In this paper, a Monte Carlo based approach for the quantification of the importance of the scattering input parameters with respect to the failure probability is presented. Using the basic idea of the alpha-factors of the First Order Reliability Method, this approach was developed to analyze correlated input variables as well as arbitrary marginal parameter distributions. Based on an efficient transformation scheme using the importance sampling principle, only a single analysis run by a plain or variance-reduced Monte Carlo method is required to give a sufficient estimate of the introduced parameter sensitivities. Several application examples are presented and discussed in the paper.
{"title":"Efficient variance-based reliability sensitivity analysis for Monte Carlo methods","authors":"Thomas Most","doi":"arxiv-2408.06664","DOIUrl":"https://doi.org/arxiv-2408.06664","url":null,"abstract":"In this paper, a Monte Carlo based approach for the quantification of the\u0000importance of the scattering input parameters with respect to the failure\u0000probability is presented. Using the basic idea of the alpha-factors of the\u0000First Order Reliability Method, this approach was developed to analyze\u0000correlated input variables as well as arbitrary marginal parameter\u0000distributions. Based on an efficient transformation scheme using the importance\u0000sampling principle, only a single analysis run by a plain or variance-reduced\u0000Monte Carlo method is required to give a sufficient estimate of the introduced\u0000parameter sensitivities. Several application examples are presented and\u0000discussed in the paper.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mixed effects models are among the most commonly used statistical methods for the exploration of multispecies data. In recent years, also Joint Species Distribution Models and Generalized Linear Latent Variale Models have gained in popularity when the goal is to incorporate residual covariation between species that cannot be explained due to measured environmental covariates. Few software implementations of such models exist that can additionally incorporate phylogenetic information, and those that exist tend to utilize Markov chain Monte Carlo methods for estimation, so that model fitting takes a long time. In this article we develop new methods for quickly and flexibly fitting phylogenetic mixed models, potentially incorporating residual covariation between species using latent variables, with the possibility to estimate the strength of phylogenetic structuring in species responses per environmental covariate, and while incorporating correlation between different covariate effects. By combining Variational approximations with a reduced rank matrix normal covariance structure, Nearest Neighbours Gaussian Processes, and parallel computation, phylogenetic mixed models can be fitted much more quickly than the current state-of-the-art. Two simulation studies demonstrate that the proposed combination of approximations is not only fast, but also enjoys high accuracy. Finally, we demonstrate use of the method with a real world dataset of wood-decaying fungi.
{"title":"Fast fitting of phylogenetic mixed effects models","authors":"Bert van der Veen, Robert Brian O'Hara","doi":"arxiv-2408.05333","DOIUrl":"https://doi.org/arxiv-2408.05333","url":null,"abstract":"Mixed effects models are among the most commonly used statistical methods for\u0000the exploration of multispecies data. In recent years, also Joint Species\u0000Distribution Models and Generalized Linear Latent Variale Models have gained in\u0000popularity when the goal is to incorporate residual covariation between species\u0000that cannot be explained due to measured environmental covariates. Few software\u0000implementations of such models exist that can additionally incorporate\u0000phylogenetic information, and those that exist tend to utilize Markov chain\u0000Monte Carlo methods for estimation, so that model fitting takes a long time. In\u0000this article we develop new methods for quickly and flexibly fitting\u0000phylogenetic mixed models, potentially incorporating residual covariation\u0000between species using latent variables, with the possibility to estimate the\u0000strength of phylogenetic structuring in species responses per environmental\u0000covariate, and while incorporating correlation between different covariate\u0000effects. By combining Variational approximations with a reduced rank matrix\u0000normal covariance structure, Nearest Neighbours Gaussian Processes, and\u0000parallel computation, phylogenetic mixed models can be fitted much more quickly\u0000than the current state-of-the-art. Two simulation studies demonstrate that the\u0000proposed combination of approximations is not only fast, but also enjoys high\u0000accuracy. Finally, we demonstrate use of the method with a real world dataset\u0000of wood-decaying fungi.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142189569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sameh Abdulah, Allison H. Baker, George Bosilca, Qinglei Cao, Stefano Castruccio, Marc G. Genton, David E. Keyes, Zubair Khalid, Hatem Ltaief, Yan Song, Georgiy L. Stenchikov, Ying Sun
