Singular Spectrum Analysis Components Research Articles

Generalized singular spectrum analysis for the decomposition and analysis of non-stationary signals

Singular spectrum analysis (SSA) has been verified to be an effective method for decomposing non-stationary signals. The decomposition and reconstruction stages can be interpreted as a zero-phase filtering process where reconstructed components are obtained by inputting a signal through moving average filters. However, mathematical analysis indicates that the use of a default rectangular window in the embedding stage would corrupt the frequency characteristics of the trajectory matrix, resulting in spectral leakage. To attenuate the effect of spectral leakage and to obtain more concentrated SSA components, this study introduces a windowing technique in SSA, called generalized singular spectrum analysis (GSSA). In GSSA, the default rectangular window is replaced with adjustable taper windows, which are widely used for attenuating spectral leakage. Through windowing, GSSA can achieve less spectral leakage, and produce more energy-concentrated reconstructed components compared with conventional SSA. Grouped spectral entropy (GSE) is used as the metric for evaluating the performance of the proposed GSSA algorithm. Results from experiments, which were conducted on a synthetic signal and two real electroencephalogram signals, show that GSSA outperforms the conventional SSA and the baseline in the reduction of spectral leakage. Compared with the baseline, the proposed GSSA achieves a lower averaged GSE, resulting in reduction of 0.4 for eigenfilters and 0.11 for reconstructed components, respectively. Our results reveal the effectiveness of GSSA in the decomposition and analysis of non-stationary signals.

Twenty-Year Spatiotemporal Variations of TWS over Mainland China Observed by GRACE and GRACE Follow-On Satellites

Terrestrial water storage (TWS) is a pivotal component of the global water cycle, profoundly impacting water resource management, hazard monitoring, and agriculture production. The Gravity Recovery and Climate Experiment (GRACE) and its successor, the GRACE Follow-On (GFO), have furnished comprehensive monthly TWS data since April 2002. However, there are 35 months of missing data over the entire GRACE/GFO observational period. To address this gap, we developed an operational approach utilizing singular spectrum analysis and principal component analysis (SSA-PCA) to fill these missing data over mainland China. The algorithm was demonstrated with good performance in the Southwestern River Basin (SWB, correlation coefficient, CC: 0.71, RMSE: 6.27 cm), Yangtze River Basin (YTB, CC: 0.67, RMSE: 3.52 cm), and Songhua River Basin (SRB, CC: 0.66, RMSE: 7.63 cm). Leveraging two decades of continuous time-variable gravity data, we investigated the spatiotemporal variations in TWS across ten major Chinese basins. According to the results of GRACE/GFO, mainland China experienced an average annual TWS decline of 0.32 ± 0.06 cm, with the groundwater storage (GWS) decreasing by 0.54 ± 0.10 cm/yr. The most significant GWS depletion occurred in the Haihe River Basin (HRB) at −2.07 ± 0.10 cm/yr, significantly substantial (~1 cm/yr) depletions occurred in the Yellow River Basin (YRB), SRB, Huaihe River Basin (HHB), Liao-Luan River Basin (LRB), and Southwest River Basin (SWB), and moderate losses were recorded in the Northwest Basin (NWB, −0.34 ± 0.03 cm/yr) and Southeast River Basin (SEB, −0.24 ± 0.10 cm/yr). Furthermore, we identified that interannual TWS variations in ten basins of China were primarily driven by soil moisture water storage (SMS) anomalies, exhibiting consistently and relatively high correlations (CC > 0.60) and low root-mean-square errors (RMSE < 5 cm). Lastly, through the integration of GRACE/GFO and Global Land Data Assimilation System (GLDAS) data, we unraveled the contrasting water storage patterns between northern and southern China. Southern China experienced drought conditions, while northern China faced flooding during the 2020–2023 La Niña event, with the inverse pattern observed during the 2014–2016 El Niño event. This study fills in the missing data and quantifies water storage variations within mainland China, contributing to a deeper insight into climate change and its consequences on water resource management.

