Parameter Weibull Distribution Research Articles

Assessment of Wind Energy Potential at Three Prime Locations in Saudi Arabia: Analysis of Sharma, Qurayyat and Sakaka Sites

Purpose: The research aims to assess the wind energy potential at three chosen sites in Saudi Arabia Sharma, Al Qurayyat, and Sakaka over a five-year period from 2019 to 2023. The main goal is to determine the most favorable locations for wind farm development based on wind speed characteristics, Weibull distribution parameters, and wind power density (WPD). Materials and Methods: The wind speed data for the chosen locations was obtained from NASA's wind website and analyzed using the Weibull distribution to determine the shape (k) and scale (c) parameters, which are indicative of wind speed distribution. Critical metrics such as mean wind speed, WPD, and the most probable wind speed were computed to evaluate the wind energy potential. The Vestas V150-4.2 MW turbine was utilized to estimate potential energy output, considering the variability and reliability of wind resources at the specified locations. The study utilized tables, figures, and wind rose diagrams to highlight important wind parameters like mean wind speed, Weibull parameters, WPD, and Vmax for each location. Flowcharts were also used to summarize the methodology and conclusions, ensuring a clear and comprehensive presentation of the data. Findings: The analysis showed that Qurayyat consistently had the highest mean wind speed and WPD, indicating excellent wind energy potential. Sharma and Sakaka also displayed significant wind resources, with high mean wind speeds and substantial WPD values. The Weibull parameters indicated that all three sites have favorable wind characteristics for energy production. The estimated energy output using the Vestas V150-4.2 MW turbine ranged from 12,809 to 17,807 MWh/yr at Site 18, 14,095 to 15,170 MWh/yr at Site 24, and 14,013 to 15,832 MWh/yr at Site 31. Implications to Theory, Practice and Policy: The study presents a thorough evaluation of the wind energy potential in the Al-Jawf and Tabuk regions of Saudi Arabia. It contributes to a deeper understanding of wind resource distribution and its significance for the development of renewable energy. The study provides valuable insights for wind farm developers and policymakers by identifying the most promising locations for investment in wind energy infrastructure. Furthermore, it emphasizes the need for detailed site-specific analyses to optimize wind energy deployment, aligning with Saudi Arabia's broader renewable energy objectives.

Performance evaluation of ten numerical methods for Weibull distribution parameter estimation applied to Nigerian wind speed data

Utilizing wind energy necessitates a thorough understanding of wind profiles as well as a precise forecast of wind speed at a study location. In this study, ten Numerical Methods (NEMs), which include the Empirical Method of Lysen (EML), Percentile Method (PCM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), Empirical Method of Justus (EMJ), Alternative Moment Method (AMM), Median and Quartiles Method (MQM), Probability Weighted Moments Based on Power Density Method (PWMBPM), Method of Mabchour (MOMAB) and Energy Variance Method (EVM) were applied to estimate the two- parameter (k and c) Weibull (Wbl) distribution in five locations (Jos, Kano, Maiduguri, Abuja, and Akure) in Nigeria. The performance of these NEMs was assessed using five different metrics and the most effective NEM was determined for each studied location. Daily wind speed data spanning 11 years for the studied locations were sourced from the Meteorological Agency in Nigeria and used in this study. The k and c parameters range from 2.91 to 5.46 and 9.95 to 10.26 (Kano); 2.31 to 4.50 and 5.63 to 6.20 (Maiduguri); 3.19 to 7.61 and 12.16 to 12.99 (Jos); 2.18 to 6.77 and 4.99 to 5.50 (Abuja), and 1.84 to 3.18 and 3.83 to 3.90 (Akure). Findings revealed that the best methods for estimating Wbl parameters for the Kano, Maiduguri, Jos, Abuja, and Akure locations were MMLM, MMLM, MQM, MQM, and EMJ, EML, and AMM, respectively, as MOMAB remained the least performing NEM for all the studied locations. The results also showed that the Vms , Vmps , and V emax varied from 3.47 m/s to 11.63 m/s, 3.40 m/s to 11.90 m/s, and 4.58 m/s to 12.59 m/s, respectively, with the most recorded for Jos. The PWPD augmented from 36.45 W/m2 (Akure) to 1000.06 W/m2 Jos), at a hub height of 10 m.Based on these results Jos was the best location for installing wind turbines while Kano was an excellent place for integrating the grid. Additionally, the Maiduguri location was determined to be suitable for a stand-alone application while Abuja and Akure were considered to be unsuitable for wind energy applications.

