Effect Of Outliers Research Articles

Robustness and Tractability for Non-convex M-estimators

We investigate two important properties of M-estimator, namely, robustness and tractability, in linear regression setting, when the observations are contaminated by some arbitrary outliers. Specifically, robustness means the statistical property that the estimator should always be close to the underlying true parameters {\em regardless of the distribution of the outliers}, and tractability indicates the computational property that the estimator can be computed efficiently, even if the objective function of the M-estimator is {\em non-convex}. In this article, by learning the landscape of the empirical risk, we show that under mild conditions, many M-estimators enjoy nice robustness and tractability properties simultaneously, when the percentage of outliers is small. We further extend our analysis to the high-dimensional setting, where the number of parameters is greater than the number of samples, $p \gg n$, and prove that when the proportion of outliers is small, the penalized M-estimators with {\em $L_1$} penalty will enjoy robustness and tractability simultaneously. Our research provides an analytic approach to see the effects of outliers and tuning parameters on the robustness and tractability for some families of M-estimators. Simulation and case study are presented to illustrate the usefulness of our theoretical results for M-estimators under Welsch's exponential squared loss.

Missing value imputation using least squares techniques in contaminated matrices

This paper describes strategies to reduce the possible effect of outliers on the quality of imputations produced by a method that uses a mixture of two least squares techniques: regression and lower rank approximation of a matrix. To avoid the influence of discrepant data and maintain the computational speed of the original scheme, pre-processing options were explored before applying the imputation method. The first proposal is to previously use a robust singular value decomposition, the second is to detect outliers and then treat the potential outliers as missing. To evaluate the proposed methods, a cross-validation study was carried out on ten complete matrices of real data from multi-environment trials. The imputations were compared with the original data using three statistics: a measure of goodness of fit, the squared cosine between matrices and the prediction error. The results show that the original method should be replaced by one of the options presented here because outliers can cause low quality imputations or convergence problems.•The imputation algorithm based on Gabriel's cross-validation method uses two least squares techniques that can be affected by the presence of outliers. The inclusion of a robust singular value decomposition allows both to robustify the procedure and to detect outliers and consider them later as missing. These forms of pre-processing ensure that the algorithm performs well on any dataset that has a matrix form with suspected contamination.

Outlier Detection and Forecasting for Bridge Health Monitoring Based on Time Series Intervention Analysis

The method of time series analysis, applied by establishing appropriate mathematical models for bridge health monitoring data and making forecasts of structural future behavior, stands out as a novel and viable research direction for bridge state assessment. However, outliers inevitably exist in the monitoring data due to various interventions, which reduce the precision of model fitting and affect the forecasting results. Therefore, the identification of outliers is crucial for the accurate interpretation of the monitoring data. In this study, a time series model combined with outlier information for bridge health monitoring is established using intervention analysis theory, and the forecasting of the structural responses is carried out. There are three techniques that we focus on: (1) the modeling of seasonal autoregressive integrated moving average (SARIMA) model; (2) the methodology for outlier identification and amendment under the circumstances that the occurrence time and type of outliers are known and unknown; (3) forecasting of the model with outlier effects. The method was tested with a case study using monitoring data on a real bridge. The establishment of the original SARIMA model without considering outliers is first discussed, including the stationarity, order determination, parameter estimation and diagnostic checking of the model. Then the time-by-time iterative procedure for outlier detection, which is implemented by appropriate test statistics of the residuals, is performed. The SARIMA-outlier model is subsequently built. Finally, a comparative analysis of the forecasting performance between the original model and SARIMA-outlier model is carried out. The results demonstrate that proper time series models are effective in mining the characteristic law of bridge monitoring data. When the influence of outliers is taken into account, the fitted precision of the model is significantly improved and the accuracy and the reliability of the forecast are strengthened.