Brain-Computer Interface: The HOL-SSA Decomposition and Two-Phase Classification on the HGD EEG Data.

An efficient processing approach is essential for increasing identification accuracy since the electroencephalogram (EEG) signals produced by the Brain-Computer Interface (BCI) apparatus are nonlinear, nonstationary, and time-varying. The interpretation of scalp EEG recordings can be hampered by nonbrain contributions to electroencephalographic (EEG) signals, referred to as artifacts. Common disturbances in the capture of EEG signals include electrooculogram (EOG), electrocardiogram (ECG), electromyogram (EMG) and other artifacts, which have a significant impact on the extraction of meaningful information. This study suggests integrating the Singular Spectrum Analysis (SSA) and Independent Component Analysis (ICA) methods to preprocess the EEG data. The key objective of our research was to employ Higher-Order Linear-Moment-based SSA (HOL-SSA) to decompose EEG signals into multivariate components, followed by extracting source signals using Online Recursive ICA (ORICA). This approach effectively improves artifact rejection. Experimental results using the motor imagery High-Gamma Dataset validate our method's ability to identify and remove artifacts such as EOG, ECG, and EMG from EEG data, while preserving essential brain activity.

Time Frequency Analysis-Based Averaging and Fusion of Features for Wearable Non-Invasive Blood Glucose Estimation

This paper proposes a Clark error grid based stacking method for fusing various time frequency averaged features together for reducing the measurement errors for performing the wearable non-invasive blood glucose estimation. Like the conventional time frequency representation based denoising methods, the most critical issue is on the selection of the time frequency components. In this paper, three time frequency representation based averaging methods are investigated. They are the discrete cosine transform (DCT) based averaging method, the singular spectrum analysis (SSA) based averaging method and the empirical mode decomposition (EMD) based averaging method. It is found that retaining 40% of the DCT coefficients as well as three SSA components and three intrinsic mode functions (IMFs) could yield the best blood glucose estimation performances. Two datasets are employed for demonstrating the effectiveness and the robustness of our proposed method. The most common blood glucose estimation method and our proposed method can yield the Pearson’s correlation coefficient (R) at 0.4308 and 0.9172, respectively, the mean absolute error (MAE) at 0.8783 and 0.8193, respectively, the root mean squares error (RMSE) at 1.0680 and 0.9621, respectively, and the mean absolute relative difference (MARD) at 0.1504 and 0.1133, respectively, as well as the percentage of the data falling in the zone A of the Clark error grid at 71.1111% and 90.5882%, respectively, for the data in the first dataset. The results obtained for the data in the second dataset are similar. Obviously, our proposed method significantly outperforms the most common blood glucose estimation methods.

On Sea-Level Change in Coastal Areas

Variations in sea-level, based on tide gauge data (GSLTG) and on combining tide gauges and satellite data (GSLl), are subjected to singular spectrum analysis (SSA) to determine their trends and periodic or quasi-periodic components. GLSTG increases by 90 mm from 1860 to 2020, a contribution of 0.56 mm/yr to the mean rise rate. Annual to multi-decadal periods of ∼90/80, 60, 30, 20, 10/11, and 4/5 years are found in both GSLTG and GSLl. These periods are commensurable periods of the Jovian planets, combinations of the periods of Neptune (165 yr), Uranus (84 yr), Saturn (29 yr) and Jupiter (12 yr). These same periods are encountered in sea-level changes, the motion of the rotation pole RP and evolution of global pressure GP, suggesting physical links. The first SSA components comprise most of the signal variance: 95% for GSLTG, 89% for GSLl, 98% for GP and 75% for RP. Laplace derived the Liouville–Euler equations that govern the rotation and translation of the rotation axis of any celestial body. He emphasized that one must consider the orbital kinetic moments of all planets in addition to gravitational attractions and concluded that the Earth’s rotation axis should undergo motions that carry the combinations of periods of the Sun, Moon and planets. Almost all the periods found in the SSA components of sea-level (GSLl and GSLTG), global pressure (GP) and polar motion (RP), of their modulations and their derivatives can be associated with the Jovian planets. The trends themselves could be segments of components with still longer periodicities (e.g., 175 yr Jose cycle).