Individual Wave Overtopping Volumes On Mound Breakwaters

Mound breakwaters are widely used to protect harbors from wave attack. Wave overtopping is a key parameter on the breakwater design, since it affects the hydraulic stability, the port operativity and also generates risks to the facilities, vehicles and pedestrians. The estimation of the mean wave overtopping rate, q[m3/s/m], has been extensively analyzed in the literature (see EurOtop, 2018 and Van Gent et al., 2007). However, the maximum individual wave overtopping volume, Vmax [m3/m], can be much larger than q[m3/s/m] and it is a better variable to evaluate the direct hazards. The prediction tools of q and Vmax are mainly based on laboratory tests, where q is registered much more easily than individual wave overtopping volumes. However, few of these studies detail the methodology to identify the number of overtopping waves and the associated individual wave overtopping volumes. The estimation of Vmax is usually made in literature (see Molines et al., 2019 and EurOtop, 2018) using a 2-parameter Weibull distribution (shape and scale factor) fitted with utility functions which consider the 10%, 30% or 50% of the highest individual wave overtopping volumes. The shape factor (b) is fitted with the laboratory measurements and the scale factor (A) is obtained by forcing the mean value of the Weibull distribution to be equal to the registered mean individual wave overtopping. In this study, a fully automatic detection methodology of the individual wave overtopping waves and volumes is developed using 2D physical tests. The performance of the 2-parameter Weibull and Exponential distributions to estimate Vmax is analyzed here with four utility functions to weight the data, f(u). The full findings of this study are available in Molines et al. (2019).

Bayesian inference for the inverse Weibull distribution based on symmetric and asymmetric balanced loss functions with application

In this study, the unified hybrid censored approach is employed to estimate the parameters of the inverse Weibull distribution, as well as the survival and hazard rate functions. Parameter estimates are obtained using both Bayesian and Maximum Likelihood approaches, with Bayesian estimates acquired through Lindley's approximation method using three distinct balanced loss functions. These encompass both symmetric and asymmetric balanced loss functions, specifically the balanced squared error (BSE) loss function, the balanced linear exponential (BLINEX) loss function, and the balanced general entropy (BGE) loss function. We conduct a simulation study to compare the effectiveness of various estimators, and a real-world data analysis is presented to illustrate practical implementation. Ultimately, our findings indicate that Bayesian parameter estimates consistently outperform their Maximum Likelihood counterparts across all methods.

A solution to multi Objective Stochastic Optimal Power Flow problem using mutualism and elite strategy based Pelican Optimization Algorithm

Wind Energy is rapidly growing into one of the most popular energy options for power generation. However, the unpredictability of wind necessitates the employment of a distribution function when describing the wind environment. In the depiction of wind speeds, the two significant parameters of the Weibull distribution, such as the scale parameter and the shape parameter are often utilized. Therefore, it is essential to choose the most appropriate approach for estimating these parameters. Furthermore, the uncertainty in delegated power from variable wind speeds makes system management challenging. In this paper, the Weibull distribution parameters were determined using the Pelican Optimization Algorithm (POA). POA has gained extensive popularity among the research community and is currently used to address a wide range of optimization problems. However, the exploration and exploitation are not properly balanced in this algorithm. To overcome this problem a novel hybrid algorithm namely Mutualism phase based POA (MPOA) is proposed with the help of mutualism phase of Symbiotic Organisms Search (SOS) algorithm. To demonstrate the efficacy and robustness MPOA is applied on multi-objective Optimal Power Flow (MO-OPF) problems. For this study IEEE 30 bus standard system and modified IEEE 30 bus system consisting of two wind farms are taken. The purpose of this work is to examine the impact of changing the Weibull parameters, reverse and penalty cost coefficients on the overall generation cost while considering all the security aspects.