Robust Object Classification Approach Using Spherical Harmonics

Point clouds produced by either 3D scanners or multi-view images are often imperfect and contain noise or outliers. This paper presents an end-to-end robust spherical harmonics approach to classifying 3D objects. The proposed framework first uses the voxel grid of concentric spheres to learn features over the unit ball. We then limit the spherical harmonics order level to suppress the effect of noise and outliers. In addition, the entire classification operation is performed in the Fourier domain. As a result, our proposed model learned features that are less sensitive to data perturbations and corruptions. We tested our proposed model against several types of data perturbations and corruptions, such as noise and outliers. Our results show that the proposed model has fewer parameters, competes with state-of-art networks in terms of robustness to data inaccuracies, and is faster than other robust methods. Our implementation code is also publicly available at <uri>https://github.com/AymanMukh/R-SCNN</uri>

A Robust Dynamic State Estimation Method for Power Systems Using Exponential Absolute Value-Based Estimator

Even though the noise model applied in power system dynamic state estimation is usually assumed to be Gaussian, this is not the case due to the unknown system inputs, influence from the communication channel noise and the outliers generated by phasor measurement units (PMUs). In this paper a robust power system dynamic state estimation (DSE) method combining a robust exponential absolute value based estimator and the unscented Kalman filter together is proposed under non-Gaussian noise. Based on the quadratic function and the exponential absolute value function, robust exponential absolute value based estimator is derived, further mitigating the effects of bad data or outliers. The influence function is utilized to calculate the state estimation error covariance of the proposed robust DSE method. The simulation results on the IEEE 39-bus system verify the robustness and effectiveness of the proposed dynamic state estimation method.

Spatiotemporal Fusion Network for Land Surface Temperature Based on a Conditional Variational Autoencoder

High spatiotemporal resolution land surface temperature (LST) data are essential for dynamic monitoring and prediction in climate change research. Due to the limitations of remote sensing instruments, the current platforms have difficulty achieving a compromise between high spatial and temporal resolutions for LST products. In this study, we propose a spatiotemporal fusion network for land surface temperature based on a conditional variational autoencoder (CVAE-LSTFM). First, an improved network is designed based on the CVAE by reconstructing an encoder and a decoder. To generate fine LST images based on dense time series, a variational inference model is formulated to integrate coarse and fine LST image pairs in variational autoencoded latent space. In addition, a new compound loss function for the proposed training method is designed to reduce the effects of noise and outliers. Then, a pretraining mechanism is adopted to optimize the network training process, and the parameters can be transferred to the training network of the CVAE to accelerate network convergence. Finally, a novel weighting strategy that considers spatiotemporal variations in LST (LST consistency weighting) is employed to solve the spatiotemporal heterogeneity problem caused by the rapid changes in LST. The method is quantitatively tested and evaluated in the Heihe River Basin using FY-4A LST and MODIS LST from September 2019. Compared with two traditional models and two deep learning-based models, CVAE-LSTFM yields lower RMSE (average&#x003C;1.26 K) and LPIPS (learning perceptual image patch similarity, average&#x003C;0.13) values and higher SSIM (structural similarity, average&#x003E;0.96). In practice, CVAE-LSTFM can generate high spatiotemporal resolution LST values (hourly LST with a 1 km spatial resolution) with high accuracy, quality, and robustness.

Additive Outliers in Open-Loop Threshold Autoregressive Models: A Simulation Study

The effect of additive outliers is studied on an adapted non-linearity test and a robust estimation method for autoregressive coefficients in open-loop TAR (threshold autoregressive) models. Through a Monte Carlo experiment, the power and size of the non-linearity test are studied. Regarding the estimation method, the bias and ratio of mean squared errors are compared between the robust estimator and least squares. Simulation exercises are carried out for different percentages of contamination and the proportion of observations on each model regime. Furthermore, the approximation of the univariate normal distribution to the empirical distribution of estimated coefficients is analyzed along with the coverage level of asymptotic confidence intervals for the parameters. Results show that the adapted non-linearity test does not have size distortions, and it has a superior power than its least-squares counterpart when additive outliers are present. On the other hand, the robust estimation method for the autoregressive coefficients has a better mean squared error than least-squares when this type of observations are present. Lastly, the use of the non-linearity test and the estimation method are illustrated through a real example.

