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At least 73 records · Page 4

Background Error Covariance Estimation Using Information from a Single Model Trajectory with Application to Ocean Data Assimilation

An attractive property of ensemble data assimilation methods is that they provide flow dependent background error covariance estimates which can be used to update fields of observed variables as well as fields of unobserved model variables. Two methods to estimate background error covariances are introduced which share the above property with ensemble data assimilation methods but do not involve the integration of multiple model trajectories. Instead, all the necessary covariance information is obtained from a single model integration. The Space Adaptive Forecast error Estimation (SAFE) algorithm estimates error covariances from the spatial distribution of model variables within a single state vector. The Flow Adaptive error Statistics from a Time series (FAST) method constructs an ensemble sampled from a moving window along a model trajectory.SAFE and FAST are applied to the assimilation of Argo temperature profiles into version 4.1 of the Modular Ocean Model (MOM4.1) coupled to the GEOS-5 atmospheric model and to the CICE sea ice model. The results are validated against unassimilated Argo salinity data. They show that SAFE and FAST are competitive with the ensemble optimal interpolation (EnOI) used by the Global Modeling and Assimilation Office (GMAO) to produce its ocean analysis. Because of their reduced cost, SAFE and FAST hold promise for high-resolution data assimilation applications.

Error Covariance↗

Background Error Covariance Estimation using Information from a Single Model Trajectory with Application to Ocean Data Assimilation into the GEOS-5 Coupled Model

An attractive property of ensemble data assimilation methods is that they provide flow dependent background error covariance estimates which can be used to update fields of observed variables as well as fields of unobserved model variables. Two methods to estimate background error covariances are introduced which share the above property with ensemble data assimilation methods but do not involve the integration of multiple model trajectories. Instead, all the necessary covariance information is obtained from a single model integration. The Space Adaptive Forecast error Estimation (SAFE) algorithm estimates error covariances from the spatial distribution of model variables within a single state vector. The Flow Adaptive error Statistics from a Time series (FAST) method constructs an ensemble sampled from a moving window along a model trajectory. SAFE and FAST are applied to the assimilation of Argo temperature profiles into version 4.1 of the Modular Ocean Model (MOM4.1) coupled to the GEOS-5 atmospheric model and to the CICE sea ice model. The results are validated against unassimilated Argo salinity data. They show that SAFE and FAST are competitive with the ensemble optimal interpolation (EnOI) used by the Global Modeling and Assimilation Office (GMAO) to produce its ocean analysis. Because of their reduced cost, SAFE and FAST hold promise for high-resolution data assimilation applications.

Data Assimilation↗

"Intelligent Ensemble" Projections of Precipitation and Surface Radiation in Support of Agricultural Climate Change Adaptation

Earth's climate is changing and will continue to change into the foreseeable future. Expected changes in the climatological distribution of precipitation, surface temperature, and surface solar radiation will significantly impact agriculture. Adaptation strategies are, therefore, required to reduce the agricultural impacts of climate change. Climate change projections of precipitation, surface temperature, and surface solar radiation distributions are necessary input for adaption planning studies. These projections are conventionally constructed from an ensemble of climate model simulations (e.g., the Coupled Model Intercomparison Project 5 (CMIP5)) as an equal weighted average, one model one vote. Each climate model, however, represents the array of climate-relevant physical processes with varying degrees of fidelity influencing the projection of individual climate variables differently. Presented here is a new approach, termed the "Intelligent Ensemble, that constructs climate variable projections by weighting each model according to its ability to represent key physical processes, e.g., precipitation probability distribution. This approach provides added value over the equal weighted average method. Physical process metrics applied in the "Intelligent Ensemble" method are created using a combination of NASA and NOAA satellite and surface-based cloud, radiation, temperature, and precipitation data sets. The "Intelligent Ensemble" method is applied to the RCP4.5 and RCP8.5 anthropogenic climate forcing simulations within the CMIP5 archive to develop a set of climate change scenarios for precipitation, temperature, and surface solar radiation in each USDA Farm Resource Region for use in climate change adaptation studies.

