Burr Distribution Research Articles

A generalized Burr mixture autoregressive models for modeling non linear time series

A more flexible type of mixture autoregressive model, namely the Burr mixture autoregressive, BMAR model is studied in this article for modeling non linear time series. The model consists of a mixture of K autoregressive components with each conditional distribution of the component following a Burr distribution. The BMAR model enjoys some nice statistical properties which allow it to capture time series with: (1) unimodal or multimodal; (2) asymmetry or symmetry conditional distribution; (3) conditional heteroscedasticity; (4) cyclical or seasonal; and (5) conditional leptokurtic distribution. Sufficient and less restrictive conditions for the ergodicity of the BMAR model are derived and discussed. A more robust constrained optimization algorithm (EM – sequential quadratic programming method) is proposed for the non linear optimization problem. From the simulation studies carried out, the parameters estimation method showed satisfying results. The variance of the estimated parameters is also addressed with the missing information principle. Real datasets from two different fields of study are used to assess the performance of the BMAR model compared to other competing models. The comparison done in the empirical examples reveals the supremacy of the BMAR model in capturing the data behavior.

Multiparametric ultrasound imaging for early-stage steatosis: Comparison with magnetic resonance imaging-based proton density fat fraction.

The prevalence of liver diseases, especially steatosis, requires a more convenient and noninvasive tool for liver diagnosis, which can be a surrogate for the gold standard biopsy. Magnetic resonance (MR) measurement offers potential, however ultrasound (US) has better accessibility than MR. This study aims to suggest a multiparametric US approach which demonstrates better quantification and imaging performance than MR imaging-based proton density fat fraction (MRI-PDFF) for hepatic steatosis assessment. We investigated early-stage steatosis to evaluate our approach. An in vivo (within the living) animal study was performed. Fat inclusions were accumulated in the animal livers by feeding a methionine and choline deficient (MCD) diet for 2 weeks. The animals (n=19) underwent US and MR imaging, and then their livers were excised for histological staining. From the US, MR, and histology images, fat accumulation levels were measured and compared: multiple US parameters; MRI-PDFF; histology fat percentages. Seven individual US parameters were extracted using B-mode measurement, Burr distribution estimation, attenuation estimation, H-scan analysis, and shear wave elastography. Feature selection was performed, and the selected US features were combined, providing quantification of fat accumulation. The combined parameter was used for visualizing the localized probability of fat accumulation level in the liver; This procedure is known as disease-specific imaging (DSI). The combined US parameter can sensitively assess fat accumulation levels, which is highly correlated with histology fat percentage (R=0.93, p-value <0.05) and outperforms the correlation between MRI-PDFF and histology (R=0.89, p-value <0.05). Although the seven individual US parameters showed lower correlation with histology compared to MRI-PDFF, the multiparametric analysis enabled US to outperform MR. Furthermore, this approach allowed DSI to detect and display gradual increases in fat accumulation. From the imaging output, we measured the color-highlighted area representing fatty tissues, and the fat fraction obtained from DSI and histology showed strong agreement (R=0.93, p-value <0.05). We demonstrated that fat quantification utilizing a combination of multiple US parameters achieved higher performance than MRI-PDFF; therefore, our multiparametric analysis successfully combined selected features for hepatic steatosis characterization. We anticipate clinical use of our proposed multiparametric US analysis, which could be beneficial in assessing steatosis in humans.

Analysis of sample size for friction test of short fiber reinforced polymer composites