Reliability Model of Medical Equipment in University of Port Teaching Hospital

In this research, on medical equipment maintainability and reliability we conducted basic statistics analysis using University of Port Harcourt Teaching Hospital as the case study; the data collected covered 18 departments, namely; Anatomical Pathology, Micro Biology, Chemical Pathology Laboratory, Radiography Department, Pediatrics, Hemodialialysis, Hematology and Blood Transfusion, Physiotherapy, Dental Department, MDR-TB unit, Pharmacy, ICU, Assisted Conception Unit, Orthopedic Ward, Care for Elderly Laboratory, Family Planning Unit, Community Medicine and Labour Ward. The results of the parametric Weibull distribution percentile suggested that the reliability of the devices tends to fail every 21 days. The reliability plot of the model indicated that the devices tend to decrease its life span with age, the Weibull model was adequately fitted following the results of Anderson adjust test of goodness of fit and the probability plot. In comparison, the Probability value of goodness of fit P(0.0000) of Weibull distribution model was compared with that of exponential distribution model P(0.034), the outcome showed that Weibull distribution is better to model the data of medical equipment in University of Port Harcourt Teaching Hospital.

State-Based Technical Condition Assessment and Prediction of Concrete Box Girder Bridges

The technical condition of bridges has become a crucial issue for organizing the maintenance and repairs in bridge management systems. It is of great practical engineering significance to construct an effective model for predicting the technical condition degradation of the bridge through the use of the historical inspection data. Based on the semi-Markov random process, this paper proposes a useful deterioration prediction model for bridges in the highway network. From the historical inspection data of the prefabricated concrete box girder bridges, the degradation curves of technical condition rating are obtained. The effect of bridge length on degradation rate of the prefabricated concrete box girder bridges is analyzed. According to the Weibull distribution parameters of different condition grades, the technical state degradation models for a bridge group and an individual bridge are proposed to predict the performance of the overall bridge and superstructure of the bridge. The results show that with the increase in bridge length, the degradation rate of bridge technical condition increases. The degradation rate of the technical condition of the superstructure is faster than that of the overall bridge. The proposed semi-Markov stochastic degradation model for the bridge group can not only predict the different condition ratings of the bridges at any time, but also predict the future deterioration trend of an individual bridge under any ratings.

Shock fatigue damage failure boundary study of BGA solder joints based on shock response spectrum

Multiple shock damage failure of ball grid array (BGA) solder joints is one of the important reasons for reusable spacecraft mission failure. Shock fatigue damage failure boundary of BGA solder joints is proposed based on shock response spectrum (SRS), which can be used to assess the damage failure state. Weibull distribution is employed to characterize the shock fatigue life for a specified SRS. A Bayesian estimation method based on the Bootstrap approach is proposed to estimate Weibull distribution parameters. By using the time-domain load synthesized method based on wavelet and the experimentally validated shock fatigue life model, the shock fatigue life data of BGA solder joints corresponding to different SRS are extended. The variation patterns of the scale and shape parameters with the SRS parameters is analyzed. Then the shock fatigue damage failure boundary based on SRS is determined for a given design life and reliability. The results show that the critical SRS amplitude increase with the rise of the inflection point frequency and the low-frequency slope, respectively. The developed method is expected to provide references for the environment-adaptive design of BGA solder joints in reusable spacecraft.

Reliability Analysis of Compressive Fatigue Life of Cement Concrete under Temperature Differential Cycling

Cement concrete, as an extensively used engineering material, is omnipresent in various infrastructure projects such as bridges and roads. However, these structures often need to operate for extended periods under varying and harsh environmental conditions, facing not only complex vehicular loads but also the effects of temperature differential cycling. Consequently, understanding how temperature differential cycling impacts the compressive fatigue life of cement concrete has become a pivotal research topic. In this study, through a comprehensive experimental design, the fatigue life of cement concrete under typical temperature difference conditions (20–60°C) and different number of temperature differential cycling (60, 120, 180, 240, 300) was tested at three stress levels (0.70, 0.75, 0.85). Statistical analysis was conducted to obtain the Weibull distribution parameters of the compressive fatigue life of cement concrete. The Pf–S–N relationship of concrete considering reliability was analyzed, and a fatigue life prediction model under different reliability probabilities was established. The results show that the fatigue life of concrete subjected to temperature differential cycling follows a two-parameter Weibull distribution well. From the Pf–N curve, it can be seen that, regardless of the stress level, the calculated fatigue life under the same reliability probability decreases with the increase of temperature differential cycling times. At a 95% reliability probability, the decrease can reach 77.5%–87.5%. Based on the exponential function, a concrete fatigue life prediction model based on different reliability levels was established. Using this model, the S–lgN curve was plotted, and it was found that, regardless of the temperature differential cycling, an increase in reliability probability could lead to a 7.3%–14.4% reduction in logarithmic fatigue life (lgN). Additionally, this study also defined a fatigue life safety factor related to the number of temperature differential cycling and reliability probability, aiming to provide a theoretical basis for the design of cement concrete materials under the coupled environment of temperature differential cycling and fatigue loading.

