Distributed estimation over complex adaptive networks with noisy links

In partial diffusion-based least mean square (PDLMS) scheme, each node shares a part of its intermediate estimate vector with its neighbors at each iteration. In this paper, besides studying the general PDLMS scheme, we figure out how the noisy links deteriorate the network performance during the exchange of weight estimates. We investigate the steady state mean square deviation (MSD) and derive a theoretical expression for it. We also derive the mean and mean-square convergence conditions for the PDLMS algorithm in the presence of noisy links. Our analysis reveals that unlike the PDLMS with ideal links, the steady-state network MSD performance of the PDLMS algorithm is not improved as the number of entries communicated at each iteration increases. Strictly speaking, the noisy links condition imposes more complexity to the MSD derivation that has a noticeable effect on the overall performance. This term violates the trade-off between the communication cost and the estimation performance of the networks in comparison with the ideal links. Our simulation results substantiate the effect of noisy links on PDLMS algorithm and verify the theoretical findings. They match well with theory.

Recently proposed distributed adaptive estimation algorithms for wireless sensor networks (WSNs) do not consider errors due to noisy links, which occur during the transmission of local estimates between sensors. In this paper, we study the effect of noisy links on the performance of distributed incremental least-mean-square (DILMS) algorithm for the case of Gaussian regressors. More specifically, we derive theoretical relations which explain how steady-state performance of DILMS algorithm (in terms of mean-square deviation (MSD), excess mean-square error (EMSE), and mean-square error (MSE)) is affected by noisy links. In our analysis, we use a spatial-temporal energy conservation argument to evaluate the steady-state performance of the individual nodes across the entire network. Our simulation results show that there is a good match between simulations and derived theoretical expressions. However, the important result is that unlike the ideal links case, the steady-state MSD, EMSE and MSE are not monotonically increasing functions of step size parameter when links are noisy. In addition, the optimal step size is found in a closed form for a special case which minimizes the steady-state values of MSD, EMSE, and MSE in each node.

Distributed estimation over networks draws much attraction in recent years. In many situations, due to imperfect information communication among nodes, the performance of traditional diffusion adaptive algorithms such as the diffusion least mean squares (DLMS) may degrade. To deal with this problem, several modified DLMS algorithms have been proposed. However, these DLMS based algorithms still suffer from biased estimation and are not robust against impulsive link noise. In this brief, we focus on improving the performance of diffusion adaptation with noisy links from two aspects: accuracy and robustness. A new algorithm called diffusion maximum total correntropy (DMTC) is proposed. The new algorithm is theoretically unbiased in Gaussian noise, and can efficiently handle the link noise in the presence of large outliers. The adaptive combination rule is applied to further improve the performance. The stability analysis of the proposed algorithm is given. Simulation results show that the DMTC algorithm can achieve good performance in both Gaussian and non-Gaussian noise environments.

In this study, the effects of noisy links are investigated on the steady-state performance of mobile adaptive networks with diffusion least mean-squares strategies. The authors derive theoretical relations which explain how the steady-state performance metrics, including the steady-state network mean-square deviation and steady-state velocity mean-square-error is affected by noisy links. The provided analysis relies on the spatial–temporal energy conservation argument. The proposed simulation results reveal that although the noisy links degrade the performance of mobile adaptive networks; however, for suitably chosen combination coefficients the mobile adaptive network with noisy links provides a bounded estimation error. Finally, the proposed simulations verify that the derived theoretical analysis closely matches the actual steady-state performance observed in a network.

This paper first constructs a tiered network model of the interbank market. Then, from the perspective of contagion risk, it studies numerically the resilience of four types of interbank market network models to shocks, namely, tiered networks, random networks, small‐world networks, and scale‐free networks. This paper studies the interbank market with homogeneous and heterogeneous banks and analyzes random shocks and selective shocks. The study reveals that tiered interbank market networks and random interbank market networks are basically more vulnerable against selective shocks, while small‐world interbank market networks and scale‐free interbank market networks are generally more vulnerable against random shocks. Besides, the results indicate that, in the four types of interbank market networks, scale‐free networks have the highest stability against shocks, while small‐world networks are the most vulnerable. When banks are homogeneous, faced with selective shocks, the stability of the tiered interbank market networks is slightly lower than that of random interbank market networks, whereas, in other cases, the stability of the tiered interbank market networks is basically between that of random interbank market networks and that of scale‐free interbank market networks.

