Optimal Variational Mode Decomposition Research Articles

A novel underwater weak signal detection method based on parameter optimized VMD and 3D chaotic system

Based on chaos theory, combined with improved variational mode decomposition, a novel method for underwater weak signal detection and frequency extraction is proposed. The aim of this research is to address the challenge of detecting underwater signals and extracting their frequencies under ultra-low Signal-to-Noise Ratio (SNR). A novel three-dimensional (3D) chaotic system with infinitely shrinking attractors is proposed, which incorporates a cosine function to achieve chaos control and enhancement. The nonlinear characteristics of the system, such as Lyapunov exponent and bifurcation, are analyzed in detail. Then, we design a frequency array and perform scale transformation on the chaotic system to construct a detection model that covering all frequencies. The model can transition from critical chaos to large period or intermittent chaos in the presence of external periodic signals by adjusting the parameters of the cosine function. Considering the relationship between the generation of intermittent chaos and the frequency of target signal, a frequency extraction algorithm based on chaos sparrow search algorithm optimization variational mode decomposition (CSSA-VMD) and Hilbert transform is designed. The algorithm can estimate the real frequency of underwater signals by performing noise reduction and envelope analysis on intermittent chaotic signals. The experimental results demonstrate that the proposed system can detect weak signals in complex marine backgrounds and extract their frequency features, with a detection SNR of -48 dB and a frequency error of less than 0.01%. Importantly, the circuit of the detection system has been implemented, verifying its feasibility for detecting and tracking underwater signals.

Strategy of Flywheel–Battery Hybrid Energy Storage Based on Optimized Variational Mode Decomposition for Wind Power Suppression

The fluctuation and intermittency of wind power generation seriously affect the stability and security of power grids. Aiming at smoothing wind power fluctuations, this paper proposes a flywheel–battery hybrid energy storage system (HESS) based on optimal variational mode decomposition (VMD). Firstly, the grid-connected power and charging–discharging power of the HESS are determined based on the sliding average algorithm. Secondly, the VMD algorithm, optimized using long short-term memory (LSTM), is used to decompose the hybrid energy storage power (HESP) into a series of sub-modes with frequencies from low to high. Then, the state of charge of the battery energy storage system and the speed of the flywheel energy storage system are monitored in real time, and the primary power of the HESS is modified according to the rules formulated by fuzzy control. Finally, through a simulation example, it is concluded that the method meets the requirements of smoothing wind power fluctuations and gives full play to the characteristics of energy storage battery and flywheel energy storage to ensure the stable operation of the energy storage system. The method presented in this paper can provide a reference for HESP configuration and control operation strategy formulation.

Carbon trading price forecasting based on parameter optimization VMD and deep network CNN–LSTM model

To meet carbon peak and neutrality targets, accurate carbon trading price forecasting is very important for enterprises making emission reduction decisions. By fusing convolutional neural network (CNN) and long short-term memory network (LSTM), the CNN–LSTM model is constructed. After variational mode decomposition (VMD), several intrinsic mode functions (IMFs) components are obtained and input into the CNN–LSTM model, thus constructing the combined sooty tern optimization algorithm (STOA)–VMD–CNN–LSTM forecasting model. To test this model, the carbon trading prices of the carbon emission trading markets of Hubei, Guangdong and Shenzhen were forecast. The prediction performance of the STOA–VMD–CNN–LSTM model is compared with ARIMA, BP, CNN and LSTM benchmark models and models combining different decomposition technologies. The international carbon trading price (EUR and CER) is used for prediction. Compared with other methods, the developed model makes fewer errors and achieves superior performance. Several important implications are provided for investors and risk managers involved in carbon financial products.

Short-term power grid load forecasting based on optimized VMD and GA-BP

Abstract The present article proposes an enhanced hybrid neural network model that combines variational mode decomposition (VMD) and genetic algorithm-backpropagation (GA-BP) to tackle the accurate prediction task of nonstationary and nonlinear power demand data. In comparison to existing methods, this study employs the North Gallic Hawk Optimization (NGO) algorithm to preliminarily ensure the optimization of the number of VMD modes, K and the penalty factor, α. Furthermore, it utilizes the envelope entropy criterion to determine the minimum optimal VMD components for predicting low-order spectral features vectors. Building upon this, the GA-BP optimization method establishes prediction models for the NGO-VMD model’s individual components. When compared with traditional methods, the GA-BP model exhibits significant advantages in terms of rapid convergence, fewer iterations and high prediction accuracy. Empirical results demonstrate that this approach holds promising prospects for practical application in power demand prediction. In comparison to traditional model predictive performance, the root mean square error is reduced by 60.36% and the mean absolute percentage error by 44.34%, providing robust support to enhance prediction accuracy and promote sustainable development of power systems.

