科學家經常使用計算機模擬模型來研究物理現象。為了研究物理現象中的不確定性會投過調整初始參數內部的隨機變量來產生多個結果。因此，每個網格點是以模擬運行的多個數據值表示，我們稱這種類型的數據集成數據集。為了深入了解物理現象，科學家需要視覺化和分析具有不確定性的聚合數據集。基於分佈數據表示是處理集成數據集和研究不確定性可視化的流行方法。但是，存儲一個時變集成數據集需要非常大量的儲存花費可以輕易地超過數硬體的儲存空間。因此我們提出了一種新的數據表示來緊湊地表示隨時間變化的科學數據，用於不確定性可視化和分析。我們的方法將時域中的數據解耦為兩種類型的分佈和存儲。分佈匯總數據時域中的值，另一個分佈描述了時域中數據值出現的概率。我們的方法可以再交低的儲存花費下提供具有不確定性分析並保存時間特徵。

Scientists often study physical phenomena using computer simulation models. The same simulation can generate different datasets because of different input parameter configurations or the internal random variables. Therefore, each grid point is represented by multiple data values from simulation runs, and we call this type of data ensemble dataset. To gain insight into the physical phenomenon, scientists often have to visualize and analyze the ensemble datasets with uncertainty. Distribution-based data representation is a popular approach to handle the ensemble dataset and support uncertainty visualization. However, storing a timevarying ensemble dataset needs hundreds or even thousands of times storage size. Given the size of the time-varying ensemble dataset, it is natural to develop storage reduced data representation to facilitate the time-varying ensemble data exploration. We propose a novel data representation to compactly represent the time-varying scientific data for uncertainty visualization and analysis. Our approach decouples data on the temporal domain into two types of distributions and stores. One distribution summarizes the data values on the temporal domain, and the other distribution describes the occurrence probability of a data value on the temporal domain. Our approach can provide time-varying ensemble scientific data analysis with uncertainty quantification and detailed temporal feature evolution with less storage requirement.

Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurelien Lucchi, Pascal Fua, and Sabine Susstrunk. Slic superpixels compared to state-of-the-art superpixel methods. ¨ IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(11):2274–2282, 2012.
Ann S. Almgren, John B. Bell, Mike J. Lijewski, Zarija Lukic, and Ethan Van Andel. ´ Nyx: A massively parallel AMR code for computational cosmology. Astrophysical Journal, 765(1), 2013.
Tushar Athawale, Elham Sakhaee, and Alireza Entezari. Isosurface visualization of data with nonparametric models for uncertainty. IEEE Transactions on Visualization and Computer Graphics, 22(1):777–786, 2016.
Chun-Ming Chen, Ayan Biswas, and Han-Wei Shen. Uncertainty modeling and error reduction for pathline computation in time-varying flow fields. In 2015 IEEE Pacific Visualization Symposium (PacificVis), pages 215–222, 2015.
Chun-Ming Chen, Ayan Biswas, and Han-Wei Shen. Uncertainty modeling and error reduction for pathline computation in time-varying flow fields. In 2015 IEEE Pacific Visualization Symposium (PacificVis), pages 215–222. IEEE, 2015.
Sheng Di and Franck Cappello. Fast error-bounded lossy hpc data compression with sz. In 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 730–739, 2016.
Soumya Dutta, Chun-Ming Chen, Gregory Heinlein, Han-Wei Shen, and Jen-Ping Chen. In situ distribution guided analysis and visualization of transonic jet engine simulations. IEEE transactions on visualization and computer graphics, 23(1):811– 820, 2016.
Anthony Y Fu, Liu Wenyin, and Xiaotie Deng. Detecting phishing web pages with visual similarity assessment based on earth mover’s distance (emd). IEEE transactions on dependable and secure computing, 3(4):301–311, 2006.
Subhashis Hazarika, Ayan Biswas, and Han-Wei Shen. Uncertainty visualization using copula-based analysis in mixed distribution models. IEEE Transactions on Visualization and Computer Graphics, 24(1):934–943, 2018.
Subhashis Hazarika, Soumya Dutta, Han-Wei Shen, and Jen-Ping Chen. Codda: A flexible copula-based distribution driven analysis framework for large-scale multivariate data. IEEE Transactions on Visualization and Computer Graphics, 25(1):1214–1224, 2019.
Wenbin He, Junpeng Wang, Hanqi Guo, Ko-Chih Wang, Han-Wei Shen, Mukund Raj, Youssef SG Nashed, and Tom Peterka. Insitunet: Deep image synthesis for parameter space exploration of ensemble simulations. IEEE transactions on visualization and computer graphics, 26(1):23–33, 2019.
 

