Best-fit Probability Distribution Research Articles

Probability Distribution Analysis of Annual Maximum Precipitation in the Niger Delta: A Critical Step towards Effective Flood Risk Management

Aims: The study aimed to analyze historical maximum precipitation data from seven cities in the Niger Delta to assess vulnerability to extreme rainfall events, identify optimal probability distribution models for future predictions, and provide insights for flood risk management strategies. Study Design: This is a retrospective analytical study based on historical precipitation data. Place and Duration of Study: The study was conducted across seven major cities in the Niger Delta region of Nigeria, utilizing data collected over several decades. Methodology: Historical maximum precipitation data were statistically analyzed to determine the best-fit probability distribution models. The Kolmogorov-Smirnov (KS) test and Mean Squared Prediction Error (MSPE) were used to validate the suitability of distributions, including Log-Normal, Normal, Log-Pearson III, and Generalized Extreme Value (GEV) for each city. Return periods of 10, 25, 50, and 100 years were calculated to predict future precipitation trends and assess flood risks. Results: Significant variations in annual maximum precipitation were observed across the cities, with Calabar recording the highest peak at 4062.7mm. The Log-Normal distribution was the best fit for Akure and Calabar, while the Normal distribution best described rainfall in Benin City. Log-Pearson III was optimal for Owerri, Umuahia, and Uyo, and GEV best fitted Port Harcourt’s data. High P-values (>0.87) indicated good model fits across the cities. Return period analysis suggested greater risks of extreme precipitation events in coastal cities like Calabar and Port Harcourt. Conclusion: The study underscores the importance of tailored, city-specific flood management strategies in the Niger Delta to mitigate the impact of extreme rainfall events, particularly in the face of climate change.

Optimal allocation of wind-based distributed generators and STATCOMs using a hierarchical stochastic programming approach

Allocating static synchronous compensators (STATCOMs) to regulate given allotted wind-based distributed generators (W-DGs) during planning stages can provide high investment returns by maximizing the installed W-DGs. Amidst rising global warming concerns, commitments to adopt renewable energy gain international momentum to curb greenhouse emissions and meet environmental targets, reducing reliance on fossil-based resources. Renewable energy's intermittency complicates distribution planning, stressing voltage devices and increasing network losses. This paper presents a new hierarchical stochastic planning model that addresses uncertainties related to W-DGs and generic loads. The model optimizes allocation with voltage constraints and reactive power while incorporating a two-stage mixed-integer nonlinear program (MINLP), maximizing net profit, and adding an economic dimension to promote renewable energy investments. Improved wind power modeling with historical data and collective evaluation metrics for selecting the best-fitted probability distribution function (PDF) enhances the accuracy of wind power integration assessment. The hierarchical approach considers relaxed voltage constraints in Stage I to allow maximum allotment of W-DGs while introducing STATCOMs and DGs' reactive power in Stage II to address voltage violations. Verification on the Canadian 41-bus network demonstrates the advantage of the hierarchical approach in allocating more W-DGs and achieving higher profits than the simultaneous planning approach. These advances significantly enhance renewable energy integration in power systems.

THE COMPARISON SOME MODELS DAILY WIND SPEED DATA IN PEKANBARU CITY USING TWO, THREE AND FOUR PARAMETERS GAMMA DISTRIBUTIONS

Accurate wind speed modeling is critical in estimating wind energy potential for harnessing wind power effectively. One of the bases for assessment of wind energy potential for a specified region is the probability distribution of wind speed, therefore wind speed data is needed to produce statistical modeling, especially in determining the best probability distribution. For this purpose, several modified distributions will be used and tested to determine the best model to describe wind speed in Pekanbaru. Six continuous probability density function such as Two Parameters Gamma (GM), Three Parameters Slashed Quasi-Gamma (SQG), Three Parameters Quasi Gamma (QG), Three Parameters Generalized Gamma (GG), Four Parameters Poly Weighted Exponentiated Gamma (PWEG) and Four Parameters Exponentiated Gamma Exponential (EGE) distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the daily wind speed data, several goodness-of-fit tests were applied, including the graphical methos test (pdf plot) and numerical criteria method test (AIC). The QG and GG distributions are found to be the best-fitted probability distribution based on goodness-of-fit tests for the daily wind speed data at the Pekanbaru station.

