Non-stationary Sequences Research Articles

Hydrological Data Projection Using Empirical Mode Decomposition: Applications in a Changing Climate

This paper demonstrates the effectiveness and superiority of Empirical Mode Decomposition (EMD) in projecting non-stationary hydrological data. The study focuses on daily Sea Surface Temperature (SST) sequences in the Niño 3.4 region and uses EMD to forecast the probability of El Niño events. Applying the Mann–Kendall test at the 5% significance level reveals a significant increasing trend in SST changes in this region, particularly noticeable after 1980. This trend is associated with the occurrence of El Niño and La Niña events, which have a recurrence interval of approximately 8.4 years and persist for over a year. The modified Oceanic Niño Index (ONI) proposed in this study demonstrates high forecast accuracy, with 97.56% accuracy for El Niño and 89.80% for La Niña events. Additionally, the EMD of SST data results in 13 Intrinsic Mode Functions (IMFs) and a residual component. The oscillation period increases with each IMF level, with IMF7 exhibiting the largest amplitude, fluctuating between ±1 °C. The residual component shows a significant upward trend, with an average annual increase of 0.0107 °C. These findings reveal that the EMD-based data generation method overcomes the limitations of traditional hydrological models in managing non-stationary sequences, representing a notable advancement in data-driven hydrological time series modeling. Practically, the EMD-based 5-year moving process can generate daily sea temperature sequences for the coming year in this region, offering valuable insights for assessing El Niño probabilities and facilitating annual updates.

Impacts of energetic particles from T Tauri flares on inner protoplanetary discs

ABSTRACT T Tauri stars are known to be magnetically active stars subject to strong flares observed in X-rays. These flares are likely due to intense magnetic reconnection events during which a part of the stored magnetic energy is converted into kinetic energy of supra-thermal particles. Since T Tauri stars are surrounded by an accretion disc, these particles may influence the disc dynamics and chemistry. This work continues on a previous stationary model, which showed that energetic particles accelerated during flares can produce a strong ionization rate at high column densities in the inner accretion disc. The present model includes non-stationary sequences of flaring events sampled by a Chandra X-ray survey of nearby young stellar objects. We calculate the averaged ionization rate expected in a radius range 0.08–0.6 au from the central star. We confirm that energetic particles produced by the flares dominate the ionization of the disc up to column densities of $10^{25}~\rm {cm^{-2}}$. We further study the main consequences of this additional source of ionization on the viscosity, the accretion rate, the volumetric heating rate, and the chemical complexity of inner protoplanetary discs.

Safety Monitoring Method for the Uplift Pressure of Concrete Dams Based on Optimized Spatiotemporal Clustering and the Bayesian Panel Vector Autoregressive Model

To establish a safety monitoring method for the uplift pressure of concrete dams, spatiotemporal information from monitoring data is needed. In the present study, the method of ordering points to identify the clustering structure is employed to spatially cluster the uplift pressure measuring points at different locations on the dam; three distance indexes and two clustering evaluation indexes are used to realize clustering optimization and select the optimal clustering results. The Bayesian panel vector autoregressive model is used to establish the uplift stress safety monitoring model for each category of monitoring point. For a nonstationary sequence, the difference method is selected to ensure that the sequence is stable, and the prediction is carried out according to the presence or absence of exogenous variables. The result is that the addition of exogenous variables increases the accuracy of the model’s forecast. Engineering examples show that the uplift pressure measurement points on the dam are divided into seven categories, and classification is based mainly on location and influencing factors. The multiple correlation coefficients of the training set and test set data of the BPVAR model are more than 0.80, and the prediction error of the validation set is lower than that of the Back Propagation neural network, XGBoost algorithm, and Support Vector Machines. The research in this paper provides some reference for seepage monitoring of concrete dams.

