For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of th...
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For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.
Accurate wind power prediction is highly significant to the safety, stability, and economic operation of power grids. Currently, typical ensemble methods for wind power forecasting are widely designed based on the mea...
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Accurate wind power prediction is highly significant to the safety, stability, and economic operation of power grids. Currently, typical ensemble methods for wind power forecasting are widely designed based on the mean square error (MSE) loss, which are very suitable for the assumption that the error distribution obeys the Gaussian distribution. However, the complex nonlinear conversion of wind energy into wind power may change the statistical characteristics of errors, thus the prediction model based on the traditional MSE loss may lead to the performance degradation of the forecasting model. To address these problems, a probability density function based adaptive ensemble learning with global convergence is proposed for wind power prediction, which comprises three modules: a data preprocessing module, a prediction module, and a combination module. Firstly, an effective feature generation mechanism is employed to extract the multi-mode characteristics of wind data. Then, an auxiliary error based adaptive global convergence model is developed as benchmark predictor in the prediction module, where an adaptive updating algorithm is derived based on the Lyapunov approach to ensure the global convergence of the model weights. Moreover, considering the asymmetric characteristic of modeling error, a probability density function (PDF) based ensemble learning is created to integrate the results of benchmark predictors. Specifically, the ensemble model parameter updating is transformed into the shape control for the modeling error PDF, which can break through the limitation of MSE loss capturing only the second moment information, and emphasize the spatial distribution of errors to make an unbiased estimate of wind power. Experimental results show that the proposed ensemble model has significant advantages over other models involved in this study.
The present work examines the applicability of the joint transported probability density function (TPDF) with the velocity field obtained from a conventional RANS k-epsilon approach in a jet-stirred reactor, where the...
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The present work examines the applicability of the joint transported probability density function (TPDF) with the velocity field obtained from a conventional RANS k-epsilon approach in a jet-stirred reactor, where the Damk & ouml;hler number is below the unity. In the TPDF equation, turbulent transport is handled with a conventional k-epsilon gradient transport assumption and the LMSE is used for the molecular mixing. Through the use of a dynamic model for the mixing time-scale, by computing the individual time-scales for the reactive scalars dynamically in each cell during the course of the simulation using the RANS-TPDF code. The chemical reaction is described by a reduced and detailed chemical propane mechanism. The modelized PDF equation constants are determined from the available experimental data conducted by using fine wire thermo-anemometry. The CFD-TPDF predictions were within engineering accuracy of experimental data. It can be summarized that the results of exercise are satisfactory, and the CPU-time and RAM memory savings encouraging.
Currently, Convolutional Neural Networks (CNNs) have demonstrated extensive success in numerous practical applications. Nevertheless, their limited interpretability remains a significant barrier to further advancement...
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Currently, Convolutional Neural Networks (CNNs) have demonstrated extensive success in numerous practical applications. Nevertheless, their limited interpretability remains a significant barrier to further advancement in certain crucial fields. Improving the interpretability of CNNs stands as an exceptionally compelling topic in the present time. This paper explores the interpretability of a basic CNN incorporating a convolution-pooled block and a fully connected layer from a statistical perspective. Assuming that the input variables adhere to a normal distribution and maintain independence from each other, the output variables subsequent to the convolution and pooling layers also conform to a normal distribution. Simultaneously, the probability density function (pdf) characterizing the final output variable belongs to an exponential family distribution. By introducing intermediate variables, the pdf of this output variable can be expressed as a linear combination of three distinct normal distributions. Furthermore, the likelihood of the predicted class label can be rewritten as a cumulative densityfunction (cdf) of the standard normal distribution. The originality of this paper lies in its provision of a more innovative and intuitive perspective for dissecting the operational mechanism of CNNs, analyzing them layer by layer to improve their interpretability. Experimental results obtained from both an artificial dataset and the image datasets CIFAR-10 and ImageNet further validate the rationality of these conclusions.
probability density function (PDF) curves are among the few charts on a Cartesian coordinate system that are commonly presented without y-axes. This design decision may be due to the lack of relevance of vertical scal...