Frequency curves of high and low flows in intermittent river basins for hydrological analysis and hydraulic design

Upper and lower percentiles of Flow Duration Curves (FDCs) of daily streamflow data were investigated to develop frequency curves. Upper percentiles with exceedance probability of 1, 5 and 10% (Q1, Q5, Q10) were used for high flows, and lower percentiles with non-exceedance probability of 10, 5 and 1% (Q90, Q95, Q99) for low flows. Median value (Q50) was covered to represent the average conditions of streamflow. A mixed frequency analysis based on the total probability theorem taking zero values into account was applied for the lower percentiles of FDC. Case studies were performed for three intermittent Streamflow Gauging Stations (SGSs) from Kucuk Menderes River Basin in western Turkey. An overall assessment of results shows that the best-fit probability distribution function does not change from one SGS to another considerably for low flows while each SGS has its own probability distribution function for high flows. Upper and lower percentiles, and median value were calculated at various return periods by using the identified probability distribution functions. The calculated values were plotted in the form of frequency curves of high flow percentiles and low flow percentiles. The frequency curves have a practically significant potential use in hydrological analysis, water resources management and hydraulic design under high and low flow conditions. They are yet open to further development for regionalization and their applicability can be extended to ungauged sites in river basins.

Evaluation of survival weighted Pareto distribution: Analytical properties and applications to industrial and aeronautics data

In statistical modeling, lifetime datasets play an important and significant role. Weighted distributions can be useful to attain the essential purposes relating to the developed distributions’ flexibility and to improve further compliances in data forming; they provide the extension of distributions. The current study suggests a new best-fit probability distribution model based on Pareto distribution. The new model is called Survival Weighted Pareto (SWP) distribution model. Furthermore, the statistical expressions for different properties, which include moments, inverse moments, quantile function, and order statistics, are reported. Moreover, to check the performance and the behavior of different estimators using different sample sizes, a simulation study, which is based on SWP distribution, is performed. According to the simulation results, it is established that the proposed model is a most useful fitted probability model. In addition, an application of the SWP model for three datasets is provided to check the goodness of fit measures of the model.

Quantifying regional low flows under data scarce conditions

Estimating low flow quantiles is essential for regulating minimum flow requirements and ensuring water availability, ecological health, and overall system sustainability. This study aims to quantify regional low flows by analyzing low-flow time series in data-limited environments. We utilized annual minimum 7-day instantaneous streamflow data collected from gaging stations. Discordancy measure assessments revealed that all sites were homogeneous, forming a single region. We employed Easy-Fit Statistical Software to select the best-fit probability distribution model and determine estimation parameter values. Through goodness-of-fit tests (GOFs), the Generalized Pareto model emerged as the most suitable, predicting low flow quantiles for 100-year returns. Rigorous application of GOFs ensures the statistical soundness of the model, capturing underlying data patterns. A correlation coefficient determination (R2) of 0.989 demonstrates the high satisfaction of the selected distribution model. The developed regression line for the region exhibited strong agreement between predicted low flows and catchment area. Thus, accurate estimation proves valuable in environmental and human-influenced decision-making processes, providing insights into low-flow behavior and mitigating drought effects on aquatic ecosystems.

Modeling Extreme Precipitation Data in a Mining Area

In recent decades, extreme precipitation events have increased in frequency and intensity in Greece and across regions of the Mediterranean, with significant environmental and socioeconomic impacts. Therefore, extensive statistical analysis of the extreme rainfall characteristics on a dense temporal scale is crucial for areas with important economic activity. For this reason, this paper uses the daily precipitation measurements of four meteorological stations in a mining area of northeastern Chalkidiki peninsula from 2006 to 2021. Three statistical approaches were carried out to develop the best-fitting probability distribution for annual extreme precipitation conditions, using the maximum likelihood method for parameter estimation: the block maxima of the generalized extreme value (GEV) distribution and the peak over threshold of the generalized Pareto distribution (GPD) based on extreme value theory (EVT), and the gamma distribution. Based upon this fitting distribution procedure, return periods for the extreme precipitation values were calculated. Results indicate that EVT distributions satisfactorily fit extreme precipitation, with GPD being the most appropriate, and lead to similar conclusions regarding extreme events.