A Distribution-Adaptive Attention Transformer for Blood Pressure Prediction

Accurately predicting intraoperative blood pressure to prevent intraoperative hypotension is of significant importance. However, the task of intraoperative blood pressure prediction often faces the challenge of substantial sample distribution disparities due to individual physiological differences among patients and the diversity of surgical conditions. These disparities can cause a decline in model performance when encountering inconsistently distributed samples, thus limiting its generalizability in real-world clinical settings. To address this challenge, this paper develops a distribution-adaptive attention strategy aimed at mitigating distributional differences by incorporating statistical information within the model's internal design. The strategy automatically adjusts the model's attention weights by learning statistical information from input windows, enhancing the capability to learn the inherent distribution patterns within non-stationary sequences. Consequently, it reduces performance fluctuations caused by distributional differences. The effectiveness of the proposed distribution-adaptive attention strategy is demonstrated through multiple comparative experiments in the experimental section.

Bearing early fault identification and condition monitoring based on dynamic difference index

Bearing condition monitoring is essential for early fault detection and early warning of large equipment, and signal processing techniques are frequently used to analyse nonlinear and nonstationary sequences. Cross-correlation integral is implemented in bearing condition monitoring because it can analyse the non-stationarity of time series in dynamic systems. This paper proposes a dynamic difference index (DDI) due to the difficulty of determining the threshold in the cross-correlation integral and the roughness and operation caused by sequence similarity of 0 or 1. It is a measure of the similarity between the fuzzy autocorrelation integral of a portion of a time series and the cross-correlation between that portion and other portions of the same time series, and it is used to determine the stationarity of the time series. When bearings begin to degrade or develop structural defects, the DDI changes dramatically. The XJTU-SY dataset and IMS Bearing Data were utilised for algorithm validation. First, the algorithm’s efficacy was demonstrated by optimising the effects of various thresholds, distance measures, and time window sizes on DDI and computational efficiency. Secondly, the comparison with common methods and state of the art shows the superiority of the algorithm in detecting early bearing faults, and reveals its appropriate use in practical engineering applications.

Load forecasting for regional integrated energy system based on complementary ensemble empirical mode decomposition and multi-model fusion

The multiple loads of the Regional Integrated Energy System (RIES) possess characteristics of randomness and relatively higher complexity. The current forecasting methods struggle to effectively handle the non-stationary sequence of these multiple loads, leading to less accurate load forecasting. To address this problem, this paper proposes a multi-model fusion prediction method based on Complementary Ensemble Empirical Mode Decomposition (CEEMD), Genetic Algorithm-Long Short Term Memory (GA-LSTM), Radial Basis Fusion-Autoencoder (RBF-AE), and Particle Swarm Optimization-Support Vector Machine (PSO-SVM). First, the load sequence is decomposed into different frequency Intrinsic Mode Functions (IMFs) components using CEEMD. The IMFs components are then grouped based on their zero-crossing rate and Sample Entropy (SE), resulting in three distinct groups: high-, medium-, and low-frequency components. Next, the high-frequency load component, which exhibit strong randomness, are predicted using GA-LSTM. The medium-frequency load component, which have weaker randomness, are predicted using RBF-AE. The smooth and periodic low-frequency load component are predicted using PSO-SVM. The prediction results from these three models are reconstructed to obtain the final predictive value. Finally, experimental results confirm that the forecasting model can effectively handle non-stationary load sequences and demonstrate the highest level of forecasting accuracy.

A novel multi-gradient evolutionary deep learning approach for few-shot wind power prediction using time-series GAN

Due to the lack of historical data, accurate prediction is a great challenge for newly constructed wind farms (NWFs). How to guarantee satisfactory prediction accuracy with limited data has become a bottleneck issue in wind power prediction (WPP). To solve the few-shot learning problem in NWFs, a novel multi-gradient evolutionary deep learning neural network (EATDLNN) prediction model is proposed, which incorporates the time-series GAN (TimeGAN) and multivariate variational mode decomposition (MVMD) method. TimeGAN adds an auto-encoder network and a supervised loss function to GAN, which enables it to capture the dynamic time series correlation characteristics of wind data and generate high-quality samples. MVMD is then employed to simultaneously decompose nonstationary sequences. In the EATDLNN prediction model, an evolution-based multi-gradient training approach is proposed to train the DLNN by integrating three gradient descent methods (i.e., Adam, MBGD-Momentum, RMSprop). Compared with the traditional single-gradient training approach, the multi-gradient training approach provides different gradient descent paths and the gradient can adaptively choose to update towards the path with the smallest error based on the evolutionary framework, which can help the model jump out of the local optimum and enhance the generalization ability. Massive experiments prove the excellent performance of the proposed method, especially in one-step prediction, the root mean square errors reduce by 79.40%∼91.05% compared with other state-of-the-art methods.