We present the design and scalable implementation of an exascale climate emulator for addressing the escalating computational and storage requirements of high-resolution Earth System Model simulations. We utilize the spherical harmonic transform to stochastically model spatio-temporal variations in climate data. This provides tunable spatio-temporal resolution and significantly improves the fidelity and granularity of climate emulation, achieving an ultra-high spatial resolution of 0.034 (approximately 3.5 km) in space. Our emulator, trained on 318 billion hourly temperature data points from a 35-year and 31 billion daily data points from an 83-year global simulation ensemble, generates statistically consistent climate emulations. We extend linear solver software to mixed-precision arithmetic GPUs, applying different precisions within a single solver to adapt to different correlation strengths. The PaRSEC runtime system supports efficient parallel matrix operations by optimizing the dynamic balance between computation, communication, and memory requirements. Our BLAS3-rich code is optimized for systems equipped with four different families and generations of GPUs, scaling well to achieve 0.976 EFlop/s on 9,025 nodes (36,100 AMD MI250X multichip module (MCM) GPUs) of Frontier (nearly full system), 0.739 EFlop/s on 1,936 nodes (7,744 Grace-Hopper Superchips (GH200)) of Alps, 0.243 EFlop/s on 1,024 nodes (4,096 A100 GPUs) of Leonardo, and 0.375 EFlop/s on 3,072 nodes (18,432 V100 GPUs) of Summit.
{"title":"Boosting Earth System Model Outputs And Saving PetaBytes in their Storage Using Exascale Climate Emulators","authors":"Sameh Abdulah, Allison H. Baker, George Bosilca, Qinglei Cao, Stefano Castruccio, Marc G. Genton, David E. Keyes, Zubair Khalid, Hatem Ltaief, Yan Song, Georgiy L. Stenchikov, Ying Sun","doi":"arxiv-2408.04440","DOIUrl":"https://doi.org/arxiv-2408.04440","url":null,"abstract":"We present the design and scalable implementation of an exascale climate\u0000emulator for addressing the escalating computational and storage requirements\u0000of high-resolution Earth System Model simulations. We utilize the spherical\u0000harmonic transform to stochastically model spatio-temporal variations in\u0000climate data. This provides tunable spatio-temporal resolution and\u0000significantly improves the fidelity and granularity of climate emulation,\u0000achieving an ultra-high spatial resolution of 0.034 (approximately 3.5 km) in\u0000space. Our emulator, trained on 318 billion hourly temperature data points from\u0000a 35-year and 31 billion daily data points from an 83-year global simulation\u0000ensemble, generates statistically consistent climate emulations. We extend\u0000linear solver software to mixed-precision arithmetic GPUs, applying different\u0000precisions within a single solver to adapt to different correlation strengths.\u0000The PaRSEC runtime system supports efficient parallel matrix operations by\u0000optimizing the dynamic balance between computation, communication, and memory\u0000requirements. Our BLAS3-rich code is optimized for systems equipped with four\u0000different families and generations of GPUs, scaling well to achieve 0.976\u0000EFlop/s on 9,025 nodes (36,100 AMD MI250X multichip module (MCM) GPUs) of\u0000Frontier (nearly full system), 0.739 EFlop/s on 1,936 nodes (7,744 Grace-Hopper\u0000Superchips (GH200)) of Alps, 0.243 EFlop/s on 1,024 nodes (4,096 A100 GPUs) of\u0000Leonardo, and 0.375 EFlop/s on 3,072 nodes (18,432 V100 GPUs) of Summit.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Piecewise deterministic Markov processes (PDMPs) are a class of continuous-time Markov processes that were recently used to develop a new class of Markov chain Monte Carlo algorithms. However, the implementation of the processes is challenging due to the continuous-time aspect and the necessity of integrating the rate function. Recently, Corbella, Spencer, and Roberts (2022) proposed a new algorithm to automate the implementation of the Zig-Zag sampler. However, the efficiency of the algorithm highly depends on a hyperparameter ($t_{text{max}}$) that is fixed all along the run of the algorithm and needs preliminary runs to tune. In this work, we relax this assumption and propose a new variant of their algorithm that let this parameter change over time and automatically adapt to the target distribution. We also replace the Brent optimization algorithm by a grid-based method to compute the upper bound of the rate function. This method is more robust to the regularity of the function and gives a tighter upper bound while being quicker to compute. We also extend the algorithm to other PDMPs and provide a Python implementation of the algorithm based on JAX.
