Singular Spectrum Research Articles

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.

Hourly significant wave height prediction via singular spectrum analysis and wavelet transform based models

Generation of significant wave height (SWH) is considered as a complex and stochastic dynamical process whose prediction is vital for effective marine, ocean engineering, sustainable development and scientific phenomena. Based on literature review, despite numerous research attempts to forecast significant wave height with short prediction time horizons, they are incapable in yielding accurate SWH predictions. To achieve this end wavelet techniques have been intensively employed as data pre-processing tools and, have been incorporated with soft computing based approaches to improve the prediction performance of developed models. However, wavelet algorithm has some limitations such as shift sensitivity, poor directionality and lack of phase information. In addition, this technique suffers from time consuming complicated mathematical procedures. In the present study, as a way of addressing the shortcomings of wavelet tool and enhancing prediction accuracy with extended time horizons, singular spectrum analysis (SSA) is proposed as a decomposition procedure. The prediction accuracy of the three distinct models is contrasted by means of diagnostic metrics, Mean Square Error (MSE), the Nash-Sutcliffe Coefficient of efficiency (CE) and determination of coefficient (R2). With its lowest MSE and highest CE value SSA-Fuzzy model clearly outperformed the stand alone Fuzzy and W-Fuzzy models in predicting hourly SWH for all stations and future time horizons. This implies that SSA technique has the utmost capability to decompose measured data effectively into its deterministic and stochastic components.

Spectral characteristics of Schrödinger operators generated by product systems

Motivated by the question of what spectral properties of dynamically defined Schrödinger operators may be preserved under periodic perturbations, we study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.

Plane-wave least-squares diffraction imaging using short-time singular spectrum analysis

Abstract Diffractions are seismic waves generated by small-scale heterogeneities in the subsurface. These are often superimposed by strong reflections so that they are not visible on the image, leading to misinterpretation and incorrect localization of the scatterers. Therefore, the separation of diffracted and reflected waves is a crucial step in identifying these small-scale diffractors. To realize the separation of diffraction and imaging, a least-squares reverse time migration method of plane waves (PLSRTM) optimized with short-time singular spectrum analysis (STSSA) was developed in this work. The proposed STSSA algorithm exploits the properties of singular spectral analysis (SSA) to separate linear signals. By establishing the Hanning window and the energy compensation function, it also compensates for the shortcomings of SSA in local dip processing and convergence of linear signals. As there is no clear boundary between reflected and diffracted waves, the energy loss during separation leads to a slow convergence rate of the diffraction wave imaging technique. We use STSSA as a constraint for PLSRTM, which greatly improves the imaging quality for diffraction waves. The tests with the Sigsbee2A model and noisy seismic data have shown that our method can effectively improve the resolution of diffraction wave imaging and that the constraint of STSSA increases the robustness to noisy data.

A combined prediction model based on secondary decomposition and intelligence optimization for carbon emission

Accurate prediction of carbon emission is critical for the development of low-carbon economy. However, most carbon emission prediction studies use a single model with low prediction accuracy, and do not consider the instability of carbon emission. Therefore, this paper proposes a combined prediction model of carbon emission. Firstly, the original data is decomposed by singular spectrum decomposition to obtain a limited amount of singular spectrum components. Secondly, high complexity components are secondarily decomposed by variational mode decomposition. Then, chameleon swarm algorithm and carnivorous plant algorithm are used to train the regularization coefficients and kernel parameters of kernel extreme learning machine and least squares support vector machine respectively, and the trained model is used to predict the decomposition components. Finally, induced ordered weighted averaging operator is used to calculate the weight of single model, and error correction is introduced to further promote the prediction accuracy. The carbon emission data of China and the United States is used to make a prediction experiment. The results indicate that the proposed model is superior to other comparative models in different indexes, which provides a new idea for carbon emission prediction.

The Frisch-Parisi conjecture II: Besov spaces in multifractal environment, and a full solution

Frisch-Parisi conjecture claims the existence of Baire function spaces in which Baire typical functions share the same multifractal behavior, prescribed in advance, and obey a multifractal formalism. In this paper, we introduce a family B of heterogeneous Besov spaces, which generalize the standard Besov spaces - they are obtained by replacing the Lebesgue measure (which plays a key role in the definition of the standard Besov spaces) by multifractal Radon measures belonging to some class constructed in the companion paper [1]. We find a characterization of the elements of B in terms of wavelet coefficients, and then describe the multifractal properties (singularity spectrum, validity of the multifractal formalism) of their Baire typical functions. This allows us to fully solve the Frisch-Parisi conjecture inside B.