NOx formation model for utility boilers using robust two-step steady-state detection and multimodal residual convolutional auto-encoder

BackgroundTo achieve carbon neutrality, thermal power plants will take on greater peak-shaving responsibilities, resulting in utility boilers running at more frequently-changing states. This variation inevitably contributes to dirtier measurement data, and the data will follow new distributions that represent asynchronous operational characteristics. MethodTo predict the NOx formation concentration in this new context, a novel modelling framework is proposed. First, a two-step steady-state detection algorithm, including a basic and a precise identification procedure, is proposed to select steady-state samples from operational data. This algorithm also combines with a bayesian optimization–variational mode decomposition (BO–VMD) approach to enhance its robustness by eliminating high-frequency noise, and with an empirical cumulative distribution based outlier detection (ECOD) to remove outliers. Therefore, a high-quality steady-state dataset is acquired. Based on these data, a multimodal residual convolutional auto-encoder is developed to extract the latent variables that vary with the operation modes. These variables are then used to construct the multimodal NOx prediction model. Significant findingsThe findings on a 660 MW utility boiler reveal that, the proposed modelling framework has a lower misdiagnosis rate in steady-state identification compared to the one-step detection technique, and achieves higher fitting and generalization accuracy than the global modelling method.

Chaotic time series wind power interval prediction based on quadratic decomposition and intelligent optimization algorithm

Wind power prediction plays a critical role in the economic deployment of power systems including wind farm. To improve the prediction accuracy of wind power, a chaotic wind power interval prediction method, which relies on quadratic decomposition and improved state transition algorithm, is proposed in this study. Firstly, the data are decomposed twice through the optimal variational mode decomposition, improved complete ensemble empirical mode decomposition and permutation entropy (OVMD-ICEEMDAN-PE) in combination, so as to deal with the volatility and complexity of wind power time series. Then, the decomposed data are used to build the prediction model for each component. In addition, the improved state transition algorithm is introduced to optimize the hyperparameters of the convolutional neural network and bidirectional long short-term memory (CNN-BiLSTM), time convolutional network (TCN) and gated recurrent unit (GRU) of the prediction models. With the optimal hyperparameters solved for each model, the linear weights are combined to achieve accurate prediction. Finally, through the combined prediction model, interval prediction is achieved. The results show that the proposed method outperforms other models in the accuracy of prediction.

Research on vessel noise signal compression sensing algorithm based on Particle Swarm Optimization and Variational Mode Decomposition

Due to the complexity of marine environmental factors and strong background noise, the collected vessel signal becomes intricate and challenging to transmit and reconstruct using conventional parameter estimation methods. Additionally, identifying vessel targets becomes difficult. In order to detect the vessel target and obtain the corresponding orientation information, it is necessary to eliminate the interference of noise as much as possible and reconstruct the original signal more accurately. In this work, we propose an optimized signal denoising method PSO-VMD (Particle Swarm Optimization–Variational Mode Decomposition). By conducting simulations using experimental data and evaluating the model, we demonstrate the superior denoising capabilities of this method. Then we apply this method to denoise the real vessel sound signal and then compress and reconstruct the signal using compressed sensing. The experimental results show that the signal processed by PSO + VMD method has the maximum SNR and the minimum RMSE and MAE. What’s more, the selected suitable denoising method can effectively suppress the environmental noise in the signal and improve the reconstruction efficiency of compressed sensing.

