Interpolation Of Missing Data Research Articles

Streaming prediction-error filters

Prediction-error filters (PEFs) are essential in seismic deconvolution and other geophysical estimation problems. We show that non-stationary multidimensional PEFs can be computed in a “streaming” manner, where the filter gets updated incrementally by accepting one new data point at a time. The computational cost of estimating a streaming PEF reduces to the cost of a single convolution. In other words, the cost of PEF design while filtering equals the cost of applying the filter. Moreover, the non-linear operation of finding and applying a streaming PEF is invertible at a similar cost, which enables a fast approach to missing data interpolation.

Missing Data Imputation Method Combining Random Forest and Generative Adversarial Imputation Network.

(1) Background: In order to solve the problem of missing time-series data due to the influence of the acquisition system or external factors, a missing time-series data interpolation method based on random forest and a generative adversarial interpolation network is proposed. (2) Methods: First, the position of the missing part of the data is calibrated, and the trained random forest algorithm is used for the first data interpolation. The output value of the random forest algorithm is used as the input value of the generative adversarial interpolation network, and the generative adversarial interpolation network is used to calibrate the position. The data are interpolated for the second time, and the advantages of the two algorithms are combined to make the interpolation result closer to the true value. (3) Results: The filling effect of the algorithm is tested on a certain bearing data set, and the root mean square error (RMSE) is used to evaluate the interpolation results. The results show that the RMSE of the interpolation results based on the random forest and generative adversarial interpolation network algorithms in the case of single-segment and multi-segment missing data is only 0.0157, 0.0386, and 0.0527, which is better than the random forest algorithm, generative adversarial interpolation network algorithm, and K-nearest neighbor algorithm. (4) Conclusions: The proposed algorithm performs well in each data set and provides a reference method in the field of data filling.

Retrospective reconstruction of four-dimensional magnetic resonance from interleaved cine imaging – A comparative study with four-dimensional computed tomography in the lung

Background and purposeImaging of respiration-induced anatomical changes is essential to ensure high accuracy in radiotherapy of lung cancer. We expanded here on methods for retrospective reconstruction of time-resolved volumetric magnetic resonance (4DMR) of the thoracic region and benchmarked the results against 4D computed tomography (4DCT). Materials and methodMR data of six lung cancer patients were collected by interleaving cine-navigator images with 2D data frame images, acquired across the thorax. The data frame images have been stacked in volumes based on a similarity metric that considers the anatomical deformation of lungs, while addressing ambiguities in respiratory phase detection and interpolation of missing data. The resulting images were validated against cine-navigator images and compared to paired 4DCTs in terms of amplitude and period of motion, assessing differences in internal target volume (ITV) margin definition. Results4DMR-based motion amplitude was on average within 1.8 mm of that measured in the corresponding 2D cine-navigator images. In our dataset, the 4DCT motion and the 4DMR median amplitude were always within 3.8 mm. The median period was generally close to CT references, although deviations up to 24 % have been observed. These changes were reflected in the ITV, which was generally larger for MRI than for 4DCT (up to 39.7 %). ConclusionsThe proposed algorithm for retrospective reconstruction of time-resolved volumetric MR provided quality anatomical images with high temporal resolution for motion modelling and treatment planning. The potential for imaging organ motion variability makes 4DMR a valuable complement to standard 4DCT imaging.

Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU).

Smart grid systems require high-quality Phasor Measurement Unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for interpolation of missing data, the computational complexity increases non-linearly with increasing data size. In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods. The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values). The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.

A time series continuous missing values imputation method based on generative adversarial networks

Generative adversarial networks (GANs) have been widely utilized in time series analysis and modeling, wherein generators and discriminators interact to generate realistic data. However, when addressing the challenge of imputing continuous missing values in time series, the generator struggles to learn meaningful features due to the loss of local information, and the discriminator’s loss function exhibits significant deviations in determining the probability, making it difficult to effectively update model parameters during training. In response to these issues, this study presents a novel imputation model called cue Wasserstein generative adversarial network with gradient penalty (CWGAIN-GP). CWGAIN-GP incorporates the contextual cue information matrixs into the generator, enabling obtain and capture the potential data evolution features that hidden beyond the missing positions. This approach constrains the generator’s generation results to be closer to the true values probability distribution. Meanwhile, it optimizes the generator and discriminator neural network structure and loss function computation strategy of the original GAN model, which not only improves the accuracy of continuous missing data interpolation, but also improves the training stability of the model. Finally, We used three imputation accuracies on two real-world datasets, including experiments on imputation accuracy for eight comparison models, ablation study, and experiments on the effect of consecutive missing lengths. The experimental results show that CWGAIN-GP achieves impressive performance in imputing continuous missing time series.

