Optimal Probability Distribution Research Articles

Probabilistic unitary synthesis with optimal accuracy

The purpose of unitary synthesis is to find a gate sequence that optimally approximates a target unitary transformation. A new synthesis approach, called probabilistic synthesis, has been introduced, and its superiority has been demonstrated over traditional deterministic approaches with respect to approximation error and gate length. However, the optimality of current probabilistic synthesis algorithms is unknown. We obtain the tight lower bound on the approximation error obtained by the optimal probabilistic synthesis, which guarantees the sub-optimality of current algorithms. We also show its tight upper bound, which improves and unifies current upper bounds depending on the class of target unitaries. These two bounds reveal the fundamental relationship of approximation error between probabilistic approximation and deterministic approximation of unitary transformations. From a computational point of view, we show that the optimal probability distribution can be computed by the semidefinite program (SDP) we construct. We also construct an efficient probabilistic synthesis algorithm for single-qubit unitaries, rigorously estimate its time complexity, and show that it reduces the approximation error quadratically compared with deterministic algorithms.

Analyzing river disruption factors and ecological flow in China's Liu River Basin amid environmental changes.

Water resources variability and availability in a basin affect river flows and sustain river ecosystems. Climate change and human activities disrupt runoff sequences, causing water environmental issues like river channel interruptions. Therefore, determining ecological flow in changing environments is challenging in hydrological research. Based on an analysis of long-term changes in hydrological and meteorological variables and interruption conditions in the semi-arid Liu River Basin (LRB), this study summarizes the controlling factors of river interruption at different temporal and spatial scales and proposes a framework to determine ecological flow under changing environments. Hydrological model and the monthly optimal probability distribution were used to determine the optimal ecological runoff of LRB. The results showed that from 1956 to 2017, precipitation and potential evapotranspiration in the basin showed no significant decreasing trend, but the streamflow significantly decreased, and the downstream interruption worsened, with an average annual interruption duration of 194days at Xinmin Station from 1988 to 2017. The controlling factors of river interruption are as follows: (1) soil and water conservation measures in the upstream significantly reduce the runoff capacity; (2) the operation mode of the controlling reservoir in the middle reaches changes from "all-year discharge" to "winter storage and spring release" to "combined storage and supply," severing the hydraulic connection between upstream and downstream; and (3) siltation in the downstream river channel coupled with over-extraction of groundwater increases the seepage capacity of the river. The monthly ecological flow of Naodehai Reservoir was determined by considering the monthly seepage losses after reconstructing the natural runoff using the SWAT model and determining the optimal probability distribution function for monthly runoff. The findings are important for downstream LRB ecological restoration and for determining the ecological flow of other river basins in changing environments.

Analysis of power dispatching decisions with energy storage systems using the optimal probability distribution model of renewable energy

Effectively managing the inherent unpredictability and fluctuation attributes of renewable energy sources within scheduling systems has emerged as a critical matter demanding immediate attention. The incorporation of energy storage technology offers notable advantages by mitigating fluctuations in wind power generation and curtailing peak shaving costs in scheduling systems through economical utilization of energy storage. This study employs the adjustment of energy storage’s reserve capacity function to modify the economic gains and losses arising from fixed spinning reserve. This approach tailors the reserve capacity function by formulating the uncertainty inherent in wind power generation. The paper presents an enhanced selection method for calculating optimal bandwidth within the framework of least squares approximation probability density function scenarios. The fusion of least squares approximation and optimal bandwidth computation permits precise refinement of bandwidth in non-parametric kernel density estimation approaches. Consequently, the study puts forth a comprehensive model for coordinated scheduling and regulation of a multi-scale energy storage power system. Finally, the stability and economy of the proposed method in the scheduling operation process were verified through simulation, and the research results can provide a theoretical basis for future uncertain wind power scheduling.

Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather

In recent years, with the intensification of global warming, extreme weather has become more frequent, intensifying the uncertainty of new energy output and load power, and seriously affecting the safe operation of power systems. Scene generation is an effective method to solve the uncertainty problem of stochastic planning of integrated systems of new energy generation. Therefore, this paper proposes a scenario generation and scenario reduction model of photovoltaic (PV) output and electric vehicle (EV) load power under extreme weather based on the copula function. Firstly, the non-parametric kernel density estimation method is used to fit a large number of sample data. The kernel density estimation expressions of PV and EV powers under extreme weather conditions are obtained and the corresponding goodness of fit tests are carried out. Then, a variety of joint distribution models based on the copula function are established to judge the goodness of fit of each model, and the optimal copula function is selected as the joint probability distribution function by combining the Kendall and Spearman correlation coefficients of each model. Finally, the optimal copula joint probability distribution is used to generate PV and EV power scenarios. The data of extremely hot weather in a certain province were selected for an example analysis. The results show that the output scenario obtained conforms to the correlation under this extreme weather, and has higher accuracy in reflecting the actual PV output and load power in this province under this extreme weather, which can provide a reference for reliability analyses of power systems and power grid planning.

Environmental parameter design for jack-up platforms based on FEM-BPNN-JPA coupled approach

This study proposes a novel coupled approach, integrating the non-linear finite element model (FEM), BP neural network (BPNN), and joint probability analysis (JPA), to conduct the ocean environmental parameter design for jack-up platforms. Two bottom constraint methods are involved in the FEM, namely the contact surface model and the hinge constraint model, to conduct the structural response analysis. Subsequently, based on the calculated structural responses from FEM, the BPNN is employed to predict the jack-up response under varying combinations of wind speed, wave height and current velocity. This FEM-BPNN coupled method proves to be an effective and efficient tool for determining the jack-up responses under different environmental load combinations for JPA. Based on the optimal univariate probability distributions of each environmental parameter, multi-dimensional copulas are adopted to construct the conditional and joint probability distributions. These distributions, in turn, yield environmental design parameters under different design criteria and corresponding horizontal displacements of the jack-up platform. Results show that environmental design parameters and corresponding horizontal displacements are smaller compared to those obtained from the univariate probability distribution for the same return period. The FEM-BPNN-JPA coupled approach proves to be effective in estimating environmental design parameters by considering parameter correlations.

Bayesian generative adversarial network emulator based end-to-end learning strategy of the probabilistic shaping for OAM mode division multiplexing IM/DD transmission.

Orbital angular momentum (OAM) mode division multiplexing (MDM) has emerged as a new multiplexing technology that can significantly increase transmission capacity. In addition, probabilistic shaping (PS) is a well-established technique that can increase the transmission capacity of an optical fiber to close to the Shannon limit. However, both the mode coupling and the nonlinear impairment lead to a considerable gap between the OAM-MDM channel and the conventional additive white Gaussian noise (AWGN) channel, meaning that existing PS technology is not suitable for an OAM-MDM intensity-modulation direct-detection (IM-DD) system. In this paper, we propose a Bayesian generative adversarial network (BGAN) emulator based on an end-to-end (E2E) learning strategy with probabilistic shaping (PS) for an OAM-MDM IM/DD transmission with two modes. The weights and biases of the BGAN emulator are treated as a probability distribution, which can be accurately matched to the stochastic nonlinear model of OAM-MDM. Furthermore, a BGAN emulator based on an E2E learning strategy is proposed to find the optimal probability distribution of PS for an OAM-MDM IM/DD system. An experiment was conducted on a 200 Gbit/s two OAM modes carrierless amplitude phase-32(CAP-32) signal over a 5 km ring-core fiber transmission, and the results showed that the proposed BGAN emulator outperformed a conventional CGAN emulator, with improvements in modelling accuracy of 29.3% and 26.3% for the two OAM modes, respectively. Moreover, the generalized mutual information (GMI) of the proposed E2E learning strategy outperformed the conventional MB distribution and the CGAN emulator by 0.31 and 0.33 bits/symbol and 0.16 and 0.2 bits/symbol for the two OAM modes, respectively. Our experimental results demonstrate that the proposed E2E learning strategy with the BGAN emulator is a promising candidate for OAM-MDM IM/DD optic fiber communication.

