Proper Probability Distribution Research Articles

A computational probabilistic procedure to quantify the time of breast cancer recurrence after chemotherapy administration

We propose a random linear differential equation with jumps to model the dynamics of breast tumor growth using real patients’ data. The model considers the effect of chemotherapy administration and tumor resection. Also, the inherent randomness of the data and the model are taken into account.The model is probabilistically solved by combining two main methods belonging to Uncertainty Quantification: the principle of maximum entropy (PME) and the random variable transformation technique (RVT). The PME is applied to assign proper probability distributions to model parameters. The RVT technique is used to determine the probability density function (PDF) of the solution of the random differential equation, which is a stochastic process.We apply computational optimization techniques to determine the PDF of model parameters so that the PDF of the solution matches the ones assigned to real patients’ data.Once random tumor dynamics has been modeled, we simulate, after surgery, different adjuvant chemotherapy strategies with the aim of delaying tumor recurrence as long as possible. Results are in concordance with medical literature, and tumor relapse is delayed as more cycles of chemotherapy are administered.Although the proposed model has a common general structure, it is shown how it can be customized to patients in order to construct projections about the tumor’s growth. To better illustrate the applicability of the proposed approach, we carefully show how our model can be tailored to two real patients treated at the Hospital Clínico de Valencia (Spain). The obtained results have been overseen by doctors from this hospital.

Exponentially Generated Modified Chen Distribution with Applications to Lifetime Dataset

In this paper, the exponentially generated system was used to modify a two-parameter Chen distribution to a four-parameter distribution with better performance. The property of complete probability distribution function was used to verify the completeness of the resulting distribution, which shows that the distribution is a proper probability distribution function. A simulation study involving varying sample sizes was used to ascertain the asymptotic property of the new distribution. Small and large sample sizes were considered which shows the closeness of the estimates to the true value as the sample size increases. Lifetime dataset were used for model comparison which shows the superiority of exponentially generated modify Chen distribution over some existing distributions. It is therefore recommended to use the four-parameter Chen distribution in place of the well-known two-parameter Chen distribution.

Downslope Windstorm Forecasting: Easier with a Critical Level, but Still Challenging for High-Resolution Ensembles

Abstract Strong downslope windstorms can cause extensive property damage and extreme wildfire spread, so their accurate prediction is important. Although some early studies suggested high predictability for downslope windstorms, more recent analyses have found limited predictability for such winds. Nevertheless, there is a theoretical basis for expecting higher downslope wind predictability in cases with a mean-state critical level, and this is supported by one previous effort to forecast actual events. To more thoroughly investigate downslope windstorm predictability, we compare archived simulations from the NCAR ensemble, a 10-member mesoscale ensemble run at 3-km horizontal grid spacing over the entire contiguous United States, to observed events at 15 stations in the western United States susceptible to strong downslope winds. We assess predictability in three contexts: the average ensemble spread, which provides an estimate of potential predictability; a forecast evaluation based upon binary-decision criteria, which is representative of operational hazard warnings; and a probabilistic forecast evaluation using the continuous ranked probability score (CRPS), which is a measure of an ensemble’s ability to generate the proper probability distribution for the events under consideration. We do find better predictive skill for the mean-state critical-level regime in comparison to other downslope windstorm–generating mechanisms. Our downslope windstorm warning performance, calculated using binary-decision criteria from the bias-corrected ensemble forecasts, performed slightly worse for no-critical-level events, and slightly better for critical-level events, than National Weather Service high-wind warnings aggregated over all types of high-wind events throughout the United States and annually averaged for each year between 2008 and 2019.

A micro-mechanically-based constitutive model for hyperelastic rubber-like materials considering the topological constraints

Hyperelastic rubber-like materials are described by their nonlinear elastic behavior subjected to large deformations. The one-dimensional strain energy function for polymeric material was introduced in Mahnken and Mirzapour (2022). This follow-up contribution presents a three-dimensional statistical micro-mechanical-based constitutive model incorporating the topological constraint’s effects. The micro strain energy corresponding to the stored energy in a single polymer chain is mainly entropic and calculated from the entropic part of the Helmholtz free energy based on the Boltzmann equation. Then, the macro strain energy corresponding to the polymer networks is obtained by homogenizing the micro strain energy over the unit volume. At the same time, it can be decomposed into cross-linked and entangled strain energies. The current model is developed on the different angles of the polymer chain. The random nature of the polymer chain’s angles is defined by proper probability distribution functions (PDFs) representing different angles’ distribution. I incorporate the entangled chains based on the statistical degree of freedom introduced as polymer angles. Numerical examples to calibrate and verify the proposed model illustrate the capability of the current proposed model to reproduce the highly nonlinear behavior of rubber-like materials under different loading conditions.