An Intelligent reference free adaptive learning algorithm for speech enhancement

Removing of noise component is an important task in all practical applications like hearing aids, speech therapy etc. In speech communication applications the speech components are contaminated with various types of noises. Separation of speech and noise component is a key issue in hearing aids, speech therapy applications. This paper demonstrates a hybrid version of singular spectrum analysis (SSA) and independent component analysis (ICA) based adaptive noise canceller (ANC) to separate noise and speech components. As ICA is not suitable for single channel sources, SSA is used to map signal channel data to multivariant data. Therefore, SSA based ICA decomposition is used to generate reference for noise cancellation process. Variable Step based adaptive learning algorithm is used to separate noise contaminations from speech signals. To reduce computational complexity of system, sign regressor operation is applied to the data vector of the proposed adaptive learning methodology. Performance measures such as Signal to noise ratio improvement, excess mean square error and misadjustment are calculated for various considered ANCs, their values for crane noise are 29.6633 dB, – 27.4854 dB and 0.2058 respectively. Among the various adaptive learning algorithms, sign regressor based step variable method performs better than the other algorithms. Hence this learning methodology is well suited for hearing aids and speech therapy applications due to its robustness, less computational complexity and filtering ability.

Hyperspectral Image Enhancement by Two Dimensional Quaternion Valued Singular Spectrum Analysis for Object Recognition

This paper proposes a two dimensional quaternion valued singular spectrum analysis based method for enhancing the hyperspectral image. Here, the enhancement is for performing the object recognition, but neither for improving the visual quality nor suppressing the artifacts. In particular, the two dimensional quaternion valued singular spectrum analysis components are selected in such a way that the ratio of the interclass separation to the intraclass separation of the pixel vectors is maximized. Next, the support vector machine is employed for performing the object recognition. Compared to the conventional two dimensional real valued singular spectrum analysis based method where only the pixels in a color plane is exploited, the two dimensional quaternion valued singular spectrum analysis based method fuses four color planes together for performing the enhancement. Hence, both the spatial information among the pixels in the same color plane and the spectral information among various color planes are exploited. The computer numerical simulation results show that the overall classification accuracy based on our proposed method is higher than the two dimensional real valued singular spectrum analysis based method, the three dimensional singular spectrum analysis based method, the multivariate two dimensional singular spectrum analysis based method, the median filtering based method, the principal component analysis based method, the Tucker decomposition based method and the hybrid spectral convolutional neural network (hybrid SN) based method.

Singular Spectrum Analysis-Based Hierarchical Multiresolution Analysis with Exploitation of Frequency Selectivities of Desirable Grouped Functions

This paper proposes a singular spectrum analysis (SSA)-based hierarchical multiresolution analysis (HMA) with the exploitation of the frequency selectivities of the desirable grouped functions. To perform the HMA, the SSA components are grouped based on the desirable grouped functions. Similar procedures are applied to the sum of the SSA components in a group in the previous level of decomposition. Computer numerical simulation results show that the SSA components in the next level of decomposition are localized within the passband of the sum of the SSA components in the corresponding group in the previous level of decomposition if its intrinsic mode functions (IMFs) or the ideal filters are employed as the desirable grouped functions. Moreover, unlike the empirical mode decomposition (EMD)-based HMA, the total number of the SSA components in each level of decomposition can be chosen.