Multi-Scale Ensemble Booster for Improving Existing TSD Classifiers

Time Series Classification (TSC) is an essential task in Time Series Data (TSD) analysis. Ensemble-based approaches now achieve the best performance on TSC tasks. However, integrating numerous different models makes them highly suffer from heavy preprocessing. Even worse, non-deep-learning ensemble-based methods suffer from substantial computational costs due to lacking GPU acceleration. Multi-scale information in TSD can improve TSC performance. However, Existing TSD classifiers employing multi-scale information struggle with heavy preprocessing and cannot help other TSD classifiers obtain multi-scale feature extraction capabilities. Inspired by these, we proposed a performance enhancement framework called multi-scale ensemble booster (MEB), helping existing TSD classifiers achieve performance leaps. In MEB, we proposed an easy-to-combine network structure without changing any of their structure and hyperparameters, only needed to set one hyperparameter, consisting of multi-scale transformation and multi-output decision fusion. Then, a probability distribution co-evolution strategy is proposed to attain the optimal label probability distribution. We conducted numerous ablation experiments of MEB on 128 univariate datasets and 29 multivariate datasets and comparative experiments with 11 state-of-the-art methods, which demonstrated the significant performance improvement ability of MEB and the most advanced performance of the model enhanced by MEB, respectively. Furthermore, to figure out why MEB can improve model performance, we provided a chain of interpretability analyses.

Optimal Probability Distribution and Applicable Minimum Time-Scale for Daily Standardized Precipitation Index Time Series in South Korea

The Standardized Precipitation Index (SPI) is a standardized measure of the variability of precipitation and is widely used for drought assessment around the world. In general, the probability distribution used to calculate the SPI in many studies is Gamma. In addition, a monthly time-scale is applied to calculate the SPI to assess drought based on atmospheric moisture supply over the medium-to-long term. However, probability distributions other than Gamma are applied in various regions, and the need for a daily time-scale is emerging as concerns about fresh drought increase. There are two main innovations of our work. The first is that we investigate the optimal probability distribution of daily SPIs rather than monthly SPIs, and the second is that we address the issue of determining the minimum time-scale that can be applied when applying a daily time-scale. In this study, we investigate the optimal probability distribution and the minimum-applicable time-scale for calculating the daily SPI using daily precipitation time series observed over 42 years at 56 sites in South Korea. Six candidate probability distributions (Gumbel, Gamma, GEV, Log-logistic, Log-normal, and Weibull) and ten time-scales (5 day, 10 day, 15 day, 21 day, 30 day, 60 day, 90 day, 180 day, 270 day, and 365 day) were applied to calculate the daily SPI. A chi-square test and AIC were applied to investigate the appropriate probability distribution for each time-scale, and the normality of the daily SPI time series derived from each probability distribution were compared. The Weibull distribution was suitable for calculating the daily SPI for short time-scales of 30 days or less, while the GEV distribution was suitable for longer time-scales of 270 days or more. However, overall, Gamma was found to be the best probability distribution. While there were some regional variations, the minimum time-scales that could be applied per season were as follows: 15 days for spring and summer, 21 days for fall, and 30 days for winter. It is shown that the minimum time-scale depends on how many zero values are included in the moving cumulative-precipitation time series, and it is shown that it is appropriate to have less than about 2.5%. Finally, the applicability of the GEV distribution is investigated.

The Remaining Useful Life Prediction Method of a Hydraulic Pump under Unknown Degradation Model with Limited Data.

This study proposes a remaining useful life (RUL) prediction method using limited degradation data with an unknown degradation model for hydraulic pumps with long service lives and no failure data in turbine control systems. The volumetric efficiency is calculated based on real-time monitoring signal data, and it is used as the degradation indicator. The optimal degradation curve is established using the degradation trajectory model, and the optimal probability distribution model is selected via the K-S test. The above process was repeated to optimize the degradation model and update parameters in different performance degradation stages of the hydraulic pump, providing quantification of the prediction uncertainty and enabling accurate online prediction of the hydraulic pump's RUL. Finally, an RUL test bench for hydraulic pumps is built for verification. The results show that the proposed method is convenient, efficient, and has low model complexity. The method enables online accurate prediction of the RUL of hydraulic pumps using only limited degradation data, with a prediction accuracy of over 85%, which meets practical application requirements.