Steady-state analysis of the deficient length incremental LMS adaptive networks with noisy links

For distributed estimation arising in the nonlinear least squares (NLLSs) problems over adaptive networks, where every node has the abilities of data processing and learning, only the incomplete local data are exploited by the traditional noncooperative method, thereby resulting in the degradation on estimation performance. In this article, a cooperative diffusion strategy is proposed by using a Gauss–Newton (GN) method in order to fully utilize the diversity of temporal–spatial data on local updates. The proposed algorithm includes two steps, i.e., aggregate then update (ATU), where the aggregating step collects in real time the global information instead of local information due to the diffusion strategy, and the updating step implements the local GN iteration. The resulting ATU diffusion algorithm is a distributed and cooperative system without any increase on communication cost, as compared with the noncooperative version. Based on the detailed convergence analysis for ATU, which is fundamental to the promotion of this algorithm, the sufficient conditions for convergence are derived and the evidences of faster convergence than the noncooperative version are provided. The simulation results confirm the obtained theoretical derivations by applying the ATU algorithm to an NLLS-based target localization problem and show the cooperation gains in many aspects, such as the convergence rate, steady-state accuracy, and robustness to noisy range, step size, node, and link failures.

Bayesian hypothesis testing detector for one bit diffusion LMS with blind missing samples

Simulation of daily maize evapotranspiration at different growth stages using four machine learning models in semi-humid regions of northwest China

Calculation of water influx into petroleum reservoir is a tedious evaluation with significant reservoir engineering applications. The classical approach developed by van Everdingen–Hurst (vEH) based on diffusivity equation solution had been the fulcrum for water influx calculation in both finite and infinite-acting aquifers. The vEH model for edge-water drive reservoirs was modified by Allard and Chen for bottom-water drive reservoirs. Regrettably, these models solution variables: dimensionless influx (WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document}) and dimensionless pressure (PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}) were presented in tabular form. In most cases, table look-up and interpolation between time entries are necessary to determine these variables, which makes the vEH approach tedious for water influx estimation. In this study, artificial neural network (ANN) models to predict the reservoir-aquifer variables WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} was developed based on the vEH datasets for the edge- and bottom-water finite and infinite-acting aquifers. The overall performance of the developed ANN models correlation coefficients (R) was 0.99983 and 0.99978 for the edge- and bottom-water finite aquifer, while edge- and bottom-water infinite-acting aquifer was 0.99992 and 0.99997, respectively. With new datasets, the generalization capacities of the developed models were evaluated using statistical tools: coefficient of determination (R2), R, mean square error (MSE), root-mean-square error (RMSE) and absolute average relative error (AARE). Comparing the developed finite aquifer models predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} with Lagrangian interpolation approach resulted in R2, R, MSE, RMSE and AARE of 0.9984, 0.9992, 0.3496, 0.5913 and 0.2414 for edge-water drive and 0.9993, 0.9996, 0.1863, 0.4316 and 0.2215 for bottom-water drive. Also, infinite-acting aquifer models (Model-1) resulted in R2, R, MSE, RMSE and AARE of 0.9999, 0.9999, 0.5447, 0.7380 and 0.2329 for edge-water drive, while bottom-water drive had 0.9999, 0.9999, 0.2299, 0.4795 and 0.1282. Again, the edge-water infinite-acting model predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and Edwardson et al. polynomial estimated WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{eD}$$\\end{document} resulted in the R2 value of 0.9996, R of 0.9998, MSE of 4.740 × 10–4, RMSE of 0.0218 and AARE of 0.0147. Furthermore, the developed ANN models generalization performance was compared with some models for estimating PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}. The results obtained for finite aquifer model showed the statistical measures: R2, R, MSE, RMSE and AARE of 0.9985, 0.9993, 0.0125, 0.1117 and 0.0678 with Chatas model and 0.9863, 0.9931, 0.1411, 0.3756 and 0.2310 with Fanchi equation. The infinite-acting aquifer model had 0.9999, 0.9999, 0.1750, 0.0133 and 7.333 × 10–3 with Edwardson et al. polynomial, then 0.9865, 09,933, 0.0143, 0.1194 and 0.0831 with Lee model and 0.9991, 0.9996, 1.079 × 10–3, 0.0328 and 0.0282 with Fanchi model. Therefore, the developed ANN models can predict WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} for the various aquifer sizes provided by vEH datasets for water influx calculation.

