Real Datasets Research Articles

Addressing the Non-Stationarity and Complexity of Time Series Data for Long-Term Forecasts

Real-life time series datasets exhibit complications that hinder the study of time series forecasting (TSF). These datasets inherently exhibit non-stationarity as their distributions vary over time. Furthermore, the intricate inter- and intra-series relationships among data points pose challenges for modeling. Many existing TSF models overlook one or both of these issues, resulting in inaccurate forecasts. This study proposes a novel TSF model designed to address the challenges posed by real-life data, delivering accurate forecasts in both multivariate and univariate settings. First, we propose methods termed “weak-stationarizing” and “non-stationarity restoring” to mitigate distributional shift. These methods enable the removal and restoration of non-stationary components from individual data points as needed. Second, we utilize the spectral decomposition of weak-stationary time series to extract informative features for forecasting. To learn features from the spectral decomposition of weak-stationary time series, we exploit a mixer architecture to find inter- and intra-series dependencies from the unraveled representation of the overall time series. To ensure the efficacy of our model, we conduct comparative evaluations against state-of-the-art models using six real-world datasets spanning diverse fields. Across each dataset, our model consistently outperforms or yields comparable results to existing models.

Classification of bioactive peptides: A systematic benchmark of models and encodings

Bioactive peptides are short amino acid chains possessing biological activity and exerting physiological effects relevant to human health. Despite their therapeutic value, their identification remains a major problem, as it mainly relies on time-consuming in vitro tests. While bioinformatic tools for the identification of bioactive peptides are available, they are focused on specific functional classes and have not been systematically tested on realistic settings. To tackle this problem, bioactive peptide sequences and functions were here gathered from a variety of databases to generate a unified collection of bioactive peptides from microbial fermentation. This collection was organized into nine functional classes including some previously studied and some unexplored such as immunomodulatory, opioid and cardiovascular peptides. Upon assessing their sequence properties, four alternative encoding methods were tested in combination with a multitude of machine learning algorithms, from basic classifiers like logistic regression to advanced algorithms like BERT. Tests on a total of 171 models showed that, while some functions are intrinsically easier to detect, no single combination of classifiers and encoders worked universally well for all classes. For this reason, we unified all the best individual models for each class and generated CICERON (Classification of bIoaCtive pEptides fRom micrObial fermeNtation), a classification tool for the functional classification of peptides. State-of-the-art classifiers were found to underperform on our realistic benchmark dataset compared to the models included in CICERON. Altogether, our work provides a tool for real-world peptide classification and can serve as a benchmark for future model development.

Empirical uncertainty evaluation for the pose of a kinematic LiDAR-based multi-sensor system

Abstract Kinematic multi-sensor systems (MSS) describe their movements through six-degree-of-freedom trajectories, which are often evaluated primarily for accuracy. However, understanding their self-reported uncertainty is crucial, especially when operating in diverse environments like urban, industrial, or natural settings. This is important, so the following algorithms can provide correct and safe decisions, i.e. for autonomous driving. In the context of localization, light detection and ranging sensors (LiDARs) are widely applied for tasks such as generating, updating, and integrating information from maps supporting other sensors to estimate trajectories. However, popular low-cost LiDARs deviate from other geodetic sensors in their uncertainty modeling. This paper therefore demonstrates the uncertainty evaluation of a LiDAR-based MSS localizing itself using an inertial measurement unit (IMU) and matching LiDAR observations to a known map. The necessary steps for accomplishing the sensor data fusion in a novel Error State Kalman filter (ESKF) will be presented considering the influences of the sensor uncertainties and their combination. The results provide new insights into the impact of random and systematic deviations resulting from parameters and their uncertainties established in prior calibrations. The evaluation is done using the Mahalanobis distance to consider the deviations of the trajectory from the ground truth weighted by the self-reported uncertainty, and to evaluate the consistency in hypothesis testing. The evaluation is performed using a real data set obtained from an MSS consisting of a tactical grade IMU and a Velodyne Puck in combination with reference data by a Laser Tracker in a laboratory environment. The data set consists of measurements for calibrations and multiple kinematic experiments. In the first step, the data set is simulated based on the Laser Tracker measurements to provide a baseline for the results under assumed perfect corrections. In comparison, the results using a more realistic simulated data set and the real IMU and LiDAR measurements provide deviations about a factor of five higher leading to an inconsistent estimation. The results offer insights into the open challenges related to the assumptions for integrating low-cost LiDARs in MSSs.