Sea-Level Change at the Brest (France) Tide Gauge and the Markowitz Component of Earth's Rotation

Le Mouel, J.-L.; Lopes, F., and Courtillot, V., 2021. Sea-level change at the Brest (France) tide gauge and the Markowitz component of Earth's rotation. Journal of Coastal Research, 37(4), 683–690. Coconut Creek (Florida), ISSN 0749-0208.This paper centers on singular spectrum analysis (SSA) of variations in sea level in Brest (France) and in Earth's axis of rotation. Brest tide gauge data (recording the Brest sea level [BSL]) are available from 1807 onward. The main features of the BSL curve are common to most stations in the Northern Hemisphere; hence, the Brest curve has recorded a gross Earth datum. Pole positions (coordinate m2), another gross Earth datum, are from the International Earth Rotation and Reference System Service (1845–2019). The first SSA components of both series, i.e. the trends, are similar, with a major acceleration event near 1900 and sea level lagging pole motion by ∼5 to 10 years. SSA components with periods of 1, 5.4, and ∼11 years are common to the two series. An important feature is a 0.5-year component that is present in sea level but absent from pole motion. The remarkable similarity of the two trends and their phase lag suggests a causal relationship opposite the one that is generally accepted.

Improving Accuracy of Low-Cost Vehicle Attitude Estimation by Denoising MIMU Based on SSA and ICA

In this article, we develop a novel method to improve the accuracy of the low-cost vehicle attitude estimation using singular spectrum analysis (SSA) and independent component analysis (ICA). The microelectromechanical system (MEMS) accelerometers and the MEMS gyros are fused by the Kalman filtering algorithm to provide a low-cost attitude estimation. The low-frequency vibration noise and the maneuvering acceleration disturbance in the measurements of the MEMS accelerometers introduce large errors to the attitude estimation. To mitigate the low-frequency vibration noise, we propose a method based on SSA to separate the quasi-periodic components in accelerometer measurements and keep the trend remained as the true attitude measurement. SSA is effective in decomposing original signal into trend, periodic or quasi-periodic components, and thus, the low-frequency vibration noise that behaves as a quasi-periodic signal can be well separated. However, the acceleration disturbance still exists in the trend. Thus, a method based on ICA is then proposed to remove the acceleration disturbance by the aid of the turning rate measurement. The proposed method is verified with driving tests under city road and off-road conditions, showing a significant improvement of the attitude accuracy.

Multiscale 2-D Singular Spectrum Analysis and Principal Component Analysis for Spatial–Spectral Noise-Robust Feature Extraction and Classification of Hyperspectral Images

In hyperspectral images (HSI), most feature extraction and data classification methods rely on corrected dataset, in which the noisy and water absorption bands are removed. This can result in not only extra working burden but also information loss from removed bands. To tackle these issues, in this article, we propose a novel spatial–spectral feature extraction framework, multiscale 2-D singular spectrum analysis (2-D-SSA) with principal component analysis (PCA) (2-D-MSSP), for noise-robust feature extraction and data classification of HSI. First, multiscale 2-D-SSA is applied to exploit the multiscale spatial features in each spectral band of HSI via extracting the varying trends within defined windows. Taking the extracted trend signals at each scale level as the input features, the PCA is employed to the spectral domain for dimensionality reduction and spatial–spectral feature extraction. The derived spatial–spectral features in each scale are separately classified and then fused at decision-level for efficacy. As our 2-D-MSSP method can extract features and simultaneously remove noise in both spatial and spectral domains, which ensures it to be noise-robust for classification of HSI, even the uncorrected dataset. Experiments on three publicly available datasets have fully validated the efficacy and robustness of the proposed approach, when benchmarked with 10 state-of-the-art classifiers, including six spatial–spectral methods and four deep learning classifiers. In addition, both quantitative and qualitative assessment has validated the efficacy of our approach in noise-robust classification of HSI even with limited training samples, especially in classifying uncorrected data without filtering noisy bands.