THE COMPARISON SOME MODELS (GAMMA, SLASHED QUASI-GAMMA, THREE PARAMETERS QUASI GAMMA, TWO PARAMETER GAMMA-EXPONENTIAL, AND MODIFIED LOG-LOGISTIC DISTRIBUTIONS) FOR DAILY TEMPERATURE DATA

With the increase in hot temperature events in recent years, there is growing interest in measuring the frequency of recurring temperature events. One of the basis for assessing the frequency of recurrence of temperature events is the probability distribution of temperature events, therefore temperature data is needed to produce statistical modeling, especially in determining the best probability distribution. The study intended to estimate the best-fitted probability model for the daily temperature at the Pekanbaru station in Indonesia from 2010 to 2020 using several statistical analyses. Five continuous probability distributions such as Gamma (GM), Slashed Quasi-Gamma (SQG), Three Parameters Quasi Gamma (TPQG), Two Parameter Gamma-Exponential (TPGE), and Modified Log-Logistic (MLL) distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the temperature data, several goodness-of-fit tests were applied, including the graphical methos test (density plot) and Numerical criteria method test (AIC, BIC, and -Log Likelihood). The GM and MLL distribution are found to be the best-fitted probability distribution based on goodness-of-fit tests for the daily temperature data at the Pekanbaru station

Determination of uncertainties of geomechanical parameters of metamorphic rocks using petrographic analyses

Geomechanical parameters of intact metamorphic rocks determined from laboratory testing remain highly uncertain because of the great intrinsic variability associated with the degrees of metamorphism. The aim of this paper is to develop a proper methodology to analyze the uncertainties of geomechanical characteristics by focusing on three domains, i.e. data treatment process, schistosity angle, and mineralogy. First, the variabilities of the geomechanical laboratory data of Westwood Mine (Quebec, Canada) were examined statistically by applying different data treatment techniques, through which the most suitable outlier methods were selected for each parameter using multiple decision-making criteria and engineering judgment. Results indicated that some methods exhibited better performance in identifying the possible outliers, although several others were unsuccessful because of their limitation in large sample size. The well-known boxplot method might not be the best outlier method for most geomechanical parameters because its calculated confidence range was not acceptable according to engineering judgment. However, several approaches, including adjusted boxplot, 2MADe, and 2SD, worked very well in the detection of true outliers. Also, the statistical tests indicate that the best-fitting probability distribution function for geomechanical intact parameters might not be the normal distribution, unlike what is assumed in most geomechanical studies. Moreover, the negative effects of schistosity angle on the uniaxial compressive strength (UCS) variabilities were reduced by excluding the samples within a specific angle range where the UCS data present the highest variation. Finally, a petrographic analysis was conducted to assess the associated uncertainties such that a logical link was found between the dispersion and the variabilities of hard and soft minerals.

Estimation of regional and at-site quantiles of extreme winds under flood index procedure

Extreme winds are becoming more common among environmental events with the most catastrophic societal consequences. A regional frequency analysis of Daily Annual Maximum Wind Speed (DAMWS) is necessary not only for a comprehensive understanding of wind hazards but also for infrastructure design and safety, wind energy potential, disaster risk reduction, insurance and risk assessment in a particular region of study. This study investigated regional frequency analysis of DAMWS of Baluchistan and Sindh provinces of Pakistan. L-moments regionalization techniques along with flood index procedure were applied to DAMWS records of 21 stations from 1990 to 2019 across the study area. We intended to find the regional frequency distribution for maximum winds and predict the returns for extreme winds events in the future. Only one station namely Lasbella was found to be discordant. With the help of cluster analysis, the remaining 20 stations were further divided into two homogeneous. Heterogeneity measures validate that both regions are homogenous with allotted stations. Regional quantiles for both regions are estimated through best-fit probability distribution among Generalized Normal (GNO), Generalized Logistic (GLO), Pearson Type 3 (P3), Generalized Pareto (GPA), and Generalized Extreme Value (GEV). Robustness of GLO distribution compared to GEV distribution is assessed through Monte Carlo simulations of relative bias and relative root mean square error. Findings clearly show that GLO distribution is the best for regional modeling. Furthermore, with the help of index flood procedure we determined at-site quantiles of all stations for various return periods. These estimated quantiles are of valuable information for various sectors, including infrastructure, energy, disaster management, and climate resilience, leading to improved planning, development, and risk reduction in the face of wind-related hazards in Sindh and Balochistan provinces of Pakistan.