On Estimating Multi- Stress Strength Reliability for Inverted Kumaraswamy Under Ranked Set Sampling with Application in Engineering

The terrible operating constraints of many real-world events cause systems to malfunction regularly. The failure of systems to perform their intended duties when they reach their lowest, highest, or both extreme operating conditions is a phenomenon that researchers rarely focus on. The multi-stress strength reliability R=P(W<X<Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R = P(W<X<Z)$$\\end{document} is deemed in this study for a component whose strength X falls between two stresses, W, and Z, where X, W, and Z are independently inverted Kumaraswamy distributed. Both maximum likelihood and maximum product spacing procedures are employed to obtain the reliability estimator under simple random sampling (SRS) and ranked set sampling (RSS) methodologies. Four scenarios for reliability estimators are considered. The reliability estimator in the first and second cases can be determined by applying the same sample design (RSS/SRS) to the strength and stress distributions. When the sample data for W and Z originate from RSS while those for X are acquired from SRS, the third reliability estimator is calculated. The drawn data of the strength and stress random variables, which are obtained from SRS and RSS, respectively, are taken into consideration in the final scenario. The effectiveness of the suggested estimators is compared using a comprehensive computer simulation. Lastly, three real data sets have been used to determine reliability estimators.

Bayesian Parameter Estimation for Geometric Process with Rayleigh Distribution

The main purpose of this study is to deal with the parameter estimation problem for the geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Rayleigh. For this purpose, maximum likelihood and Bayesian parameter estimation methods are discussed. Lindley and MCMC approximation methods are used in Bayesian calculations. Additionally, a novel method called the Modified-Lindley approximation has been proposed as an alternative to the Lindley approximation. An extensive simulation study was conducted to compare the performances of the prediction methods. Finally, a real data set is analyzed for illustrative purposes.

STBGRN: A Traffic Prediction Model Based on Spatiotemporal Bidirectional Gated Recurrent Units and Graph Convolutional Residual Networks

With the development of society and the advancement of urbanization, the development of intelligent transportation system has attracted much attention. Efficient and accurate traffic flow prediction is one of the core tasks in the research of intelligent transportation system. Most of the existing spatiotemporal traffic flow prediction models do not make full use of various periodic spatiotemporal dynamic characteristics of traffic flow, and it is difficult to effectively capture the complex spatiotemporal changes of traffic flow. To accurately predict the traffic flow of complex urban network, this paper proposes a traffic prediction model (STBGRN) based on spatiotemporal bidirectional gated cycle unit (ST-BiGRU) and graph convolution residual network (GCN-ResNet). The spatiotemporal information in traffic data is learned using the spatiotemporal bidirectional GRU coupled with the forward and backward information of the current time step, and the topology structure of the traffic network is captured by the graph convolution residual network, which is combined to complete the prediction task of the urban traffic network. In this paper, STBGRN was tested on two real data sets, SZ-taxi and Los-loop, and compared with other baseline methods, the RMSE index reached 3.857 and 5.461, and the MAE index reached 2.702 and 3.781, respectively. Accuracy index is 72.66% and 91.35%, R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}^{2}$$\\end{document} index is 0.858 and 0.845, var index is 0.858 and 0.849. The experimental results show that STBGRN model can obtain spatiotemporal dependence from complex traffic data, and can be used in spatiotemporal traffic data processing.