Removal of EOG artifacts and separation of different cerebral activity components from single channel EEG—An efficient approach combining SSA–ICA with wavelet thresholding for BCI applications

The electroencephalogram (EEG) signals are usually interfered by many sources of noise like electrooculogram (EOG), which degrades the signals of interest. It causes the poor performance of the Brain–Computer Interface (BCI) systems. In this work, the problem of removal of EOG artifacts and separation of different cerebral activities found in a single-channel contaminated EEG is addressed. For this purpose, a novel model, based on the combined use of Singular Spectrum Analysis (SSA) and Independent Component Analysis (ICA) with a Stationary Wavelet Transform (SWT) is presented. ICA technique is a highly efficient method, which deals with the multichannel EEG signals. But it is difficult to apply the ICA on single channel EEG. Hence, using SSA, the single channel contaminated EEG signals are converted into multivariate information. Then, the multivariate information is fed to ICA, which separates the source signals as different independent components (ICs). Despite the fact that the ICA method performs excellent source separation, still, some required EEG signal content is present in the IC representing itself as an artifact, and thus dropping it would cause loss of EEG signal content. To avoid this problem, SWT is applied on the artifact IC, which performs the thresholding, to separate the actual artifact and preserve the EEG signal content. Matlab simulations have been done on both synthetically generated and real-life EEG signals and the proposed model is compared with the existing works. It is demonstrated that the proposed model has the best artifact separation performance than all the existing techniques, which is shown in terms of the metrics, RRMSE (Relative Root Mean Square Error) and MAE (Mean Absolute Error).

Effectiveness analysis of bio-electronic stimulation therapy to Parkinson’s diseases via joint singular spectrum analysis and discrete fourier transform approach

This paper studies the effectiveness of applying the bio-electronic stimulation therapy to suppress the syndromes of the Parkinson’s diseases. Here, eight patients suffering from the Parkinson’s diseases are invited for performing the investigation. The patients wear the wearable devices before and after applying the bio-electronic stimulation therapy for acquiring the corresponding resting tremor signals. First, the DC means of these signals are removed. Then, these zero mean signals are normalized to the unit energy signals. Second, the singular spectrum analysis is performed and the sets of singular spectrum analysis components are obtained. Next, the singular spectrum analysis components corresponding to the voluntary movements and the noises from various sources are removed. Third, the fast Fourier transforms are computed on the sums of the rest of singular spectrum analysis components. Then, the signal components between 3 Hz and 8 Hz are extracted out. Next, the energies of the signal components are computed for performing the analysis. The computer numerical simulation results show that the energies of the tremors after applying the bio-electronic stimulation therapy are lower than those before applying the bio-electronic stimulation therapy for all these eight patients. Compare to the processing without performing the singular spectrum analysis, the energies of the tremors after applying the bio-electronic stimulation therapy are lower than those before applying the bio-electronic stimulation therapy only for seven patients. This verifies that the singular spectrum analysis approach is effective for performing the analysis and the bio-electronic stimulation therapy is effective for suppressing the syndromes of the Parkinson’s diseases.

Crustal movement derived by GNSS technique considering common mode error with MSSA

Common mode errors (CMEs) are a kind of space–time related error. In order to estimate the influence of CMEs on the estimation of station velocity and analyze the crustal movement, GPS data around the world from 2011 to 2016 were precisely processed to obtain precise coordinate time series with one optimal sub-network scheme. The multi-channel singular spectrum analysis (MSSA) and principal component analysis (PCA) are used to extract the CMEs. In order to justify the reliability of extracting CMEs by MSSA, the results of extracting CMEs by PCA are compared. The comparison results show that MSSA can extract CMEs more effectively. The accuracy of the reference frame origin was considered. The trend of GPS site movement determined by MSSA is linearly fitted with the least squares method to estimate the velocity. In order to analyze the influence of CMEs on the estimation of velocity, the velocity is estimated from the coordinate with/without CMEs andtheassociateduncertainties are analyzed. The results indicate that the standard deviation of uncertainty in N, E and U decreased by 11.76%, 4.76% and 28.17%, respectively. The results of IGS stations in the study are compared with the velocity released by the International Terrestrial Reference Frame (ITRF), and the results show that after removing the CMEs by MSSA, the standard deviation of differences in N, E and U decreased by 44.59%, 32.31% and 40.70%, respectively. The velocities before and after removing the CMEs were compared with the ITRF2014 plate motion model, and the results show that the standard deviation of differences in N, E and U decreased by 25.00%, 17.83% and 22.10%, respectively. Finally, a detailed analysis of the global crustal movement is made.

