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In recent years, power analysis has become widely used in applied sciences, with the increasing importance of the replicability issue. When distribution-free methods, such as Partial Least Squares (PLS)-based approaches, are considered, formulating power analysis turns out to be challenging. In this study, we introduce the methodological framework of a new procedure for performing power analysis when PLS-based methods are used. Data are simulated by the Monte Carlo method, assuming the null hypothesis of no effect is false and exploiting the latent structure estimated by PLS in the pilot data. In this way, the complex correlation data structure is explicitly considered in power analysis and sample size estimation. The paper offers insights into selecting statistical tests for the power analysis procedure, comparing accuracy-based tests and those based on continuous parameters estimated by PLS. Simulated and real datasets are investigated to show how the method works in practice.

Uncertainty is a key feature of any machine learning model and is particularly important in neural networks, which tend to be overconfident. This overconfidence is worrying under distribution shifts, where the model performance silently degrades as the data distribution diverges from the training data distribution. Uncertainty estimation offers a solution to overconfident models, communicating when the output should (not) be trusted. Although methods for uncertainty estimation have been developed, they have not been explicitly linked to the field of explainable artificial intelligence (XAI). Furthermore, literature in operations research ignores the actionability component of uncertainty estimation and does not consider distribution shifts. This work proposes a general uncertainty framework, with contributions being threefold: (i) uncertainty estimation in ML models is positioned as an XAI technique, giving local and model-specific explanations; (ii) classification with rejection is used to reduce misclassifications by bringing a human expert in the loop for uncertain observations; (iii) the framework is applied to a case study on neural networks in educational data mining subject to distribution shifts. Uncertainty as XAI improves the model's trustworthiness in downstream decision-making tasks, giving rise to more actionable and robust machine learning systems in operations research.

This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

This work develops a machine learned structural design model for continuous beam systems from the inverse problem perspective. After demarcating between forward, optimisation and inverse machine learned operators, the investigation proposes a novel methodology based on the recently developed influence zone concept which represents a fundamental shift in approach compared to traditional structural design methods. The aim of this approach is to conceptualise a non-iterative structural design model that predicts cross-section requirements for continuous beam systems of arbitrary system size. After generating a dataset of known solutions, an appropriate neural network architecture is identified, trained, and tested against unseen data. The results show a mean absolute percentage testing error of 1.6% for cross-section property predictions, along with a good ability of the neural network to generalise well to structural systems of variable size. The CBeamXP dataset generated in this work and an associated python-based neural network training script are available at an open-source data repository to allow for the reproducibility of results and to encourage further investigations.

Knowledge sharing about emerging threats is crucial in the rapidly advancing field of cybersecurity and forms the foundation of Cyber Threat Intelligence (CTI). In this context, Large Language Models are becoming increasingly significant in the field of cybersecurity, presenting a wide range of opportunities. This study surveys the performance of ChatGPT, GPT4all, Dolly, Stanford Alpaca, Alpaca-LoRA, Falcon, and Vicuna chatbots in binary classification and Named Entity Recognition (NER) tasks performed using Open Source INTelligence (OSINT). We utilize well-established data collected in previous research from Twitter to assess the competitiveness of these chatbots when compared to specialized models trained for those tasks. In binary classification experiments, Chatbot GPT-4 as a commercial model achieved an acceptable F1 score of 0.94, and the open-source GPT4all model achieved an F1 score of 0.90. However, concerning cybersecurity entity recognition, all evaluated chatbots have limitations and are less effective. This study demonstrates the capability of chatbots for OSINT binary classification and shows that they require further improvement in NER to effectively replace specially trained models. Our results shed light on the limitations of the LLM chatbots when compared to specialized models, and can help researchers improve chatbots technology with the objective to reduce the required effort to integrate machine learning in OSINT-based CTI tools.