{"title":"Automated Techniques for Efficient Sampling of Piecewise-Deterministic Markov Processes","authors":"Charly Andral, Kengo Kamatani","doi":"arxiv-2408.03682","DOIUrl":"https://doi.org/arxiv-2408.03682","url":null,"abstract":"Piecewise deterministic Markov processes (PDMPs) are a class of\u0000continuous-time Markov processes that were recently used to develop a new class\u0000of Markov chain Monte Carlo algorithms. However, the implementation of the\u0000processes is challenging due to the continuous-time aspect and the necessity of\u0000integrating the rate function. Recently, Corbella, Spencer, and Roberts (2022)\u0000proposed a new algorithm to automate the implementation of the Zig-Zag sampler.\u0000However, the efficiency of the algorithm highly depends on a hyperparameter\u0000($t_{text{max}}$) that is fixed all along the run of the algorithm and needs\u0000preliminary runs to tune. In this work, we relax this assumption and propose a\u0000new variant of their algorithm that let this parameter change over time and\u0000automatically adapt to the target distribution. We also replace the Brent\u0000optimization algorithm by a grid-based method to compute the upper bound of the\u0000rate function. This method is more robust to the regularity of the function and\u0000gives a tighter upper bound while being quicker to compute. We also extend the\u0000algorithm to other PDMPs and provide a Python implementation of the algorithm\u0000based on JAX.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Steven D. Barnett, Lauren J. Beesley, Annie S. Booth, Robert B. Gramacy, Dave Osthus
Gaussian processes (GPs) are canonical as surrogates for computer experiments because they enjoy a degree of analytic tractability. But that breaks when the response surface is constrained, say to be monotonic. Here, we provide a mono-GP construction for a single input that is highly efficient even though the calculations are non-analytic. Key ingredients include transformation of a reference process and elliptical slice sampling. We then show how mono-GP may be deployed effectively in two ways. One is additive, extending monotonicity to more inputs; the other is as a prior on injective latent warping variables in a deep Gaussian process for (non-monotonic, multi-input) non-stationary surrogate modeling. We provide illustrative and benchmarking examples throughout, showing that our methods yield improved performance over the state-of-the-art on examples from those two classes of problems.
高斯过程（GPs）是计算机实验的典型代表，因为它们具有一定程度的可分析性。但是，当响应面受到约束，例如必须是单调的时候，这种可分析性就不复存在了。在这里，我们为单一输入提供了一种单 GP 结构，即使计算是非解析的，它也非常高效。其关键要素包括推理过程的转换和椭圆切片采样。然后，我们展示了如何以两种方式有效地部署单GP。一种是加法，将单调性扩展到更多输入；另一种是作为深高斯过程中注入式潜翘变量的先验，用于（非单调、多输入）非稳态代理建模。我们通篇提供了说明性和基准示例，表明我们的方法在这两类问题的示例中取得了优于最先进方法的性能。
{"title":"Monotonic warpings for additive and deep Gaussian processes","authors":"Steven D. Barnett, Lauren J. Beesley, Annie S. Booth, Robert B. Gramacy, Dave Osthus","doi":"arxiv-2408.01540","DOIUrl":"https://doi.org/arxiv-2408.01540","url":null,"abstract":"Gaussian processes (GPs) are canonical as surrogates for computer experiments\u0000because they enjoy a degree of analytic tractability. But that breaks when the\u0000response surface is constrained, say to be monotonic. Here, we provide a\u0000mono-GP construction for a single input that is highly efficient even though\u0000the calculations are non-analytic. Key ingredients include transformation of a\u0000reference process and elliptical slice sampling. We then show how mono-GP may\u0000be deployed effectively in two ways. One is additive, extending monotonicity to\u0000more inputs; the other is as a prior on injective latent warping variables in a\u0000deep Gaussian process for (non-monotonic, multi-input) non-stationary surrogate\u0000modeling. We provide illustrative and benchmarking examples throughout, showing\u0000that our methods yield improved performance over the state-of-the-art on\u0000examples from those two classes of problems.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christophe Andrieu, Nicolas Chopin, Ettore Fincato, Mathieu Gerber
In this paper we propose a novel, general purpose, algorithm to optimize functions $lcolon mathbb{R}^d rightarrow mathbb{R}$ not assumed to be convex or differentiable or even continuous. The main idea is to sequentially fit a sequence of parametric probability densities, possessing a concentration property, to $l$ using a Bayesian update followed by a reprojection back onto the chosen parametric sequence. Remarkably, with the sequence chosen to be from the exponential family, reprojection essentially boils down to the computation of expectations. Our algorithm therefore lends itself to Monte Carlo approximation, ranging from plain to Sequential Monte Carlo (SMC) methods. The algorithm is therefore particularly simple to implement and we illustrate performance on a challenging Machine Learning classification problem. Our methodology naturally extends to the scenario where only noisy measurements of $l$ are available and retains ease of implementation and performance. At a theoretical level we establish, in a fairly general scenario, that our framework can be viewed as implicitly implementing a time inhomogeneous gradient descent algorithm on a sequence of smoothed approximations of $l$. This opens the door to establishing convergence of the algorithm and provide theoretical guarantees. Along the way, we establish new results for inhomogeneous gradient descent algorithms of independent interest.