Aircraft routing using dynamic programming and reinforcement learning: A customer-centric approach

Over the past few decades, the airline industry has evolved into a sophisticated business, with conventional objectives such as operational efficiency, effective fleet assignment, flight punctuality, and equitable crew scheduling being the primary focus. Nevertheless, given the growing accessibility to customer feedback (either through public forums or private data), it is crucial to make business decisions that cater to the customers’ requirements in addition to conventional goals. Within aircraft operations, there is a series of decisions to be made, such as flight scheduling, fleet assignment, aircraft routing, and crew scheduling. After designating a fleet type to each flight leg, airlines ascertain the sequence of flight legs, or strings, to be operated by individual aircrafts, while complying with obligatory maintenance checks. This constitutes the aircraft routing problem.We introduce a novel approach to aircraft routing problem, by infusing traditional methodologies with insightful customer feedback to create a more responsive and customer-centric model. Unlike conventional techniques of constructing routes, in our work, we construct maintenance feasible strings. Traditional aircraft routing methodologies typically concentrate on minimizing propagated delay. However, by integrating customer feedback, more effective prioritization to minimize delays for flight legs with higher customer dissatisfaction can be achieved. We propose an optimization model that strives to minimize propagated delay while prioritizing flights based on customer feedback, thus resulting in customer-aware routes. To solve the proposed optimization model, we design an iterative alternating optimization scheme where feasible strings are first constructed from the pool of available flight legs using a string generation procedure, followed by solving an integer linear optimization problem over only the feasible set of strings obtained in the first stage. To construct feasible strings of flights at each iteration, we use two different methods. The first method is based on a dynamic programming approach which is useful for mean delay information. To handle practical cases where delays are uncertain, we propose a reinforcement learning (RL) approach to construct the strings. We provide experiments of our proposed methods on synthetic and real data sets and demonstrate the effectiveness of our RL model in comparison to dynamic programming based model. Our experiments affirm that routing decisions informed by customer feedback prioritize minimizing delays for specific flight legs, resulting in routes that are inherently customer-friendly.

Framework for Ranking Machine Learning Predictions of Limited, Multimodal, and Longitudinal Behavioral Passive Sensing Data: Combining User-Agnostic and Personalized Modeling.

Passive mobile sensing provides opportunities for measuring and monitoring health status in the wild and outside of clinics. However, longitudinal, multimodal mobile sensor data can be small, noisy, and incomplete. This makes processing, modeling, and prediction of these data challenging. The small size of the data set restricts it from being modeled using complex deep learning networks. The current state of the art (SOTA) tackles small sensor data sets following a singular modeling paradigm based on traditional machine learning (ML) algorithms. These opt for either a user-agnostic modeling approach, making the model susceptible to a larger degree of noise, or a personalized approach, where training on individual data alludes to a more limited data set, giving rise to overfitting, therefore, ultimately, having to seek a trade-off by choosing 1 of the 2 modeling approaches to reach predictions. The objective of this study was to filter, rank, and output the best predictions for small, multimodal, longitudinal sensor data using a framework that is designed to tackle data sets that are limited in size (particularly targeting health studies that use passive multimodal sensors) and that combines both user agnostic and personalized approaches, along with a combination of ranking strategies to filter predictions. In this paper, we introduced a novel ranking framework for longitudinal multimodal sensors (FLMS) to address challenges encountered in health studies involving passive multimodal sensors. Using the FLMS, we (1) built a tensor-based aggregation and ranking strategy for final interpretation, (2) processed various combinations of sensor fusions, and (3) balanced user-agnostic and personalized modeling approaches with appropriate cross-validation strategies. The performance of the FLMS was validated with the help of a real data set of adolescents diagnosed with major depressive disorder for the prediction of change in depression in the adolescent participants. Predictions output by the proposed FLMS achieved a 7% increase in accuracy and a 13% increase in recall for the real data set. Experiments with existing SOTA ML algorithms showed an 11% increase in accuracy for the depression data set and how overfitting and sparsity were handled. The FLMS aims to fill the gap that currently exists when modeling passive sensor data with a small number of data points. It achieves this through leveraging both user-agnostic and personalized modeling techniques in tandem with an effective ranking strategy to filter predictions.