Evaluation of CMIP6 model performance and extreme precipitation prediction in the Awash basin

Extreme rainfall and its accompanying hydrological extremes are happening more frequently as a result of global warming's alteration of regional and local weather patterns. This poses a serious risk to ecosystem, environment and the community livelihoods. The Awash basin in Ethiopia is especially vulnerable to these events, posing significant threats to the region. There are, however, limited information's available that could be used to characterize the condition of extreme precipitation in the basin. Therefore, this study aims to evaluate the performance of CMIP6 models in simulating extreme precipitation in the Awash basin. Additionally, the study calculated extreme precipitation using best-fit probability distribution functions (PDFs) for the period from 1985 to 2014. The Climate Hazards Group Infrared Precipitation with station data (CHIRPS) were used to evaluate the global climate models. Simulated data were interpolated using bilinear techniques. Four statistical indices (percentage of bias, root mean square error, mean absolute error, and Pearson correlation) assessed GCM performance in simulating precipitation extremes. Graphical approaches, numerical methods, and empirical distribution functions were employed to evaluate the performance of various probability distribution functions (PDFs). The study identified MIROC6, CESM2-WACCM, and Ensemble as well-performing models with PBIAS and RMSE of 6.6 %, −10.2 %, −17.2 %, and 11.5, 10, 9.7 respectively, while MPI-ESM1-2-HR and EC-Earth3 struggled with extreme rainfall simulation. The generalized extreme values distribution was found to be a good fit for extreme rainfall estimation. GFDL-ESM4 and BCC-CSM2-MR models estimated the highest extreme rainfall of 90 mm/day and 80 mm/day, respectively, however these models overestimated the return period. Conversely, MRI-ESM2-0, NorESM2-MM, ACCESS ESM1-5, and CMCC-ESM2 models underestimated the return periods. Spatially, GFDL-ESM4 and ACCESS-ESM1-5 models exhibited uniform peak rainfall values over a large area. Overall, the study suggests that employing the generalized extreme value distribution could effectively inform risk assessment and management of extreme events in the Awash basin.

Probability analysis and rainfall forecasting using ARIMA model

A 34-year rainfall data from 1976 to 2009 of ten sub-basins of the Vaigai River in Tamil Nadu were collected and analysed statistically using various probability distribution functions. The best-fit probability distributions for the annual, monthly and seasonal rainfall for the study area were found using two goodness-of-fit tests. The Box-Jenkins Autoregressive Integrated Moving Average (ARIMA) methodology has been adopted for model identification, diagnostic checking and forecasting the study area's annual rainfall. The best ARIMA models were selected for each sub-basin and the average annual precipitation for 2010, 2015, 2020 and 2025 has been forecasted. The forecasted result compared well with observed dataup to 2020, which indicates the appropriateness of the model.

Geostatistical Modeling of Wind Speed Distribution in Uganda Using Ordinary Kriging Interpolation Technique

Wind power is one of the thriving renewable energy technologies lately in the world. Therefore, the assessment of wind speed data is imperative for a specific site. This study focused on geostatistical modeling of wind speed distribution in Uganda. Wind speed data from 1981 to 2019, recorded at a height of 10 m above mean sea level was captured from NASA POWER Data Access Viewer and analyzed. The study area consisted of 35 stations evenly distributed over the country. Probabilistic assessment was performed using Minitab® statistical software to determine the best-fitting probability distribution function to the data sets. Experimental semi-variograms were calculated using exceedance probability data sets of 20%, 50%, 80%, and 95% obtained from probabilistic analysis. The theoretical models were then fitted to the experimental semi-variograms. Jack-knifing cross-validation approach was employed to assess the performance and validity of the theoretical semi-variogram models and their parameters. Kriging maps for the wind speed of pre-defined probabilities were then generated using JeoStat geostatistical software and Surfers 13®. Of the different theoretical semi-variogram models tested, the spherical model showed the best fit to all experimental semi-variograms. Cross-validation proved that the theoretical models obtained were in good agreement with the experimental data used. The kriging maps revealed that areas around Lake Victoria and Mt. Elgon experience higher wind speeds compared to other parts. Therefore, Kriging maps can be used to assess spatial distribution and magnitude of wind speed as well as the representativeness of geographical locations of observation stations.