Taylor, Patrick C.↗

A Framework for Evaluating Climate Model Performance Metrics

The CMIP5 archive contains future climate projections from over 50 models provided by dozens of modeling centers from around the world. Individual model projections, however, are subject to biases created by structural model uncertainties. As a result, ensemble averaging of multiple models is often used to add value to model projections: consensus projections have been shown to consistently outperform individual models. Previous reports for the IPCC establish climate change projections based on an equal-weighted average of all model projections. However, certain models reproduce climate processes better than other models. Should models be weighted based on performance? Unequal ensemble averages have previously been constructed using a variety of mean state metrics. What metrics are most relevant for constraining future climate projections? This project develops a framework for systematically testing metrics in models to identify optimal metrics for unequal weighting multi-model ensembles. A unique aspect of this project is the construction and testing of climate process-based model evaluation metrics. A climate process-based metric is defined as a metric based on the relationship between two physically related climate variables?e.g., outgoing longwave radiation and surface temperature. Metrics are constructed using high-quality Earth radiation budget data from NASA's Clouds and Earth's Radiant Energy System (CERES) instrument and surface temperature data sets. It is found that regional values of tested quantities can vary significantly when comparing weighted and unweighted model ensembles. For example, one tested metric weights the ensemble by how well models reproduce the time-series probability distribution of the cloud forcing component of reflected shortwave radiation. The weighted ensemble for this metric indicates lower simulated precipitation (up to .7 mm/day) in tropical regions than the unweighted ensemble: since CMIP5 models have been shown to overproduce precipitation, this result could indicate that the metric is effective in identifying models which simulate more realistic precipitation. Ultimately, the goal of the framework is to identify performance metrics for advising better methods for ensemble averaging models and create better climate predictions.

Noel C Baker↗

Ensemble variational Fokker-Planck methods for data assimilation

Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Itˆo process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Itˆo process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean- Vlasov-Itˆo processes. Here, the effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz ’63, Lorenz ’96 and the quasi-geostrophic equations.

97 MATHEMATICS AND COMPUTING↗

Behavior of Filters and Smoothers for Strongly Nonlinear Dynamics

The Kalman filter is the optimal filter in the presence of known gaussian error statistics and linear dynamics. Filter extension to nonlinear dynamics is non trivial in the sense of appropriately representing high order moments of the statistics. Monte Carlo, ensemble-based, methods have been advocated as the methodology for representing high order moments without any questionable closure assumptions. Investigation along these lines has been conducted for highly idealized dynamics such as the strongly nonlinear Lorenz model as well as more realistic models of the means and atmosphere. A few relevant issues in this context are related to the necessary number of ensemble members to properly represent the error statistics and, the necessary modifications in the usual filter situations to allow for correct update of the ensemble members. The ensemble technique has also been applied to the problem of smoothing for which similar questions apply. Ensemble smoother examples, however, seem to be quite puzzling in that results state estimates are worse than for their filter analogue. In this study, we use concepts in probability theory to revisit the ensemble methodology for filtering and smoothing in data assimilation. We use the Lorenz model to test and compare the behavior of a variety of implementations of ensemble filters. We also implement ensemble smoothers that are able to perform better than their filter counterparts. A discussion of feasibility of these techniques to large data assimilation problems will be given at the time of the conference.

Zhu, Yanqui↗

Application of artificial intelligence methods in the international roughness index prediction of rigid and composite pavements: a systematic review