Thomas Hollt, Ahmed Magdy, Peng Zhan, Guoning Chen, Ganesh Gopalakrishnan, ¨ Ibrahim Hoteit, Charles D. Hansen, and Markus Hadwiger. Ovis: A framework for visual analysisof ocean forecast ensembles. IEEE Transactions on Visualization and Computer Graphics, 20(8):1114–1126, 2014.
Teng-Yok Lee and Han-Wei Shen. Efficient local statistical analysis via integral histograms with discrete wavelet transform. IEEE Transactions on Visualization and Computer Graphics, 19(12):2693–2702, 2013.
Guan Li, Jiayi Xu, Tianchi Zhang, Guihua Shan, Han-Wei Shen, Ko-Chih Wang, Shihong Liao, and Zhonghua Lu. Distribution-based particle data reduction for in-situ analysis and visualization of large-scale n-body cosmological simulations. In 2020 IEEE Pacific Visualization Symposium (PacificVis), pages 171–180. IEEE, 2020.
Shaomeng Li, Nicole Marsaglia, Christoph Garth, Jonathan Woodring, John Clyne, and Hank Childs. Data reduction techniques for simulation, visualization and data analysis. In Computer Graphics Forum, volume 37, pages 422–447. Wiley Online Library, 2018.
Shusen Liu, Joshua A. Levine, Peer-Timo Bremer, and Valerio Pascucci. Gaussian mixture model based volume visualization. In IEEE Symposium on Large Data Analysis and Visualization (LDAV), pages 73–77, 2012.
Alison Luo, David Kao, and Alex Pang. Visualizing spatial distribution data sets. In VisSym, volume 3, pages 29–38, 2003.
Kwan-Liu Ma. In situ visualization at extreme scale: Challenges and opportunities. IEEE Computer Graphics and Applications, 29(6):14–19, 2009.
Kenneth Moreland, Christopher Sewell, William Usher, Li-ta Lo, Jeremy Meredith, David Pugmire, James Kress, Hendrik Schroots, Kwan-Liu Ma, Hank Childs, et al. Vtk-m: Accelerating the visualization toolkit for massively threaded architectures. IEEE computer graphics and applications, 36(3):48–58, 2016.
Patrick O’Leary, James Ahrens, Sebastien Jourdain, Scott Wittenburg, David H Rogers, ´ and Mark Petersen. Cinema image-based in situ analysis and visualization of mpasocean simulations. Parallel Computing, 55:43–48, 2016.
Kai Pothkow and Hans-Christian Hege. Nonparametric models for uncertainty visu- ¨ alization. In Computer Graphics Forum, volume 32, pages 131–140. Wiley Online Library, 2013.
A.L. Read. Linear interpolation of histograms. Nuclear Instruments and Methods in Physics Research. Section A, Accelerators, Spectrometers, Detectors and Associated Equipment, 425(1):357–360, 1999.
Jibonananda Sanyal, Song Zhang, Jamie Dyer, Andrew Mercer, Philip Amburn, and Robert Moorhead. Noodles: A tool for visualization of numerical weather model ensemble uncertainty. IEEE Transactions on Visualization and Computer Graphics, 16(6):1421–1430, 2010.
Ronell Sicat, Jens Kruger, Torsten M ¨ oller, and Markus Hadwiger. Sparse pdf volumes ¨ for consistent multi-resolution volume rendering. IEEE Transactions on Visualization and Computer Graphics, 20(12):2417–2426, 2014.
Cheng Sun and Ko-Chih Wang. Dla-vps: Deep learning assisted visual parameter space analysis of cosmological simulations. IEEE Computer Graphics and Applications, 2022.
David Thompson, Joshua A. Levine, Janine C. Bennett, Peer-Timo Bremer, Attila Gyulassy, Valerio Pascucci, and Philippe P. Pebay. Analysis of large-scale scalar data ´ using hixels. In 2011 IEEE Symposium on Large Data Analysis and Visualization, pages 23–30, 2011.
Anna Tikhonova, Carlos D. Correa, and Kwan-Liu Ma. Visualization by proxy: A novel framework for deferred interaction with volume data. IEEE Transactions on Visualization and Computer Graphics, 16(6):1551–1559, 2010.
Junpeng Wang, Subhashis Hazarika, Cheng Li, and Han-Wei Shen. Visualization and visual analysis of ensemble data: A survey. IEEE Transactions on Visualization and Computer Graphics, 25(9):2853–2872, 2019.
Junpeng Wang, Xiaotong Liu, Han-Wei Shen, and Guang Lin. Multi-resolution climate ensemble parameter analysis with nested parallel coordinates plots. IEEE Transactions on Visualization and Computer Graphics, 23(1):81–90, 2017.
Ko-Chih Wang, Kewei Lu, Tzu-Hsuan Wei, Naeem Shareef, and Han-Wei Shen. Statistical visualization and analysis of large data using a value-based spatial distribution. In 2017 IEEE Pacific Visualization Symposium (PacificVis), pages 161–170, 2017.
Ko-Chih Wang, Naeem Shareef, and Han-Wei Shen. Image and distribution based volume rendering for large data sets. In 2018 IEEE Pacific Visualization Symposium (PacificVis), pages 26–35, 2018.
Jonathan Woodring, James P. Ahrens, J. Figg, Joanne Wendelberger, Salman Habib, and Katrin Heitmann. In-situ sampling of a large-scale particle simulation for interactive visualization and analysis. Computer Graphics Forum, 30, 2011.
Hao-Yi Yang, Zhi-Rong Lin, and Ko-Chih Wang. Efficient and portable distribution modeling for large-scale scientific data processing with data-parallel primitives. Algorithms, 14(10):285, 2021.
Hongfeng Yu, Chaoli Wang, Ray W. Grout, Jacqueline H. Chen, and Kwan-Liu Ma. In situ visualization for large-scale combustion simulations. IEEE Computer Graphics and Applications, 30(3):45–57, 2010.