Hybrid model development and nonlinear model predictive control implementation for continuous dry granulation process

This study focuses on the development of a hybrid model to integrate roll compaction and ribbon milling operations to design and control a continuous dry granulation process. The proposed hybrid model has three features: (1) it compensates for underestimated roll gap measurements with knurled rolls, (2) it represents the bimodal size distribution of granules using five fitting parameters of the bimodal Weibull distribution instead of only using specific size percentiles, and (3) it considers the impact of rotor-screen gaps on granules. Furthermore, the hybrid model facilitates the implementation of nonlinear model predictive control in a roller compactor Alexanderwerk WP120. Compared to widely applied open-loop operations in the pharmaceutical industry, nonlinear model predictive control demonstrates better performance, indicated by lower integral absolute errors in controlling mass throughput and ribbon solid fraction, in which real-time measurements can be obtained using a near-infrared sensor and a novel spectra selection approach.

Time-to-onset Analysis of Rhabdomyolysis due to Different Proton Pump Inhibitors Using a Pharmacovigilance Database.

Recent research has increasingly demonstrated an association between proton pump inhibitors (PPIs) and serious adverse events. This study aimed to evaluate the association between PPI and rhabdomyolysis (RM), examining its time-to-onset profiles using the Japanese Adverse Drug Event Report (JADER) database. Data spanning from April 2004 to March 2022 were used. The association between PPIs and RM was evaluated using the reporting odds ratio (ROR), adjusted for sex and age. Subsequent analyses were conducted after excluding cases involving concomitant use of statins or fibrates. Furthermore, the onset time of RM and Weibull distribution parameters were calculated to evaluate the expression profile of RM, and the outcomes were examined. RM was associated with the use of esomeprazole, omeprazole, and rabeprazole, even in the absence of concomitant statin or fibrate use. The median time to RM onset varied among PPIs, ranging from 6.5 to 127 d. The Weibull distribution parameters indicated that the hazard types of nearly all orally administered PPIs were classified as early failure or close to random failure. Regarding outcomes, cases of death were reported for all PPIs except vonoprazan. The findings suggest the need for vigilant monitoring of RM during PPI administration, particularly in the early stages, considering the varying onset times.

Optimizing Weibull Distribution Parameters for Improved Earthquake Modeling in Japan: A Comparative Approach

The Weibull distribution is considered one of the important distributions used in reliability and in the distribution of survival times and in neutrosophic prediction. This paper contained an estimate of a two-parameter Weibull distribution using three estimation methods. One of these methods for estimating parameters is the traditional method, which represents the estimation of the greatest possibility, and the other two methods are estimation using the sine algorithm. and cosine (SCA) and the Ant Colony Algorithm (ACO). The simulation method was used to compare the methods, and it was found that the best method for estimating the parameters of the Weibull distribution is the sine and cosine algorithm (SCA) method. Then, real data was used, represented by the intensity of the earthquake in Japan (Richter) for the period July 1, 2023, to July 16, 2023, to estimate the rate of earthquake intensity in Japan, since Japan is one of the countries most exposed to earthquakes, and it was shown from The results are that the average magnitude of the earthquake in Japan in the period studied is 3.328593, which can be said. Weak buildings may be greatly damaged, but strong buildings are not greatly damaged. Also, a neutrosophic simulation of the same set of data will be suggested for future applications.