Probabilistic Twin Support Vector Machine for Solving Unclassifiable Region Problem

Support Vector Machine classifiers are widely used in many classification tasks. However, they have two considerable weaknesses, Unclassifiable Region (UR) in multiclass classification and outliers. In this research, we address these problems by introducing Probabilistic Least Square Twin Support Vector Machine (PLS-TSVM). The proposed algorithm introduces continuous and probabilistic outputs over the model obtained by Least-Square Twin Support Vector Machine (LS-TSVM) method with both linear and polynomial kernel functions. PLS-TSVM not only solves the unclassifiable region problem by introducing a continuous output value membership function, but it also reduces the adverse effects of noisy data and outliers. For showing the superiority of our proposed method, we have conducted experiments on various UCI datasets. In the most cases, higher or competitive accuracy to other methods have been obtained such that in some UCI datasets, PLS-TSVM could obtain up to 99.90% of classification accuracy. Moreover, PLS-TSVM has been evaluated against ”one-against-all” and ”one-against-one” approaches on several well-known video datasets such as Weizmann, KTH, and UCF101 for human action recognition task. The results show the higher accuracy of PLS-TSVM compared to its counterparts. Specifically, the proposed algorithm could improve respectively about 12.2%, 2.8%, and 12.1% of classification accuracy in three video datasets compared to the standard SVM and LS-TSVM classifiers. The final results indicate that the proposed algorithm could achieve better overall performances than the literature.

Observer-based H∞ consensus for linear multi-agent systems subject to measurement outliers

The consensus problem has a crucial role in theoretical and practical aspects of interconnected systems, for instance in smart computing field. This paper is concerned with the observer-based H∞ consensus problem for a class of linear multi-agent systems subject to measurement outliers. In the addressed observer-based H∞ consensus problem, measurement outliers can affect the estimation accuracy and thus affect the consensus performance. To assuage the impact of measurement outliers, a control protocol based on an observer containing a saturation function with variable saturation limits is proposed. The purpose of this paper is to find a solution to the addressed H∞ consensus problem for a class of linear multi-agent systems subject to measurement outliers by designing an observer-based control protocol such that multi-agent systems can fulfill the H∞ consensus performance over a finite horizon. With the aid of Lyapunov theory, the sufficient condition is established to guarantee that the consensus error dynamic system satisfies the H∞ consensus performance. Then, the linear matrix inequality (LMI) approach is used to obtain the desired parameters of the observer-based control protocol. Finally, the effectiveness of the proposed control protocol is verified in a simulation environment. Aim: The purpose of this study is to find a solution to the addressed H∞ consensus problem for a class of linear multi-agent systems subject to measurement outliers. Methods: By choosing a suitable Lyapunov function, the sufficient condition is obtained, which can guarantee the consensus error dynamic system satisfying the given H∞ consensus performance. The LMI approach is employed to design the desired controller. With Matlab LMI toolbox, a numerical example was conducted to demonstrate the effectiveness of the proposed observer-based control protocol in the simulation environment. Results: A satisfactory consensus performance can be guaranteed for multi-agent systems subject to measurement outliers under the proposed observer-based control protocol. The constructed saturation function with variable saturation limits can mitigate the effect of measurement outliers by dynamically regulating saturation limits. Compared with the traditional observer-based control protocol, in this paper, the proposed observer-based control protocol shows robustness against measurement outliers. Conclusion: A solution to the addressed multi-agent consensus problem subject to measurement outliers was found by designing an observer-based consensus controller. The obtained results can be extended to sensor networks, neural networks, and nonlinear multi-agent systems.