AbstractAccording to the recommendations of literature and standards, the minimum sample size for the friction test of thermoplastics is usually 3–5. While this size is valid for the friction test of neat polymer resins with homogeneous structures, its applicability to short fiber reinforced polymer composites with inhomogeneous structures remains unclear because the uneven distribution of discontinuous fibers or fillers in the matrix leads to the performance of the composites being dispersed. In this work, as a typical example, 100 friction samples made of short carbon fiber reinforced polyetherimide (SCF/PEI) composite with a typical content of 10 vol% SCFs are prepared, and their friction coefficients are measured. The maximum likelihood estimation and goodness of fit methods are employed to assess the fitting degree between the overall friction coefficient results and five mathematical distributions. The results show that the overall friction coefficient data distribution is in good agreement with the Lognormal, Generalized extreme value (GEV), and Burr distributions, respectively. Based on the fitting result, the minimum sample size is analyzed by Monte Carlo Simulation. It shows that the minimum sample size strongly depends on the property dispersion parameter (COV: coefficient of variation). Based on this parameter, the minimum sample size is given. In a practical test, the performance dispersion degree of a composite cannot be known in advance, hence a sample size determination method is proposed to yield a reasonable performance description parameter by applying the results of this work, and this method is validated by the overall test data.Highlights Proposed a sample size determination method for practical friction tests. The minimum sample size for the friction test is yielded statistically. The minimum sample size strongly depends on a property dispersion parameter.

The Inverse Burr-Generalized Family of Distributions: Theory and Applications

This paper presents a new class of generalized distributions based on the inverse Burr (Burr III) distribution. Statistical properties of the proposed family of distributions such as the density and cumulative distribution functions, survival and hazard rate functions, quantile, moments, moment generating function, probability weighted moments, Renyi entropy and distribution of order statistics are derived. The maximum likelihood estimation method is employed to obtain the parameter estimates of the family of distributions. A Monte Carlo simulation study is conducted in order to investigate the asymptotic behaviour of the parameter estimates of a sub-model from the proposed family of distributions. Finally, the utility of proposed family of distributions in lifetime data fittings is illustrated using two real data sets and the results obtained were compared to some existing non-nested models. Based on some model selection criteria and goodness of fit test statistics, it was evident that the sub-model from the proposed family of distributions performed reasonably better than the competitor distributions in fitting the two data sets.

Server load estimation by Burr distribution mixture analysis of TCP SYN response time

Server load estimation is key in balancing traffic between servers when optimizing data center resources. Intrusive methods are sometimes difficult or impossible to implement. Therefore, non-intrusive estimation methods are the best alternative in these cases. The objective of this paper is to present a server load estimation method based on external network traffic measurements obtained in a vantage point close to the server. Statistical distributions of TCP SYN response time, that is, the time from SYN to SYN+ACK segments at the server side, are used to fit Burr Type XII heavy tail distribution mixtures. The fitting algorithm, based on maximum likelihood estimation, is developed in detail in this paper. Experimental data shows that the median of the fitted distribution correlates within the 95% confidence interval of the server load figures and, thus, it can be used as a non-intrusive and accurate method to measure it. This new method can be applied to almost any existing load balancing algorithm, as it does not make any assumption about the server, which is considered a black box.

An Integrated Model of Maintenance Policies and Economic Design of X-bar Control Chart Under Burr XII Shock Model

The combination of different methods of maintenance and economic design of control charts has received much attention from researchers in recent years. These maintenance policies increase the reliability of a system. So far, only a small number of failure distributions, with fixed or increasing hazard rates, have been used as a shock model in the integrated model of maintenance and economic design of X-bar control chart. A proper alternative for this situation may be the Burr distribution, in which the hazard rate has general form. In this paper, for the economic design of the X-bar control chart under the Burr XII shock model, we have presented an integrated model using preventive maintenance and incomplete maintenance. The obtained integrated model has been implemented using a numerical example and its sensitivity analysis has been performed. The results show that, the use of higher maintenance levels causes further reduction of the average cost per unit time of the quality cycle. The results of sensitivity analysis also show that our proposed model is not sensitive to the misspecification of the Burr XII distribution parameters.