An ILP model for stochastic placement of [formula omitted]PMUs with limited voltage and current channels in a reconfigurable distribution network

Micro phasor measurement units (μPMUs) play a major role in the evolution of distribution networks toward the smart grids by providing precise and synchronized measurements. In this study, an integer linear programming (ILP) model for stochastic optimal μPMU placement (SOMPP) problem is formulated in order that the network would remain observable following a network reconfiguration. Furthermore, the current measurement channels are considered limited and it is also extended to voltage measurement channels, in order to be employed in cases that measurement channels are insufficient (e.g. three-phase networks). The uncertain quantities including the variability of load demands and the unreliability of lines are characterize by proper probability distribution functions (PDFs) instead of constant values, and employed in a Monte Carlo simulation (MCS) process. The most probable scenarios are obtained by a scenario reduction method, and the most probable topologies within each scenario are identified by multiple execution of reconfiguration for the reduced scenarios using the genetic algorithm (GA). Then the SOMPP model is solved for each reduced scenario using the concept of equivalent network that is also introduced in present work. A formulation for the deterministic form of the problem is also developed in order to compare the results of the SOMPP model with the deterministic one, by which the observability of all possible topologies is guaranteed with a limited number of channels. Furthermore, using the concept of voltage channels limit, the formulation of the SOMPP model is extended to include the unbalanced networks. The simulations are performed on 33-bus and 240-bus distribution networks. The numerical results revealed the relative superiority of the SOMPP model over the deterministic version and previous works in terms of decreasing the required number of μPMUs. Moreover, the effectiveness of the model in handling the random variables with unknown PDFs is also determined. The success of the reduced scenarios in representing the most general scenarios is also demonstrated by evaluating a large number of scenarios. The validation of the concept of equivalent network is also examined for multiple simulation cases.

A stochastic simulation-based risk assessment method for water allocation under uncertainty

Abstract There are a lot of uncertainties in the water resources system, which makes the water allocation plan very risky. In order to analyze the risks of water resources allocation under uncertain conditions, a new methodology called the stochastic simulation-based risk assessment approach is developed in this paper. First, the main hydrological stochastic variable is fitted by a proper probability distribution. Second, suitable two-stage stochastic programming is constructed to obtain the expected benefit and optimized water allocation targets. Third, the Monte Carlo method is used to obtain a suitable stochastic sample of the hydrological variable. Fourth, a pre-allocated water optimization model is proposed to obtain optimized actual benefit. The methodology can give a way for risk analysis of water allocation plans obtained by uncertain optimization models, which provides reliable assistance to water managers in decision-making. The proposed methodology is applied to the Zhanghe Irrigation District and the risk of the water allocation plan obtained under the randomness of annual inflow is assessed. In addition, three different division methods of the annual inflow are applied in the first step, namely three levels, five levels and seven levels, respectively. From the results, the risk of the water allocation scheme obtained by the TSP model is 0.372–0.411 and decreases with the increase of the number of hydrological levels. Considering both the risk and model complexity, seven hydrological levels are recommended when using the TSP model to optimize water allocation under stochastic uncertainty.

Assignment of unimodal probability distribution models for quantitative morphological phenotyping