TDOA-Based Robust Sound Source Localization With Sparse Regularization in Wireless Acoustic Sensor Networks

Time difference of arrival (TDOA) measurements, which are contaminated by large values of error, known as outliers, would have a significant impact on the accuracy of sound source localization (SSL) in wireless acoustic sensor networks (WASNs). Few techniques are reported in the literature to tackle SSL in WASNs by taking TDOA outliers into consideration. To mitigate the effect of outliers on the accuracy of SSL, we propose outlier-resistant robust sound source localization (RSSL) algorithms based on sparse regularization using an unsynchronized network of microphone arrays. The TDOA errors are divided into two components: a) energy-bounded inliers and b) outliers. Assuming that outliers are sparse in the measurement set, we formulate the RSSL problem as that of minimizing the number of outliers, mathematically, a <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm optimization problem with non-convex constraints. Five sub-optimal RSSL solvers are derived, among which the first two solvers are applicable to the scenario involving only outliers while the last three solvers concentrate on the scenario incorporating both outliers and inliers. In common, these solvers exploit a convex approximation technique called Concave Convex Procedure to dispose of the non-convex constraints. Differently, the first solver approximates the original <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm cost function with the <inline-formula><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> norm while a concave surrogate function is adopted in the second solver to yield a tighter approximation to the <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm. Apart from the application of these two approximation techniques, the third and fourth solvers relax the non-convex <inline-formula><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> norm constraint with the <inline-formula><tex-math notation="LaTeX">$\ell _{\infty }$</tex-math></inline-formula> norm. The fifth solver is dedicated to the <inline-formula><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> norm regularization problem with the Lasso formulation, which is equivalent to the M-estimator of Huber&#x2019;s function solved via the iteratively reweighted least squares paradigm. Experimental results validate the effectiveness and robustness of the proposed algorithms.

Prediction from Transmuted Rayleigh Distribution in the Presence of Outliers

The quality of the procedures used in statistical analysis depends largely on the assumed probability distribution. However, there are still many problems with data that do not follow any of the classical distributions; therefore, researchers have developed many standardized probability distributions by generalizing or transforming them. Transmuted Rayleigh distribution extends the Rayleigh distribution in the analysis of data and provides larger flexibility in modeling real data. In this article, Bayesian predictive intervals for order statistics of future observations from this distribution are obtained in the presence of outliers when the scale parameter is unknown. The slippage outlier model is utilized in addition to the two‐sample prediction scheme. We shall consider two cases: (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample. Numerical computations are obtained to illustrate the effect of outliers on the Bayesian predictive intervals.

A Weighted k-Nearest Neighbours Ensemble With Added Accuracy and Diversity

Ensembles based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> NN models are considered effective in reducing the adverse effect of outliers, primarily, by identifying the closest observations to a test point in a given training data. Class label of the test point is estimated by taking a majority vote of the nearest observations’ class labels. While identifying the closest observations, certain training patterns might possess high regulatory power than the others. Therefore, assigning weights to observations and then calculating weighted distances are deemed important. This paper proposes a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> NN ensemble that identifies nearest observations based on their weighted distance in relation to the response variable via support vectors. This is done while building a large number of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> NN models each on a bootstrap sample from the training data along with a randomly selected subset of features from the given feature space. The estimated class of the test observation is decided via majority voting based on the estimates given by all the base <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> NN models.The ensemble is assessed on 14 benchmark and simulated datasets against other classical methods, including <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> NN based models using Brier score, classification accuracy and Kappa as performance measures. Average values of the given measures along with boxplots are given for the proposed and the other competitors. On both benchmark and simulated datasets, the proposed ensemble outperformed the other competitive methods in majority of the cases. The suggested ensemble gave overall better classification performance than the other methods on 8 datasets. The ensemble performed better than the other methods in simulation study that considered the presence of noisy features in a dataset. The analyses reveal that the proposed method is effective in classification problems that involve noisy features in the data. Furthermore, feature weighting and randomization also make the method robust to the choice of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , i.e., the number of nearest observations in a base model.

A robust approach for outlier imputation: Singular spectrum decomposition

Singular spectrum analysis (SSA) is a nonparametric method for separating time series data into a sum of small numbers of interpretable components (signal + noise). One of the steps of the SSA method, which is referenced to Embedding, is extremely sensitive to contamination of outliers which are often founded in time series analysis. To reduce the effect of outliers, SSA based on Singular Spectrum Decomposition (SSD) method is proposed. In this article, the ability of SSA based on SSD and basic SSA are compared in time series reconstruction in the presence of outliers. It is noteworthy that the matrix norm used in Basic SSA is the Frobenius norm or L 2-norm. There is a newer version of SSA that is based on L 1-norm and called L 1-SSA. It was confirmed that L 1-SSA is robust against outliers. In this regard, this research is also introduced a new version of SSD based on L 1-norm which is called L 1-SSD. A wide empirical study on both simulated and real data verifies the efficiency of basic SSA based on SSD and L 1-norm in reconstructing the time series where polluted by outliers.

