroencephalogram (EEG). Experimental results show that, with the aid of the proposed method, the computational cost of the eigentriple selection in SSA can be effectively reduced.

Detail-preserving pulse wave extraction from facial videos using consumer-level camera.

With the popularity of smart phones, non-contact video-based vital sign monitoring using a camera has gained increased attention over recent years. Especially, imaging photoplethysmography (IPPG), a technique for extracting pulse waves from videos, conduces to monitor physiological information on a daily basis, including heart rate, respiration rate, blood oxygen saturation, and so on. The main challenge for accurate pulse wave extraction from facial videos is that the facial color intensity change due to cardiovascular activities is subtle and is often badly disturbed by noise, such as illumination variation, facial expression changes, and head movements. Even a tiny interference could bring a big obstacle for pulse wave extraction and reduce the accuracy of the calculated vital signs. In recent years, many novel approaches have been proposed to eliminate noise such as filter banks, adaptive filters, Distance-PPG, and machine learning, but these methods mainly focus on heart rate detection and neglect the retention of useful details of pulse wave. For example, the pulse wave extracted by the filter bank method has no dicrotic wave and approaching sine wave, but dicrotic waves are essential for calculating vital signs like blood viscosity and blood pressure. Therefore, a new framework is proposed to achieve accurate pulse wave extraction that contains mainly two steps: 1) preprocessing procedure to remove baseline offset and high frequency random noise; and 2) a self-adaptive singular spectrum analysis algorithm to obtain cyclical components and remove aperiodic irregular noise. Experimental results show that the proposed method can extract detail-preserved pulse waves from facial videos under realistic situations and outperforms state-of-the-art methods in terms of detail-preserving and real time heart rate estimation. Furthermore, the pulse wave extracted by our approach enabled the non-contact estimation of atrial fibrillation, heart rate variability, blood pressure, as well as other physiological indices that require standard pulse wave.

The Decomposition and Forecasting of Mutual Investment Funds Using Singular Spectrum Analysis.

Singular spectrum analysis (SSA) is a non-parametric method that breaks down a time series into a set of components that can be interpreted and grouped as trend, periodicity, and noise, emphasizing the separability of the underlying components and separate periodicities that occur at different time scales. The original time series can be recovered by summing all components. However, only the components associated to the signal should be considered for the reconstruction of the noise-free time series and to conduct forecasts. When the time series data has the presence of outliers, SSA and other classic parametric and non-parametric methods might result in misleading conclusions and robust methodologies should be used. In this paper we consider the use of two robust SSA algorithms for model fit and one for model forecasting. The classic SSA model, the robust SSA alternatives, and the autoregressive integrated moving average (ARIMA) model are compared in terms of computational time and accuracy for model fit and model forecast, using a simulation example and time series data from the quotas and returns of six mutual investment funds. When outliers are present in the data, the simulation study shows that the robust SSA algorithms outperform the classical ARIMA and SSA models.

Removal of EMG Artifacts from Multichannel EEG Signals Using Combined Singular Spectrum Analysis and Canonical Correlation Analysis.

Electroencephalography (EEG) signals collected from human scalps are often polluted by diverse artifacts, for instance electromyogram (EMG), electrooculogram (EOG), and electrocardiogram (ECG) artifacts. Muscle artifacts are particularly difficult to eliminate among all kinds of artifacts due to their complexity. At present, several researchers have proved the superiority of combining single-channel decomposition algorithms with blind source separation (BSS) to make multichannel EEG recordings free from EMG contamination. In our study, we come up with a novel and valid method to accomplish muscle artifact removal from EEG by using the combination of singular spectrum analysis (SSA) and canonical correlation analysis (CCA), which is named as SSA-CCA. Unlike the traditional single-channel decomposition methods, for example, ensemble empirical mode decomposition (EEMD), SSA algorithm is a technique based on principles of multivariate statistics. Our proposed approach can take advantage of SSA as well as cross-channel information. The performance of SSA-CCA is evaluated on semisimulated and real data. The results demonstrate that this method outperforms the state-of-the-art technique, EEMD-CCA, and the classic technique, CCA, under multichannel circumstances.

Vibration feature extraction based on the improved variational mode decomposition and singular spectrum analysis combination algorithm

Extraction of the vibration characteristics of a flood discharge structure under the influence of intensive background noise is one of the main challenges in vibration-based damage identification. A novel algorithm called normalized central frequency difference spectrum is proposed to improve the variational mode decomposition algorithm for high-frequency noise filtering. To eliminate the errors caused by end effect, the waveform matching extension algorithm is used to further improve the variational mode decomposition. However, the vibration signal is still coupled in low-frequency noise. Thereupon, the singular spectrum analysis algorithm is applied to filter the low-frequency noise. In this article, a simulated signal and the measured signals from a dam model are analyzed by the proposed algorithm. The results indicate that the proposed algorithm is robust to noise and has high denoising precision. In addition, this algorithm can offer clues for damage identification and localization of a flood discharge structure.

Gap-Filling of MODIS Time Series Land Surface Temperature (LST) Products Using Singular Spectrum Analysis (SSA)

Land surface temperature (LST) is a basic parameter in energy exchange between the land and the atmosphere, and is frequently used in many sciences such as climatology, hydrology, agriculture, ecology, etc. Time series of satellite LST data have usually deficient, missing, and unacceptable data caused by the presence of clouds in images, the presence of dust in the atmosphere, and sensor failure. In this study, the singular spectrum analysis (SSA) algorithm was used to resolve the problem of missing and outlier data caused by cloud cover. The region studied in the present research included an image frame of the Moderate Resolution Imaging Spectroradiometer (MODIS) with horizontal number 22 and vertical number 05 (h22v05). This image involved a large part of Iran, Turkmenistan, and the Caspian Sea. In this study, MODIS LST products (MOD11A1) were used during 2015 with approximately 1 km × 1 km spatial resolution and day/night LST data (daily temporal resolution). On average, the data have 36.37% gaps in each pixel profile with 730 day/night LST data. The results of the SSA algorithm in the reconstruction of LST images indicated a root mean square error (RMSE) of 2.95 Kelvin (K) between the original and reconstructed LST time series data in the study region. In general, the findings showed that the SSA algorithm using spatio-temporal interpolation can be effectively used to resolve the problem of missing data caused by cloud cover.

Removal of Electrocardiogram Interference from Diaphragmatic Electromyogram Signals using Sliding Singular Spectrum Analysis.

Over recent years, Singular Spectrum Analysis (SSA) has gained popularity as an effective means to denoise biologically sourced single channel signals, especially Electromyogram (EMG) and Electrocardiogram (ECG) signals amongst others. There are numerous applications whereby the signal acquisition process results in the mixing of both types of signals along with body motion artifacts and the inevitable electromagnetic interference. Both ECG and EMG signals are very useful to physicians, though preferably in isolation, though they rarely present themselves in this manner. Simple filtering techniques are ineffective in their separation as both signal spectra overlap in the frequency domain. In this paper, we propose a technique based on a sliding SSA algorithm which proves to be more successful in separating real mixed EMG and ECG signals than traditional block based approaches on single channel data. SSA is a non-parametric technique that decomposes the original time series into a number of additive components, each of which can then be readily identified based on statistical analysis as belonging to EMG or ECG signals. This approach could be applied equally to other signal types using different statistical methods as required, moreover, this technique is relatively straight-forward to implement and does not require any reference signals or training.