Multivariate two dimensional singular spectrum analysis based fusion method for four view image based object classification

Multivariate two dimensional singular spectrum analysis based fusion method for four view image based object classification

The Frisch-Parisi conjecture I: Prescribed multifractal behavior, and a partial solution

In this work and its companion [1], we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family Ed of almost-doubling fully supported capacities on Rd with prescribed singularity spectra is constructed. With each μ∈Ed we associate a Baire function space Bμ(Rd) (a generalisation of Hölder-Zygmund spaces) in which typical functions share the same singularity spectrum as μ. This yields a partial solution to the conjecture. In [1], we introduce and study a family B={Bqμ,p(Rd)}μ∈Ed,(p,q)∈[1,+∞]2 of heterogeneous Besov spaces that contains {Bμ(Rd)}μ∈Ed and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B.

Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.

Evolving cycles and self-organised criticality in social dynamics

In many complex systems, self-organised criticality (SOC) provides a mechanism for the diversity of spatiotemporal scales that optimises the system’s response to omnipresent driving forces. Signatures of SOC are increasingly more evidenced in collective social behaviours. However, the spontaneous occurrence of critical states and their role in maintaining the system’s functional properties still need to be better understood; the reason can be related to the complexity of human interactions and the ubiquitous presence of cycles in social dynamics. In this work, we shed new light on these issues based on a critical survey and the extensive data analysis of online social dynamics. Firstly, we highlight prominent features of human activity patterns, conditioned by circadian cycles and content-related interactions, that can affect the course of the dynamics from the elemental to the global scale. We then analyse the prototypal time series of emotion-driven communications in the online social network MySpace to demonstrate the coexistence of SOC states with the modulated cyclical trends. Precisely, we determine avalanches of emotional comments exhibiting multifractal scaling, scale-invariant inter-avalanching behaviours and temporal correlations coexist with the cyclical trends of broad singularity spectra. We demonstrate that similar multi-harmonic cycles occur in entirely different datasets, particularly the negative emotion-driven Diggs and the infection-rate data from recent epidemics. Our results reveal the dynamical regime where the modulated cycles coexist with self-organised critical states; in contrast, in the cycles-dominated regime, exemplified by the infection time series, the nature of collective dynamics remains hidden behind the cycle modulation.

Data‐Driven Gap Filling and Spatio‐Temporal Filtering of the GRACE and GRACE‐FO Records

AbstractGravity Recovery And Climate Experiment and Follow On (GRACE/‐FO) global monthly measurements of Earth's gravity field have led to significant advances in quantifying mass transfer. However, a significant temporal gap between missions hinders evaluating long‐term mass variations. Moreover, instrumental and processing errors translate into large non‐physical North‐South stripes polluting geophysical signals. We use Multichannel Singular Spectrum Analysis (M‐SSA) to overcome both issues by exploiting spatio‐temporal information of Level‐2 GRACE/‐FO solutions, filtered using the DDK7 decorrelation and a new complementary filter, built based on the residual noise between fully processed data and a parametric fit to observations. Using an iterative M‐SSA on Equivalent Water Height (EWH) time series processed by Center of Space Research, GeoForschungsZentrum, Institute of Geodesy at Graz University of Technology, and Jet Propulsion Laboratory, we replace missing data and outliers to obtain a combined evenly sampled solution. Then, we apply M‐SSA to retrieve common signals between each EWH time series and its same‐latitude neighbors to further reduce residual spatially uncorrelated noise. Comparing GRACE/‐FO M‐SSA solution with Satellite Laser Ranging and Swarm low‐degree Earth's gravity field and hydrological model demonstrates its ability to satisfyingly fill missing observations. Our solution achieves a noise level comparable to mass concentration (mascon) solutions over oceans (3.0 mm EWH), without requiring a priori information nor regularization. While short‐wavelength signals are challenging to capture using highly filtered spherical harmonics or mascons solutions, we show that our technique efficiently recovers localized mass variations using well‐documented mass transfers associated with reservoir impoundments.