Analysis Of Rainfall Characteristics In Balikpapan City Based On Data From Balikpapan Meteorological Station

This study objected to finding the best-fit probability distribution method using four distributions: Normal Distribution, Gumbel Distribution, Log Normal Distribution, and Log Pearson Type III Distribution. According to the score of the goodness-of-fit test, Log Pearson Type III was selected as the best-fit probability distribution. Design rainfall and rainfall intensity were also calculated for the return period of 2 years, 5 years, 10 years, 15 years, and 25 years. In relation to the rainfall intensity, the IDF curve was created. The IDF curve shows higher intensity in the first 5 hours of rainfall, and then the trend decreases gradually. Moreover, it shows that the higher the return period, the higher the rainfall intensity. The movement of curves is essential for designing drainage, flood intensity, and planting period.

Daily maximum rainfall estimation by best-fit probability distribution in the source region of Indus River

Daily maximum rainfall estimation by best-fit probability distribution in the source region of Indus River

Assessment of the Uncertainty Associated with Statistical Modeling of Precipitation Extremes for Hydrologic Engineering Applications in Amman, Jordan

Estimates of extreme precipitation are commonly associated with different sources of uncertainty. One of the primary sources of uncertainty in the statistical modeling of precipitation extremes comes from extreme data series (i.e., sampling uncertainty). Therefore, this research aimed to quantify the sampling uncertainty in terms of confidence intervals. In addition, this article examined how the data record length affects predicted extreme precipitation estimates and data set statistics. A nonparametric bootstrap resample was utilized to quantify the precipitation quantile sampling distribution at a particular non exceedance probability. This sampling distribution can provide a point estimation of the precipitation quantile and the confidence interval at a particular non exceedance probability. It has been shown that the different types of probability distributions fit the extreme precipitation data series of various weather stations. Therefore, the uncertainty analysis should be conducted using the best-fit probability distribution for extreme precipitation data series rather than a predefined single probability distribution for all stations based on modern extreme value theory. According to the 95% confidence intervals, precipitation quantiles are subject to significant uncertainty and the band of the uncertainty intervals increases with the return period. These uncertainty bounds need to be integrated into any frequency analysis from historical data. The average, standard deviation, skewness and kurtosis are highly affected by the data record length. Thus, a longer record length is desirable to decrease the sampling uncertainty and, therefore, decrease the error in the predicted quantile values. Moreover, the results suggest that a series of at least 40 years of data records is needed to obtain reasonably accurate estimates of the distribution parameters and the precipitation quantiles for 100 years return periods and higher. Using only 20 to 25 years of data to obtain estimates of the higher return period quantile is risky, since it created high sampling variability relative to the full data length.

The effect of fast charging and equalization on the reliability and cycle life of the lead acid batteries

Three-wheeled e-rickshaws driven by lead-acid batteries are a common means of transport in Asian countries due to their low cost, and durability. Conventional charging is time-consuming and lacks proper controls for the termination of the charge. Though fast charging can significantly reduce the charging time to about one-third or less, it is prone to temperature rise, excessive gassing, and reduction in the useful life of the battery. A new fast-charging procedure along with periodic equalizing is proposed in this paper. A multistage constant current fast charger with depolarization pulse was used to charge a 48 V e-rickshaw battery pack in 3 h 37 min from 20 % to 80 % state of charge. The effect of the said fast charging procedure on the coulombic efficiency, end voltage pattern, capacity degradation, reliability, and useful life of the lead-acid batteries is investigated. Experimental results for 150 charging-discharging cycles show a temperature rise up to 5–6 °C, average coulombic efficiency of 93 %, and a maximum top-of-charge voltage of 2.6 Volts Per Cell (VPC). Attainment of such a low-temperature rise coupled with high coulombic efficiency during fast charge is the most competitive result reported for commercial batteries.A regression model and best-fit probability distribution (Weibull 3-parameter distribution) are used to estimate the degradation rate, and Cycle-To-Failure (CTF). The expected life of the batteries subjected to such a fast charging and equalizing charge is predicted to be 1296 cycles, which is about 2 times the current life of the battery. The proposed fast charger along with periodic equalizing is, therefore, a promising and sustainable option for charging lead-acid batteries.

