Best-fit Probability Distribution Research Articles

Investigating traffic-related PM exposure on and under pedestrian bridges: A case study in Xi'an, China

We measured particulate matter concentrations (PMCs), including PM 10 , PM 2.5 , and PM 1.0 fractions, on and under pedestrian bridges over two different types of roads: an urban expressway and an arterial road. The objectives of the study were to 1) assess the variability of PMCs measured at pedestrian bridges, 2) identify best-fit probability distribution curves of different measures of PMCs, and 3) quantify pedestrian exposure (by respiratory deposition dose (RDD) rate). Results showed that the average on-bridge PM concentrations were significantly higher than the under-bridge values over the urban expressway, although no similar trend observed over the arterial road. More specifically, average PM 1.0 concentration on the bridge was 1.29 times of that under the bridge over the urban expressway. Gaussian distribution “best” fit the 30s average PMCs for PM 10 on and under the pedestrian bridge over the urban expressway, and LogNormal distribution “best” fit the PMCs for PM 10 on and under the bridge for the arterial road. The average RDD rates for PM 10 (143.85 μg h −1 ), PM 2.5 (9.41 μg h −1 ), and PM 1.0 (3.78 μg h −1 ) on the bridge over the urban expressway (OBUE) were higher by 47%, 41%, and 11%, respectively, than those on the bridge over the arterial road (OBAR). While only minor differences were noticed in the average RDDs of PM between on and under the pedestrian bridge over the arterial road, a significant difference in the average RDD rate was found between on and under pedestrian bridges over the urban expressway. • Measurements of particulate matter concentrations (PMCs) on and under pedestrian bridges were performed. • The best-fit probability distribution curves were identified to understand the dynamics of PMC on the bridges. • The pedestrian exposures on the pedestrian bridges were quantified in terms of respiratory deposition dose (RDD) rate.

Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh

The study of frequency analysis is important to find the most suitable model that could anticipate extreme events of certain natural phenomena e.g., rainfall, floods, etc. The goal of this study is to determine the best-fit probability distributions in the case of maximum monthly rainfall using 30 years of data (1984–2013) from 35 locations in Bangladesh by using different statistical analysis and distribution types. Commonly used frequency distributions were applied. Parameters of these distributions were estimated by the method of moments and L-moments estimators. Three goodness-of-fit test statistics were applied. The best-fit result of each station was taken as the distribution with the lowest sum of the rank scores from each of the three test statistics. Generalized Extreme Value, Pearson type 3 and Log-Pearson type 3 distributions showed the largest number of best-fit results. Among the best score results, Generalized Extreme Value yielded the best-fit for 36% of the stations and Pearson type 3 and Log-Pearson type 3 each yielded the best-fit for 26% of the stations. The more practical result of this paper was that the 10-year, 25-year, 50-year and 100-year return periods of maximum monthly rainfall were calculated for all locations. The result of this study can be used to develop more accurate models of flooding risk and damage.

Effect of Extraordinary Large Floods on at-site Flood Frequency

The peak flow of extraordinary large floods that occur during a period of systematic record is a controversial problem for flood frequency analysis (FFA) using traditional methods. The present study suggests that such floods be treated as historic flood data even though their historical period is unknown. In this paper, the extraordinary large flood peak was first identified using statistical outlier tests and normal probability plots. FFA was then applied with and without the extraordinary large floods. In this step, two goodness-of-fit tests including mean absolute relative deviation and mean squared relative deviation were used to identify the best-fit probability distributions. Next, the generalized extreme value (GEV), three-parameter lognormal (LN3), log-Pearson type III (LP3), and Wakeby (WAK) probability distributions were used to incorporate and adjust the extraordinary large floods with other systematic data. Finally, procedures with and without historical adjustment were compared for the extraordinary large floods in terms of goodness-of-fit and flood return-period quantiles. The results of this comparison indicate that historical adjustment from an operational perspective was more viable than without adjustment procedure. Furthermore, the results without adjustment were unreasonable (subject to over- and under-estimation) and produced physically unrealistic estimates that were not compatible with the study area. The proposed approach substantially improved the probability estimation of rare floods for efficient design of hydraulic structures, risk analysis, and floodplain management.