A wind speed forecasting technique, using deep learning architectures based on long short-term memory (LSTM) model and bidirectional long short-term memory (BiLSTM) model is presented in this work. The coastal belts of the Indian peninsula are vulnerable to natural disasters like storm surges and inundations due to cyclones each year. The wind speed is a major parameter for analyzing extreme weather events. Prediction using numerical models is not efficient enough due to the irregular patterns in the data and, thus, deep neural network models involving many layers have been tested. The shallow feed-forward model has also been considered along with deep learning models to estimate future values from past data. The present work employs a comparison study of different models to forecast wind speed time series at two locations in the Bay of Bengal and the Arabian Sea, respectively, having different dynamics and randomness. For training the models, daily wind speed data are considered for the period 2006–2017 and an independent validation set is chosen comprising 2018 wind speed data to check the accuracy. To evaluate forecast efficiency among different network models fitted to given time series, mean square error (MSE) and root mean square error (RMSE) have been computed. Multiple experiments are conducted with different hidden unit values and epoch values to obtain the minimum error. Regression equations generated may be used for forecasting future time series. The BiLSTM model connecting hidden states of opposite directions proved to be most efficient for the wind speed forecasting in different regions.

The minimum spanning tree: An unbiased method for brain network analysis

A variable step-size diffusion LMS algorithm over networks with noisy links

CBR is a crucial metric used to assess the durability of base course materials and subgrade soils in various types of pavements. In this research, the machine learning (ML) approach has been implemented using random forest (RF), decision tree (DT), linear regression (LR), and artificial neural network (ANN) models to estimate CBR (California bearing ratio) values of the soil based on seven predictors such as maximum dry density, soil classification, optimum moisture content, liquid limit, plastic limit, plastic index, and swell, which can be easily determined from the laboratory. AASHTO M 145 was used to categorize 252 soil samples that formed the basis of an experimental data set. In this model study, the data were split into 20% test data and 80% training data. Standard statistical measures including coefficient of determination, correlations, and errors were used to assess the effectiveness of the models such as MSE (mean squared error), MAE (mean absolute error), and RMSE (root mean square error). From these evaluation metrics, the random forest algorithm gets a smaller error and larger relative error (R2) value to compare with other algorithms. Therefore, it can be concluded that a random forest algorithm based on the analysis findings can accurately forecast the soil’s CBR.

Spatial prediction of soil organic carbon is a crucial proxy to manage and conserve natural resources, monitoring CO2 and preventing soil erosion strategies within the landscape, regional, and global scale. The objectives of this study was to evaluate capability of artificial neural network and multivariate linear regression models in order to predict soil organic carbon using terrain attributes. A study area of 24 km2 in hilly regions of Zargham Ababd in south of Semirom under natural rangeland uses, was selected and then 125 soil samples (0-10 cm depth) were collected. Soil organic carbon was measured for the collected soil samples. Topographic attributes were calculated by a digital elevation model with 10 m spacing. Finally, multiple linear regression (MLR) analysis and ANN models were developed for soil organic carbon estimation in the study area and then the developed modeless were validated by additional samples (25 points). The results showed that the MLR and ANN models explained 60 and 89 % of the total variability of SOC, respectively, in the study area using terrain attributes. Sensitivity analysis based upon the ANN models, revealed that the profile curvature, stream power index, slope, sediment transport index, wetness index, plan curvature and aspect were identified as the important topographic attributes influencing the SOC distribution within the selected hillslope. The overall results indicated that topographic attributes and hydrological process control a significant variability of SOC. Prediction of the statistical studied models in the study area resulted in mean error and root mean square error values of 0.25, 0.3 in MLR equation and 0.006, 0.027 in ANN, respectively. Therefore, the ANN model could provide superior predictive performance when compared with developed MLR model. Keywords: Soil organic carbon, Terrain attributes, Linear regression, Artificial neural network