Uncertainty quantification for granular materials with a stochastic discrete element method

Significant uncertainty exists in granular materials, as demonstrated by experimental and simulation studies. Quantifying this uncertainty by integrating refined discrete element analysis with direct stochastic simulation is challenging due to computational cost constraints. To this end, the probability density evolution method (PDEM) is introduced to propose the methodology of stochastic discrete element analysis. The objective of this paper is to develop a formulation of the stochastic discrete element method (DEM) on uncertainty quantification for granular materials. First, the uncertainty characterization of inter-particle friction coefficients (μ) is performed via experimental data, resulting in the identification of an optimal probability distribution. Second, the 100 representative deterministic points of basic random variables are selected for subsequent DEM shearing simulation until the critical state. Third, the PDEM is developed to obtain the probability density function (PDF) surface representing the system’s performance, thus offering a comprehensive prediction of uncertainty for granular materials. The results show that the uncertainty of μ propagates to the behavior of granular materials. Utilizing the stochastic DEM methodology clarifies which behavior parameters exhibit significant randomness and which possess negligible randomness. Notable uncertainties are observed in large strain macro-scale behavior and the proportion of inactive particles. It can be inferred that conventional methods that neglect probability distributions might produce spurious results. Moreover, the influence of uncertainties on granular material behavior varies depending on the its shearing state and related soil parameters. In contrast, limited uncertainties are identified for the randomness of small strain stiffness, and the randomness on the particle scale stress transmission shape is also limited. This study marks the initial application of the PDEM in geotechnical engineering, suggesting that considering probability density evolution could enhance geotechnical engineering design. The current study highlights the necessity of using statistical methods to assess key geotechnical parameters.

Determining optimal probability distributions for gridded precipitation data based on L-moments

Understanding the probability distributions of precipitation is crucial for predicting climatic events and constructing hydraulic facilities. To overcome the inadequacy of precipitation data, regional frequency analysis was commonly used by “trading space for time”. However, with the increasing availability of gridded precipitation datasets with high spatial and temporal resolutions, the probability distributions of precipitation for these datasets have been less explored. We used L-moments and goodness-of-fit criteria to identify the probability distributions of annual, seasonal, and monthly precipitation for a 0.5° × 0.5° dataset across the Loess Plateau (LP). We examined five 3-parameter distributions, namely General Extreme Value (GEV), Generalized Logistic (GLO), Generalized Pareto (GPA), Generalized Normal (GNO), and Pearson type III (PE3), and evaluated the accuracy of estimated rainfall using the leave-one-out method. We also presented pixel-wise fit-parameters and quantiles of precipitation as supplements. Our findings indicated that precipitation probability distributions vary by location and time scale, and the fitted probability distribution functions are reliable for estimating precipitation under various return periods. Specifically, for annual precipitation, GLO was prevalent in humid and semi-humid areas, GEV in semi-arid and arid areas, and PE3 in cold-arid areas. For seasonal precipitation, spring precipitation mainly conforms to GLO distribution, summer precipitation around the 400 mm isohyet prevalently follows GEV distribution, autumn precipitation primarily meets GPA and PE3 distributions, and winter precipitation in the northwest, south, and east of the LP mainly conforms to GPA, PE3 and GEV distributions, respectively. Regarding monthly precipitation, the common distribution functions are PE3 and GPA for the less-precipitation months, whereas the distribution functions of precipitation for more-precipitation months vary substantially across different regions of the LP. Our study contributes to a better understanding of precipitation probability distributions in the LP and provides insights for future studies on gridded precipitation datasets using robust statistical methods.

A robust reliability evaluation model with sequential acceleration method for power systems considering renewable energy temporal-spatial correlation

As the penetration of renewable energy increases, it becomes critical to evaluate the power system reliability, while considering the correlations among renewable energy sources (RESs) and their uncertainties. In this paper, the robust reliability evaluation model based on the minimum volume enclosing ellipsoid (MVEE) algorithm is established, and the sequential acceleration method is proposed to improve the convergence. First, the robust reliability evaluation model is built, where the robust state analysis is performed under each sampling. Second, the sequential Cross-Entropy-Latin Hypercube Sampling (CE-LHS) acceleration method is presented, which first obtains the optimal probability distribution of system parameters, and then forces the sampling values to be evenly distributed to overcome the truncated tail effect. Correspondingly, the reliability indices are improved. Third, for state analysis, the robust multi-period optimal load shedding (RMPOLS) model is established, where the dynamic performance of energy storage and system ramping rate constraints are described in detail. Therefore, the periods are strongly coupled, and the impact of RES uncertainties is carefully considered. Besides, to address the temporal-spatial correlation of RESs, the MVEE algorithm is introduced to generate the convex hull of RESs scenarios. Therefore, the RMPOLS model can be transformed into a second-order cone form. Finally, to further reduce the computational complexity, the branch flow constraints scanning strategy is proposed, which can quickly remove inactive constraints in advance. Numerical results prove that the MVEE-based robust reliability evaluation model considers the temporal-spatial correlation of RES and thus improves the system reliability by almost 30 %. Besides, branch flow constraint scanning can eliminate more than 65 % of the SOC constraints. Moreover, CE-LHS can improve the convergence by about 8 times compared with Monte Carlo Sampling (MCS) while the error is less than 1 %.