Estimation of rainfall and streamflow missing data under uncertainty for Nile basin headwaters: the case of Ghba catchments

The datasets in semi-arid and arid regions of developing countries are known for their long period of missing data. To approximate any missing data interpolation or estimation techniques are frequently used. The current study focuses on applying the concept of data uncertainty by setting out different percentages of missing data from the known dataset at all target stations calibrated and validated against the ensemble simulation interpolation and reconstitution approach by contrasting findings with voluntarily excluded data that will be re-estimated using the various approaches selected. While testing the nearby stations (6 to 50 km) for filling the missing data were used. The tests on the selected stations shows that the better estimates of precipitation and streamflow are the ID and the NR methods. Thus, the methods that revealed a better estimation were used for filling in missing data of the respective stations to undergo a water resource study.

Interpolation of GNSS Position Time Series Using GBDT, XGBoost, and RF Machine Learning Algorithms and Models Error Analysis

The global navigation satellite system (GNSS) position time series provides essential data for geodynamic and geophysical studies. Interpolation of the GNSS position time series is necessary because missing data will produce inaccurate conclusions made from the studies. The spatio-temporal correlations between GNSS reference stations cannot be considered when using traditional interpolation methods. This paper examines the use of machine learning models to reflect the spatio-temporal correlation among GNSS reference stations. To form the machine learning problem, the time series to be interpolated are treated as output values, and the time series from the remaining GNSS reference stations are used as input data. Specifically, three machine learning algorithms (i.e., the gradient boosting decision tree (GBDT), eXtreme gradient boosting (XGBoost), and random forest (RF)) are utilized to perform interpolation with the time series data from five GNSS reference stations in North China. The results of the interpolation of discrete points indicate that the three machine learning models achieve similar interpolation precision in the Up component, which is 45% better than the traditional cubic spline interpolation precision. The results of the interpolation of continuous missing data indicate that seasonal oscillations caused by thermal expansion effects in summer significantly affect the interpolation precision. Meanwhile, we improved the interpolation precision of the three models by adding data from five stations which have high correlation with the initial five GNSS reference stations. The interpolated time series for the North, East, and Up (NEU) are examined by principal component analysis (PCA), and the results show that the GBDT and RF models perform interpolation better than the XGBoost model.

Missing data interpolation and multi‐sensors integration and its application in accelerated degradation data

AbstractMultiple sensors are commonly used for accelerated degradation monitoring. Since different sensors may be sensitive at different stages of the accelerated degradation process and each sensor dataset may contain only partial information of the unit degradation, then integration approaches of the accelerated degradation data from multiple sensors can effectively improve degradation modeling and life prediction accuracy. We present a non‐parametric approach that assigns weights to each sensor based on the dynamic clustering of the sensors' observations. Missing data are common in degradation data acquisition, especially when multiple sensors are used. We provide two approaches for data interpolation: the nonlinear Brownian bridge and the inverse Gaussian bridge when the underlying degradation paths follow the Brownian motion process and inverse Gaussian process, respectively. The stochastic bridges capture the nonlinearity and uncertainties of the degradation processes. The data integration model and stochastic bridge models are validated with real accelerated fatigue crack growth data monitored with multiple NDT sensors. The proposed models provide an accurate accelerated degradation path and reliability prediction.

Accelerated submillimeter wave-encoded magnetic resonance imaging via deep untrained neural network.

Wave gradient encoding can adequately utilize coil sensitivity profiles to facilitate higher accelerations in parallel magnetic resonance imaging (pMRI). However, there are limitations in mainstream pMRI and a few deep learning (DL) methods for recovering missing data under wave encoding framework: the former is prone to introduce errors from the auto-calibration signals (ACS) signal acquisition and is time-consuming, while the latter requires a large amount of trainingdata. To tackle the above issues, an untrained neural network (UNN) model incorporating wave-encoded physical properties and deep generative model, named WDGM, was proposed with additional ACS- and trainingdata-free. Generally, the proposed method can provide powerful missing data interpolation capability using the wave physical encoding framework and designed UNN to characterize the MR image (k-space data) priors. Specifically, the MRI reconstruction combining physical wave encoding and elaborate UNN is modeled as a generalized minimization problem. The designation of UNN is driven by the coil sensitivity maps (CSM) smoothness and k-space linear predictability. And then, the iterative paradigm to recover the full k-space signal is determined by the projected gradient descent, and the complex computation is unrolled to the network with optimized parameters by the optimizer. Simulated wave encoding and in vivo experiments are exploited to demonstrate the feasibility of the proposed method. The best quantitative metrics RMSE/SSIM/PSNR of 0.0413, 0.9514, and 37.4862 gave competitive results in all experiments with at least six-fold acceleration, respectively. In vivo experiments of human brains and knees showed that the proposed method can achieve comparable reconstruction quality and even has superiority relative to the comparison, especially at a high resolution of 0.67 mm and fewer ACS. In addition, the proposed method has a higher computational efficiency achieving a computation time of 9.6 s/perslice. The model proposed in this work addresses two limitations of MRI reconstruction in the wave encoding framework. The first is to eliminate the need for ACS signal acquisition to perform the time-consuming calibration process and to avoid errors such as motion during the acquisition procedure. Furthermore, the proposed method has clinical application friendly without the need to prepare large training datasets, which is difficult in the clinical. All results of the proposed method demonstrate more confidence in both quantitative and qualitative metrics. In addition, the proposed method can achieve higher computationalefficiency.