BackgroundCell morphology is a complex and integrative readout, and therefore, an attractive measurement for assessing the effects of genetic and chemical perturbations to cells. Microscopic images provide rich information on cell morphology; therefore, subjective morphological features are frequently extracted from digital images. However, measured datasets are fundamentally noisy; thus, estimation of the true values is an ultimate goal in quantitative morphological phenotyping. Ideal image analyses require precision, such as proper probability distribution analyses to detect subtle morphological changes, recall to minimize artifacts due to experimental error, and reproducibility to confirm the results.ResultsHere, we present UNIMO (UNImodal MOrphological data), a reliable pipeline for precise detection of subtle morphological changes by assigning unimodal probability distributions to morphological features of the budding yeast cells. By defining the data type, followed by validation using the model selection method, examination of 33 probability distributions revealed nine best-fitting probability distributions. The modality of the distribution was then clarified for each morphological feature using a probabilistic mixture model. Using a reliable and detailed set of experimental log data of wild-type morphological replicates, we considered the effects of confounding factors. As a result, most of the yeast morphological parameters exhibited unimodal distributions that can be used as basic tools for powerful downstream parametric analyses. The power of the proposed pipeline was confirmed by reanalyzing morphological changes in non-essential yeast mutants and detecting 1284 more mutants with morphological defects compared with a conventional approach (Box–Cox transformation). Furthermore, the combined use of canonical correlation analysis permitted global views on the cellular network as well as new insights into possible gene functions.ConclusionsBased on statistical principles, we showed that UNIMO offers better predictions of the true values of morphological measurements. We also demonstrated how these concepts can provide biologically important information. This study draws attention to the necessity of employing a proper approach to do more with less.

Probabilistic flood hazard assessment method considering local intense precipitation at NPP sites

In recent years, the flooding risk of major facilities has increased significantly due to heavy rainfall events. These facilities must evaluate both external and internal flooding risk such as from heavy rainfall, flash floods, watershed flooding, river flooding, and coastal flooding. This study first estimates flooding from extreme rainfall with local intense precipitation and analyzes the resulting impacts on buildings and roads at a specific nuclear power plant (NPP) site, with which a roughness coefficient according to the landuse condition is estimated. A two-dimensional external flooding hazard analysis is then carried out with tidal levels as the external boundary conditions, and based on the results, new hazard curves for the inundation depth with frequency and duration are developed for the NPP site to show the relationships among rainfall, flood depth, and annual exceedance probability. To match the proper probability distribution types to the flood hazard curves, the fit was analyzed through Akaike’s information criterion verification. After analyzing the correlation between the flood depths, the mode values were calculated through Monte Carlo simulation with the verified probability distribution types. Finally, probabilistic flood hazard curves for the NPP site were obtained based on the calculated mode values. Representatively, when a 106-year return period rainfall occurs at the nuclear power plant site, the mode flood depth was found to be 1.07 m for power plant 1 and 0.61 m for power plant 2. In this way, the approach of this study is expected to support the waterproof design of critical facilities, flood prevention function design, the advancement of flood prevention measures and procedures, and the evaluation of flood mitigation functions.

FREPD: A Robust Federated Learning Framework on Variational Autoencoder

Federated learning is an ideal solution to the limitation of not preserving the users’ privacy information in edge computing. In federated learning, the cloud aggregates local model updates from the devices to generate a global model. To protect devices’ privacy, the cloud is designed to have no visibility into how these updates are generated, making detecting and defending malicious model updates a challenging task. Unlike existing works that struggle to tolerate adversarial attacks, the paper manages to exclude malicious updates from the global model’s aggregation. This paper focuses on Byzantine attack and backdoor attack in the federated learning setting. We propose a federated learning framework, which we call Federated Reconstruction Error Probability Distribution (FREPD). FREPD uses a VAE model to compute updates’ reconstruction errors. Updates with higher reconstruction errors than the average reconstruction error are deemed as malicious updates and removed. Meanwhile, we apply the Kolmogorov-Smirnov test to choose a proper probability distribution function and tune its parameters to fit the distribution of reconstruction errors from observed benign updates. We then use the distribution function to estimate the probability that an unseen reconstruction error belongs to the benign reconstruction error distribution. Based on the probability, we classify the model updates as benign or malicious. Only benign updates are used to aggregate the global model. FREPD is tested with extensive experiments on independent and identically distributed (IID) and non-IID federated benchmarks, showing a competitive performance over existing aggregation methods under Byzantine attack and backdoor attack.