Studying the function spaces defined by neural networks helps to understand the corresponding learning models and their inductive bias. While in some limits neural networks correspond to function spaces that are reproducing kernel Hilbert spaces, these regimes do not capture the properties of the networks used in practice. In contrast, in this paper we show that deep neural networks define suitable reproducing kernel Banach spaces. These spaces are equipped with norms that enforce a form of sparsity, enabling them to adapt to potential latent structures within the input data and their representations. In particular, leveraging the theory of reproducing kernel Banach spaces, combined with variational results, we derive representer theorems that justify the finite architectures commonly employed in applications. Our study extends analogous results for shallow networks and can be seen as a step towards considering more practically plausible neural architectures.

Markov networks are probabilistic graphical models that employ undirected graphs to depict conditional independence relationships among variables. Our focus lies in constraint-based structure learning, which entails learning the undirected graph from data through the execution of conditional independence tests. We establish theoretical limits concerning two critical aspects of constraint-based learning of Markov networks: the number of tests and the sizes of the conditioning sets. These bounds uncover an exciting interplay between the structural properties of the graph and the amount of tests required to learn a Markov network. The starting point of our work is that the graph parameter maximum pairwise connectivity, $\kappa$, that is, the maximum number of vertex-disjoint paths connecting a pair of vertices in the graph, is responsible for the sizes of independence tests required to learn the graph. On one hand, we show that at least one test with the size of the conditioning set at least $\kappa$ is always necessary. On the other hand, we prove that any graph can be learned by performing tests of size at most $\kappa$. This completely resolves the question of the minimum size of conditioning sets required to learn the graph. When it comes to the number of tests, our upper bound on the sizes of conditioning sets implies that every $n$-vertex graph can be learned by at most $n^{\kappa}$ tests with conditioning sets of sizes at most $\kappa$. We show that for any upper bound $q$ on the sizes of the conditioning sets, there exist graphs with $O(n q)$ vertices that require at least $n^{\Omega(\kappa)}$ tests to learn. This lower bound holds even when the treewidth and the maximum degree of the graph are at most $\kappa+2$. On the positive side, we prove that every graph of bounded treewidth can be learned by a polynomial number of tests with conditioning sets of sizes at most $2\kappa$.

Surrogate modelling techniques have seen growing attention in recent years when applied to both modelling and optimisation of industrial design problems. These techniques are highly relevant when assessing the performance of a particular design carries a high cost, as the overall cost can be mitigated via the construction of a model to be queried in lieu of the available high-cost source. The construction of these models can sometimes employ other sources of information which are both cheaper and less accurate. The existence of these sources however poses the question of which sources should be used when constructing a model. Recent studies have attempted to characterise harmful data sources to guide practitioners in choosing when to ignore a certain source. These studies have done so in a synthetic setting, characterising sources using a large amount of data that is not available in practice. Some of these studies have also been shown to potentially suffer from bias in the benchmarks used in the analysis. In this study, we present a characterisation of harmful low-fidelity sources using only the limited data available to train a surrogate model. We employ recently developed benchmark filtering techniques to conduct a bias-free assessment, providing objectively varied benchmark suites of different sizes for future research. Analysing one of these benchmark suites with the technique known as Instance Space Analysis, we provide an intuitive visualisation of when a low-fidelity source should be used and use this analysis to provide guidelines that can be used in an applied industrial setting.

Software testing is essential for the reliable development of complex software systems. A key step in software testing is fault localization, which uses test data to pinpoint failure-inducing combinations for further diagnosis. Existing fault localization methods, however, are largely deterministic, and thus do not provide a principled approach for assessing probabilistic risk of potential root causes, or for integrating domain and/or structural knowledge from test engineers. To address this, we propose a novel Bayesian fault localization framework called BayesFLo, which leverages a flexible Bayesian model on potential root cause combinations. A key feature of BayesFLo is its integration of the principles of combination hierarchy and heredity, which capture the structured nature of failure-inducing combinations. A critical challenge, however, is the sheer number of potential root cause scenarios to consider, which renders the computation of posterior root cause probabilities infeasible even for small software systems. We thus develop new algorithms for efficient computation of such probabilities, leveraging recent tools from integer programming and graph representations. We then demonstrate the effectiveness of BayesFLo over state-of-the-art fault localization methods, in a suite of numerical experiments and in two motivating case studies on the JMP XGBoost interface.

Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin.