Error Density Function Research Articles

A multi-factor combination prediction model of carbon emissions based on improved CEEMDAN.

As the global greenhouse effect intensifies, carbon emissions are gradually becoming a hot topic of discussion. Accurate carbon emissions prediction is an important foundation to realize carbon neutrality and peak carbon dioxide emissions. To accurately predict carbon emissions, a multi-factor combination prediction model based on improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), bidirectional long short-term memory optimized by lemurs optimizer (LOBiLSTM) and least squares support vector machine optimized by lemurs optimizer (LOLSSVM), named ICEEMDAN-LOBiLSTM-LOLSSVM, is proposed. Firstly, the influencing factors of carbon emissions are selected by Spearman correlation coefficient, and carbon emissions are decomposed into intrinsic mode functions (IMFs) by ICEEMDAN. Secondly, the influencing factors and IMFs are input into LOBiLSTM and LOLSSVM respectively for prediction. Then, the point prediction results are obtained by weighting the prediction results of LOBiLSTM and LOLSSVM. Finally, probability density function of point prediction error is calculated by kernel density estimation, and the interval prediction results are calculated according to different confidence intervals. Carbon emissions of China and Germany are selected to verify the superiority of ICEEMDAN-LOBiLSTM-LOLSSVM. The experiment shows that RMSE, MAE, MAPE, and R2 of the proposed model are 0.4468, 0.3612, 0.0120, and 0.9839 respectively for China, which is the best among the nine models, as well as for Germany.

Constructions of statistical estimates of digital measurement error in the case of small samples

The paper presents a computational-experimental method for the construction of statistical estimates of the error of digital measurements carried out on complex technical objects with metrological support. The algorithm of construction of the estimation of the distribution density function of random error of measurements on the basis of complex application of methods of the theory of characteristic functions, the theory of Lie operator series and methods of statistical modeling in the processing of a limited amount of statistical information in the conditions of small samples is described. The solution of the practical problem on the construction of the estimation of the measurement error distribution density and the problem on the zero mark of the measuring instrument in the probabilistic formulation is given. The presented computational and experimental method can be used to construct statistical estimates for the probability distribution density of the general type, including those given by implicit functions, characteristic functions and operator series.

Anomaly detection for maritime navigation based on probability density function of error of reconstruction

Abstract Anomaly detection is a fundamental problem in data science and is one of the highly studied topics in machine learning. This problem has been addressed in different contexts and domains. This article investigates anomalous data within time series data in the maritime sector. Since there is no annotated dataset for this purpose, in this study, we apply an unsupervised approach. Our method benefits from the unsupervised learning feature of autoencoders. We utilize the reconstruction error as a signal for anomaly detection. For this purpose, we estimate the probability density function of the reconstruction error and find different levels of abnormality based on statistical attributes of the density of error. Our results demonstrate the effectiveness of this approach for localizing irregular patterns in the trajectory of vessel movements.

Estimating Earth Pressure during Tunneling in Gravel-cobble Stratum by Considering Sensor-particle Size Effect

When tunneling in gravel-cobble stratum with earth balanced shield, the ratio between the diameters of the pressure sensor embodied in muck chamber and cut gravels is generally small. The small sensor-particle size ratio leads to considerable difference between the measured and actual earth pressure. This makes the configuration of earth pressure a difficult task and furtherly affects tunneling efficiency and safety. This work characterizes the variation of stress measurement error in a granular packing with consideration of the sensor-particle size ratio, the particle size distribution as well as the contact force distribution. The probability density function of stress measurement error is built according to a general probability distribution of inter-particle contact force. After then, a probability-based regressive procedure is proposed to estimate the applied stress with the measured stress. For stress samples with a same target stress but different biases, the stress estimated by the probability-based procedure has a minor variation and thus is more reliable than the arithmetic mean stress. Finally, the probability-based procedure is adopted to estimate the actual earth pressure of a well-tunneled section in Chengdu gravel-cobble stratum, according to which the lateral pressure coefficient of Chengdu gravel-cobble stratum is suggested as 0.20 – 0.28.