On the wind resource in Algeria: Probability distributions evaluation

The present study aims to present a contribution to the wind resource assessment in Algeria using ERA-Interim reanalysis. Firstly, the ERA-Interim reanalysis 10 m wind speed data are considered for the elaboration of the mean annual 10 m wind speed map for a period starting from 01-01-2000 to 31-12-2017. Moreover, the present study intends to highlight the importance of the descriptive statistics other than the mean in wind resource assessment. On the other hand, this study aims also to select the proper probability distribution for the wind resource assessment in Algeria. Therefore, nine probability distributions were considered, namely: Weibull, Gamma, Inverse Gaussian, Log Normal, Gumbel, Generalized Extreme Value (GEV), Nakagami, Generalized Logistic and Pearson III. Furthermore, in combination with the distribution, three parameter estimation methods were considered, namely, Method of Moment, Maximum Likelihood Method and L-Moment Method. The study showed that Algeria has several wind behaviours due to the diversified topographic, geographic and climatic properties. Moreover, the annual mean 10 m wind speed map showed that the wind speed varies from 2.3 to 5.3 m/s, where 73% of the wind speeds are above 3 m/s. The map also showed that the Algerian Sahara is windiest region, while, the northern fringe envelopes the lowest wind speeds. In addition, it has been shown that the study of the mean wind speeds for the evaluation of the wind potential alone is not enough, and other descriptive statistics must be considered. On the other hand, among the nine considered distribution, it appears that the GEV is the most appropriate probability distribution. Whereas, the Weibull distribution showed its performance only in regions with high wind speeds, which, implies that this probability distribution should not be generalized in the study of the wind speed in Algeria.

On Probability Distributions of the Time Deviation Law of Container Liner Ships Under Interference Uncertainty

AbstractContainer liner shipping is a kind of transportation mode that is operated according to a schedule. Although the goal is to operate container liner ships on time, the actual arrival time and handling time often deviate from the schedule due to uncertain factors. The identification of a proper probability distribution to describe time deviation law will have a significant impact on accurately recognizing the uncertainty of the operation of container liner ships. In view of this problem, this paper discusses the basic characteristics of container liner ships’ operation time, analyses the properties of relevant probability distributions, and selects representative container ports around the world to collect data on the container liner ships’ operation time for statistical verification. The results show that under schedule constraints and interference uncertainty, the time deviation presents a specific state between a fixed length and random distribution that conforms to the properties of an Erlang distribution. Given that container liner shipping follows the same operation rules worldwide, it is reasonable to deduce that the time deviation law could be generalized to other container ports. Finally, the practical value of this study is demonstrated through quantitative evaluation of port congestion degree under various probabilistic models.

Multistep-Ahead Prediction of Urban Traffic Flow Using GaTS Model

The mathematical models for traffic flow have been widely investigated for a lot of application, like planning transportation and easing traffic pressure by using statistics and machine learning methods. However, there remains a lot of challenging problems for various reasons. In this research, we mainly focused on three issues: (a) the data of traffic flow are nonnegative, and hereby, finding a proper probability distribution is essential; (b) the complex stochastic property of the traffic flow leads to the nonstationary variance, i.e., heteroscedasticity; and (c) the multistep-ahead prediction of the traffic flow is often of poor performance. To this end, we developed a Gamma distribution-based time series (GaTS) model. First, we transformed the original traffic flow observations into nonnegative real-valued data by using the Box-Cox transformation. Then, by specifying the generalized linear model with the Gamma distribution, the mean and variance of the distribution are regressed by the past data and homochronous terms, respectively. A Bayesian information criterion is used to select the proper Box-Cox transformation coefficients and the optimal model structures. Finally, the proposed model is applied to the urban traffic flow data achieved from Dalian city in China. The results show that the proposed GaTS has an excellent prediction performance and can represent the nonstationary stochastic property well.

Optimal energy management and sizing of renewable energy and battery systems in residential sectors via a stochastic MILP model

Energy supply through integrated renewable energy sources (RESs) and battery systems will be of higher importance for future residential sectors. Optimal energy management and sizing for the components of residential systems can enhance efficiency, self-suffiency, and meanwhile can be cost-effective by reducing investment as well as operating costs. Accordingly, this paper proposes an exhaustive optimization model for determining the capacity of RESs, namely: wind turbines and photovoltaic (PV) systems. In this study, batteries and electric vehicles (EVs) are utilized in line with other sources to capture fluctuations of RESs. To model the uncertainties of RESs, energy prices, and load demands a linearized stochastic programming framework is applied. The proposed framework involves long-term and efficient resource development alongside with short-term management and utilization of these resources for supplying the demand load. In our study, we utilize the roulette wheel mechanism (RWM) method as well as proper probability distribution functions (PDFs) to generate scenarios for all sources of uncertainties, including wind turbines, PV systems, demand, and electricity market price. The approach is verified in two different cases, including an individual home and a larger micro-grid (MG). The results of multiple numerical simulations demonstrate the effectiveness of the proposed stochastic model.