Nonstationary frames of translates and frames from the Weyl–Heisenberg group and the extended affine group *

In this work, we analyze Gabor frames for the Weyl–Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we prove the existence of Gabor frames from the Weyl–Heisenberg group and wavelet frames for the extended affine group. We present a representation of functions in the closure of the linear span of a Gabor frame sequence in terms of the Fourier transform of window functions. We show that the canonical dual of frames of translates has the same structure. An approximation of the inverse of the frame operator of nonstationary frames of translates is presented. It is shown that a nonstationary frame of translates is a Riesz basis if it is linearly independent and satisfies the approximation of the inverse frame operator. Finally, we give equivalent conditions for a nonstationary sequence of translates to be linearly independent.

Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised kth order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised kth order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

On the rate of convergence in the invariance principle for weakly dependent random variables

UDC 519.21 We consider nonstationary sequences of φ -mixing random variables. By using the Levy–Prokhorov distance, we estimate the rate of convergence in the invariance principle for nonstationary φ -mixing random variables. The obtained results extend and generalize several known results for nonstationary φ -mixing random variables.

A Novel End-to-end Network Based on a bidirectional GRU and a Self-Attention Mechanism for Denoising of Electroencephalography Signals

Electroencephalography (EEG) signals are nonlinear and non-stationary sequences that carry much information. However, physiological signals from other body regions may readily interfere with EEG signal capture, having a significant unfavorable influence on subsequent analysis. Therefore, signal denoising is a crucial step in EEG signal processing. This paper proposes a bidirectional gated recurrent unit (GRU) network based on a self-attention mechanism (BG-Attention) for extracting pure EEG signals from noise-contaminated EEG signals. The bidirectional GRU network can simultaneously capture past and future information while processing continuous time sequence. And by paying different levels of attention to the content of varying importance, the model can learn more significant feature of EEG signal sequences, highlighting the contribution of essential samples to denoising. The proposed model is evaluated on the EEGdenoiseNet data set. We compared the proposed model with a fully connected network (FCNN), the one-dimensional residual convolutional neural network (1D-ResCNN), and a recurrent neural network (RNN). The experimental results show that the proposed model can reconstruct a clear EEG waveform with a decent signal-to-noise ratio (SNR) and the relative root mean squared error (RRMSE) value. This study demonstrates the potential of BG-Attention in the pre-processing phase of EEG experiments, which has significant implications for medical technology and brain-computer interface (BCI) applications.

EXAMPLES OF INFORMATION TECHNOLOGIES FOR RECONSTRUCTION FROM THE DATA OF THE SPECTRUM OF SOME CLASSES OF RANDOM FUNCTIONS

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of "strings", i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator A, namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a (x). The cases a (x) = 0, a (x) = const, a (x) = x and a (x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator A promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Ant Colony Algorithm-Based Audit Supervision to Promote the Optimization of New Infrastructure Investment Environment.

Infrastructure investment has the characteristics of high start-up cost, high initial cost, long cycle, slow recovery, high risk, and high public welfare. Maintenance after completion requires stable financial support, maintenance costs are easily affected by government agencies, and investment efficiency is significantly reduced. This article focuses on the ant colony algorithm audit and supervision to promote the optimization of the new infrastructure investment environment and understands the relevant theories of infrastructure investment on the basis of literature data, and then, the audit supervision based on the ant colony algorithm promotes the optimization of the new infrastructure investment environment. The model is constructed, and the constructed model is tested. The test results show that, first, there is a dynamic equilibrium relationship between total infrastructure investment and GDP, even if it cannot explain the potential relationship between the two. In essence, infrastructure investment has a positive impact on economic growth. As the economic level increases, the demand for infrastructure will increase and infrastructure investment will inevitably increase. Second, the level values of the first order are all nonstationary sequences; after the difference, they are all stable and have the same single integral order. The model meets the preconditions of the cointegration test.