Evaluation of Extreme Precipitation Indices in Brazil’s Semiarid Region from Satellite Data

Brazil’s semiarid region (SAB) has a heterogeneous precipitation distribution, with the occurrence of periodic droughts and occasional extreme rainfall events. The precipitation monitoring system in this region is insufficient, but remote sensing products can provide information on rainfall in areas with low data coverage. Thus, the main objective of this study was to evaluate 12 extreme precipitation indices calculated using Integrated Multi-Satellite Retrievals for Global Precipitation Measurement (IMERG) data in comparison with indices calculated from data measured by rain gauges for different SAB locations. To evaluate the IMERG product, we used rainfall data measured by 56 rain gauges during the period from 1 January 2000 to 31 December 2020. The satellite product was evaluated through juxtaposition between the IMERG and actual rainfall data, by calculating the statistical indices bias, root-mean-squared error, Spearman correlation, and probability density function. The results showed that most of the extreme precipitation indices were well represented by the satellite data, except for the simple precipitation intensity index (SDII), in which case the correlation coefficient was 0.2. This result can be explained as this index is calculated from the exact value of daily precipitation, while the other indices are estimated by rainfall values above some thresholds. On the other hand, total annual precipitation and precipitation above 1 mm presented Spearman correlation reaching 0.97 in some locations. We conclude that the IMERG database is adequate to represent the maximum precipitation in the Brazilian semiarid region, and the extreme precipitation indices had good performance according to the region where the maximum rain occurred.

An Anomaly Detection Method of Time Series Data for Cyber-Physical Integrated Energy System Based on Time-Frequency Feature Prediction

An integrated energy system (IES) is vulnerable to network attacks due to the coupling features of multi-energy systems, as well as the deep integration between a physical system and an information system. The anomaly detection of the time-series data in an IES is a key problem to defend against network attacks and ensure the cyber-physical security of IES. Aiming at false data injection attacks (FDIAs) on IES, this paper proposes an anomaly detection method for time-series data in a cyber-physical integrated energy system based on time-frequency feature prediction. The time-frequency features of the time-series data are extracted based on three time-frequency transform methods (DWT, EMD, and EWT). Then the extracted time-frequency features are input to the autoencoder (AE) to capture the hidden features and nonlinear structure of the original data in the frequency domain. The time-domain data within the detected time period are predicted by applying regression prediction on the top-layer features of AE. Considering the uncertainty of regression prediction, kernel density estimation (KDE) is used to estimate the probability density function of prediction error and the interval of the predicted data is estimated accordingly. The estimated lower boundary value of the predicted data is selected as the attack judgment threshold for anomaly detection. The results of the case study verify the advantages of the proposed method in reducing the false positive rate and improving the anomaly detection accuracy.

Data-driven reliable prediction of production indicators in the blast furnace using TS fuzzy neural network based on bat algorithm

ABSTRACT Operational optimisation of the blast furnace (BF) is significant in facilitating smooth operation and reducing the production cost. Operation indicators are pivotal parameters used to measure operating status and reflect the quality of molten iron of the BF. In this study, a data-driven method for the prediction of production indicators is proposed by a novel fuzzy neural network of Takagi-Sugeno with bat algorithm (BA-TS-FNN). In practice, a BF usually works under varying operating conditions due to internal or external factors, redundant features with high dimensionally lead to poor computation time and prediction performance. To solve this problem, mutual information is applied to enhance understanding of the data. Considering that the standard TS-FNN cannot commendably cope with rapid convergence, local optimal, and sensitivity to initial weight problems during learning the prediction model, a bat optimisation algorithm is proposed to search global optimal parameters and improve the convergence rate. Finally, the probability density function (PDF) of modelling error is estimated by the kernel density estimation (KDE) as a criterion to measure the performance of the method. Experiments with industrial data from BF have demonstrated that the proposed method produces higher estimating accuracy than other modelling methods.

Random forest and nature-inspired algorithms for mapping groundwater nitrate concentration in a coastal multi-layer aquifer system