Multi-Scale Response Analysis and Displacement Prediction of Landslides Using Deep Learning with JTFA: A Case Study in the Three Gorges Reservoir, China

Reservoir water and rainfall, leading to fluctuations groundwater levels, are the main triggering factors that induce landslides in the Three Gorges Reservoir area. This study investigates the response mechanism of landslide deformation under reservoir water and rainfall variations through long-time on-site observations. To address the non-stationary characteristics of the time-series records, joint time-frequency analysis (JTFA) is first introduced into our landslide prediction model. This model employs optimal variational mode decomposition (VMD) to obtain specific signal components with clear physical meaning, such as trend component and periodic components. Then, multi-scale response analysis between the displacement and external factors three wavelet methods was conducted. The analysis results show a 1 year primary cycle of the time series associated with the landslide evolution. The reservoir water level and rainfall show anti-phase fluctuations. The periodic displacement correlates significantly with rainfall, lagging by about two months. The reservoir water is anti-phase with the landslide displacement, preceding it by approximately three months (−51 ± 8° phase difference). For landslide displacement prediction, the gated recurrent units (GRU) neural network model is integrated into the deep learning forecasting architecture. The model takes into account the correlation and hysteresis effect of input variables. Through six experiments, we investigate the effect of data volume on model predictions to determine the optimal model. The results demonstrate that our proposed model ensures high performance in landslide prediction. Moreover, a comparison with six other intelligent algorithms shows the advantages of our model in terms of time-effectiveness and long-sequence forecasting.

Coal-rock drilling states recognition of drilling robot for rockburst prevention based on multi-sensor information fusion

Accurate recognition of coal-rock drilling sates is a prerequisite for achieving intelligent drilling pressure relief. In this paper, a novel coal-rock drilling states recognition method of drilling robot for rockburst prevention is proposed. Firstly, different coal-rock drilling signals are collected and processed by using improved antlion optimization (ALO) algorithm and variational mode decomposition (VMD). Meanwhile, the elite opposition-based learning (EOL) strategy is used to improve the global search ability and optimization performance of ALO, and the EOL-ALO is developed and employed to automatically search the optimal key parameters of VMD. Subsequently, the root mean square of frequency and kurtosis are used to extract the feature information from the decomposed signals and the singular value decomposition method is employed to reduce the dimensionality of high-dimensional feature vectors. Furthermore, an improved D-S evidence theory is developed to fuse the recognition results of support vector machine through a single sensor information and the fusion recognition framework of coal-rock drilling states is designed. Finally, a coal-rock drilling experimental platform is established and some experimental analysis is carried out. The experimental results indicate the feasibility and superiority of proposed coal-rock drilling states recognition method.

Short-Term Multi-Step Wind Direction Prediction Based on OVMD Quadratic Decomposition and LSTM

Accurate and reliable wind direction prediction is important not only for enhancing the efficiency of wind power conversion and ensuring safe operation, but also for promoting sustainable development. Wind direction forecasting is a challenging task due to the random, intermittent and unstable nature of wind direction. This paper proposes a short-term wind direction prediction model based on quadratic decomposition and long short-term memory (LSTM) to improve the accuracy and efficiency of wind direction prediction. Firstly, the model adopts a seasonal-trend decomposition procedure based on the loess (STL) method to divide the wind direction series into three subsequences: trend, seasonality and the remainder, which reduces the impact of the original sequence’s complexity and non-stationarity on the prediction performance. Then, the remainder subsequence is decomposed by the optimal variational mode decomposition (OVMD) method to further explore the potential characteristics of the wind direction sequence. Next, all the subsequences are separately input into the LSTM model, and the prediction results of each subsequence from the model are superimposed to obtain the predicted value. The practical wind direction data from a wind farm were used to evaluate the model. The experimental results indicate that the proposed model has superior performance in the accuracy and stability of wind direction prediction, which also provides support for the efficient operation of wind turbines. By developing advanced wind prediction technologies and methods, we can not only enhance the efficiency of wind power conversion, but also ensure a sustainable and reliable supply of renewable energy.