Best-Fit Probability Distribution Model for Rainfall Frequency Analysis of Three Cities in South Eastern Nigeria

Rainfall frequency analysis is the estimation of how often rainfall of specified magnitude will occur. Such analyses are helpful in defining policies relating to water resources management. It serves as the source of data for flood hazard mitigation and the design of hydraulic structures aimed at reducing losses due to floods action. In this study rainfall frequency analysis for three (3) cities in South Eastern Nigeria were carried out using annual maximum series of daily rainfall data for the stations. The objective of the study was to select the probability distribution model from among six commonly used probability distribution models namely: Generalized Extreme value distribution (GEV), Extreme value type I distribution (EVI), Generalized Pareto distribution (GPA), Pearson Type III (PIII), log Normal (LN) and Log Pearson Type III (LP111) distributions. These distributions were applied to annual maximum series of daily precipitation data at each station using the parameters of the distributions estimated by the method of moments. The best fit probability distribution model at each location was selected based on the results of seven goodness of fit tests entry: root mean square error (RMSE), relative root mean square error (RRMSE), mean absolute deviation index (MADI) and probability plot correlation coefficient (PPCC), Maximum Absolute Error (MAE), Chi square test and D- Index and a scoring and ranking scheme. Our results indicate that the best fit probability distribution model at all study locations is GEV and this was used to forecast rainfall return values for the stations for return periods of between 5years and 500years. The values obtained are useful for planning, design and management of hydraulic structures for flood mitigation and prevention of flood damage at the location.

Evaluation of Best-Fit Probability Distribution Models for the Prediction of Inflows of Kainji Reservoir, Niger State, Nigeria

The analysis of time series is essential for building mathematical models to generate synthetic hydrologic records, to forecast hydrologic events, to detect intrinsic stochastic characteristics of hydrologic variables, as well as to fill missing and extend records. To this end, various probability distribution models were fitted to river inflows of Kainji Reservoir in New Bussa, Niger State, Nigeria. This is to evaluate the probability function that is best suitable for the prediction of their values and subsequently using the best model to predict for both the expected maximum and minimum monthly inflows at some specific return periods. Three models, ie, Gumbel extreme value type I (EVI), log-normal (LN), and normal (N), were evaluated for the inflows and the best model was selected based on the statistical goodness-of-fit test. The values of goodness-of-fit test for Kainji hydropower dam are as follows: r = 0.96, R2 = 0.99, SEE = 0.0087, χ2 = 0.0054, for Gumbel (EVI); r = 0.79, R2 = 0.85, SEE = 0.02, χ2 = 0.31 for LN; and r = 0.0.75, R2 = 0.0.68, SEE = 0.056, χ2 = 1376.39 for N. For the Kainji hydropower dams, the Gumbel (EVI) model gave the best fit. These probability distribution models can be used to predict the near-future reservoir inflow at the Kainji hydropower dams.

Structural characteristics of annual precipitation in Lake Urmia basin

Precipitation as the main process that brings evaporated water from the oceans to the land’s surface is a critical role player in Lake Urmia basin (Iran). As a hyper-saline lake declared as UNESCO’s biosphere reserve in Ramsar Convention, it is dealing with gradual atrophy. In this study, characteristics of annual precipitation in the Lake Urmia basin are investigated by means of several statistical measures and tests. Data in 53 meteorological stations widespread across the basin for a period of 31 years from 1981 to 2011 are considered for analysis. Fundamental statistical characteristics of the data like mean, maximum, minimum, standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, auto-correlation and cross-correlation coefficients of the annual precipitation are calculated. Entropy in each station is also calculated with respect to the long-run mean precipitation of the basin. Results of the analysis are plotted in contour maps. Several tests for consistency, randomness, trend and best-fit probability distribution function are applied to investigate characteristics of the annual precipitation. Heterogeneity and dependence on local conditions are the main results revealed by this study while consistency and dependency of precipitation on North West and West of the basin are considered as the most effective among other regions. Due to the North-South oriented mountains, a relatively sharp decline in the precipitation from West to East can be compared to the gradual decline in precipitation from North to South due to smooth change in the terrain. It is also seen that such characteristics as probability distribution, consistency, randomness, trend, and uncertainty of annual precipitation in the Lake Urmia basin become more complex as crossing from West to East than crossing from North to South on the basin.