Accurate prediction of groundwater nitrate concentrations is critical for pollution control and sustainable groundwater resource management. However, Ant optimization (Ant), Firefly, Multiple Objective Evolutionary (MOE), and Grey Wolf Optimization (GWO), are novel data-driven hydrid algorithms that are barely used in groundwater pollution studies. Yet, areas susceptible to groundwater nitrate concentration have not been evaluated in the Mekong Delta. In this paper, we develop five novel hybrid algorithms: Ant, Firefly, MOE, GWO and Particle Swarm Optimization (PSO) which are integrated with the random forest (RF) algorithm for mapping groundwater nitrate concentrations in the coastal multi-aquifers of the Mekong Delta. The geographic information system (GIS) environment was used to construct 24 conditional factors believed to affect groundwater nitrate contamination. In 216 wells, nitrate concentrations were collected and quantified, and the results were utilized as a dependent parameter in modeling. To assess the effectiveness of these proposed hybrid models, evaluation measures such as statistical error indices, confusion matrices, Taylor diagram, the probability density function of error, and scatter plots were used. The RF-Firefly model (RMSE = 6.25, AUC = 0.961) performed the best based on testing datasets, followed by the RF-MOE (RMSE = 8.67, AUC = 0.878), the RF-Ant (RMSE = 7.27, AUC = 0.874), the RF-GWO ( RMSE = 7.35, AUC = 0.87), RF-PSO (RMSE = 10.24, AUC = 0.864) and RF ( RMSE = 15.85, AUC = 0.816) models. The results of mapping groundwater nitrate concentrations also showed that the south-central region had a very high nitrate concentration (> 45 mg/L) compared to other places. It was also found that extensive agricultural activities, wells near farming areas, and overexploitation of groundwater were the main causes of nitrate pollution in the study. The proposed models show their capability and application in predicting groundwater nitrate pollution in various deltaic locations.

Development of Regional Climate Model (RCM) for Cameron Highlands based on Representative Concentration Pathways (RCP) 4.5 and 8.5

Development of local climate is important for climate hazard assessment. Cameron Highlands was chosen as study area (longitude from 101°21’ to 101°30’ and latitude from 4°19’ to 4°37’) for climate downscaling. This paper presents the work of downscaling techniques and regional climate model (RCM) development. The second-generation Canadian Earth System Model (CanESM2) based on the representative concentration pathways (RCP) 4.5 and 8.5 was applied to develop the local climate model for the period 2020-2100. The climatic parameters chosen were temperature (maximum and minimum) and rainfall. The simulated RCMs are then analysed using statistical reliability including Pearson correlation coefficient, linear regression, root mean square error (RMSE) and probability density function (PDF). The result showed that the simulated maximum, minimum temperature, and rainfall are most likely to follow RCP 8.5 scenario. Precipitation threshold for occurrence of flood event was estimated using intensity duration frequency (IDF) relationship generated by maximum precipitation. Return period of two years and four hours rainfall duration is used for threshold estimation as the rainfall is convective. The daily rainfall threshold for flood occurrence is estimated to be 11.3 mm/hr.

A Novel Atmosphere-Informed Data-Driven Predictive Channel Modeling for B5G/6G Satellite-Terrestrial Wireless Communication Systems at Q-Band

This paper proposes a novel atmosphere-informed predictive satellite channel model for beyond the fifth-generation (B5G)/the sixth-generation (6G) satellite-terrestrial wireless communication systems at Q-band to model/predict channel attenuation at any specific time. The proposed channel model is a data-driven model based on either of two deep learning networks, i.e., multi-layer perceptron (MLP) and long short-term memory (LSTM). The accuracy of the proposed channel model is measured by cumulative density function (CDF) of absolute error and mean square error (MSE) between modeled/predicted and measured channel attenuation. The complexity of the proposed channel model is assessedby the training time, loading time, and test time of deep learning networks. To further improve the accuracy of the proposed channel model, weather classification is developed at the stage of database construction. Based on our established channel and weather measurement campaign, the performance of the proposed data-driven channel model based on different deep learning networks, e.g., MLP and LSTM, with or without the weather classification is investigated and analyzed comprehensively. Finally, the close agreement is achieved between the channel attenuation modeled/predicted from the proposed atmosphere-informed predictive satellite channel model and the one from real channel measurements, verifying the utility of proposed channel model.

Global Gravitational Search Algorithm-Aided Kalman Filter Design for Volterra-Based Nonlinear System Identification