Longitudinal Exposure-Response Modeling of Multiple Indicators of Alzheimer's Disease Progression.

Progression in Alzheimer's disease manifests as changes in multiple biomarker, cognitive, and functional endpoints. Disease progression modeling can be used to integrate these multiple measures into a synthesized metric of where a patient lies within the disease spectrum, allowing for a more dynamic measure over the range of the disease. This study aimed to combine modeling techniques from psychometric research (e.g., item response theory) and pharmacometrics (e.g., hierarchical models) to describe the multivariate longitudinal disease progression for patients with mild-to-moderate Alzheimer's disease. Additionally, we aimed to extend the subsequent model to make it suitable for clinical trial simulation, with the inclusion of covariates, to explain variability in latent progression (i.e., disease progression) and to aid in the assessment of enrichment strategies. Multiple longitudinal endpoints in the Alzheimer's Disease Neuroimaging Initiative database were modeled. This model was validated internally using visual predictive checks, and externally by comparing data from the placebo arms of two Phase 2 crenezumab studies, ABBY (NCT01343966) and BLAZE (NCT01397578). The Alzheimer's Disease Neuroimaging Initiative began in 2004: the initial 5-year study (ADNI-1) was extended by 2 years in 2009 by a Grand Opportunities grant (ADNI-GO), and in 2011 and 2016 by further competitive renewals of the ADNI-1 grant (ADNI-2 and ADNI-3, respectively). This work studies natural progression data from patients with confirmed Alzheimer's disease. The Phase 2 ABBY and BLAZE trials evaluated the safety and efficacy of crenezumab in patients with mild-to-moderate Alzheimer's disease. From the Alzheimer's Disease Neuroimaging Initiative database, 305 subjects who had a baseline diagnosis of mild-to-moderate Alzheimer's disease were included in modeling. From the ABBY and BLAZE studies, 158 patients were included from the studies' placebo arms. Longitudinal cognitive and functional assessments modeled included the Clinical Dementia Rating (both as Sum of Boxes and individual item scores), the Mini-Mental State Examination, the Alzheimer's Disease Assessment Scale - Cognitive Subscale, the Functional Activities Questionnaire, the Montreal Cognitive Assessment, and the Rey Auditory Verbal Learning Test. Also included were the imaging variable fluorodeoxyglucose-positron emission tomography and the following magnetic resonance imaging volumetrics: entorhinal, fusiform, hippocampal, intra-cranial, mid-temporal, ventricular, and whole brain. Applying item response theory approaches in this longitudinal setting showed clinical assessments informing a common disease scale in the following order (from early disease to late disease): Rey Auditory Verbal Learning Test, Functional Activities Questionnaire, Montreal Cognitive Assessment, Alzheimer's Disease Assessment Scale - Cognitive Subscale 12, Clinical Dementia Rating - Sum of Boxes, and Mini-Mental State Examination. The Clinical Dementia Rating communication and home-and-hobbies items were most informative at earlier disease stages, while memory, orientation, and personal care informed the disease status at later stages. A clinical trial simulation model was developed and accurately described within-sample longitudinal distribution of endpoints. Simplifying the model to use only baseline age, MMSE, and APOEε4 status as predictors, out-of-sample mean progression of ADAS-Cog and CDR Sum of Boxes in the ABBY and BLAZE placebo arms was accurately described; however, the variability in these endpoints was underpredicted and suggests possibility for further model refinement when extrapolating from the ADNI sample to trial data. Clinical trial simulations were performed to exemplify use of the model to investigate hypothetical disease modification effects on the multivariate, longitudinal progression on the Alzheimer's Disease Assessment Scale - Cognitive Subscale and the Clinical Dementia Rating - Sum of Boxes. The latent variable structure of item response theory can be extended to capture a variety of scales that are common assessments and indicators of disease status in mild-to-moderate Alzheimer's disease. These models are not intended to support causal inferences, but they do successfully characterize the observed correlation between endpoints over time and result in concise numerical indices of disease status that reflect the totality of evidence from considering the endpoints jointly. As such, the models have utility for a variety of tasks in clinical trial design, including simulation of hypothetical drug effects, interpolation of missing data, and assessment of in-sample information.

AtmoDist: Self-supervised representation learning for atmospheric dynamics

Abstract Representation learning has proven to be a powerful methodology in a wide variety of machine-learning applications. For atmospheric dynamics, however, it has so far not been considered, arguably due to the lack of large-scale, labeled datasets that could be used for training. In this work, we show how to sidestep the difficulty and introduce a self-supervised learning task that is applicable to a wide variety of unlabeled atmospheric datasets. Specifically, we train a neural network on the simple yet intricate task of predicting the temporal distance between atmospheric fields from distinct but nearby times. We demonstrate that training with this task on the ERA5 reanalysis dataset leads to internal representations that capture intrinsic aspects of atmospheric dynamics. For example, when employed as a loss function in other machine-learning applications, the derived AtmoDist distance leads to improved results compared to the $ {\mathrm{\ell}}_2 $ -loss. For downscaling one obtains higher resolution fields that match the true statistics more closely than previous approaches and for the interpolation of missing or occluded data the AtmoDist distance leads to results that contain more realistic fine-scale features. Since it is obtained from observational data, AtmoDist also provides a novel perspective on atmospheric predictability.

Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

Background: Smart grid systems require high-quality phasor measurement unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for the interpolation of missing data, the computational complexity increases non-linearly with increasing data size. Methods: In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods. Results: The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values). Conclusions: The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.

Facial Animation with Disentangled Identity and Motion using Transformers

AbstractWe propose a 3D+time framework for modeling dynamic sequences of 3D facial shapes, representing realistic non‐rigid motion during a performance. Our work extends neural 3D morphable models by learning a motion manifold using a transformer architecture. More specifically, we derive a novel transformer‐based autoencoder that can model and synthesize 3D geometry sequences of arbitrary length. This transformer naturally determines frame‐to‐frame correlations required to represent the motion manifold, via the internal self‐attention mechanism. Furthermore, our method disentangles the constant facial identity from the time‐varying facial expressions in a performance, using two separate codes to represent neutral identity and the performance itself within separate latent subspaces. Thus, the model represents identity‐agnostic performances that can be paired with an arbitrary new identity code and fed through our new identity‐modulated performance decoder; the result is a sequence of 3D meshes for the performance with the desired identity and temporal length. We demonstrate how our disentangled motion model has natural applications in performance synthesis, performance retargeting, key‐frame interpolation and completion of missing data, performance denoising and retiming, and other potential applications that include full 3D body modeling.

Missing Data Interpolation with Variational Bayesian Inference for Socio-economic Statistics Applications

<p>The information integrity is needed to solving socio-economic statistical problems. However, the information integrity is destroyed by missing data which is caused by various subjective and objective reasons. So the missing data interpolation is used to supplement missing data. In this paper, missing data interpolation with variational Bayesian inference is proposed. This method is combined with Gaussian model to approximate the posterior distribution to obtain complete data. The experiments include two datasets (artificial dataset and actual dataset) based on three missing ratios separately. The missing data interpolation performance of variational Bayesian method is compared with that which is obtained by mean interpolation and K-nearest neighbor interpolation methods separately in MSE (Mean Square Error) and MAPE (Mean Absolute Percentage Error). The experimental results show that the proposed variational Bayesian method is better in MSE and MAPE.</p> <p> </p>

Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

Background: Smart grid systems require high-quality phasor measurement unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for interpolation of missing data, the computational complexity increases non-linearly with increasing data size. Methods: In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods. Results: The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values). Conclusions: The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.