Lipid control in adult Chinese patients with type 2 diabetes: a retrospective analysis of time trends and geographic regional differences.

Lipid control in adult Chinese patients with type 2 diabetes: a retrospective analysis of time trends and geographic regional differences.

A robust hybrid predictive model of mixed oil length with deep integration of mechanism and data

Accurate estimation of mixed oil length is highly required in multi-product pipelines because it can guide the operator to correctly handle the mixed oil segment and effectively reduce the loss of petroleum product quality. In previous study, a hybrid model combined with machine learning algorithm with existing mechanism has been developed and has good predictive accuracy. Unfortunately, due to incorrect measurement and improper recording, outliers are widely present in industrial datasets and may render the predictive performance of the previous model quite disappointing, while the effect of outliers on predictive models for the mixed oil length is rarely discussed. In order to deal with such issues, this paper first proposes a way to define the outlier sample and explicitly studies its impact on the performance of the predictive model for mixed oil prediction. Subsequentially, various new hybrid modeling methods are developed driven by both operation data (exploited by the Gradient Boosting Decision Tree algorithm) and the mechanism (based on the Austin-Palfrey equation) in different arrangements. Extensive experiments are conducted on real-life transportation pipelines, and the results show that with the clean training set, the R2 index of the proposed serial-parallel hybrid model (SPHM) is 0.96, which is higher than that of mechanism model and the existing hybrid model. Even with all the outliers added, advantage in prediction accuracy of the SPHM is still noticed, demonstrating feasibility and robustness of the hybrid modeling approach for prediction of mixed oil length.

A novel Sustainable Development Goal 7 composite index as the paradigm for energy sustainability assessment: A case study from Europe

The Sustainable Development Goal 7 endeavors to ameliorate the energy system towards sustainability. Monitoring the country’s progress to the goal will be of utmost for the government to take suitable actions and thus, constructing a performance monitoring index for Sustainable Development Goal 7 would tune the pace of implementation. This study aims to develop a novel Sustainable Development Goal 7 or Energy Sustainability Composite Index to assess the energy sustainability performance. Since Europe tends to have diverged efforts towards energy sustainability, assessing them with the proposed Sustainable Development Goal 7 composite index would provide the evidence needed for effective sustainable development strategies. By describing Europe, the authors signify 40 European countries and the selection of country depends on the availability of all the data that are required for the energy sustainability assessment. In this study, the analyzed energy sustainability aspects include clean energy conversion, energy security, energy accessibility, energy intensity and carbon intensity. The results show that Iceland, Norway, and Sweden tops in energy sustainability aspects with scores of 0.7313, 0.6967, and 0.6313 (on a scale of 0 to 1), respectively. The proposed Sustainable Development Goal 7 composite index is also compared with the actual Sustainable Development Goal 7 index, which comprises the indicators defined by the United Nations. The prime difference between the proposed Sustainable Development Goal 7 composite index and the actual Sustainable Development Goal 7 index resides in the consideration of energy security and carbon intensity indicators and in the framework designed to evaluate the clean energy prevalence. The evaluated actual Sustainable Development Goal 7 index scores of Germany and France are 0.4915 and 0.4656, respectively. On comparing with the proposed Sustainable Development Goal 7 composite index scores, the scores decreased by 20.9% for Germany and increased by 7.2% for France. The robustness of the proposed composite index relies on reducing the effect of outliers by using a modified min–max methodology, namely Aggregated Normalization based on Maximum and Minimum Outliers for normalization and the subsequent weightage allocation criteria utilized in Analytic Hierarchy Process methodology. Sensitivity analyses highlighted that the clean energy indicator is the most influencing indicator for the designed composite index. Nevertheless, uncertainty analysis indicates that the weightage scenario has a more prominent influence than various normalization and aggregation methods.