This paper proposes an efficient global gravitational search (GGS) algorithm-assisted Kalman filter (KF) design, called a GGS-KF technique, for accurate estimation of the Volterra-type nonlinear systems. KF is a well-known estimation technique for the dynamic states of the system. The best estimate is achieved if the system dynamics and noise statistical model parameters are available at the beginning. However, to estimate the real-time problems, these parameters are unstipulated or partly known. Due to this limitation, the performance of the KF degrades or sometimes diverges. In this work, two steps have been proposed for unknown system identification while overcoming the difficulty encountered in KF. The first step is to optimise the parameters of the KF using the GGS algorithm by considering a properly balanced fitness function. The second step is to estimate the unknown coefficients of the system by using the basic KF method with the optimally tuned KF parameters obtained from the first step. The proposed GGS-KF technique is tested on five different Volterra systems with various levels of noisy (10 dB, 15 dB and 20 dB) and noise-free input conditions. The simulation results confirm that the GGS-KF-based identification approach results in the most accurate estimations compared to the conventional KF and other reported techniques in terms of parameter estimation error, mean-squared error (MSE), fitness percentage (FIT%), mean-squared deviation (MSD), and cumulative density function (CDF). To validate the practical applicability of the proposed technique, two benchmark systems have also been identified based on the original data sets.

Targeted design for adaptive clinical trials via semiparametric model

Precision medicine approach that assigns treatment according to an individual's personal (including molecular) profile is revolutionizing health care. Existing statistical methods for clinical trial design typically assume a known model to estimate characteristics of treatment outcomes, which may yield biased results if the true model deviates far from the assumed one. This article aims to achieve model robustness in a phase II multi-stage adaptive clinical trial design. We propose and study a semiparametric regression mixture model in which the mixing proportions are specified according to the subjects' profiles, and each sub-group distribution is only assumed to be unimodal for robustness. The regression parameters and the error density functions are estimated by semiparametric maximum likelihood and isotonic regression estimators. The asymptotic properties of the estimates are studied. Simulation studies are conducted to evaluate the performance of the method after a real data analysis.

Unified data‐driven stochastic and robust service restoration method using non‐parametric estimation in distribution networks with soft open points

The service restoration is a vital measurement to enhance the resilience of the electrical distribution network. With the integration of distributed generations (DGs) in the distribution network, the uncertainties are introduced, which have a considerable impact on service restoration. Meanwhile, the soft open points (SOPs) are electronic devices installed in the tie-line of the distribution networks. They have the ability to control the branch power flow, which can be utilised to assist the service restoration. This study proposes a data-driven stochastic and robust service restoration method to handle the uncertainties of DGs outputs, which is modelled as a two-stage conic optimisation programming problem. The non-parametric estimation method is introduced to estimate the probability density function of forecasted error based on historical data. Furthermore, the nonlinear constraints of the SOPs are reformulated into conic forms so that the proposed model could be solved efficiently. Numerical tests are carried out on the IEEE 33-bus and 123-bus distribution networks to show the effectiveness of the proposed method, and the results show the superiority of the proposed method.

Machine-learning-based high-resolution DOA measurement and robust directional modulation for hybrid analog-digital massive MIMO transceiver

At hybrid analog-digital (HAD) transceiver, an improved HAD estimation of signal parameters via rotational invariance techniques (ESPRIT), called I-HAD-ESPRIT, is proposed to measure the direction of arrival (DOA) of a desired user, where the phase ambiguity due to HAD structure is dealt with successfully. Subsequently, a machine-learning (ML) framework is proposed to improve the precision of measuring DOA. Meanwhile, we find that the probability density function (PDF) of DOA measurement error (DOAME) can be approximated as a Gaussian distribution by the histogram method in ML. Then, a slightly large training data set (TDS) and a relatively small real-time set (RTS) of DOA are formed to predict the mean and variance of DOA/DOAME in the training stage and real-time stage, respectively. To improve the precisions of DOA/DOAME, three weight combiners are proposed to combine the-maximum-likelihood-learning outputs of TDS and RTS. Using the mean and variance of DOA/DOAME, their PDFs can be given directly, and we propose a robust beamformer for directional modulation (DM) transmitter with HAD by fully exploiting the PDF of DOA/DOAME, especially a robust analog beamformer on RF chain. Simulation results show that: (1) the proposed I-HAD-ESPRIT can achieve the HAD Cramer-Rao lower bound (CRLB); (2) the proposed ML framework performs much better than the corresponding real-time one without training stage; (3) the proposed robust DM transmitter can perform better than the corresponding non-robust ones in terms of secrecy rate.