Modelling and experimental validation of burr control in micro milling of metals

Classical deformation theories are limited to predicting microscopic deformation when the machining parameters are comparable to the dimension of a single grain. Micro-cutting is one such process where the physical phenomenon is significantly influenced by the microstructure characteristics and initial state of stress. The micro-features such as burrs caused owing to these cutting processes, are undesired and affect the quality of the product. In this study, an analytical methodology is proposed, supported by FEM pre-processing, to determine the burr morphology during the micro-milling process for Ti-6Al-4V, Al-6061 and OFHC-Cu. At first, the flow stress is calculated with the help of strain gradient and dynamic recrystallization methods; contact pressure on the cutting edge and minimum chip thickness are introduced through an FE simulation, this assists to find out the axial and radial plastic strain which eventually yields the dimensions of individual burrs. Peening of the material surface prior to micro-milling provided 67 % less formation of top burr. The average burr width and height are measured and validated with predicted results. The grain structure and dynamic recrystallization of the micro-milled surface are revealed by electron backscatter diffraction. Further, the burr distribution from FE modelling is verified with an SEM image.

Distribution cycle of wind speed: A case study in the Southern Part of Malaysia

Wind energy has historically helped many people and has been utilized for farming and seafaring. Today, modern usage is more comprehensive such as generating electricity. However, its usage in Malaysia remains at an all-time low. Numerous factors contribute to this result. Consequently, a full assessment of wind energy in a particular region is deemed necessary. This study investigates wind speed distribution in Mersing. It also illustrates the period during which the regional distribution statistics changed. The researcher used Autofit 5.6 and the Maximum Likelihood Method to figure out the parameter value of each distribution for wind speed data from 2012 to 2016. This study utilized three distinct methods of Goodness of fit (GOF) to estimate the optimal distribution ranking, such as Chi-Square, Anderson-Darling, and Kolmogorov-Smirnov. The results reveal that the Burr, Gamma, and Lognormal distributions correlate to the wind speed data in Mersing and the Burr distribution is the most appropriate. In addition, the findings indicate that distribution changes occur at a minimum rate of 8.89%. In conclusion, these findings reveal that there has been no significant change over the five years. This information enables stakeholders to determine whether they will earn a profit owing to changes in the distribution of a particular area, reducing risk.

Second Order Expansions for Extreme Quantiles of Burr Distributions and Asymptotic Theory of Record Values

In this paper we investigate the Burr distributions family which contains twelve members. Second order expansions of quantiles of the Burr’s distributions are provided on which may be based statistical methods, in particular in extreme value theory. Beyond the proper interest of these expansions, we apply them to characterize the asymptotic laws of their records of Burr’s distributions, lead to new statistical tests.

Envelope distribution and Doppler spectrum of V2V channels at 5.9 GHz in mountainous roads

Land transportation of people and goods often relies on strategic routes that pass through mountains. Despite the importance of these routes, the characteristics of the vehicular radio communication channel in such environments have received little attention. To close the gap, this paper presents the results of a measurement campaign at 5.9 GHz in mountainous roads. Data collected during the measurement campaign were used to quantify the envelope distribution and Doppler spectrum of small-scale vehicle-to-vehicle (V2V) communication channels. The measurements were taken in three different locations, two in San Luis Potosí (SLP), México, and one near Provo, Utah, USA. The locations in SLP are shrubland areas, whereas that in Utah is a coniferous forest. The diverse mountainous environments observed during the experiments were classified into five reference propagation scenarios. For each reference scenario the empirical channel envelope distribution was computed and compared with previously-published theoretical models based on the Rice, Nakagami, Weibull, and Gamma distributions. The Rice model provided the best fitting in 72.3% of the analyzed cases, whereas the Nakagami, Weibull and Gamma distributions were the best option in the remaining 11.3%, 14.9% and 1.3%, respectively. Likewise, the empirical distributions of the instantaneous mean Doppler shift (IMDS) and instantaneous Doppler spread (IDS) were computed. The results show that the IMDS follows the Generalized Extreme Value and Logistic distributions, whereas the Birnbaum-Saunders, Exponential, Burr and Nakagami distributions are proper models for the IDS. Our analysis also shows that the line-of-sight (LOS) path is the dominant factor for V2V communication in mountainous terrain, especially in roads surrounded by rocky structures with sparse vegetation. However, the contribution of non-LOS paths is not negligible, and it appears capable of supporting short intervals of communication. The results presented here can be used as guidelines for the analytical modeling and computer simulation of small-scale V2V channels in mountainous environments.