Temporal analysis and the joint probability distribution of rainfall indexes in the Taihu Lake Basin, China

Taking the Taihu Lake Basin as an example, in this study, the characteristics of the rainfall factors in the study area were analyzed using daily rainfall data from 1955 to 2019. Three factors, i.e., the contribution rate of the rainfall in the flood season, the rainfall frequency, and the maximum daily rainfall, were selected to determine the optimal probability distribution function for each single factor. Furthermore, the root mean square error goodness-of-fit test was used to determine the optimal copula function for the three-dimensional joint probability distribution characteristics of the rainfall factors. The research results show that the three-dimensional copula joint probability method contains much more information than the results of the single variable probability calculation. The copula function can be used to analyze the multi-dimensional joint distribution of rainfall factors, which can fill the gap in research on multiple rainfall factors.

Stochastic resonance in Bayesian estimation and CRLB for nonlinear system

In this work, noise enhanced parameter estimation problems are investigated for a general nonlinear system, where an additive noise is added to the nonlinear system input and a Bayesian estimator is developed based on the noise modified output. The optimal probability distribution of the additive noises is formulated successively for minimizing the mean square error (MSE) of the optimal Bayesian estimation and the Cramer–Rao lower bound (CRLB). Then the optimal additive noises for the two different noise enhanced optimization problems are explicitly derived as constant vectors, which implies the randomization of constant vectors is not beneficial to these optimizations. Finally, numerical results are presented to illustrate the theoretical results.

Stochastic models of tropical cyclone genesis in the Philippine Sea

Climate change has influenced the anomalies of extreme weather events, including tropical cyclones (TCs), strong wind, flood and landslide. The investigation of TC genesis is of great significance in evaluating TC activity in order to mitigate TC – induced risks. This study focuses on developing the stochastic models to simulate TC geneses over the Philippine Sea, considering seasonal effects. The historical TC data, including TC number and genesis positions, were obtained from the Japanese Meteorology Agency (JMA) over the period 1951 – 2020. To select the optimal probability distributions to fit the observed TC data, the good-of-fit tests were performed throughout three metrics, namely Mean Absolute Error; Normalized Mean Squared Error and Correlation Distance. Moreover, the Chi-square test and the Kolmogorov – Smirnov test were utilized to evaluate the homogeneity of the selected probability distributions to the observed TC data. The statistical analysis on TC characteristics presented that most TC geneses occurred in the latitudinal belt 8°N - 30°N from July to October, called the active season (AS). Meanwhile, there was a low TC genesis frequency distributed in the region south of 15°N from November to June, called the inactive TC season (IAS). The stochastic models showed the distributions of the observed TC number followed Negative binomial (NB) distribution for the two seasons. Generalized extreme value (Gev) distribution fitted TC genesis longitude well for the inactive season and fitted TC genesis latitude well for the active season. In the active season, the kernel distribution was appropriate for genesis longitude, whereas in the inactive season, it was suitable for genesis latitude. Monte Carlo simulations for 1000 years indicated the similarities in the geographical distributions of the simulated TC geneses and the historical ones for the inactive and active seasons. Data Availability StatementThe Best Track data can be accessed from the Japanese Meteorology Agency. Data of the Philippine Sea boundary can be accessed from Marine Regions, governed by the Flanders Marine Institute.

