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Recurrent Neural Networks (RNNs) are classical models for processing sequential data, demonstrating excellent performance in tasks such as natural language processing and time series prediction. However, during the training of RNNs, the issues of vanishing and exploding gradients often arise, significantly impacting the model’s performance and efficiency. In this paper, we investigate why RNNs are more prone to gradient problems compared to other common sequential networks. To address this issue and enhance network performance, we propose a method for gradient normalization of network weights. This method suppresses the occurrence of gradient problems by altering the statistical properties of RNN weights, thereby improving training effectiveness. Additionally, we analyze the impact of weight gradient normalization on the probability-distribution characteristics of model weights and validate the sensitivity of this method to hyperparameters such as learning rate. The experimental results demonstrate that gradient normalization enhances the stability of model training and reduces the frequency of gradient issues. On the Penn Treebank dataset, this method achieves a perplexity level of 110.89, representing an 11.48% improvement over conventional gradient descent methods. For prediction lengths of 24 and 96 on the ETTm1 dataset, Mean Absolute Error (MAE) values of 0.778 and 0.592 are attained, respectively, resulting in 3.00% and 6.77% improvement over conventional gradient descent methods. Moreover, selected subsets of the UCR dataset show an increase in accuracy ranging from 0.4% to 6.0%. The gradient normalization method enhances the ability of RNNs to learn from sequential and causal data, thereby holding significant implications for optimizing the training effectiveness of RNN-based models.

The use of precast prestressed concrete bridge piers is rapidly evolving and widely applied. Nevertheless, the probabilistic behavior of the bending performance of precast prestressed concrete bridge piers has often been overlooked. This study aims to address this issue by utilizing actual precast bridge piers as the engineering context. Through the implementation of the Monte-Carlo simulation and Gradient Boosted Regression Trees (GBRT) algorithm, the stochastic distribution of the bending performance and their critical factors are identified. The results show that the normal distribution is the most suitable for the random distribution of bending performance indicators. The variability of the elastic modulus of ordinary steel bars, initial strain of prestressed steel hinge wires, and constant load axial force has little effect on the bending moment performance, while the yield stress of ordinary steel bars, elastic modulus of concrete, compressive strength of unrestrained concrete, and elastic modulus of prestressed steel hinge wires have a greater impact on the bending performance. Additionally, the compressive strength of unrestrained concrete has a significant influence on the equivalent bending moment of the cross-section that concerns designers.

This chapter contains sections titled: Observation of data patterns Skewness and mode tests to identify five types of probability distributions Procedure for discovering probability distribution change data characteristics for attacks Distribution change attack characteristics Summary References

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.

This book uses the EM (expectation maximization) algorithm to simultaneously estimate the missing data and unknown parameter(s) associated with a data set.The parameters describe the component distributions of the mixture; the distributions may be continuous or discrete.

Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff–Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff–Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

We investigate the consumption consequences of the 2006–9 housing collapse using the highly unequal geographic distribution of wealth losses across the United States. We estimate a large elasticity of consumption with respect to housing net worth of 0.6 to 0.8, which soundly rejects the hypothesis of full consumption risk-sharing. The average marginal propensity to consume (MPC) out of housing wealth is 5–7 cents with substantial heterogeneity across ZIP codes. ZIP codes with poorer and more levered households have a significantly higher MPC out of housing wealth. In line with the MPC result, ZIP codes experiencing larger wealth losses, particularly those with poorer and more levered households, experience a larger reduction in credit limits, refinancing likelihood, and credit scores. Our findings highlight the role of debt and the geographic distribution of wealth shocks in explaining the large and unequal decline in consumption from 2006 to 2009.

Multivariate analysis of flood frequency was used extensively in water resources research. Often the only flood peak or volume is analyzed with statistical distributions, but for a perfect and exact result, the four main characteristics of a flood event, as well as peak, volume, duration, and time-to-peak, are needed. For this reason, multivariate statistical approaches like copula functions developed. This research aims to define and use the bivariate copula (2-copula) probability distribution functions (PDF) for flood characteristics multivariate analysis. When the joint distribution of characteristics such as volume and peak is known, it is possible to define the probability of simultaneous occurrence of design volume and peak flow values.

This study introduces the Wrapped Epanechnikov Exponential Distribution (WEED), a novel circular distribution derived from the Epanechnikov exponential distribution. The probability density function and cumulative distribution function are presented, together with a comprehensive analysis of its properties and parameters, including the characteristic function and trigonometric moments. Parameters are estimated using maximum likelihood estimation (MLE). A simulation study with 10,000 samples demonstrates the consistency of the MLE method, with bias decreasing from 0.14221 to 0.03203 and MSE improving from 0.03456 to 0.00163 for β = 1 as sample size increases from N  = 30 to N  = 500. Applications to real-world datasets confirm WEED’s superior flexibility compared to established models, achieving lower AIC values across multiple datasets (Wind direction: 100.72 vs. 112.907; Turtle orientation: 142.764 vs. 145.254; Fisher-B5: 77.6998 vs. 79.833) when compared with the Wrapped Exponential Distribution (WED). Kolmogorov–Smirnov tests further support WEED’s improved goodness-of-fit, with consistently lower test statistics across all datasets. This work contributes to the field of circular statistics by providing a promising tool for modeling asymmetric circular data with enhanced flexibility and accuracy.

The class of infinitely divisible exponential distribution is determined by property of Levy measure from its canonical representation of characteristic functions. Levy measure of the exponential distribution is obtained belonging to a class of completely monotones and measurable function. The class of this Levy measure is governed from exponential distribution similarly to the class of Thorin.