Hybrid intelligent deep learning model for solar radiation forecasting using optimal variational mode decomposition and evolutionary deep belief network - Online sequential extreme learning machine

Accurate prediction of solar radiation is of great significance to improve the utilization of solar energy for photovoltaic power generation on the roofs of urban buildings. Therefore, a hybrid model, namely the OVMD-PACF-ISSA-DBN-OSELM model, is being proposed in this study for solar radiation prediction. Firstly, the data is decomposed by optimal variational mode decomposition (OVMD) technology. Secondly, the partial autocorrelation function (PACF) technology selected features for components decomposed by OVMD technology. Thirdly, a deep belief network-online sequential extreme learning machine (DBN-OSELM) prediction model is constructed, and sparrow search algorithm (SSA) is utilized to optimize the model hyperparameters. Aiming at the shortcomings of SSA algorithm, Tent map and differential evolution strategy are introduced to improve the SSA algorithm. Finally, the model is used to predict each component, and the prediction results of each component are summarized to obtain the prediction results of solar radiation. To verify that the proposed method can obtain accurate prediction results, experiments are conducted using solar radiation data for January, May and October 2022, and compared with other models. Taking January as an example, compared with shallow model BP, the RMSE, MAE and R of the proposed model are increased by 66.83%, 58.65% and 5.33%, respectively. Moreover, the experiments of the proposed model in other months have excellent results. The current findings demonstrate that the suggested model is a strong contender in the arena of solar radiation prediction, and can consequently provide valuable support in enhancing the efficiency of photovoltaic equipment installed on urban building roofs.

A novel rolling bearing fault diagnosis method based on parameter optimization variational mode decomposition with feature weighted reconstruction and multi-target attention convolutional neural networks under small samples.

To enhance the precision of rolling bearing fault diagnosis, an intelligent hybrid approach is proposed in this paper for signal processing and fault diagnosis in small samples. This approach is based on advanced techniques, combining parameter optimization variational mode decomposition weighted by multiscale permutation entropy (MPE) with maximal information coefficient and multi-target attention convolutional neural networks (MTACNN). First, an improved variational mode decomposition (VMD) is developed to denoise the raw signal. The whale optimization algorithm was used to optimize the penalty factor and mode component number in the VMD algorithm to obtain several intrinsic mode functions (IMFs). Second, separate MPE calculations are performed for both the raw signal and each of the IMF components obtained from the VMD decomposition; the results are used to calculate the maximum information coefficient (MIC). Subsequently, each MIC is normalized and converted to a weight coefficient for signal reconstruction. Ultimately, the reconstructed signals serve as input to the MTACNN for diagnosing rolling bearing faults. Results demonstrate that the signal processing approach exhibits superior noise reduction capability through simple processing. Furthermore, compared to several similar approaches, The method proposed for fault diagnosis achieves superior performance levels in the fault pattern recognition target and the fault severity recognition target.

Hybrid solar radiation forecasting model with temporal convolutional network using data decomposition and improved artificial ecosystem-based optimization algorithm

Solar energy is highly economical and widespread in new energy applications, and analyzing solar radiation information is an important part of solar photovoltaic power applications. However, because of its data complexity and difficulty to measure, solar radiation data needs to be predicted. Temporal Convolutional Network (TCN) model is used to extract features and Artificial Ecosystem-based Optimization (AEO) algorithm is used to optimize the parameters of TCN. Out of consideration for the phenomenon of strong fluctuations and complex features of solar radiation data, the optimal variational mode decomposition (OVMD) method is incorporated into the model. First, the signal decomposition is performed on the original data to obtain several subsequences, and then aggregated by fuzzy entropy to reduce the number of sequences, after which the data are fed into the TCN model and the model parameters are optimized using the improved AEO algorithm, and finally the results of the model prediction are the output. Four months of solar radiation data are selected for testing, it is finally concluded that the OVMD-IAEO-TCN model can be used for solar radiation prediction with higher accuracy and reliability than the other nine comparison models.

Hybrid Prediction Model Based on Decomposed and Synthesized COVID-19 Cumulative Confirmed Data