A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan

AbstractThis study was designed to find the best-fit probability distribution of annual maximum rainfall based on a twenty-four-hour sample in the northern regions of Pakistan using four probability distributions: normal, log-normal, log-Pearson type-III and Gumbel max. Based on the scores of goodness of fit tests, the normal distribution was found to be the best-fit probability distribution at the Mardan rainfall gauging station. The log-Pearson type-III distribution was found to be the best-fit probability distribution at the rest of the rainfall gauging stations. The maximum values of expected rainfall were calculated using the best-fit probability distributions and can be used by design engineers in future research.

The Hydrometeorology of the Kariba Catchment Area Based on the Probability Distributions

Abstract In this paper, monthly, maximum seasonal, and maximum annual hydrometeorological (i.e., evaporation, lake water levels, and rainfall) data series from the Kariba catchment area of the Zambezi River basin, Zimbabwe, have been analyzed in order to determine appropriate probability distribution models of the underlying climatology from which the data were generated. In total, 16 probability distributions were considered and the Kolmogorov–Sminorv (KS), Anderson–Darling (AD), and chi-square (χ2) goodness-of-fit (GoF) tests were used to evaluate the best-fit probability distribution model for each hydrometeorological data series. A ranking metric that uses the test statistic from the three GoF tests was formulated and used to select the most appropriate probability distribution model capable of reproducing the statistics of the hydrometeorological data series. Results showed that, for each hydrometeorological data series, the best-fit probability distribution models were different for the different time scales, corroborating those reported in the literature. The evaporation data series was best fit by the Pearson system, the Lake Kariba water levels series was best fit by the Weibull family of probability distributions, and the rainfall series was best fit by the Weibull and the generalized Pareto probability distributions. This contribution has potential applications in such areas as simulation of precipitation concentration and distribution and water resources management, particularly in the Kariba catchment area and the larger Zambezi River basin, which is characterized by (i) nonuniform distribution of a network of hydrometeorological stations, (ii) significant data gaps in the existing observations, and (iii) apparent inherent impacts caused by climatic extreme events and their corresponding variability.

Big Data–Based Estimation for Ship Safety Distance Distribution in Port Waters

The water area of a large container port such as Singapore's has high ship traffic density because of continuously increasing international seaborne trade. The behavior of ships sailing in the port's waters exhibits high diversity. Ships must maintain a minimum safety distance when moving in and out of port waters to avoid collisions. This study estimated the probability distributions for ship safety distance by using automatic identification system (AIS) data collected in Singapore port waters. Thirty-six navigation scenarios classified by ship type and size, visibility (daytime and night), and direction of movement (crossing, head-on, and overtaking) were investigated. Safety distances for various ship types and sizes were first examined with nonparametric statistical tests. A tangible approach incorporating the maximum likelihood estimation and Kolmogorov–Smirnov test techniques was designed for determining the best-fitted probability distribution with the parameters calibrated by AIS data for ship safety distance. It was found that the lognormal and gamma distributions could well fit the ship safety distance in Singapore port waters according to the collected AIS data.