Performance of bituminous paving mixtures containing recycled concrete aggregate, fly ash and rice husk ash: A probabilistic approach

Due to the rapid increase in construction activities, the depletion of natural resources is accelerating. Extensive research has focused on finding alternative materials to conserve resources and minimize costs. This study explores the utilization of recycled concrete aggregate (RCA) as a substitute for coarse aggregate, along with waste materials like fly ash (FA) and rice husk ash (RHA) as fillers in bituminous paving mixtures. The aim is to address environmental concerns and waste disposal issues. The research evaluates the performance of different filler types in non-conventional bituminous mixtures using VG 30 grade bitumen as a binder, specifically for lower bituminous layers. A typical bitumen concentration of 5% by weight of dense bituminous macadam (DBM) mixtures is chosen based on existing literature. The study assesses various performance characteristics, including Marshall parameters, indirect tensile strength, and moisture susceptibility. A probabilistic approach is employed to determine the most suitable distribution model, focusing on retained stability (RS) as a measure of moisture susceptibility. For this, fifteen distribution functions were considered with three Goodness-of-fit (GOF) tests. To identify the best suitable function, 50 numbers of samples were prepared to find RS with two types of fillers and two types of coarse aggregate. Experimental results indicate that mixtures with natural aggregate exhibit slightly superior performance, but those with RCA meet all the requirements specified in relevant Indian codes and specifications. The probability analysis confirms that the Burr distribution best fits the experimental data, particularly for moisture susceptibility characteristics.

Scattering angle resolved optical coherence tomography measures morphological changes in Bacillus subtilis colonies.

An unmet need is recognized for early detection and diagnosis of neurological diseases. Many psychological markers emerge years after disease onset. Mitochondrial dysfunction and corresponding neurodegeneration occur before onset of large-scale cell and tissue pathology. Early detection of subcellular morphology changes could serve as a beacon for early detection of neurological diseases. This study is on bacterial colonies, Bacillus subtilis, which are similar in size to mitochondria. This study investigates whether morphological changes can be detected in Bacillus subtilis using scattering angle resolved optical coherence tomography (SAR-OCT). The SAR-OCT was applied to detect scattering angle distribution changes in Bacillus subtilis. The rod-to-coccus shape transition of the bacteria was imaged, and the backscattering angle was analyzed by recording the distribution of the ratio of low- to medium angle scattering (L/M ratio). Bacillus orientation at different locations in colonies was analytically modeled and compared with SAR-OCT results. Significant differences in the distribution of backscattering angle were observed in Bacillus subtilis transitioning from rod-to-coccus shapes. In Bacillus subtilis, the -parameter of the Burr distribution of the SAR-OCT-derived L/M ratio was significantly smaller in coccus compared with rod-shaped bacteria. SAR-OCT-derived L/M ratio varied with bacterial position in the colony and is consistent with predicted orientations from previous studies. Study results support the potential of utilizing SAR-OCT to detect bacterial morphological changes.

Frequency Characteristic Analysis of Acoustic Emission Signals of Pipeline Leakage

The leakage detection of a water distribution system (WDS) needs the support of a large number of field data. This paper collected over 6800 leak detection signals from cast iron pipelines used in a WDS. We found that 3280 signals indicated leakage, and the remaining indicated no leakage. The characteristics of the signals were extracted and analyzed from three perspectives: the central frequency of the power spectrum, the spectral roll-off rate, and the spectral flatness. Significant statistical distributions were found. The central frequencies of the leakage signals followed the normal distribution, and their spectral roll-off rates demonstrated the Burr distribution; the Birnbaum–Saunders distribution could describe the spectral flatness of the signals. Based on these characteristics, the recognition rate of the ML model for leak detection was improved. The Random Forest model was used to classify the leakage detection signals. The recall rate was 100%, and the false positive rate was 8.27%.