{"title":"Gradient-free optimization via integration","authors":"Christophe Andrieu, Nicolas Chopin, Ettore Fincato, Mathieu Gerber","doi":"arxiv-2408.00888","DOIUrl":"https://doi.org/arxiv-2408.00888","url":null,"abstract":"In this paper we propose a novel, general purpose, algorithm to optimize\u0000functions $lcolon mathbb{R}^d rightarrow mathbb{R}$ not assumed to be\u0000convex or differentiable or even continuous. The main idea is to sequentially\u0000fit a sequence of parametric probability densities, possessing a concentration\u0000property, to $l$ using a Bayesian update followed by a reprojection back onto\u0000the chosen parametric sequence. Remarkably, with the sequence chosen to be from\u0000the exponential family, reprojection essentially boils down to the computation\u0000of expectations. Our algorithm therefore lends itself to Monte Carlo\u0000approximation, ranging from plain to Sequential Monte Carlo (SMC) methods. The algorithm is therefore particularly simple to implement and we illustrate\u0000performance on a challenging Machine Learning classification problem. Our\u0000methodology naturally extends to the scenario where only noisy measurements of\u0000$l$ are available and retains ease of implementation and performance. At a\u0000theoretical level we establish, in a fairly general scenario, that our\u0000framework can be viewed as implicitly implementing a time inhomogeneous\u0000gradient descent algorithm on a sequence of smoothed approximations of $l$.\u0000This opens the door to establishing convergence of the algorithm and provide\u0000theoretical guarantees. Along the way, we establish new results for\u0000inhomogeneous gradient descent algorithms of independent interest.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Léa Loisel, Vincent Raquin, Maxime Ratinier, Pauline Ezanno, Gaël Beaunée
Arboviruses represent a significant threat to human, animal, and plant health worldwide. To elucidate transmission, anticipate their spread and efficiently control them, mechanistic modelling has proven its usefulness. However, most models rely on assumptions about how the extrinsic incubation period (EIP) is represented: the intra-vector viral dynamics (IVD), occurring during the EIP, is approximated by a single state. After an average duration, all exposed vectors become infectious. Behind this are hidden two strong hypotheses: (i) EIP is exponentially distributed in the vector population; (ii) viruses successfully cross the infection, dissemination, and transmission barriers in all exposed vectors. To assess these hypotheses, we developed a stochastic compartmental model which represents successive IVD stages, associated to the crossing or not of these three barriers. We calibrated the model using an ABC-SMC (Approximate Bayesian Computation - Sequential Monte Carlo) method with model selection. We systematically searched for literature data on experimental infections of Aedes mosquitoes infected by either dengue, chikungunya, or Zika viruses. We demonstrated the discrepancy between the exponential hypothesis and observed EIP distributions for dengue and Zika viruses and identified more relevant EIP distributions . We also quantified the fraction of infected mosquitoes eventually becoming infectious, highlighting that often only a small fraction crosses the three barriers. This work provides a generic modelling framework applicable to other arboviruses for which similar data are available. Our model can also be coupled to population-scale models to aid future arbovirus control.