The International Roughness Index (IRI) is a widely adopted metric for quantifying pavement roughness, directly influencing vehicle safety, ride comfort, and overall roadway performance. In recent years, the use of Machine Learning (ML) models for IRI prediction has gained momentum, with the goal of improving the allocation of maintenance and rehabilitation resources by enabling accurate assessments of pavement conditions. Most prior reviews, however, have concentrated on flexible pavements, leaving a notable gap regarding rigid and composite pavements. To address this gap, the present study conducts a systematic review of Artificial Intelligence (AI) methods applied to IRI prediction for rigid and composite pavements. Literature published between 2004 and 2025 is synthesized to highlight prevailing trends, methodological contributions, and directions for future research. Particular attention is given to the types of models employed, the datasets used for training and validation, and the role of input variables and data-processing strategies. Across the included studies, ensemble learning methods (especially gradient boosting variants such as XGBoost), artificial neural networks, and hybrid architectures frequently achieved high predictive skill, with several models reporting test-set coefficients of determination approaching 0.9–0.96, indicating strong potential for capturing the influence of traffic, pavement structure, and climatic factors. Since these results are obtained from heterogeneous datasets and evaluation protocols, they are interpreted qualitatively rather than as strict cross-study rankings. Analysis of input variables revealed that pavement age and initial IRI were included in 91% (21 of 23) and 78% (18 of 23) of studies, respectively. Climatic variables such as the freezing index appeared in 57% (13 of 23), while traffic-related factors were considered in 65% (15 of 23). The findings underscore the importance of standardized, high-quality datasets, such as those from the Long-Term Pavement Performance (LTPP) program, along with data consistency, model interpretability, computational efficiency, and replicability in enhancing IRI prediction. Future research should focus on incorporating input variable selection techniques to identify the most influential predictors, thereby improving accuracy and robustness. Integrating these approaches with advanced non-linear data-driven models, coupled with robust hyperparameter optimization, holds considerable promise for strengthening the reliability of IRI prediction and supporting resilient pavement management strategies.

42 ENGINEERING↗

Ensemble Simulations on Leadership Computing Systems

Scientific productivity can be enhanced through workflow management tools, relieving large High Performance Computing (HPC) system users from the tedious tasks of scheduling and designing the complex computational execution of scientific applications. This paper presents a study on the usage of ensemble workflow tools to accelerate science using the Summit and Frontier supercomputing systems. The research aims to connect science domain simulations using Oak Ridge Leadership Computing Facility (OLCF) supercomputing platforms with ensemble workflow methods in order to accelerate HPC-enabled discovery and boost scientific impact. We present the coupling, porting and optimization of Radical-Cybertools on three applications: Chroma, NAMD and LAMMPS. The tools augment traditional HPC monolithic runs with a pilot scheduler. Lessons-learned are discussed for physics, biology and materials science applications. We discuss intrinsic limitations of coupling and porting ensemble workflow tools to applications that run on large HPC systems. The origins of technical challenges and their solutions developed during the implementation process are discussed. Data management strategies, OLCF’s policies for ensembles, and natively supported workflow tools are also summarized.

Georgiadou, Antigoni [ORNL] (ORCID:000000020977631↗

The Behavior of Filters and Smoothers for Strongly Nonlinear Dynamics

The Kalman filter is the optimal filter in the presence of known Gaussian error statistics and linear dynamics. Filter extension to nonlinear dynamics is non trivial in the sense of appropriately representing high order moments of the statistics. Monte Carlo, ensemble-based, methods have been advocated as the methodology for representing high order moments without any questionable closure assumptions (e.g., Miller 1994). Investigation along these lines has been conducted for highly idealized dynamics such as the strongly nonlinear Lorenz (1963) model as well as more realistic models of the oceans (Evensen and van Leeuwen 1996) and atmosphere (Houtekamer and Mitchell 1998). A few relevant issues in this context are related to the necessary number of ensemble members to properly represent the error statistics and, the necessary modifications in the usual filter equations to allow for correct update of the ensemble members (Burgers 1998). The ensemble technique has also been applied to the problem of smoothing for which similar questions apply. Ensemble smoother examples, however, seem to quite puzzling in that results of state estimate are worse than for their filter analogue (Evensen 1997). In this study, we use concepts in probability theory to revisit the ensemble methodology for filtering and smoothing in data assimilation. We use Lorenz (1963) model to test and compare the behavior of a variety implementations of ensemble filters. We also implement ensemble smoothers that are able to perform better than their filter counterparts. A discussion of feasibility of these techniques to large data assimilation problems will be given at the time of the conference.