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probability density function (PDF) curves are among the few charts on a Cartesian coordinate system that are commonly presented without y-axes. This design decision may be due to the lack of relevance of vertical scaling in normal PDFs. In fact, as long as two normal PDFs have the same means and standard deviations (SDs), they can be scaled to occupy different amounts of vertical space while still remaining statistically identical. Because unfixed PDF height increases as SD decreases, visualization designers may find themselves tempted to vertically shrink low-SD PDFs to avoid occlusion or save white space in their figures. Although irregular vertical scaling has been explored in bar and line charts, the visualization community has yet to investigate how this visual manipulation may affect reader comparisons of PDFs. In this paper, we present two preregistered experiments (n = 600, n = 401) that systematically demonstrate that vertical scaling can lead to misinterpretations of PDFs. We also test visual interventions to mitigate misinterpretation. In some contexts, we find including a y-axis can help reduce this effect. Overall, we find that keeping vertical scaling consistent, and therefore maintaining equal pixel areas under PDF curves, results in the highest likelihood of accurate comparisons. Our findings provide insights into the impact of vertical scaling on PDFs, and reveal the complicated nature of proportional area comparisons.
Groundwater storage in aquifers has become a vital water source due to water scarcity in recent years. However, aquifer systems are full of uncertainties, which inevitably propagate throughout the modeling computation...
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Groundwater storage in aquifers has become a vital water source due to water scarcity in recent years. However, aquifer systems are full of uncertainties, which inevitably propagate throughout the modeling computations, mainly reducing the reliability of the model output. The fundamental science problem this study addresses is the development of a two-dimensional stochastic confined groundwater flow model, which determines the time-- space evolution of the ensemble mean and ensemble variance of the flow field over a model domain under uncertain parameters and uncertain sink/source conditions. This is achieved by linking the stochastic partial differential equation that governs the confined aquifer flow to a non-local Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE). In the form of the LEFPE, the resulting deterministic governing equation describes the spatio-temporal evolution of the probability density function of the state variables in the confined groundwater flow process by one single numerical realization instead of requiring thousands of simulations in the Monte Carlo approach. As will be shown in the paper's text, the time-space evolving ensemble mean and ensemble variance of the flow process are then obtained from the pdf of the state variable (hydraulic head) of the process that is determined from the solution of the LEFPE of the process under specified initial and boundary conditions. As such, in the developed methodology, no assumption is made on the distribution of the time-space varying pdf of the flow process, which is obtained from the solution of the LEFPE of the process under specified initial and boundary conditions. Consequently, the groundwater flow process's mean and standard deviation behavior can be modeled under uncertainty in the transmissivity field and recharge and/or pumping conditions. In addition, an appropriate numerical method for LEFPE's solution is subsequently devised. Then, its solution is presented, discussed,
The unscented Kalman filter (UKF) has demonstrated its effectiveness for state estimation in highly nonlinear systems over the past two decades. However, the UKF's assumption of a single Gaussian probability densi...
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The unscented Kalman filter (UKF) has demonstrated its effectiveness for state estimation in highly nonlinear systems over the past two decades. However, the UKF's assumption of a single Gaussian probability density function (PDF) adversely affects its filtering accuracy. To address this limitation, this article proposes a variant of the UKF, referred to as MaxUKF, which enhances estimation accuracy by incorporating the maximum point of the PDF, denoting the point getting maximum value of the PDF. First, two mathematical theorems are presented, providing direct ways to accurately predict the maximum point of the PDF of the state after nonlinear transformation. Subsequently, a Gaussian sum PDF is constructed to accurately match the mean, covariance, and maximum point of the PDF after nonlinear transformation. Then, the MaxUKF algorithm is presented in a manner analogous to the UKF, following a two-step process: time-update step and measurement-update step, during which the non-Gaussian PDF of the state is represented by the constructed Gaussian sum PDF. Finally, the effectiveness of the proposed MaxUKF is verified through a series of numerical simulations. The results demonstrate that the MaxUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time.
Controlling the volume and geometrical ratio of microdroplets offers numerous advantages, particularly in fields like pharmaceutics, photonics, soft electronics, and robotics. For microdroplets generated via microflui...