Since 2020, COVID-19 has repeatedly arisen around the world, which has had a significant impact on the global economy and culture. The prediction of the COVID-19 epidemic will help to deal with the current epidemic and similar risks that may arise in the future. So, this paper proposes a hybrid prediction model based on particle swarm optimization variational mode decomposition (PSO-VMD), Long Short-Term Memory Network (LSTM) and AdaBoost algorithm. To address the issue of determining the optimal number of modes K and the penalty factor (α) in the variational mode decomposition (VMD), an adaptive value for particle swarm optimization (PSO) is proposed. Specifically, the weighted average sample entropy of the relevant coefficients is utilized to determine the adaptive value. First, the epidemic data are decomposed into multiple modal components, known as intrinsic mode functions (IMFs), using PSO-VMD. These components, along with policy-based factors, are integrated to form a multivariate forecast dataset. Next, each IMF is predicted using AdaBoost-LSTM. Finally, the prediction results of all the IMF components are reconstructed to obtain the final prediction result. Our proposed method is validated by the cumulative confirmed data of Hubei and Hebei provinces. Specifically, in the case of cumulative confirmation data, the coefficient of determination (R2) of the mixed model is increased compared to the control model, and the average mean absolute error (MAE) and root mean square error (RMSE) decreased. The experimental results demonstrate that the VMD–AdaBoost–LSTM model achieves the highest prediction accuracy, thereby offering a new approach to COVID-19 epidemic prediction.

Chaotic time series wind power prediction method based on OVMD-PE and improved multi-objective state transition algorithm

As the global wind power generation capacity is constantly increasing, the problems of safe operation and utilization after grid connection are becoming more and more prominent. Aiming at the problem of low accuracy and stability of wind power time series with chaotic characteristics, a time series prediction method combining chaotic characteristic processing and neural networks is proposed. First, optimal variational mode decomposition with permutation entropy (OVMD-PE) is used to decompose the original wind power time series and overcome the disadvantage of insufficient mode aliasing encountered by empirical and integrated empirical modes. Second, an improved multi-objective state transition algorithm is proposed to determine the weight coefficients among the neural networks and improve the accuracy of the reconstructed predictive neural networks. Finally, the combined prediction method is used to study and analyze the wind power data from a wind farm in Xinjiang, China, from the perspectives of multiple scenarios and multiple time scales. The experimental results show that OVMD-PE can successfully deal with chaotic characteristics and the improved algorithm has improved the prediction accuracy. Compared with other traditional prediction models, the combined prediction model has higher robustness and stability.

Short term power load forecasting based on BES-VMD and CNN-Bi-LSTM method with error correction

Aiming at the strong non-linear and non-stationary characteristics of power load, a short-term power load forecasting method based on bald eagle search (BES) optimization variational mode decomposition (VMD), convolutional bi-directional long short-term memory (CNN-Bi-LSTM) network and considering error correction is studied to improve the accuracy of load forecasting. Firstly, a decomposition loss evaluation criterion is established, and the VMD optimal decomposition parameters under the evaluation criterion are determined based on BES to improve the decomposition quality of the signal. Then, the original load sequence is decomposed into different modal components, and the corresponding CNN-Bi-LSTM network prediction models are established for each modal component. In addition, considering the influence of various modal components, holiday and meteorological factors on the error, an error correction model considering short-term factors is established to mine the hidden information contained in the error to reduce the inherent error of the model. Finally, the proposed method is applied to a public dataset provided by a public utility in the United States. The results show that this method can better track the changes of load and effectively improve the accuracy of short-term power load forecasting.

A Novel Lower-Limb Coordination Assessment Scheme Using Multi-Scale Nonlinear Coupling Characteristics With sEMG

Motor dysfunction (e.g., incoordination of upper or lower limb) significantly limits the individuals’ ability of daily living, and thus the provisioning of a motion coordination assessment method becomes of vital importance. As a quantitative indicator, inter-muscular coupling strength could assess limb coordination. Since surface electromyography (sEMG) signals cover nonlinear coupling characteristics, in this paper, a lower-limb motion coordination assessment scheme with intermuscular coherence (IMC) analysis and optimization variational mode decomposition (VMD) is proposed. First, sEMG signals are decomposed into several intrinsic mode functions (IMFs) using optimized VMD. Then, the nonlinear coupling feature vector in the beta-band frequency domian (15-35Hz) is extracted by the G-P algorithm from optimal IMF, thereby estimating intermuscular coupling strength by IMC. In particular, differential evolution (DE) algorithm’s global optimization capability and envelope entropy (EE)’s sparsity are adopted to provide the basis for optimization VMD and screening optimal IMF, respectively. Finally, signals are collected from four muscle pairs of the 12 male and 12 female subjects to assess lower-limb motions coordination. Simultaneously, the independent sample T test is leveraged to compare the effects of gender on groups’ characteristics (e.g., age, height, and body mass). Results demonstrate not only the effectiveness of the proposed approach (i.e., p < 0.01 with both “S-walking” and “Running” motions, p < 0.05 with “Q-walking”), but also the difference of motion coordination relationship between male and female adults as well as the significance of muscle selection.