Monitoreo de la confiabilidad del sistema hombre-máquina del área de mecanizado mediante la distribución de Weibull

This publication presents the development of a method that seeks to monitor the parameters β (shape) and η (scale) for each component-subsystem combination following the Weibull distribution, necessary for the calculation of the reliability of the man-machine system in the machining area. This system defines the workshops of the metal-mechanic, with high-mix and low-volume batch production where conventional and Computerized Numerical Control (CNC) machines are involved, which share the manufacturing of parts that sometimes are unique, or their manufacturing period is short. The design of the man-machine system is based on the analysis of the failures of non-conforming parts in the machining area and on the failure rates, which the statistical model is developed for its evaluation, considering the 2-parameter Weibull distribution, and a redundant system with series-parallel configuration. The results obtained were based on the theoretical-practical, using mathematical and statistical models, as well as the Study Case. With the use of mathematical and statistical models, it is demonstrated that the probability of failure (risk) of the man-machine system is time-dependent and is generated by mechanical type stresses, which occur in the manufacture of parts.

Probabilistic modeling of 10-min mean wind speed and its application in analytical simulation of snowdrift on building roofs

The typical resolution for long-term wind speed records that are publicly available in China is daily, this is too coarse for a sound analytical simulation of snowdrift on building roofs. Take Harbin, China as an example, an algorithm was proposed in this study to address this issue, where the commonly-used 2-parameter Weibull distribution was applied to fit the distribution of 10-min mean wind speed. A parameter estimation method, which combines the method of moment and cumulative probability, was proposed to estimate the parameters of Weibull distribution using very limited information on wind speed. The fitted probability model was validated using high-resolution wind speed data by comparing the snowdrift estimated by the modeled wind speed and that estimated by the actual wind speed. Finally, an analytical simulation of snowdrift on a flat roof was carried out to illustrate the application of the proposed model, and the probabilistic characteristic of the derived ground-to-roof conversion factors for snow loads were analyzed. It is indicated that the proposed model is easy to implement and provides a good estimation of the snowdrift on building roofs, and the derived conversion factors could be satisfactorily modeled using a Generalized Extreme Value (GEV) distribution or a normal distribution.

Estimation based on Ranked Set Sampling for Farlie--Gumbel--Morgenstern Bivariate Weibull Distribution Parameters with an application to medical data

Estimation based on Ranked Set Sampling for Farlie--Gumbel--Morgenstern Bivariate Weibull Distribution Parameters with an application to medical data

Experimental Study on the Compressive Strength and Fatigue Life of Cement Concrete under Temperature Differential Cycling

Concrete, as an engineering material with extremely wide applications, is widely used in various infrastructure projects such as bridges, highways, and large buildings. However, structures such as highways and bridges often need to be situated in variable and harsh service environments for long periods. They not only face cyclic reciprocating vehicle loads but also have to contend with the effects of temperature cycling. Therefore, studying the impact and mechanism of temperature differential cycling on the compressive strength and fatigue life of cement concrete has certain theoretical significance and practical value. This study employed a comprehensive experimental design to investigate cement concrete specimens subjected to typical temperature variations (20–60 °C) and different numbers of temperature differential cycling (0, 60, 120, 180, 240, 300). Axial compressive strength tests, ultrasonic tests, and compressive fatigue tests were conducted. The axial compressive strength test measured the compressive strength of the cement concrete. It was found that with an increase in the number of temperature differential cycling, the compressive strength exhibited a trend of an initial increase followed by a decrease: at 60 cycles, the strength increased by 10.8%, gradually declined; returned to near-initial strength at 120 cycles, and continued decreasing, reaching a decline of 19.4% at 300 cycles. The ultrasonic test measured the ultrasound velocity of the concrete specimens after different temperature differential cycling. It revealed a decreasing trend in ultrasound velocity with an increase in times of temperature differential cycling, showing a strong linear relationship between the ultrasound velocity loss and strength loss, confirming the correlation between the degree of concrete strength degradation and internal damage. The compressive fatigue test analyzed the fatigue life variation in cement concrete under different times of temperature differential cycling and stress levels, showing good adherence to the Weibull distribution pattern. Based on the approximation assumptions of log-normal distribution and the Weibull distribution, the Weibull distribution parameters for the compressive fatigue life of cement concrete under temperature differential cycling were obtained.