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Controlling the volume and geometrical ratio of microdroplets offers numerous advantages, particularly in fields like pharmaceutics, photonics, soft electronics, and robotics. For microdroplets generated via microfluidics, precise control of their length-to-width ( Lw) ratio enhances their overall performance and contributes to a comprehensive understanding of the results. Advanced experimental tools, including droplet generation systems, visualization techniques, digital processing algorithms, and analysis, are crucial, especially for in situ control of microdroplet size during generation. This study investigates the variation in microdroplet Lw ratio and their generation frequencies using two novel approaches: the probability density function (derived both from the Gaussian assumption and directly from experimental data distribution) and the fast Fourier transform. The configuration utilizing one syringe pump (1mp) and syringes (made entirely of plastic) demonstrated the best monodispersity in microdroplet generation with volume differences ranged from 40 +/- 2.2 nl to 65 +/- 6.7 nl, corresponding to coefficient of variation values of 5.5 % and 11 %, respectively. The analysis of similar to 7 million Lw data points revealed different types of oscillation frequencies. Specifically, a frequency of around 1 mHz originated from the syringe pump, while frequencies between similar to 0.5 and 3 Hz were attributed to the syringe materials and the interaction of a combination of syringe pumps. These results contribute to the phenomenological understanding and classification of droplet generation within the T-junction microfluidic system. Furthermore, they present an enhanced method for visualizing the geometrical variations of the generated microdroplets.
The response analysis of random vibration systems holds significant importance in comprehending intricate natural phenomena, bolstering the reliability of engineering design, and optimizing structural performance. Tra...
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The response analysis of random vibration systems holds significant importance in comprehending intricate natural phenomena, bolstering the reliability of engineering design, and optimizing structural performance. Traditional analytical approaches frequently encounter hurdles, particularly in deriving parametric expressions for the probabilitydensity of system response. This paper aims to propose a novel data-driven methodology based on the Buckingham Pi theorem to identify the parametric expression of probabilitydensity of slowly-varying processes from state data. The proposed data-driven method is carried out by two successive steps. The first step involves identifying the parametric expressions of invariants of the corresponding conservative system within the framework of the Koopman operator theory in conjunction with the Pi theorem. This process yields the expressions for slowly-varying processes. In the second step, the parametric expression of probabilitydensity of the slowly-varying processes is identified from the data of slowly-varying processes. This identification is based on the principle of maximum entropy and the Pi theorem. The data of slowly-varying processes can be calculated from the random state data with the expression identified in the first step. Sparse optimizations are carried out in both steps. The applicability and effectiveness of the proposed method are illustrated through three typical systems, including the Duffing-van der Pol system, Coulomb friction, and a two-degree-of-freedom system. The extensionality of the parametric expression for the probabilitydensity of slowly-varying processes is demonstrated. The proposed data-driven method is capable of reducing system dimensions due to the identified slowly-varying processes with lower dimensions compared to the system state. Additionally, leveraging the Pi theorem results in a smaller number of involved dimensionless groups in the identification compared to the original parameters.
The random movement of a non-stabilized camera platform during the shooting process can lead to jitter in the captured video sequences. We propose a digital image stabilization algorithm based on time-varying filter-b...
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The random movement of a non-stabilized camera platform during the shooting process can lead to jitter in the captured video sequences. We propose a digital image stabilization algorithm based on time-varying filter-based empirical mode decomposition (TVF-EMD) and probability density function (PDF) evaluation criterion. First, an accelerated robust feature algorithm obtains the global motion sequence. Then, TVF-EMD decomposes the global motion sequence to get the intrinsic mode functions (IMFs) and separates the intentional and jitter motion-dominated IMFs using PDFs. Finally, the deliberate motion sequence is reconstructed by summing the intentional motion-dominated IMFs. The experimental results demonstrate that, compared with existing methods, the video sequences reconstructed using the algorithm proposed in this paper exhibit a higher peak signal-to-noise ratio and structural similarity index, as well as a lower gradient magnitude similarity deviation, which proves the effectiveness of the technique.
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