Ship Navigation Behavior Prediction Based on AIS Data

Real-time and accurate ship navigation dynamic prediction can effectively improve maritime supervision’s intelligence and precision level and ensure ship navigation safety. To further enhance the accuracy of ship navigation dynamic prediction, this paper uses ship AIS as the data source and proposes an improved LSTM navigation dynamic prediction model based on the attention mechanism. Firstly, a set of pre-processing means, including navigation data extraction, abnormal data processing, and missing data interpolation, is proposed to solve the problems of information loss and inaccuracy in AIS data and incomplete retention of dynamic navigation features; secondly, combining the dynamic characteristics of ship navigation in AIS data with time series, using longitude, latitude, heading, speed, a ship heading and time increment as input to establish a dynamic forecasting model for navigation based on LSTM; The existing navigation sequence coding distortion and space-time data incoherence problem, an optimized Attention-LSTM neural network navigation dynamic prediction method is proposed, and the accuracy and robustness of the model are verified by simulation analysis. The results show that this method can achieve high-precision prediction of the ship’s longitude, latitude, heading, and speed.

Missing Data Interpolation of Alzheimer’s Disease Based on Column-by-Column Mixed Mode

Research on clinical data sets of Alzheimer’s disease can predict and develop early intervention treatment. Missing data is a common problem in medical research. Failure to deal with more missing data will reduce the efficiency of the test, resulting in information loss and result bias. To address these issues, this paper designs and implements the missing data interpolation method of mixed interpolation according to columns by combining the four methods of mean interpolation, regression interpolation, support vector machine (SVM) interpolation, and multiple interpolation. By comparing the effects of the mixed interpolation method with the above four interpolation methods and giving the comparison results, the experiment shows that the results of the mixed interpolation method under different data missing rates have better performance in terms of root mean square error (RMSE), mean absolute error (MSE), and error rate, which proves the effectiveness of the interpolation mechanism. The characteristics of different variables might lead to different interpolation strategy choices, and column-by-column mixed interpolation can dynamically select the best method according to the difference of features. To a certain extent, it selects the best method suitable for each feature and improves the interpolation effect of the data set as a whole, which is beneficial to the clinical study of Alzheimer’s disease. In addition, in the processing of missing data, a combination of deletion method and interpolation method is adopted with reference to expert knowledge. Compared with the direct interpolation method, the data set obtained by this method is more accurate.

Covid-19 Pandemic and the Market Performance Analysis: Evidence From Indonesia

This paper empirically identified the impact of the Covid-19 pandemic on market performance in Indonesia. We use a cointegrating regression model of FM-OLS and D-OLS along with Cubic Spline missing data interpolation, summary unit root test, unit root test with Dickey-Fuller breakpoint selection, and Johansen cointegration test. Daily time series data of Covid-19 cumulative case, exchange rate, IDX composite, and gold commodity price were analyzed around 119 days after the first announced case in Indonesia. The finding from FM-OLS and D-OLS analysis showed that the Covid-19 pandemic has a positive and significant impact on exchange rate and gold commodity price. The Covid-19 pandemic impact is appeared to be negative and significant in explaining IDX composite price. It seems that during the Covid-19 pandemic, people prefer to do transactions in the commodity market than neither capital nor money market. The diagnostic of cointegrating regression showed that all variables are integrated of order one I (1), and the long-run cointegration among variables in each equation exists. From the unit root test with break point selection, this study revealed that Covid-19 pandemic has an extreme impact on Indonesian market performance only in the first month after the case was announced. The government responses to mitigate the spread of the Covid-19 pandemic and to minimize the possibly profound impact of market performance are appeared to be successful. Besides, it seems that people's fear is diminished, quite high in the first month, and began to shrink in the following month.

Atakrig: An R package for multivariate area-to-area and area-to-point kriging predictions

Geostatistical interpolation methods are used in diverse disciplines, such as environmental science, ecology, and hydrology. With the increasing availability of areal spatial data, area-to-area and area-to-point interpolations have great application potential. In this study, based on the variogram deconvolution algorithm proposed by Goovaerts (2008), an open-source area-to-area kriging package atakrig is developed in the R environment. In atakrig, point-scale variogram and cross-variogram can be automatically deconvoluted from spatial areal samples. It provides a general framework for area-to-area and area-to-point ordinary kriging and cokriging. Two applications show that the package works well in river runoff prediction and missing data interpolation for remote sensing aerosol optical depth. The package can be deployed on different operating systems and computer hardware platforms.

Measuring the equivalent dose rate over the Shelter object after completion of the New Safe Confinement

An experiment on measuring the equivalent dose rate over the Shelter object after the completion of the construction of the New Safe Confinement is described. Measurements of the gamma radiation were performed in the time of commissioning of the New Safe Confinement with the help of the sensor installed on its Main Cranes System. The carriage with a sensor was moved by a chaotic trajectory so the method of missing data interpolation was proposed. As a result, a cartogram of the distribution of the gamma field at the level of movement of the bogies of the Main Cranes System of the New Safe Confinement is provided in this paper.