Gaussian convolution decomposition for non-Gaussian shaped pulsed LiDAR waveform

The full waveform decomposition technique is significant for LiDAR ranging. It is challenging to extract the parameters from non-Gaussian shaped waveforms accurately. Many parametric models (e.g. the Gaussian distribution, the lognormal distribution, the generalized normal distribution, the Burr distribution, and the skew-normal distribution) were proposed to fit sharply-peaked, heavy-tailed, and negative-tailed waveforms. However, these models can constrain the shape of the waveform components. In this article, the Gaussian convolution model is established. Firstly, a set of Gaussian functions is calculated to characterize the system waveform so that asymmetric and non-Gaussian system waveforms can be included. The convolution result of the system waveform and the target response is used as the model for fitting the overlapped echo. Then a combination method of the Richardson–Lucy deconvolution, layered iterative, and Gaussian convolution is introduced to estimate the initial parameters. The Levenberg–Marquardt algorithm is used for the optimization fitting. Through experiments on synthetic data and practical recorded coding LiDAR data, we compare the proposed method with two decomposition approaches (Gaussian decomposition and skew-normal decomposition). The experiment results revealed that the proposed method could precisely decompose the overlapped non-Gaussian heavy-tailed waveforms and provide the best ranging accuracy, component fitting accuracy, and anti-noise performance. However, the traditional Gaussian and skew-normal decomposition methods can not fit the components well, resulting in inaccurate range estimates.

An NHPP software reliability model with a fault detection rate function of the Burr type XII distribution

The software reliability model is developed considering the number of software failures and the interval time between failures, and is a very useful model for predicting software failures that may occur in the future. In most existing software reliability model, the mean value function is considered to follow a non-homogeneous Poisson process (NHPP). NHPP software reliability model has a mean value function that considers the failure intensity, detection rate, number of remaining failures, and various environments. The Burr distribution can fit a wide range of empirical data. In addition, it was proposed several types of the cumulative functions and has stressed the advantages obtained by the direct use of the cumulative function. The Burr type XII distribution is a continuous probability distribution for a non-negative random variable. In this paper, we propose a new software reliability model with a fault detection rate function of the Burr type XII distribution based on NHPP. We show that the NHPP software reliability model proposed through various criteria shows much better the goodness-of-fit than other existing NHPP software reliability models.

Appropriate Description of Probability Distribution of Prostrate Specific Antigen (PSA): An Aid to Early Detection of Prostrate Cancer

Prostrate cancer has been observed to be a worrisome ailment among adult males across the globe over some decades. Just like any other ailment, its early detection and treatment gives the patient a better chance of survival. Measuring the Prostrate specific antigen (PSA) of the patients at specified intervals is one of the techniques of detecting the onset of prostrate cancer. This paper attempts to obtain the most appropriate probability distribution of prostrate specific antigens (PSA) that gives the best estimates of the parameters of the distribution. The age specific probabilities of patients with prostrate specific antigen (PSA)&gt;4.0ng/ml were calculated. The results show that the most appropriate distribution among the distributions fitted is the Burr distribution. The probability of observing a PSA greater than 4.0 ng/ml in adult males above the age of 45 years was seen to be moderately high (0.68) with a median PSA of 8.30 ng/ml and an inter quartile range of 16.40. The results also show that PSA of 60.0 ng/ml and above were exceptional cases that require urgent attention. It was observed that men whose ages are between 60 and 80 years have higher chances of having PSA values greater 4.0ng/ml with men between 60 and 75 years having the highest chances. It is therefore suggested that men with PSA higher than 4.0ng/ml between 60 and 80 years of age may not be subjected to needle biopsies but be referred directly for digital rectal examinations.