A regional Bayesian hierarchical model for flood frequency analysis

In this study, we propose a regional Bayesian hierarchical model for flood frequency analysis. The Bayesian method is an alternative to the traditional regional flood frequency analysis. Instead of relying on the delineation of implicit homogeneous regions, the Bayesian hierarchical method describes the spatial dependence in its inner structure. Similar to the classical Bayesian hierarchical model, the process layer of our model presents the spatial variability of the parameters by considering different covariates (e.g., drainage area, elevation, precipitation). Beyond the three classical layers (data, process, and prior) of the Bayesian hierarchical model, we add a new layer referred to as the “L-moments layer”. The L-moments layer uses L-moments theory to select the best-fit probability distribution based on the available data. This new layer can overcome the subjective selection of the distribution based on extreme value theory and determine the distribution from the data instead. By adding this layer, we can combine the merits of regional flood frequency and Bayesian methods. A standard process of covariates selection is also proposed in the Bayesian hierarchical model. The performance of the Bayesian model is assessed by a case study over the Willamette River Basin in the Pacific Northwest, U.S. The uncertainty of different flood percentiles can be quantified from the posterior distributions using the Markov Chain Monte Carlo method. Temporal changes for the 100-year flood percentiles are also examined using a 20- and 30-year moving window method. The calculated shifts in flood risk can aid future water resources planning and management.

Laboratory investigation to study the corrosion initiation of rebars in fly ash concrete

The objective of this study is to detect reliably the onset of corrosion in fly ash concrete using an embedded reference electrode. The half-cell potential of rebars in fly ash concrete subjected to 10% sodium chloride solution is monitored using an embedded reference electrode and an alternating current impedance spectroscope, along with other impedance parameters such as polarisation resistance and specific concrete resistance. At initiation of corrosion, the values of chloride threshold levels are obtained and the effect of long curing for fly ash concrete on chloride threshold values is analysed. The combination of two corrosion monitoring systems allows the estimation of corrosion initiation in fly ash concrete. It is found that chloride threshold values for fly ash concrete given a longer curing period approach the values for Portland cement concrete. Probabilistic analysis of the experimental data is carried out to find a best-fit probability distribution for time to initiate corrosion and chloride threshold values of fly ash concrete.

Analyses of rainfall using probability distribution and Markov chain models for crop planning in Daspalla region in Odisha, India

Rainfed agriculture plays and will continue to play a dominant role in providing food and livelihoods for an increasing world population. Rainfall analyses are helpful for proper crop planning under changing environment in any region. Therefore, in this paper, an attempt has been made to analyse 16 years of rainfall (1995–2010) at the Daspalla region in Odisha, eastern India for prediction using six probability distribution functions, forecasting the probable date of onset and withdrawal of monsoon, occurrence of dry spells by using Markov chain model and finally crop planning for the region. For prediction of monsoon and post-monsoon rainfall, log Pearson type III and Gumbel distribution were the best-fit probability distribution functions. The earliest and most delayed week of the onset of rainy season was the 20th standard meteorological week (SMW) (14th–20th May) and 25th SMW (18th–24th June), respectively. Similarly, the earliest and most delayed week of withdrawal of rainfall was the 39th SMW (24th–30th September) and 47th SMW (19th–25th November), respectively. The longest and shortest length of rainy season was 26 and 17 weeks, respectively. The chances of occurrence of dry spells are high from the 1st–22nd SMW and again the 42nd SMW to the end of the year. The probability of weeks (23rd–40th SMW) remaining wet varies between 62 and 100 % for the region. Results obtained through this analysis would be utilised for agricultural planning and mitigation of dry spells at the Daspalla region in Odisha, India.

A new procedure to best-fit earthquake magnitude probability distributions: including an example for Taiwan

Since the year 1973, more than 54,000 M w ≥ 3.0 earthquakes have occurred around Taiwan, and their magnitude–frequency relationship was found following with the Gutenberg–Richter recurrence law with b value equal to 0.923 from the least-square calculation. However, using this b value with the McGuire–Arabasz algorithm results in some disagreement between observations and expectations in magnitude probability. This study introduces a simple approach to optimize the b value for better modeling of the magnitude probability, and its effectiveness is demonstrated in this paper. The result shows that the optimal b value can better model the observed magnitude distribution, compared with two customary methods. For example, given magnitude threshold = 5.0 and maximum magnitude = 8.0, the optimal b value of 0.835 is better than 0.923 from the least-square calculation and 0.913 from maximum likelihood estimation for simulating the earthquake’s magnitude probability distribution around Taiwan.