Quantifying uncertainty of marine water quality forecasts for environmental management using a dynamic multi-factor analysis and multi-resolution ensemble approach

Unpredictable climate change and human activities pose enormous challenges to assessing the water quality components in the marine environment. Accurately quantifying the uncertainty of water quality forecasts can help decision-makers implement more scientific water pollution management strategies. This work introduces a new method of uncertainty quantification driven by point prediction for solving the engineering problem of water quality forecasting under the influence of complex environmental factors. The constructed multi-factor correlation analysis system can dynamically adjust the combined weight of environmental indicators according to the performance, thereby increasing the interpretability of data fusion. The designed singular spectrum analysis is utilized to reduce the volatility of the original water quality data. The real-time decomposition technique cleverly avoids the problem of data leakage. The multi-resolution-multi-objective optimization ensemble method is adopted to absorb the characteristics of different resolution data, so as to mine deeper potential information. Experimental studies are conducted using 6 actual water quality high-resolution signals with 21,600 sampling points from the Pacific islands and corresponding low-resolution signals with 900 sampling points, including temperature, salinity, turbidity, chlorophyll, dissolved oxygen, and oxygen saturation. The results illustrate that the model is superior to the existing model in quantifying the uncertainty of water quality prediction.

Carbon trading and COVID-19: a hybrid machine learning approach for international carbon price forecasting.

Accurate carbon price forecasting can better allocate carbon emissions and thus ensure a balance between economic development and potential climate impacts. In this paper, we propose a new two-stage framework based on processes of decomposition and re-estimation to forecast prices across international carbon markets. We focus on the Emissions Trading System (ETS) in the EU, as well as the five main pilot schemes in China, spanning the period from May 2014 to January 2022. In this way, the raw carbon prices are first separated into multiple sub-factors and then reconstructed into factors of 'trend' and 'period' with the use of Singular Spectrum Analysis (SSA). Once the subsequences have been thus decomposed, we further apply six machine learning and deep learning methods, allowing the data to be assembled and thus facilitating the prediction of the final carbon price values. We find that from amongst these machine learning models, the Support vector regression (SSA-SVR) and Least squares support vector regression (SSA-LSSVR) stand out in terms of performance for the prediction of carbon prices in both the European ETS and equivalent models in China. Another interesting finding to come out of our experiments is that the sophisticated algorithms are far from being the best performing models in the prediction of carbon prices. Even after accounting for the impacts of the COVID-19 pandemic and other macro-economic variables, as well as the prices of other energy sources, our framework still works effectively.

Three dimensional seismic data reconstruction based on truncated nuclear norm

The rank reduction method is widely used to reconstruct three dimensional (3D) seismic data. The traditional multichannel singular spectrum analysis (MSSA) utilizes truncated singular value decomposition (TSVD) to approximately solve the rank function of the block Hankel structure matrix constructed by frequency slices of seismic data. However, the TSVD algorithm discards all nonzero singular values except a few largest singular values and ignores the useful seismic information in small nonzero singular values. To further utilize the seismic data information contained in these small singular values, this paper proposes a method to recover 3D seismic data with truncated nuclear norm (TNN). The proposed method imposes different constraints on large singular values and small singular values. Essentially, the TNN is closer to the rank function than the TSVD. Finally, the proposed model is addressed by the alternating direction method of multipliers, and closed-form solutions are used to update the optimization variables. The experimental results demonstrate that the proposed method achieves superior reconstruction results than the traditional MSSA method.

APPLICATION OF SINGULAR SPECTRUM ANALYSIS METHOD IN FORECASTING INDONESIA COMPOSITE DATA

The wellbeing of the public is a key state objective. To attain this objective, developments are required, including economic development. Economic development can be initiated by enhancing a state's economic growth, as it describes the state's economic conditions. Forecasting future economic conditions is one of the things that may be done to ensure economic stability. Investment business can be utilized as indicators, with the Indonesia Composite Index (ICI) being one of them. Singular Spectrum Analysis (SSA) is one of the available techniques for forecasting. Due to the fact that SSA is non-parametric, no assumptions must be met, resulting in high performance and adaptability. Thus, SSA will be utilized for forecasting ICI. The ICI data utilized is obtained from Yahoo Finance. On the basis of the forecasting result for the closing price of ICI from March 2, 2020 to March 28, 2022 using SSA, which yielded MAPE values of 1.59% for training data and 4.84% for testing data, it can be inferred that this method is accurate. The outcome also revealed that the tendency tends to rise over the next few periods.

Financial cycles: how long and how certain?