Zhu, Yanqiu↗

Evaluating probabilistic deep learning methods for uncertainty quantification of temperature downscaling

Deep learning (DL) has emerged as a promising tool for downscaling coarse-resolution climate data to high-resolution outputs, enabling improved regional climate predictions. A critical aspect of DL-based downscaling is the incorporation of uncertainty quantification (UQ), which enhances the interpretability and reliability of predictions—key factors for climate risk assessment and decision-making. This study develops a DL model to downscale 2 m temperature across the contiguous United States using reanalysis datasets. We systematically evaluate three epistemic UQ methods—deep ensembles (DEns), Monte Carlo dropout (MCD), and Flipout—based on their probabilistic accuracy, downscaling performance, sensitivity to geographical features, and computational efficiency. Results indicate that MCD generally outperforms Flipout and DEns in terms of calibration and downscaling accuracy. However, DEns demonstrate lower calibration errors in coastal regions, indicating its higher confidence within these areas. Flipout, in contrast, is more sensitive to elevation gradients and exhibits higher calibration errors in mountainous regions. Hence, the choice of UQ method for this task depends on the specific requirements of the application. For applications that prioritize overall calibration, downscaling accuracy, and computational efficiency, MCD is a strong candidate. These findings highlight the importance of selecting UQ methods based on application-specific requirements, such as geographical context and computational constraints. By addressing the trade-offs between UQ methods, this study provides actionable insights for improving the reliability, scalability, and utility of DL-based downscaling in climate science.

Environmental sciences↗

Development of an Inertial Sensor-based Methodology for Spacesuited Geology Task Assessments during Simulated Lunar Extravehicular Activities

Lunar surface exploration during Artemis missions will require the specific skill set of geology sampling. Apollo astronauts had extensive training and used specialized tools to collect lunar rocks, core samples, pebbles, sand, and dust. The inflexibility of the pressurized Apollo spacesuits forced sampling to be taken at a standstill posture. However, new exploration spacesuits are expected to incorporate advanced materials and joint bearings, allowing for greater mobility and a wider range of functional postures. Thus, science and exploration during Artemis missions will likely involve a variety of standing, squatting, and kneeling postures. In preparation for future lunar exploration missions, NASA provides geologic training to astronauts and other mission personnel. This professional training with a spacesuit in simulated lunar environments will enhance performance and reduce risk of injury to astronauts on the lunar surface. However, anecdotally, untrained or newly trained people wearing prototype planetary spacesuits have been observed to performing motions differently than a trained geologist would when conducting the same geology sampling tasks. Therefore, a tool for evaluating geology postures at extravehicular activity (EVA) training facilities becomes required. In this paper, we introduce a novel inertial measurement unit (IMU)-based method of geology task assessments in spacesuited conditions during simulated lunar EVAs. As a case study, two subjects (one geologist and one non-geologist) participated and donned the Mark III prototype planetary spacesuit during offloading with the spreader bar gimbal in NASA’s Active Response Gravity Offload System (ARGOS). For automated geology task assessments, the spacesuit was instrumented with three wireless IMUs (APDM Opal, OR, USA): one on the chest and one each on the left and right ankle bearings. Then subjects performed geology tasks using various tools (rake, trench, hammer chisel, scoop, and drive tube) for 45 minutes each. The chest IMU measured the torso tilt angle in the sagittal plane. We used an ensemble learning method with the ankle IMUs to discriminate between standing and kneeling activities. IMU data were processed using custom MATLAB (Mathworks, MA, USA) software. In our case study, the developed method was able to discriminate differences in standing and kneeling activity levels between subjects who were all highly experienced with spacesuited testing. Our preliminary data showed one subject maintained the constant and lower range of the upper body tilt angle while both standing and kneeling, while the other subject showed more variation of the upper body tilt angle and preferred bending the upper body rather than changing from standing to kneeling posture and vice versa. While geology experience may be a factor, these results need further investigation as suit sizing and ARGOS offloading configurations have been proven to have a significant influence on suited ARGOS tasks. Also, more subjects will be needed to complete these tasks for validation. IMU-based geology task assessments can provide useful information for geology training programs. Additionally, our IMU-based posture analysis can provide new insights into how to evaluate spacesuited geology task characteristics of astronauts during simulated lunar EVAs.