Entropy-Based Pattern Learning Based on Singular Spectrum Analysis Components for Assessment of Physiological Signals

Measures of predictability in physiological signals based on entropy metrics have been widely used in the application domain of medical assessment and clinical diagnosis. In this paper, we propose a new entropy-based pattern learning by a combination of singular spectrum analysis (SSA) and entropy measures for assessment of physiological signals. Physiological signals are first represented as a series of SSA components, and then well-established entropy measures are extracted from the resulting SSA components that can help to facilitate the features extraction from physiological signals. The entropy measures of notable SSA components are used to form input features and fed into pattern classifier. To demonstrate its validity, applicability, and versatility, the proposed entropy-based pattern learning is used to perform medical assessments with three kinds of classical physiological signals, that is, electroencephalogram (EEG), electromyogram (EMG), and RR-interval signals. Experiments demonstrate that in all cases, the proposed entropy-based pattern learning can effectively capture specific biosignal patterns of physiological signals and achieve excellent identification performances for the assessments of EEG, EMG, and RR-interval signals. Besides, through the comparison of the identification performances for entropy-based pattern learning based on the physiological signals themselves and the SSA components, it is concluded that the discriminating power of entropy-based pattern learning based on the SSA components is much stronger than that based on the physiological signals themselves. Since it can be easily extended to any other physiological signal analysis, the proposed entropy-based pattern learning may use as an efficient approach to reveal biosignal patterns for medical assessment of physiological signals.

Suppressing the Spikes in Electroencephalogram via an Iterative Joint Singular Spectrum Analysis and Low-Rank Decomposition Approach

The novelty and the contribution of this paper consists of applying an iterative joint singular spectrum analysis and low-rank decomposition approach for suppressing the spikes in an electroencephalogram. First, an electroencephalogram is filtered by an ideal lowpass filter via removing its discrete Fourier transform coefficients outside the wave band, the wave band, the wave band, the wave band and the wave band. Second, the singular spectrum analysis is performed on the filtered electroencephalogram to obtain the singular spectrum analysis components. The singular spectrum analysis components are sorted according to the magnitudes of their corresponding eigenvalues. The singular spectrum analysis components are sequentially added together starting from the last singular spectrum analysis component. If the variance of the summed singular spectrum analysis component under the unit energy normalization is larger than a threshold value, then the summation is terminated. The summed singular spectrum analysis component forms the first scale of the electroencephalogram. The rest singular spectrum analysis components are also summed up together separately to form the residue of the electroencephalogram. Next, the low-rank decomposition is performed on the residue of the electroencephalogram to obtain both the low-rank component and the sparse component. The low-rank component is added to the previous scale of the electroencephalogram to obtain the next scale of the electroencephalogram. Finally, the above procedures are repeated on the sparse component until the variance of the current scale of the electroencephalogram under the unit energy normalization is larger than another threshold value. The computer numerical simulation results show that the spike suppression performance based on our proposed method outperforms that based on the state-of-the-art methods.

Solar turbulence from sunspot records

ABSTRACTIt is generally assumed that coupling between the turbulent flow and the magnetic field at the top of the Sun's convection zone leads to a Kolmogorov cascade of kinetic to magnetic energy. An inertial range and a slope value close to −5/3 have been recognized in a log–log diagram of power spectral density versus frequency (or period). However, published values of the slope have large uncertainties and the inertial period range is limited to 0.1 s to 2 yr. We have applied an adapted version of the singular spectrum analysis (SSA) method to the series of (quasi-) daily sunspot numbers ISSN (an indirect way of monitoring solar activity) from 1868 to 2019. The log–log diagram of ISSN variance of SSA components versus frequency displays an inertial slope value of −1.66 ± 0.16 and an inertial range from about 4 to 100 yr. This is consistent with the existence of Kolmogorov turbulent behaviour in the Sun's convection zone.