{"title":"Within-vector viral dynamics challenges how to model the extrinsic incubation period for major arboviruses: dengue, Zika, and chikungunya","authors":"Léa Loisel, Vincent Raquin, Maxime Ratinier, Pauline Ezanno, Gaël Beaunée","doi":"arxiv-2408.00409","DOIUrl":"https://doi.org/arxiv-2408.00409","url":null,"abstract":"Arboviruses represent a significant threat to human, animal, and plant health\u0000worldwide. To elucidate transmission, anticipate their spread and efficiently\u0000control them, mechanistic modelling has proven its usefulness. However, most\u0000models rely on assumptions about how the extrinsic incubation period (EIP) is\u0000represented: the intra-vector viral dynamics (IVD), occurring during the EIP,\u0000is approximated by a single state. After an average duration, all exposed\u0000vectors become infectious. Behind this are hidden two strong hypotheses: (i)\u0000EIP is exponentially distributed in the vector population; (ii) viruses\u0000successfully cross the infection, dissemination, and transmission barriers in\u0000all exposed vectors. To assess these hypotheses, we developed a stochastic\u0000compartmental model which represents successive IVD stages, associated to the\u0000crossing or not of these three barriers. We calibrated the model using an\u0000ABC-SMC (Approximate Bayesian Computation - Sequential Monte Carlo) method with\u0000model selection. We systematically searched for literature data on experimental\u0000infections of Aedes mosquitoes infected by either dengue, chikungunya, or Zika\u0000viruses. We demonstrated the discrepancy between the exponential hypothesis and\u0000observed EIP distributions for dengue and Zika viruses and identified more\u0000relevant EIP distributions . We also quantified the fraction of infected\u0000mosquitoes eventually becoming infectious, highlighting that often only a small\u0000fraction crosses the three barriers. This work provides a generic modelling\u0000framework applicable to other arboviruses for which similar data are available.\u0000Our model can also be coupled to population-scale models to aid future\u0000arbovirus control.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent advancements in understanding the brain's functional organization related to behavior have been pivotal, particularly in the development of predictive models based on brain connectivity. Traditional methods in this domain often involve a two-step process by first constructing a connectivity matrix from predefined brain regions, and then linking these connections to behaviors or clinical outcomes. However, these approaches with unsupervised node partitions predict outcomes inefficiently with independently established connectivity. In this paper, we introduce the Supervised Brain Parcellation (SBP), a brain node parcellation scheme informed by the downstream predictive task. With voxel-level functional time courses generated under resting-state or cognitive tasks as input, our approach clusters voxels into nodes in a manner that maximizes the correlation between inter-node connections and the behavioral outcome, while also accommodating intra-node homogeneity. We rigorously evaluate the SBP approach using resting-state and task-based fMRI data from both the Adolescent Brain Cognitive Development (ABCD) study and the Human Connectome Project (HCP). Our analyses show that SBP significantly improves out-of-sample connectome-based predictive performance compared to conventional step-wise methods under various brain atlases. This advancement holds promise for enhancing our understanding of brain functional architectures with behavior and establishing more informative network neuromarkers for clinical applications.
{"title":"Supervised brain node and network construction under voxel-level functional imaging","authors":"Wanwan Xu, Selena Wang, Chichun Tan, Xilin Shen, Wenjing Luo, Todd Constable, Tianxi Li, Yize Zhao","doi":"arxiv-2407.21242","DOIUrl":"https://doi.org/arxiv-2407.21242","url":null,"abstract":"Recent advancements in understanding the brain's functional organization\u0000related to behavior have been pivotal, particularly in the development of\u0000predictive models based on brain connectivity. Traditional methods in this\u0000domain often involve a two-step process by first constructing a connectivity\u0000matrix from predefined brain regions, and then linking these connections to\u0000behaviors or clinical outcomes. However, these approaches with unsupervised\u0000node partitions predict outcomes inefficiently with independently established\u0000connectivity. In this paper, we introduce the Supervised Brain Parcellation\u0000(SBP), a brain node parcellation scheme informed by the downstream predictive\u0000task. With voxel-level functional time courses generated under resting-state or\u0000cognitive tasks as input, our approach clusters voxels into nodes in a manner\u0000that maximizes the correlation between inter-node connections and the\u0000behavioral outcome, while also accommodating intra-node homogeneity. We\u0000rigorously evaluate the SBP approach using resting-state and task-based fMRI\u0000data from both the Adolescent Brain Cognitive Development (ABCD) study and the\u0000Human Connectome Project (HCP). Our analyses show that SBP significantly\u0000improves out-of-sample connectome-based predictive performance compared to\u0000conventional step-wise methods under various brain atlases. This advancement\u0000holds promise for enhancing our understanding of brain functional architectures\u0000with behavior and establishing more informative network neuromarkers for\u0000clinical applications.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we examine the Sample Average Approximation (SAA) procedure within a framework where the Monte Carlo estimator of the expectation is biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to enhance the computational efficiency of solving optimization problems. In this context, we conduct a thorough analysis, exploiting Cram'er's large deviation theory, to establish uniform convergence, quantify the convergence rate, and determine the sample complexity for both standard Monte Carlo and MLMC paradigms. Additionally, we perform a root-mean-squared error analysis utilizing tools from empirical process theory to derive sample complexity without relying on the finite moment condition typically required for uniform convergence results. Finally, we validate our findings and demonstrate the advantages of the MLMC estimator through numerical examples, estimating Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested expectation framework.