Performance Analysis of Multi-User NOMA Wireless-Powered mMTC Networks: A Stochastic Geometry Approach

In this paper, we aim to improve the connectivity, scalability, and energy efficiency of machine-type communication (MTC) networks with different types of MTC devices (MTCDs), namely Type-I and Type-II MTCDs, which have different communication purposes. To this end, we propose two transmission schemes called connectivity-oriented machine-type communication (CoM) and quality-oriented machine-type communication (QoM), which take into account the stochastic geometry-based deployment and the random active/inactive status of MTCDs. Specifically, in the proposed schemes, the active Type-I MTCDs operate using a novel Bernoulli random process-based simultaneous wireless information and power transfer (SWIPT) architecture. Next, utilizing multi-user power-domain non-orthogonal multiple access (PD-NOMA), each active Type-I MTCD can simultaneously communicate with another Type-I MTCD and a scalable number of Type-II MTCDs. In the performance analysis of the proposed schemes, we prove that the true distribution of the received power at a Type-II MTCD in the QoM scheme can be approximated by the Singh-Maddala distribution. Exploiting this unique statistical finding, we derive approximate closed-form expressions for the outage probability (OP) and sum-throughput of massive MTC (mMTC) networks. Through numerical results, we show that the proposed schemes provide a considerable sum-throughput gain over conventional mMTC networks.

MODELING STOCK-CLOSING PRICES: EVIDENCE FROM THE BANKING SECTOR OF JORDAN

This study aims to find the best-fitted distribution of the average closing price (ACP) for the Amman Stock Exchange banking sector using various distributions models, namely, Normal, Gamma, Burr, Weibull, Bareto, lognormal, Log logistic, Exponential, and Gumbel distribution. This will enable financial analysts to forecast stock prices to build their buying and selling decisions. The methodology was applied according to the comparison between maximum likelihood estimation, moment matching, quantile matching and maximum goodness-of-fit estimation. The result of computing the different goodness of fit statistics (K.S), (CvM) and (A.D) and the other statistics based on the log-likelihood (AIC and BIC). The best-fitted distributions have the smaller values for each goodness of fit statistics. The results emphasized that the Gamma distribution is the most appropriate one, which has the lowest values of AIC (364.6), and BIC (373.9).

Probabilistic assessment of ACI 318 minimum thickness requirements for two-way slabs

ABSTRACT Following the ACI 318 design code and to control deflections, structural engineers must adhere to minimum thickness requirements for reinforced concrete (RC) slabs. However, these requirements do not account for various influencing variables such as span length, aspect ratio and applied live load. This paper assesses the adequacy of the ACI 318 minimum thickness requirements for two-way RC slabs through a probabilistic approach using Monte Carlo simulation (MCS). The probability distribution of the span-to-depth ratio (SDR) was fitted to the Burr distribution, and the goodness-of-fit test was applied to assess the fitting. Then, the effect of span length, aspect ratio, and live load on the maximum allowable SDR was evaluated, considering all material properties and applied loads as random variables. The analyses have shown a high correlation between the studied variables and the SDR. Therefore, multiple regression analysis was carried out, and four prediction models were proposed to conservatively predict the SDR for two-way slabs. The accuracy of the prediction models was statistically tested by comparing calculated to predicted SDR values, and a 10% maximum relative error was exhibited. The proposed prediction models were accurate for all practical ranges of span length, aspect ratio and live load in two-way slabs.

Quantifying bus travel time variability and identifying spatial and temporal factors using Burr distribution model

Travel time variability (TTV) is the key indicator used in assessing the service quality of bus transit system. This study explores the most appropriate model to describe the day-to-day TTV of bus section. By investigating a 7-month travel time data for 10 bus routes in Klang Valley, Malaysia, this study demonstrates that Burr distribution is the most promising model in describing bus TTV. Bus TTV is found to be sensitive to both temporal and spatial effect. This means that TTV service varies for weekdays and weekends (temporal). Also, it differs for the five operating environments (spatial) investigated in this study. The Burr regression analysis conducted in the second part of this study further confirmed that bus section length and traffic signal density are the major contributing factors to bus TTV. However, both factors have varying levels of impact under different spatiotemporal effect. For example, in the suburban and residential areas, these factors cause higher TTV on weekends but lesser during weekdays, while a vice versa impact is observed in the Central Business District. This distinguishes from earlier studies which purely assumed normality in the regression analysis while not emphasizing the importance of spatiotemporal factors on TTV. Thus, this study serves as an analysis tool that could be used in the planning of bus routes and schedules under varying bus operating environments and operation times.