Understanding and measuring financial cycles is important to academics and policy-makers, particularly those in charge of macroprudential policy. While business cycle periodic components have been largely estimated with different methods, empirical evidence related to financial cycles’ periodicities is scarce. Worse, ad hoc filters, are commonly used by policy-makers to calculate the credit gap and make important policy decisions based on the amplitude of this gap. To address the issue of depicting financial cycles, we estimate the credit gap and credit periodic components in 28 countries using Bayesian Structural Time Series Methods (STM) and Singular Spectrum Analysis (SSA). We find financial cycles of 13 to 20 years in most countries, but some present periodicities close to those of the business cycle. We also find discrepancies between our estimated cycles and the ad hoc filter calibration usually adopted by policy-makers to measure the credit gap, which conceal uncertainty and can be misleading.

Irregularly sampled seismic data interpolation with self-supervised learning

Supervised convolutional neural networks (CNNs) are commonly used for seismic data interpolation, in which a recovery network is trained over corrupted (input)/complete (label) pairs. However, the trained model always suffers from poor generalization when the target test data are significantly different from the training data sets. To address this issue, we have developed a self-supervised deep learning method for interpolating irregularly missing traces, which uses only the corrupted seismic data for training. This approach is based on the receptive field characteristic of CNNs, and the training pairs are extracted from the corrupted seismic data through a random trace mask. After network training, all target data are recovered using the trained model. This self-supervised learning interpolation (SSLI) method can be easily integrated into commonly used CNNs. Synthetic and field examples demonstrate that SSLI not only significantly outperforms traditional multichannel singular spectrum analysis and unsupervised deep seismic prior methods but also competes with supervised learning methods.

Sliding mode control for longitudinal oscillating combustion

Considerable research has been reported on developing effective active control means to suppress oscillating combustion. The typical pressure oscillation can be divided into linear growth, transition and saturation stages. In this study, a sliding mode control strategy, consisting of a state estimate model, disturbance observers and a sliding mode controller, is proposed to suppress the longitudinal oscillating combustion. The control strategy is first tested with a nonlinear 0D state space model as the controlled plant. Results show that the state estimate model combined with the singular spectrum analysis (SSA) method can accurately estimate the system state quantities by grouping the SSA modes according to the frequency difference and calculating mode envelopes. To ensure the estimate accuracy, the number of truncated SSA modes varies according to the oscillation stage. The disturbance observers are designed to improve the robustness of the controller by introducing broadband spectrum disturbance to account for the external noise in the observed values. The sliding mode controller can limit the disturbance amplitude, and effectively suppress the pressure oscillation. A 1D Rijke tube acoustic network is also tested to further validate the controller adaptability. With this controller, the Rijke tube pressure oscillation can be effectively eliminated when control starts at the linear growth, transition, or saturation stages.

Prediction of air quality index based on the SSA-BiLSTM-LightGBM model

The air quality index (AQI), as an indicator to describe the degree of air pollution and its impact on health, plays an important role in improving the quality of the atmospheric environment. Accurate prediction of the AQI can effectively serve people’s lives, reduce pollution control costs and improve the quality of the environment. In this paper, we constructed a combined prediction model based on real hourly AQI data in Beijing. First, we used singular spectrum analysis (SSA) to decompose the AQI data into different sequences, such as trend, oscillation component and noise. Then, bidirectional long short-term memory (BiLSTM) was introduced to predict the decomposed AQI data, and a light gradient boosting machine (LightGBM) was used to integrate the predicted results. The experimental results show that the prediction effect of SSA-BiLSTM-LightGBM for the AQI data set is good on the test set. The root mean squared error (RMSE) reaches 0.6897, the mean absolute error (MAE) reaches 0.4718, the symmetric mean absolute percentage error (SMAPE) reaches 1.2712%, and the adjusted R2 reaches 0.9995.

A RUL prediction method for lithium-ion batteries based on improved singular spectrum analysis and CSA-KELM

Accurate and efficient capacity prediction of lithium-ion batteries plays an important role in improving performance and ensuring safe operation. An improved singular spectral analysis (ISSA) based on the fractal dimension (FD) is proposed to eliminate capacity regeneration during battery degradation. Firstly, FD is used to determine whether the signal decomposition component belongs to the main component or the noise is a useless component. Secondly, the kernel extreme learning machine (KELM) is introduced to predict the main components. In addition, a new heuristic optimization algorithm, the circle search algorithm (CSA), is used to optimize the regularization coefficients and kernel function parameters of KELM. Finally, experimental simulations are performed based on four batteries from two open lithium-ion battery datasets. The results show that the ISSA algorithm has better decomposition efficiency than the empirical mode decomposition (EMD) and the variational mode decomposition (VMD). Compared with other intelligent optimization algorithms, the CSA is superior and can achieve a faster convergence rate.