在本文中，我们在蒙特卡罗期望估计器有偏差的框架下研究了样本平均逼近（SAA）程序。我们还在 SAA 设置中引入了多级蒙特卡罗（MLMC），以提高解决优化问题的计算效率。在此背景下，我们利用克拉姆（Cram'er）的大偏差理论（large deviationtheory）进行了深入分析，为标准蒙特卡罗和 MLMC 范式建立了均匀收敛性、量化了收敛速率并确定了样本复杂度。此外，我们还利用经验过程理论的工具进行了均方根误差分析，得出了样本复杂度，而无需依赖均匀收敛结果通常需要的有限矩条件。最后，我们通过数值示例验证了我们的发现，并证明了 MLMC 估计器的优势，即在几何布朗运动和嵌套期望框架下估计条件风险值（CVaR）。
{"title":"Multilevel Monte Carlo in Sample Average Approximation: Convergence, Complexity and Application","authors":"Devang Sinha, Siddhartha P. Chakrabarty","doi":"arxiv-2407.18504","DOIUrl":"https://doi.org/arxiv-2407.18504","url":null,"abstract":"In this paper, we examine the Sample Average Approximation (SAA) procedure\u0000within a framework where the Monte Carlo estimator of the expectation is\u0000biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to\u0000enhance the computational efficiency of solving optimization problems. In this\u0000context, we conduct a thorough analysis, exploiting Cram'er's large deviation\u0000theory, to establish uniform convergence, quantify the convergence rate, and\u0000determine the sample complexity for both standard Monte Carlo and MLMC\u0000paradigms. Additionally, we perform a root-mean-squared error analysis\u0000utilizing tools from empirical process theory to derive sample complexity\u0000without relying on the finite moment condition typically required for uniform\u0000convergence results. Finally, we validate our findings and demonstrate the\u0000advantages of the MLMC estimator through numerical examples, estimating\u0000Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested\u0000expectation framework.","PeriodicalId":501215,"journal":{"name":"arXiv - STAT - Computation","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Naichen Shi, Hao Yan, Shenghan Guo, Raed Al Kontar
In this paper, we present a generic physics-informed generative model called MPDM that integrates multi-fidelity physics simulations with diffusion models. MPDM categorizes multi-fidelity physics simulations into inexpensive and expensive simulations, depending on computational costs. The inexpensive simulations, which can be obtained with low latency, directly inject contextual information into DDMs. Furthermore, when results from expensive simulations are available, MPDM refines the quality of generated samples via a guided diffusion process. This design separates the training of a denoising diffusion model from physics-informed conditional probability models, thus lending flexibility to practitioners. MPDM builds on Bayesian probabilistic models and is equipped with a theoretical guarantee that provides upper bounds on the Wasserstein distance between the sample and underlying true distribution. The probabilistic nature of MPDM also provides a convenient approach for uncertainty quantification in prediction. Our models excel in cases where physics simulations are imperfect and sometimes inaccessible. We use a numerical simulation in fluid dynamics and a case study in heat dynamics within laser-based metal powder deposition additive manufacturing to demonstrate how MPDM seamlessly integrates multi-idelity physics simulations and observations to obtain surrogates with superior predictive performance.
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