Reliability and maintainability optimization of load haul dump machines using genetic algorithm and particle swarm optimization

PurposeReliability and maintainability estimation of any system depends on the identification of the best-fitted probability distribution of failure and repair rates. The parameters of the best-fitted probability distribution are also contributing significantly to reliability estimation. In this work, a case study of load haul dump (LHD) machines is illustrated that consider the optimization of failure and repair rate parameters using two well established metaheuristic approaches, namely, genetic algorithm (GA) and particle swarm optimization (PSO). This paper aims to analyze the aforementioned points.Design/methodology/approachThe data on time between failures (TBF) and time to repairs (TTR) are collected for a LHD machine. The descriptive statistical analysis of TBF & TTR data is performed, trend and serial correlation tested and using Anderson–Darling (AD) value best-fitted distributions are identified for repair and failure times of various subsystems. The traditional methods of estimation like maximum likelihood estimation, method of moments, least-square estimation method help only in finding the local solution. Here, for finding the global solution two well-known metaheuristic approaches are applied.FindingsThe reliability of the LHD machine after 60 days on the real data set is 28.55%, using GA on 250 generations is 17.64%, and using PSO on 100 generations and 100 iterations is 30.25%. The PSO technique gives the global best value of reliability.Practical implicationsThe present work will be very convenient for reliability engineers, researchers and maintenance managers to understand the failure and repair pattern of LHD machines. The same methodology can be applied in other process industries also.Originality/valueIn this case study, initially likelihood function of the best-fitted distribution is optimized by GA and PSO. Reliability and maintainability of LHD machines evaluated by the traditional approach, GA and PSO are compared. These results will be very helpful for maintenance engineers to plan new maintenance strategies for better functioning of LHD machines.

Greedy copula segmentation of multivariate non-stationary time series for climate change adaptation

Non-stationary climate data are often encountered in dealing with natural hazards, climate change and disaster reduction. With drought, for instance, it is common to encounter such non-stationary data sets (time series). The objectives of this work are to formulate a rational data-driven approach that can consider non-stationary and time series on multiple random variables that can have generalized underlying probability distributions and dependence structures. The methodology proposed seeks to divide up the data into non-overlapping segments, each of which is treated as stationary with some underlying probability and dependence structure, while the long time series yields multiple such segments that are mutually independent. The Greedy Copula Segmentation (GCS) algorithm developed employs best-fit probability distributions and copula functions after data-driven time series segmentation. Validation of the proposed methodology is demonstrated using a benchmark problem as well as a single-site realistic drought example. The proposed GCS approach has potential use in climate change adaptation (CCA) and disaster risk reduction (DRR) for any climate-related hazards involving non-stationary time series data.

Best fit probability distribution analysis of precipitation and potential evapotranspiration of India’s highly dense population state - Bihar

Planning of water resources and its management with the ambiguity and non-uniformity accompanying with precipitation and other meteorological physical characteristics may perhaps effect on agricultural production in Bihar where the farmers mostly depend on precipitation. The precipitation and potential evapotranspiration temporal distribution of the state is irregular due to geomorphology, climatic and other anthropogenic factors of the state. In the present study, attempt is taken to expose the best-fit probability distribution among the various available probability distribution of annual average precipitation and potential evapotranspiration based on 102 year of past records of all 37 districts of the state. On the basis of ranks of goodness of fit tests such as Kolmogorov Smirnov, Anderson Darling and Chi-Squared, the normal distribution was observed the best-fit probability distribution for 11 districts followed by Weibull (3P) for 9 districts, the Beta distribution for 5 districts and other distribution for rest districts for precipitation. Whereas Cauchy distribution was come out with the best-fit probability distribution for potential evapotranspiration for all districts and the second best was Gamma (3P) covering almost 60% of the total districts followed by General Extreme Value distribution (32%). The results can be used in future hydraulic design, hydrological study for estimation of return period and water resource planners for policy development.