H-adaptive topology optimization considering variations of material properties and energy error density recovery

PurposeThe purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.Design/methodology/approachThe new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.FindingsTwo-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.Originality/valueThis paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Construction and simulation of performance evaluation index system of Internet of Things based on cloud model

Due to the lack of corresponding scientific evaluation standards and effective evaluation methods for the overall performance of the Internet of Things system, the Internet of Things industry and the market are in a chaotic state, and the quality of various products is uneven. It is difficult for users to select a more suitable application system, which seriously restricts the development process of the industrialization of Internet of Things technology. Therefore, it is very necessary for us to conduct research on the performance evaluation methods of IoT systems. In this paper, the performance of IoT system is selected as the performance evaluation index of the system, and the evaluation system is constructed. The performance evaluation method of the Internet of Things system based on the cloud model is proposed. The comprehensive evaluation of the reliability of the IoT system for the cloud model will be based on the AHP. Based on the combination of attribute decision-making methods, a comprehensive evaluation method based on indicators is proposed, and the system’s indicator model is established. The reliability of the Internet of Things system can be comprehensively evaluated by the value of the indicators. Secondly, the probability density function of tracking error is studied, and its optimal probability density distribution is given by optimal control. Based on the optimal probability density distribution, the performance evaluation index of the system is constructed. In order to compare the current probability density function with the reference probability density function, the tracking error function is first vector, then the performance evaluation index is constructed by gray correlation, and the general steps of performance evaluation based on this index are given. Finally, the effectiveness of the proposed system performance evaluation algorithm is verified by simulation comparison. The data processing and experimental simulation are used to achieve the purpose of evaluating the performance of the Internet of Things system.

Evaluation of roundness error uncertainty by a Bayesian dynamic model

The majority of existing measurement and evaluation methods of geometric tolerance of products are for the static processes and thus are not suitable for problems of measurement information traceability, dynamic management and control of measurement process. In this paper, a Bayesian dynamic model for evaluating the uncertainty of roundness error is proposed. The least square error evaluation is carried out first based on coordinate measuring machine (CMM) information, and then the probability density function (PDF) of roundness error is described by the maximum entropy principle. By using the prior distribution of Bayesian theory to calculate the synthetic posterior distribution, the measurement information is fused, and then the uncertainty is evaluated dynamically. Through digital experiment and test analysis of bearing outer ring, and compared with ISO standard method, the results show that Bayesian dynamic model can correctly and effectively evaluate the uncertainty dynamically. The overall estimation process is stable.

State Uncertainty Normality Detection

Two fundamentally different approaches of determining normality of the probability density function of the state estimation error are compared by application to a range of test cases. The first method is the Henze-Zirkler test, which operates on a random particle sample. The variability of its result is quantified. Using this method, departure from normality has been found to occur in three stages which are detailed. The second test compares the offset in whitened space of the predicted state to the predicted covariance mean obtained from the unscented transform. This test is much more efficient than the random particle based approach and can be applied using any perturbations model. The comparison is performed on the state estimation error in Cartesian space and using two-body motion without process noise. The more efficient, unscented transform based approach shows excellent agreement with the Henze-Zirkler test for constructed test cases. Application to orbit determination results from passive optical observations assessed with a Batch-Least-Squares orbit determination however reveals some discrepancies which have yet to be understood and underline the importance of rigorous testing.

Estimation of a functional single index model with dependent errors and unknown error density

The problem of error density estimation for a functional single index model with dependent errors is studied. A Bayesian method is utilized to simultaneously estimate the bandwidths in the kernel-form error density and regression function, under an autoregressive error structure. For estimating both the regression function and error density, empirical studies show that the functional single index model gives improved estimation and prediction accuracies than any nonparametric functional regression considered. Furthermore, estimation of error density facilitates the construction of prediction interval for the response variable.

Anisotropic Mesh Refinement Considering a Recovery-Based Error Estimator and Metric Tensors

This paper proposes a new anisotropic h-adaptive technique applied to the finite element method considering triangular finite elements. The proposed technique, named anisotropic error density recovery, utilizes the concept of the energy error density function and the analytical solution of an optimization problem by Lagrange’s multipliers to obtain a metric tensor that represents the dimensions of the new elements. This process aims to limit and equidistribute discretization errors throughout the mesh utilizing finite elements with large aspect ratios. The anisotropic h-adaptive technique is computationally implemented and tested in two-dimensional elliptic problems. The results demonstrate the ability of this approach to generate meshes of elements with high aspect ratios and oriented according to the principal directions of discretization errors. Thus, when compared with isotropic refinement techniques, the anisotropic error density recovery approach shows a significant reduction in the number of vertices for the same permissible error value.