Kyoung Jae Kim↗

Transcriptomics-based Machine Learning Analysis Predicts Space-Exposed Murine Livers

Limited sample sizes, high data dimensionality, and sensitivity to technical and biological variability of next generation sequencing (NGS), has typically limited machine learning (ML) in space studies and further study of radiation effects. However, pooling smaller studies while addressing intra- and inter-study variabilities allows for ML predictive modeling. Here, integration methods were applied to whole transcriptome shotgun sequencing (RNAseq) data from 6 mouse liver GeneLab datasets (GLDS) with a total of 113 spaceflight and ground-control samples to determine top features relevant to spaceflight including the effect of radiation exposure. Data was normalized within each study, then merged and scaled across all datasets. Data dimensionality was reduced using a minimum redundancy maximum relevance (MRMR) methodology. The top MRMR features were used to predict spaceflight vs. ground-control samples using a Random Forest (RF) classifier with 5-fold cross validation (CV). The ML-based gene sets were further compared against differential gene expression results from individual GLDS. CV training using the top 100 MRMR genes show averages of 86% accuracy and 0.95 AUC value on the validation set over 5 folds (Figure 1A). Baseline set analysis on differentially expressed genes (DEGs) identified using padj ≤ 0.05 show 811 or 68 DEGs overlapping between at least 2 or 3 studies, respectively (Figure 1B). Over-representation analysis showed overlapping biological processes related to fatty acid and lipid metabolism. Set analysis between the MRMR features and the DEGs showed 60 or 8 genes overlapping with at least 1 or 2 studies, respectively. MRMR feature selection and ensemble ML methods (e.g. RF) improve performance relative to a Naïve Bayes classifier when NGS data sets are analyzed. A challenge of applying ML methods across heterogeneous NGS data is accounting for signal:noise ratio. Here, signal validation across studies was shown by intersecting sets between top MRMR genes and DEGs from RNASeq analysis. Non-intersecting sets introduce opportunity to explore spaceflight relevant genes and implementing ML methods across existing NGS datasets may overcome sample size limitations. ML coupled with existing analytical methods enhances understanding of disease by revealing common underlying pathways across datasets.

Machine Learning↗

Uncertainty quantification of a physics-informed model based on sparse identification of a Thermal Energy Distribution System

Integrated energy systems (IES)s are crucial for enhancing the economy and efficiency of power generation sources (e.g., nuclear energy) necessary to unleash American energy dominance. These systems can be integrated with thermal energy storage (TES) and intermittent renewable energies to optimize overall energy use, peak-load regulation, and demand-side responses. However, the stabilization of energy generation, transport, and utilization introduces operational complexities that exceed the challenges of managing each sub-component individually. Currently, though IESs rely on human operators for efficiency and stability, reducing human error risk and enhancing performance through automation is highly desirable. Recent advances at Idaho National Laboratory have demonstrated successful control of the Thermal Energy Distributed System (TEDS). However, the automatic control system depends on a deterministic Sparse Identification of Nonlinear Dynamics with Control (SINDyC) model, which are trained based on simulation data from physics-based simulations. Because of uncertainties in physics-based simulation, SINDyC model results in large discrepancies against experimental data and cannot be reliably used in automatic control. In this paper, we present an innovative approach to address these discrepancies by quantifying uncertainties and developing a more robust model. We first generated trajectories by using first-principles physics codes to encapsulate the experiment. Next, we trained thousands of models by randomly sampling these trajectories. We then collapsed all those models into one probabilistic SINDyC by fitting a multivariate Gaussian distribution onto the resulting coefficient’s distribution. Despite its simplicity, our approach successfully produced 95% confidence intervals that captured the experimental trajectories. It even did so with a higher probability and better U-pooling score across six of the seven relevant quantities of interest (QoIs), as compared to other classical approaches. In conclusion, ongoing research is focusing on generating new experimental trajectories to validate this approach, and on employing Bayesian calibration to refine parametric uncertainties and guide future model development efforts.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