A study on selection of probability distributions for at-site flood frequency analysis in Australia

The most direct method of design flood estimation is at-site flood frequency analysis, which relies on a relatively long period of recorded streamflow data at a given site. Selection of an appropriate probability distribution and associated parameter estimation procedure is of prime importance in at-site flood frequency analysis. The choice of the probability distribution for a given application is generally made arbitrarily as there is no sound physical basis to justify the selection. In this study, an attempt is made to investigate the suitability of as many as fifteen different probability distributions and three parameter estimation methods based on a large Australian annual maximum flood data set. A total of four goodness-of-fit tests are adopted, i.e., the Akaike information criterion, the Bayesian information criterion, Anderson–Darling test, and Kolmogorov–Smirnov test, to identify the best-fit probability distributions. Furthermore, the L-moments ratio diagram is used to make a visual assessment of the alternative distributions. It has been found that a single distribution cannot be specified as the best-fit distribution for all the Australian states as it was recommended in the Australian rainfall and runoff 1987. The log-Pearson 3, generalized extreme value, and generalized Pareto distributions have been identified as the top three best-fit distributions. It is thus recommended that these three distributions should be compared as a minimum in practical applications when making the final selection of the best-fit probability distribution in a given application in Australia.

Application of L-moments for IDF determination in an Austrian basin

The aim of this paper is to evaluate a precise intensity-duration-frequency (IDF) relation for a small catchment area which is situated at the north part of Graz in Austria. In order to determine the probability of occurrence of any rainfall event, the frequency distribution, which can fit past characteristics on the magnitude and the probability of occurrence of such rainfalls, should be known. In this paper, in order to find the best-fit probability distribution model, some parameter estimation techniques such as L-moments and maximum likelihood models were used and for goodness of fit test, three methods were used as chi-square, Kolmogorov-Smirnov and the root mean square error (RMSE). In this paper, a comparison between four commonly used rainfall frequency distributions were carried out such as generalised extreme value (GEV), Gumbel, log-Pearson type 3 (LP3) and three-parameter log-normal (LN3). Finally, using the best-fit probability distribution, the IDF for this catchment area using a simple graphical method was determined.

Advanced Stochastic Schedule Simulation System

This paper introduces an automated tool named Advanced Stochastic Schedule Simulation System (AS4). It integrates critical path method (CPM) schedule data exported from Primavera Project Planner (P3) and historical activity duration data obtained from a project data warehouse, computes the best-fit probability distribution functions (PDFs) of historical activity durations, assigns the PDFs identified to respective activities, computes the optimum number of simulation runs, simulates the schedule network for the optimum number of iteration runs, and estimates the best-fit PDF of project completion times (PCTs) without user intervention at any time. AS4 improves the reliability of simulation-based scheduling by effectively dealing with the uncertainties of the activities’ durations, increases the usability of the schedule data obtained from commercial CPM software, and effectively handles the variability of the PCTs by finding the best-fit PDF of PCTs. AS4 encourages the use of simulation-based scheduling because it is simple to use and it simplifies the tedious and burdensome process involved in finding the PDFs of the many activities’ durations, in assigning the PDFs to the many activities of a new network under modelling, and in estimating the best-fit PDF of PCTs.