Precision Enhancement by Compensation of Hemispherical Resonator Gyroscope Dynamic Output Errors

In order to compensate for the hemispherical resonator gyroscope (HRG) dynamic output drift errors, a novel compensation method is presented. Through deriving the resonator motion equations and analyzing the mechanism of HRG dynamic output drift errors, it is found that the resonator defect nonuniformity is the main cause of the HRG dynamic output errors. By analyzing the mechanisms of these errors, the influence of these error sources is interacted and coupled, and it is difficult to analyze HRG output error mechanism and establish an error model comprehensively. We proposed a compensation prediction model for the HRG dynamic output to suppress the influence of these errors comprehensively. To establish the high precision HRG dynamic output compensation model, we proposed an improved variational mode decomposition (VMD) method to achieve an accurate HRG output. First, we used VMD to decompose the HRG output, and the optimal VMD decomposition parameters are achieved by genetic algorithm. In order to better achieve the effective information, the decomposed components of HRG drift are classified using the sample entropy. Then the mixed components are further processed using time frequency peak filtering. Finally, the accurate HRG output is achieved by reconstructing all the effective components. The accurate HRG output is used to construct the training set, and the high-precision HRG dynamic output prediction model is established by long short-term memory. The software compensation of the HRG dynamic output drift is realized by the prediction model. The experiment shows that the HRG dynamic output drift is mainly produced by the resonator nonuniformity, and it is strongly correlated with the standing wave azimuth, which verifies the correctness of the mechanism analysis. The HRG dynamic drift peak-peak value decreases to 5.715×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> °/s, which verifies the effectiveness of the compensation method.

A Bayesian Optimized Variational Mode Decomposition-Based Denoising Method for Measurement While Drilling Signal of Down-the-Hole Drilling

Measurement while drilling (MWD) emerges as a reliable technique for assessing rock mass properties. However, the measured MWD signals are often contaminated with noise, leading to distorted signals. To address this issue, this paper proposes a denoising method that utilizes variational mode decomposition (VMD) and wavelet soft thresholding (WST). The proposed method employs Bayesian optimization to adaptively determine the optimal VMD parameters. The decomposed modes are then further denoised using WST with appropriate thresholding and optimal wavelet decomposition levels. To validate the effectiveness of the proposed method, four synthetic signals with varying input signal-to-noise ratios (SNR) and two sets of measured MWD signals were evaluated. The results demonstrate the surpassing denoising efficiency of the proposed technique in terms of higher output SNR, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and lower RMSE values. Moreover, the denoised MWD signals show promising results, with reduced amplitude and volatility while preserving the characteristics of the original signals.

Hybrid the long short-term memory with whale optimization algorithm and variational mode decomposition for monthly evapotranspiration estimation

The sustainability of artificial sand-binding vegetation is determined by the water balance between evapotranspiration (ET) and precipitation in desert regions. Consequently, accurately estimating ET is a critical prerequisite for determing the types and spatial distribution of artificial vegetation in different sandy areas. For this purpose, a novel hybrid estimation model was proposed to estimate monthly ET by coupling the deep learning long short term memory (LSTM) with variational mode decomposition (VMD) and whale optimization algorithm (WOA) (i.e., VMD-WOA-LSTM) to estimate the monthly ET in the southeast margins of Tengger Desert. The superiority of LSTM was selected due to its capability of automatically extracting the nonlinear and nonstationary features from sequential data, WOA was employed to optimize the hyperparameters of LSTM, and VMD was used to extract the intrinsic traits of ET time series. The estimating results of VMD-WOA-LSTM has been compared with actual ET and estimation of other hybrid models in terms of standard performance metrics. The results reveale that VMD-WOA-LSTM provide more accurate and reliable estimating results than that of LSTM, the support vector machine (SVM), and the variants of those models. Therefore, VMD-WOA-LSTM could be recommended as an essential auxiliary method to estimate ET in desert regions.