Linear phase properties of the singular spectrum analysis components for the estimations of the RR intervals of electrocardiograms

Denoising is the first step in both the QRS complex detection and the beat classification. However, infinite impulse response filters usually exhibit nonlinear phase responses. As a result, the group delays of the output signals based on the infinite impulse response filtering are different at different time instants. This causes the inaccuracies of the estimations of the RR intervals of the electrocardiograms. In this paper, the denoising is performed based on the singular spectrum analysis approach. The linear phase properties of the singular spectrum analysis components are investigated. Finally, the RR intervals of the denoised electrocardiograms are estimated and those obtained based on the infinite impulse response filtering approach are compared. The computer numerical simulation results show that the estimation performance based on the singular spectrum analysis approach outperforms that based on the infinite impulse response filtering approach.

A Solar Signature in Many Climate Indices

AbstractWe first apply singular spectrum analysis (SSA) to the international sunspot number (1849–2015) and the count of polar faculae (1906–2006). The SSA method finds 22‐, 11‐, and 5.5‐year components as the first eigenvectors of these solar activity proxies. We next apply SSA to the 10 Madden‐Julian oscillation (MJO; 1978–2016) indices. The first, most intense component SSA finds in all MJO indices has either a period of 5.5 or 11 years. The longer‐term modulation of amplitude is on the order of one third of the total variation. The 5.5‐year SSA component 1 of most MJO indices moreover follows the decreasing amplitude of solar cycles. We then apply SSA to climate indices Pacific Decadal Oscillation, El Nino Southern Oscillation precipitation index, Arctic Oscillation, Atlantic Multidecadal oscillation, Tropical Southern Atlantic oscillation, Western Hemisphere Warm Pool, and Brazil and Sahel rainfalls. We find that the first SSA eigenvectors are all combinations of rather pure 11, 5.5, and 3.6‐year pseudo‐cycles. The 5.5‐year component is frequently observed and is particularly important and sharp in the series in which it appears. All these periods have long been attributed to solar activity, and this by itself argues for the existence of a strong link between solar activity and climate. The mechanisms of coupling must be complex and probably nonlinear but they remain to be fully understood (UV radiation, solar wind, and galactic cosmic rays being the most promising candidates). We propose as a first step a Kuramoto model of nonlinear coupling that generates phase variations compatible with the observed ones.

Hybrid SSA-TSR-ARIMA for water demand forecasting

Water supply management effectively becomes challenging due to the human population and their needs have been growing rapidly. The aim of this research is to propose hybrid methods based on Singular Spectrum Analysis (SSA) decomposition, Time Series Regression (TSR), and Automatic Autoregressive Integrated Moving Average (ARIMA), known as hybrid SSA-TSR-ARIMA, for water demand forecasting. Monthly water demand data frequently contain trend and seasonal patterns. In this research, two groups of different hybrid methods were developed and proposed, i.e. hybrid methods for individual SSA components and for aggregate SSA components. TSR was used for modeling aggregate trend component and Automatic ARIMA for modeling aggregate seasonal and noise components separately. Firstly, simulation study was conducted for evaluating the performance of the proposed methods. Then, the best hybrid method was applied to real data sample. The simulation showed that hybrid SSA-TSR-ARIMA for aggregate components yielded more accurate forecast than other hybrid methods. Moreover, the comparison of forecast accuracy in real data also showed that hybrid SSA-TSR-ARIMA for aggregate components could improve the forecast accuracy of ARIMA model and yielded better forecast than other hybrid methods. In general, it could be concluded that the hybrid model tends to give more accurate forecast than the individual methods. Thus, this research in line with the third result of the M3 competition that stated the accuracy of hybrid method outperformed, on average, the individual methods being combined and did very well in comparison to other methods.