Stochastic Project Financing Analysis System for Construction

This paper presents an automated tool named the stochastic project financing analysis system (SPFA). The system makes use of critical path method (CPM) schedule data exported from Primavera Project Planner (P3), computes the best-fit probability distribution functions (PDFs) of historical activity durations, assigns to respective activities the probability density functions identified, simulates the schedule network, computes the deterministic and stochastic project cashflows, plots the corresponding cash-flow diagrams, and estimates the best-fitPDFs of overdraft and net profit of a project without user intervention at any time. The SPFA improves the reliability of project cash-flow analysis by effectively dealing with the uncertainties of the activities’ durations and costs, increases the usability of the schedule data obtained from commercial CPM software, allows the user to incorporate the contractual terms of payment into the cashflow, and effectively handles the variability of the overdrafts and net profits by finding their best-fitPDF. It is implemented as an easy-to-use computer tool programmed in matrix laboratory (MATLAB). The SPFA allows a financial manager to estimate the extent of cash-flow problems in a specific period by using both deterministic and stochastic modes. Two test cases verify the usability and validity of the system in practice

Efficient Method Based on Optimization and Simulation for the Probabilistic Strength Computation of the Ship Hull

The accurate computation of the strength of a ship hull is vital for ensuring its reliability. Taking into account the uncertainties in the material properties and fabrication details requires a probabilistic approach for the determination of the strength of the ship hull. This study proposes an efficient approach for the determination of the probabilistic strength of the ship hull. First, a novel deterministic method for the fast and accurate calculation of the strength of the ship hull is presented. In this method, stresses in the hull section are determined using the constitutive models of the stiffened panels. These models take into account the various possible failure modes and initial imperfections. The ultimate strength is found by an optimization search algorithm. The method is shown to be as accurate as the rigorous incremental curvature method but with significantly less computational time. The method is then applied with a Latin-hypercube sampling simulation, and the output sample is tested against several potential distributions. The best-fit probability distribution of the ultimate strength of the ship hull is provided.

Statistical Modeling of Combat Mortality Events by Using Subject Matter Expert Opinions and Operation Iraqi Freedom Empirical Results from the Navy-Marine Corps Combat Trauma Registry

In 12 Navy—Marine Corps medical logistics studies and analyses conducted by the Naval Health Research Center (NHRC) and Teledyne Brown Engineering (TBE) over the past 5 years, estimates of battlefield mortality have been a major metric of interest to medical planners. An ongoing concern is how mortality is related to delays in treatment for medical logistics reasons. In this paper, we describe how NHRC and TBE have been developing a statistically based model for mortality since 2003, first starting with panel results from a group of military medical doctors and continuing here with an analysis of empirical injury data from Operation Iraqi Freedom (OIF). The panel results and statistical analysis of life-threatening injury data in OIF from early 2004 to mid-2006 indicate that the Weibull distribution describes the timing of high risk of mortality events in a reasonable manner within surgical medical treatment facilities. The quest for a best-fitting probability distribution with parameters dependent on the casualty flow chain of treatment and evacuation in the theater is ongoing. In reality, combining analytical and subject matter expert (SME) results to model mortality is necessary given the breadth of theater medical delivery systems and general paucity of adequate empirical data in many of the segments of patient flow.

Statistical Distributions of Hydraulic Conductivity from Reliability Analysis Data

This paper describes a first-order reliability-based analysis to identify the best-fit probability distributions for hydraulic conductivity. The analysis involved the use of existing hydraulic conductivity model developed from laboratory data and applied to lateritic soils, considering variations in soil parameters. Plots of reliability indices versus coefficients of variation were first made for hydraulic conductivity as well as for initial degree of saturation, plasticity index and clay content, considering three compactive efforts and log-normally distributed hydraulic conductivity. The traditional two-parameter log-normal distribution was compared to four alternative distributions: normal, gamma, Gumbel (extreme value type I-EVT-I) and Weibull (extreme value type III-EVT-III). The analysis showed that the Weibull and normal are the best-fit probability distributions for the hydraulic conductivity based reliability data. Hydraulic conductivities predicted from reliability analysis were used to demonstrate the possibility of applying the results obtained in this research by practising engineers. Experimentally-determined hydraulic conductivities were shown to be in good agreement with predicted values.