GSA-Fuzz: Optimize Seed Mutation with Gravitational Search Algorithm

Mutation-based fuzzing is currently one of the most effective techniques to discover software vulnerabilities. It relies on mutation strategies to generate interesting seeds. As a state-of-the-art mutation-based fuzzer, AFL follows a mutation strategy with high randomization, which uses randomly selected mutation operators to mutate seeds at random offsets. Its strategy may ignore some efficient mutation operators and mutation positions. Therefore, in this paper, we propose a solution named GSA-Fuzz to improve the efficiency of seed mutation strategy with the gravitational search algorithm (GSA). GSA-Fuzz uses GSA to learn the optimal selection probability distributions of operators and mutation positions and designs a position-sensitive strategy to guide seed mutation with learned distributions. Besides, GSA-Fuzz also provides a flip mode to calculate the efficiencies of the deterministic stage and indeterministic stage and implements switching between the two stages to further improve the efficiency of seed mutation. We compare GSA-Fuzz with the state-of-the-art fuzzers AFL, MOPT-AFL, and EcoFuzz on 10 open-source programs. GSA-Fuzz finds 145% more paths than AFL, 66% more paths than EcoFuzz, and 43% more paths than MOPT-AFL. In addition, GSA-Fuzz also outperforms other fuzzers in bug detection and line coverage.

Optimal Discrete Constellation Inputs for Aggregated LiFi-WiFi Networks

In this paper, we investigate the performance of a practical aggregated LiFi-WiFi system with the discrete constellation inputs from a practical view. We derive the achievable rate expressions of the aggregated LiFi-WiFi system for the first time. Then, we study the rate maximization problem via optimizing the constellation distribution and power allocation jointly. Specifically, a multilevel mercy-filling power allocation scheme is proposed by exploiting the relationship between the mutual information and minimum mean-squared error (MMSE) of discrete inputs. Meanwhile, an inexact gradient descent method is proposed for obtaining the optimal probability distributions. To strike a balance between the computational complexity and the transmission performance, we further develop a framework that maximizes the lower bound of the achievable rate where the optimal power allocation can be obtained in closed forms and the constellation distributions problem can be solved efficiently by Frank-Wolfe method. Extensive numerical results show that the optimized strategies are able to provide significant gains over the state-of-the-art schemes in terms of the achievable rate.

Accelerated failure time models in analyzing duration of employment

Parametric accelerated failure time models (AFT), that assess the relationship between the event times and dependent variables constitute an essential class of regression models. In this research we have used the accelerated failure time models: in particular, we have taken into consideration exponential distribution, Weibull distribution, log-logistic model and lognormal distribution, to analyze the duration of employment for employees that have been in their current job or otherwise known as job tenure. A variety of commercial and public companies in Albania were surveyed for this study. Initially the goal of the research is to find the optimal probability distribution to suit the data. Then, the accelerated failure time model is used to assess the impact of the predictors, such as: the employee's age, wages, the employee's age when he has started the job, sex, profession, academic degree, marital status and years of experience prior to this position, in job termination. The log-normal AFT model was the most accurate model for predicting the outcomes of the study, and from this model we can conclude that the employee's age, the employee's age when he has started the job, wages, academic degree and position in the company can affect how long someone stays employed.

Statistical Characteristics of Recycled Aggregate Permeable Concrete

In view of the randomness of mechanical properties and durability of recycled aggregate permeable concrete, a method combining the goodness of fit test with Bayes theory was proposed to analyze its statistical characteristics probability model. Firstly, the optimal probability distribution type of recycled aggregate permeable concrete strength was identified by goodness of fit method. Then, Bayes method was used to estimate the parameters of probability model. Finally, the reliability theory was used to evaluate the durability of recycled aggregate permeable concrete. This method has good theoretical significance and practical value for reasonable evaluation of mechanical properties and durability of recycled aggregate permeable concrete.

Asymptotic behavior of free energy when optimal probability distribution is not unique

Bayesian inference is a widely used statistical method. The free energy and the generalization loss, which are used to estimate the accuracy of Bayesian inference, are known to be small in singular models that do not have a unique optimal parameter. However, their characteristics have not yet been clarified when there are multiple optimal probability distributions. In this paper, we theoretically derive the asymptotic behaviors of the generalization loss and the free energy in the case that the optimal probability distributions are not unique and show that they contain asymptotically different terms from those of the conventional asymptotic analysis.