Wind power density characterization in arid and semi-arid Taita-Taveta and Garissa counties of Kenya

Wind Power Density (WPD) is a crucial parameter that can be used in assessing the potential of a given site for energy development and determining the suitability of wind turbine installation. A 7-year long-term data (2014–2020) of temperature, relative humidity, and wind speeds were obtained from Voi and Garissa synoptic station with a 3-h resolution. The objective of the study was to characterize wind power density in selected arid regions in Kenya. Analysis was performed using Weibull distribution parameters statistical tools i.e. Moment of Methods, Empirical Method (Justus), and Empirical Method (Lyssen), and error analysis using Mean Absolute Percentage Error, Mean Absolute Deviation (MAD), Coefficient of determination (R2) and Root Mean squared Error to determine the WPD accurate characteristics. Results show that Moment of Methods (MoM) performed better compared to other statistical tools, while the Taita Taveta had a better coefficient of Variance (CoV) ranging between 0.20 and 0.28% compared to 0.28–0.43% in Garissa. Based on the wind power density, the sites were found to be within Class II on the wind power classification from IEC and thus not viable for commercial power purposes. Results imply that power produced can be used in supplementing Kenya Offgrid Solar Access Project (KoSAP) which supplements power production used in gazetted marginalized counties by Kenya Power.

A Weibull process monitoring with AEWMA control chart: an application to breaking strength of the fibrous composite

In recent times, there has been a growing focus among researchers on memory-based control charts. The Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) charts and the adaptive control charting approaches got the attention. Control charts are commonly employed to oversee processes, assuming the monitored variable follows a normal distribution. However, it's worth noting that this assumption does not hold true in many real-world situations. The use of the algebraic expression for normalization, which can be used for all kinds of skewed distributions with a closed-form distribution function, using the proposed continuous function to adapt a smoothing constant, motivates this study. In the present manuscript, we design an EWMA statistic-based adaptive control chart to monitor the irregular variations in the mean of two parametric Weibull distribution and use Hasting approximation for normalization. The adaptive control charts are used to update the smoothing constant according to the estimated shift. Here we use the proposed continuous function to adapt the smoothing constant. The average run length and standard deviation of run length are calculated under different parameter settings. The effectiveness of the proposed chart is argued in terms of ARLs over the considered EWMA chart through Monte-Carlo (MC) simulation method. The proposed chart is examined, followed by a real data set to demonstrate the design and application procedures.

Random Furrowing for a Stochastic Unit Commitment Solution

The Unit Commitment Problem involves the inherent difficulty of obtaining optimal combinatorial power generation schedules over a future short term period. The formulation of the generalized Unit Commitment Schedule formulation involves the specific combination of generation units at several de-rated capacities during each hour of the planning horizon, the load demand profile, load indeterminateness and several other operating constraints. This largely deterministic schedule continues to find favor with several plant operators, keeping in mind the close operating time-periods involved. However, the deterministic nature of the load profile is sought to be phased out by a stochastic pattern that is realistic and mirrors real-life situations, owing to modern trends in Demand side management. This shift is in tune with the ongoing power restructuring activities of electricity power reforms. The stochastic profile is obtained by a suitably tuned 2-parameter Weibull distribution that uses appropriate shape and scale parameters. The resulting band of generated load profiles are used to evaluate net power and penal costs associated with a set of pervasive randomized probability indices. The exact UCS comprises of a specific unit absolute state corresponding to a certain time period within the planning horizon. Subsequently, regression analysis is applied to establish the correlation between the absolute states and the cumulative randomized load demand against the intervals within the planning horizon. This method is analogous to random furrowing of probabilistic demand profile.

From Zhentong Gao Method to Generalized Zhentong Gao Method

It is found that the data of structural fatigue life are mostly in accordance with Weibull distribution rather than Gaussian distribution, but the biggest obstacle affecting the application of Weibull distribution is that the estimation of its three- parameters will encounter difficulties in mathematical derivation and complicated calculation. Based on the estimation of correlation coefficient, the author has proposed Zhentong Gao method (Z.T. Gao method) which can avoid these difficulties. Further research found that the idea of Z.T. Gao method can be used to directly calculate the maximum value of the likelihood function of three-parameter Weibull distribution and estimate the three parameters simultaneously, which is called Generalized Zhentong Gao method. At the same time, it is also found that Nonlinear Programming method and Boltzmann Annealing method in machine learning can estimate the three parameters of Weibull distribution by directly calculating the maximum of likelihood function, so they can be regarded as Generalized Z. T. Gao methods in a sense. The results obtained by these methods are compared and some enlightening conclusions are obtained.