Novel algorithm for reconstruction of a distribution by fitting its first-four statistical moments

• A parameterized probability distribution (PPD) system is developed for recovering an unknown PDF/CDF. • A two-step strategy is proposed to select an appropriate PPD model. • The proposed method is exemplified by fitting several theoretical distributions, as well as a real dataset, with very favorable results. • Typical moments-based reliability problems are studied to validate the efficacy of the proposed strategy. This paper proposes a novel algorithm to reconstruct an unknown distribution by fitting its first-four moments to a proper parametrized probability distribution (PPD) model. First, a PPD system containing three previously developed PPD models is suggested to approximate the unknown distribution, rather than empirically adopting a single distribution model. Then, a two-step algorithm based on the moments matching criterion and the maximum entropy principle is proposed to specify the appropriate (final) PPD model in the system for the distribution. The proposed algorithm is first verified by approximating several commonly used analytical distributions, along with a set of real dataset, where the existing measures are also employed to demonstrate the effectiveness of the proposed two-step algorithm. Further, the effectiveness of the algorithm is demonstrated through an application to three typical moments-based reliability problems. It is found that the proposed algorithm is a robust tool for selecting an appropriate PPD model in the system for recovering an unknown distribution by fitting its first-four moments.

Enhanced configurational sampling with hybrid non-equilibrium molecular dynamics-Monte Carlo propagator.

Molecular dynamics (MD) trajectories based on classical equations of motion can be used to sample the configurational space of complex molecular systems. However, brute-force MD often converges slowly due to the ruggedness of the underlying potential energy surface. Several schemes have been proposed to address this problem by effectively smoothing the potential energy surface. However, in order to recover the proper Boltzmann equilibrium probability distribution, these approaches must then rely on statistical reweighting techniques or generate the simulations within a Hamiltonian tempering replica-exchange scheme. The present work puts forth a novel hybrid sampling propagator combining Metropolis-Hastings Monte Carlo (MC) with proposed moves generated by non-equilibrium MD (neMD). This hybrid neMD-MC propagator comprises three elementary elements: (i) an atomic system is dynamically propagated for some period of time using standard equilibrium MD on the correct potential energy surface; (ii) the system is then propagated for a brief period of time during what is referred to as a "boosting phase," via a time-dependent Hamiltonian that is evolved toward the perturbed potential energy surface and then back to the correct potential energy surface; (iii) the resulting configuration at the end of the neMD trajectory is then accepted or rejected according to a Metropolis criterion before returning to step 1. A symmetric two-end momentum reversal prescription is used at the end of the neMD trajectories to guarantee that the hybrid neMD-MC sampling propagator obeys microscopic detailed balance and rigorously yields the equilibrium Boltzmann distribution. The hybrid neMD-MC sampling propagator is designed and implemented to enhance the sampling by relying on the accelerated MD and solute tempering schemes. It is also combined with the adaptive biased force sampling algorithm to examine. Illustrative tests with specific biomolecular systems indicate that the method can yield a significant speedup.

Joint Blind Equalization and Decoding in OFDM Systems Using Particle Filtering

In this paper a sequential algorithm is proposed for joint blind channel equalization and decoding for orthogonal frequency-division multiplexing (OFDM) in frequency selective channels. This algorithm offers a recursive method to sequentially calculate the posterior probability for maximum a posteriori detection. Recursive calculations are done along the indexes in each OFDM symbol using a particle filter. By defining an appropriate importance function, and a proper prior probability distribution function for the channel tap coefficients (and marginalizing it), an efficient method is presented for joint equalization and channel decoding in OFDM based systems. Performance of the proposed detector is evaluated using computer simulations and its bit error rate is compared with the trained turbo equalizer and a conventional particle filter-based method. The results show that the proposed method outperforms the previously presented particle filter-based method without a need for training data.