\(\ell^{\infty}\) Poisson invariance principles from two classical Poisson limit theorems and extension to non-stationary independent sequences

The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to associated data in the first place.

An Exponential Curve Model and Its Application in Forecasting Private Car Ownership of China

To solve problems of modeling and prediction of nonstationary sequences caused by deterministic factors in the real world, a new type of exponential curve model with first‐order polynomial term is proposed in this paper based on the classical exponential curve model and the modified exponential curve model. According to the characteristics of the new model, the piecewise summation method is adopted to determine the analytical expressions of parameters. Moreover, the model is applied to the ownership of private cars in China. Comparing the calculated results with the classical exponential curve model and the modified exponential curve model, it can be seen that the new exponential curve model has higher accuracy in the analysis of the ownership of private cars in China.

RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0, a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Almost sure invariance principle for random dynamical systems via Gouëzel's approach

We extend the spectral approach of S Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behaviour of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.

Analysis and Economic Prediction of Stock Index Based on Index Tracking and ARIMA Model

Investors buy assets in the target index, if all, the cost is too high, generally not feasible. This paper carries on the stationarity test and the unit root test to the time series, establishes the ARIMA model of the stationary sequence obtained after the difference of the non-stationary sequence, determines the constant of the ARIMA model according to the ACF diagram and the PACF diagram. By analyzing the current exponential fluctuation, the time series model is used to predict the exponential fluctuation in the next year. Through the evaluation and analysis of the prediction results, reasonable investment suggestions and strategies can be obtained.

A New Approach for the DFT NIST Test Applicable for Non-Stationary Input Sequences

The National Institute of Standards and Technology (NIST) document is a list of fifteen tests for estimating the probability of signal randomness degree. Test number six in the NIST document is the Discrete Fourier Transform (DFT) test suitable for stationary incoming sequences. But, for cases where the input sequence is not stationary, the DFT test provides inaccurate results. For these cases, test number seven and eight (the Non-overlapping Template Matching Test and the Overlapping Template Matching Test) of the NIST document were designed to classify those non-stationary sequences. But, even with test number seven and eight of the NIST document, the results are not always accurate. Thus, the NIST test does not give a proper answer for the non-stationary input sequence case. In this paper, we offer a new algorithm or test, which may replace the NIST tests number six, seven and eight. The proposed test is applicable also for non-stationary sequences and supplies more accurate results than the existing tests (NIST tests number six, seven and eight), for non-stationary sequences. The new proposed test is based on the Wigner function and on the Generalized Gaussian Distribution (GGD). In addition, this new proposed algorithm alarms and indicates on suspicious places of cyclic sections in the tested sequence. Thus, it gives us the option to repair or to remove the suspicious places of cyclic sections (this part is beyond the scope of this paper), so that after that, the repaired or the shortened sequence (original sequence with removed sections) will result as a sequence with high probability of random degree.

AQI prediction based on CEEMD-WOA-Elman neural network

Aiming at the problem of predicting AQI, this paper proposes a hybrid prediction model. The specific method is to use Complementary Ensemble Empirical Mode Decomposition (CEEMD) to preprocess the non-stationary sequence, and then use the Elman neural network optimized by WOA to predict. The construction of CEEMD-WOA-Elman model mainly includes four parts: preprocessing, optimization, prediction, and evaluation. In order to evaluate the effectiveness and generalization ability of the model, we introduced four evaluation indicators to comprehensively evaluate the prediction model proposed in this paper. The analysis results show that compared with other models, the hybrid prediction model proposed in this paper has higher prediction accuracy and the predicted results obtained are more excellent.