Ensemble Monte Carlo calculations with five novel moves

We introduce five novel types of Monte Carlo (MC) moves that brings the number of moves of ensemble MC calculations from three to eight. So far such calculations have relied on affine invariant stretch moves that were originally introduced by Christen (2007), walk moves by Goodman and Weare (2010) and quadratic moves by Militzer (2023). Ensemble MC methods have been very popular because they harness information about the fitness landscape from a population of walkers rather than relying on expert knowledge. Here we modified the affine method and employed a simplex of points to set the stretch direction. We adopt the simplex concept to quadratic moves. We also generalize quadratic moves to arbitrary order. Finally, we introduce directed moves that employ the values of the probability density while all other types of moves rely solely on the location of the walkers. We apply all algorithms to the Rosenbrock density in 2 and 20 dimensions and to the ring potential in 12 and 24 dimensions. We evaluate their efficiency by comparing error bars, autocorrelation time, travel time, and the level of cohesion that measures whether any walkers were left behind. Our code is open source.

97 MATHEMATICS AND COMPUTING↗

Stacked reverberation mapping of high-redshift quasars in DESI. I. Feasibility analysis

The broad-line region of quasars has long been probed by reverberation mapping techniques that measure time lags between continuum and broad emission-line variations. Stacked reverberation mapping has been proposed as a less observationally expensive alternative to traditional methods. This ensemble approach also reduces biases from small-number statistics. The Dark Energy Spectroscopic Instrument (DESI) is conducting the most extensive spectroscopic survey of quasars to date. We create mock light curves emulating expected DESI quasar observations at redshifts $1.48\lt z\lt 5.2$ and luminosities $44.68 \le \log \lambda L_{1350 \mathring{\rm A}{}} / \mathrm{erg\, s^{-1}} \le 45.99$ to test stacked reverberation mapping feasibility using sparse spectroscopic data paired with well-sampled photometric data. The pipeline, using the lag estimation code JAVELIN (Just Another Vehicle for Estimating Lags In Nuclei), successfully recovers the simulated C IV lags within 1σ of the true values using spectroscopic light curves composed of only a few spectral epochs (2–10) with irregular cadences. We investigate how observational factors, including C IV flux error magnitude, number of stacked quasars, and spectral epoch count, affect performance. This work motivates a pathway for future stacked reverberation mapping projects with large-scale spectroscopic surveys of quasars having $\ge 2$ spectroscopic observations. Our results suggest an economical alternative for constraining and extending the radius–luminosity relation to higher redshifts and luminosities. Subsequently, this relation can be employed more reliably in single-epoch black hole mass measurements and quasar cosmology in these distant regimes.

quasars: general, quasars: supermassive black hole↗

pnnl/SNAP

In this work, we detail two uncertainty quantification (UQ) methods that provide complementary information. Readout ensembling, by finetuning only the readout layers of an ensemble of foundation models, provides information about model uncertainty. Amending the final readout layer to predict upper and lower quantiles replaces point predictions with distributional predictions, which provide information about uncertainty within the underlying training data. We demonstrate our approach with the MACE-MP-0 model, applying UQ to both the foundation model and a series of finetuned models. The uncertainties produced by the ensemble and quantile methods are demonstrated to be distinct measures by which the quality of the NNP output can be judged.

Pope, Jenna (Bilbrey) [Pacific Northwest National ↗

Modeling the effects of high-G stress on pilots in a tracking task

Air-to-air tracking experiments were conducted at the Aerospace Medical Research Laboratories using both fixed and moving base dynamic environment simulators. The obtained data, which includes longitudinal error of a simulated air-to-air tracking task as well as other auxiliary variables, was analyzed using an ensemble averaging method. In conjunction with these experiments, the optimal control model is applied to model a human operator under high-G stress.

Korn, J.↗

Online Bagging and Boosting

Bagging and boosting are two of the most well-known ensemble learning methods due to their theoretical performance guarantees and strong experimental results. However, these algorithms have been used mainly in batch mode, i.e., they require the entire training set to be available at once and, in some cases, require random access to the data. In this paper, we present online versions of bagging and boosting that require only one pass through the training data. We build on previously presented work by presenting some theoretical results. We also compare the online and batch algorithms experimentally in terms of accuracy and running time.

Oza, Nikunji C.↗