Application of tests of goodness of fit in determining the probability density function for spacing of steel sets in tunnel support system

One of the conventional methods for temporary support of tunnels is to use steel sets with shotcrete. The nature of a temporary support system demands a quick installation of its structures. As a result, the spacing between steel sets is not a fixed amount and it can be considered as a random variable. Hence, in the reliability analysis of these types of structures, the selection of an appropriate probability distribution function of spacing of steel sets is essential. In the present paper, the distances between steel sets are collected from an under-construction tunnel and the collected data is used to suggest a proper Probability Distribution Function (PDF) for the spacing of steel sets. The tunnel has two different excavation sections. In this regard, different distribution functions were investigated and three common tests of goodness of fit were used for evaluation of each function for each excavation section. Results from all three methods indicate that the Wakeby distribution function can be suggested as the proper PDF for spacing between the steel sets. It is also noted that, although the probability distribution function for two different tunnel sections is the same, the parameters of PDF for the individual sections are different from each other.

Fuzzy Piecewise Logistic Growth Model for Innovation Diffusion: A Case Study of the TV Industry

The logistic model is adopted in order to fit growth trends of innovative products for a single growth process. In the current competitive environment, we are incapable of predicting a product’s life cycle such that it can be described as a smooth S curve. Given this, we propose the use of a fuzzy piecewise regression model as a revision of the traditional logistic model. While no proper probability distribution for market share data currently exists, the proposed method is not only able to detect change-points, but can also identify predicted intervals when the growth trend of an analyzed generation is affected by other product generations. The market shares of four television technologies are used in order to demonstrate the performance of the proposed model. The results show that the proposed model outperforms the logistic model, providing both the best and worst possible market shares for the corresponding generation, and highlighting the time of impact of external influences by identifying change-points.

Efficient hybrid non-equilibrium molecular dynamics--Monte Carlo simulations with symmetric momentum reversal.

Hybrid schemes combining the strength of molecular dynamics (MD) and Metropolis Monte Carlo (MC) offer a promising avenue to improve the sampling efficiency of computer simulations of complex systems. A number of recently proposed hybrid methods consider new configurations generated by driving the system via a non-equilibrium MD (neMD) trajectory, which are subsequently treated as putative candidates for Metropolis MC acceptance or rejection. To obey microscopic detailed balance, it is necessary to alter the momentum of the system at the beginning and/or the end of the neMD trajectory. This strict rule then guarantees that the random walk in configurational space generated by such hybrid neMD-MC algorithm will yield the proper equilibrium Boltzmann distribution. While a number of different constructs are possible, the most commonly used prescription has been to simply reverse the momenta of all the particles at the end of the neMD trajectory ("one-end momentum reversal"). Surprisingly, it is shown here that the choice of momentum reversal prescription can have a considerable effect on the rate of convergence of the hybrid neMD-MC algorithm, with the simple one-end momentum reversal encountering particularly acute problems. In these neMD-MC simulations, different regions of configurational space end up being essentially isolated from one another due to a very small transition rate between regions. In the worst-case scenario, it is almost as if the configurational space does not constitute a single communicating class that can be sampled efficiently by the algorithm, and extremely long neMD-MC simulations are needed to obtain proper equilibrium probability distributions. To address this issue, a novel momentum reversal prescription, symmetrized with respect to both the beginning and the end of the neMD trajectory ("symmetric two-ends momentum reversal"), is introduced. Illustrative simulations demonstrate that the hybrid neMD-MC algorithm robustly yields a correct equilibrium probability distribution with this prescription.

Smart microgrid energy and reserve scheduling with demand response using stochastic optimization

Demand side participation is one of the important resources that help the operator to schedule generation and consumption with lower cost and higher security. Customers can participate in both energy and reserve operational scheduling and earn benefit from reducing or shifting their consumption. In this paper, a novel stochastic energy and reserve scheduling method for a microgrid (MG) which considers various type of demand response (DR) programs is proposed. In the proposed approach, all types of customers such as residential, commercial and industrial ones can participate in demand response programs which will be considered in either energy or reserve scheduling. Also, the uncertainties related to renewable distributed generation are modeled by proper probability distribution functions and are managed by reserve provided by both DGs and loads. The proposed method was tested on a typical MG system comprising different type of loads and distributed generation units. The results demonstrate that the adoption of demand response programs can reduce total operation costs of a MG and determine a more efficient use of energy resources.