Robust SURE estimates of profitability in the Egyptian insurance market

This paper proposes three robust estimators (M-estimation, S-estimation, and MM-estimation) for handling the problem of outlier values in seemingly unrelated regression equations (SURE) models. The SURE model is one of regression multivariate cases, which have especially assumption, i.e., correlation between errors on the multivariate linear models; by considering multiple regression equations that are linked by contemporaneously correlated disturbances. Moreover, the effects of outliers may permeate through the system of equations; the primary aim of SURE which is to achieve efficiency in estimation, but this is questionable. The goal of robust regression is to develop methods that are resistant to the possibility that one or several unknown outliers may occur anywhere in the data. In this paper, we study and compare the performance of robust estimations with the traditional non-robust (ordinary least squares and Zellner) estimations based on a real dataset of the Egyptian insurance market during the financial year from 1999 to 2018. In our study, we selected the three most important insurance companies in Egypt operating in the same field of insurance activity (personal and property insurance). The effect of some important indicators (exogenous variables) issued by insurance corporations on the net profit has been studied. The results showed that robust estimators greatly improved the efficiency of the SURE estimation, and the best robust estimation is MM-estimation. Moreover, the selected exogenous variables in our study have a significant effect on the net profit in the Egyptian insurance market.

High leverage detection in general functional regression models with spatially correlated errors

AbstractThe presence of curves that deviate markedly from the core of a set of curves can greatly affect inference and forecasting in a functional regression model. Thus their detection is key to increase the accuracy of the required estimates. This work introduces the concepts of high leverage in general functional regression models with independent and spatially correlated errors. The projection matrix, also known as Hat matrix, plays a crucial role in classical model diagnosis, since it provides a measure of leverage. We propose a generalisation of the projection matrix in both the functional and the spatial functional frameworks under two settings, when the response variable is a scalar, and when it is a function itself, the so‐called total model. Commonly used influence measures are also proposed as functions of the generalised functional leverages and residuals. An application of the proposed procedures for investigating the effect of outliers on the relationship between transformation of the banking industry and the size of cooperative banks in Italy over a period of 14 years is presented.

New Robust Part-Based Model with Affine Transformations for Facial Landmark Localization and Detection in Big Data

In this paper, we developed a new robust part-based model for facial landmark localization and detection via affine transformation. In contrast to the existing works, the new algorithm incorporates affine transformations with the robust regression to tackle the potential effects of outliers and heavy sparse noises, occlusions and illuminations. As such, the distorted or misaligned objects can be rectified by affine transformations and the patterns of occlusions and outliers can be explicitly separated from the true underlying objects in big data. Moreover, the search of the optimal parameters and affine transformations is cast as a constrained optimization programming. To mitigate the computations, a new set of equations is derived to update the parameters involved and the affine transformations iteratively in a round-robin manner. Our way to update the parameters compared to the state of the art of the works is relatively better, as we employ a fast alternating direction method for multiplier (ADMM) algorithm that solves the parameters separately. Simulations show that the proposed method outperforms the state-of-the-art works on facial landmark localization and detection on the COFW, HELEN, and LFPW datasets.

Adaboost-based ensemble of polynomial chaos expansion with adaptive sampling

In the study, we propose a new polynomial chaos expansion surrogate modeling method based on Adaboost (Adaboost-PCE) for uncertainty quantification. Adaboost is an ensemble learning technique originating from the machine learning field. The idea of Adaboost-PCE is to construct many times the “weak” PCE model with assigned weights to each sample point. Each time, the weight of a particular sample point in the training set depends on the performance of the surrogate models on that sample. In this way, weighted least-squares approximation is employed to exploit the weights of each sample in order to reduce the effect of outliers. The Adaboost-PCE is appealing since it is possible to estimate the ensemble weights without using any explicit error metrics as in most existent ensemble methods. Moreover, it has an expectation of the prediction error that enables the efficient adaptive sampling. The proposed method is validated with a numerical comparison of their performance on a series of numerical tests including partial differential equations with high-dimensional inputs.