High-Speed Surface Property Recognition with a 140 GHz Frequency

In the field of integrated sensing and communication, there is a growing need for advanced environmental perception. The terahertz (THz) frequency band, significant for ultra-high-speed data connections, shows promise in environmental sensing, particularly in detecting surface textures crucial for autonomous systems’ decision-making. However, traditional numerical methods for parameter estimation in these environments struggle with accuracy, speed, and stability, especially in high-speed scenarios like vehicle-to-everything communications. This study introduces a deep learning approach for identifying surface roughness using a 140-GHz setup tailored for such conditions. A high-speed data acquisition system was developed to mimic real-world scenarios, and a diverse set of rough surface samples was prepared for realistic high-speed datasets to train the models. The model was trained and validated in three challenging scenarios: random occlusions, sparse data, and narrow-angle observations. The results demonstrate the method’s effectiveness in high-speed conditions, suggesting terahertz frequencies’ potential in future sensing and communication applications.

Child face recognition at scale: synthetic data generation and performance benchmark

We address the need for a large-scale database of children’s faces by using generative adversarial networks (GANs) and face-age progression (FAP) models to synthesize a realistic dataset referred to as “HDA-SynChildFaces”. Hence, we proposed a processing pipeline that initially utilizes StyleGAN3 to sample adult subjects, which is subsequently progressed to children of varying ages using InterFaceGAN. Intra-subject variations, such as facial expression and pose, are created by further manipulating the subjects in their latent space. Additionally, this pipeline allows the even distribution of the races of subjects, allowing the generation of a balanced and fair dataset with respect to race distribution. The resulting HDA-SynChildFaces consists of 1,652 subjects and 188,328 images, each subject being present at various ages and with many different intra-subject variations. We then evaluated the performance of various facial recognition systems on the generated database and compared the results of adults and children at different ages. The study reveals that children consistently perform worse than adults on all tested systems and that the degradation in performance is proportional to age. Additionally, our study uncovers some biases in the recognition systems, with Asian and black subjects and females performing worse than white and Latino-Hispanic subjects and males.

Alpha–beta-power family of distributions with applications to exponential distribution

This article proposes a new way to increase the flexibility of a family of statistical distributions by adding two additional parameters. The newly proposed family is called the alpha–beta power transformation family of distributions. A specific model, the alpha–beta power exponential distribution, was thoroughly investigated. The hazard rates of the proposed distribution can be decreasing, bathtub-shaped, and unimodal. The derived structural features of the proposed model include explicit formulations for the quantiles, the moments, the moment-generating function, and the incomplete and conditional moments. In addition, estimates of maximum likelihood, least squares, weighted least squares, and minimum distance of Cramér von Mises, von Anderson–Darling, and right-tail Anderson–Darling are obtained for the unknown parameters. Two real data sets were examined to demonstrate the usefulness of the proposed approach.

Understanding the Impact of Changes in Application Characteristics on SRGM

Due to the increasing reliance on technology in nearly every industry over the past three decades, it has become necessary to evaluate the performance of a software product prior to its formal release in the market. The properties of a software application, such as its complexity and lines of code, are subject to change over time as a result of factors namely the testing environment, allocation of resources, testing efficiency, and testing team’s expertise. The assumption of constant Fault Detection Rate (FDR) may not accurately anticipate the potential number of bugs correctly. Keeping all these considerations in mind, a framework is developed to incorporate change point in the development of a testing effort-based Software Reliability Growth Model (SRGM) that takes into account the effect of application characteristics under both perfect and imperfect debugging settings. In addition, these outcomes are compared to the model without a change point. The proposed model is validated on two real-life software fault datasets. The results demonstrate that the proposed model performs better than the model without a change point.

Maximal size constraint community search over bipartite graphs

Community search over bipartite graphs is a crucial issue with significant applications in various domains, including group recommendations, fraud detection and representation learning. Current research primarily focuses on the structural cohesiveness constraint between two sets of vertices, often neglecting the community’s size constraint. Unfortunately, this oversight may lead to communities of sizes ranging from excessively small to substantially large, thereby affecting their applicability. Therefore, in this paper, we focus on the maximal size constraint community search over bipartite graphs, which not only takes into account the (α,β) degree constraints for each set of vertices but also ensures that the number of vertices in each set is limited to ξ and ζ, respectively. To solve this problem, we first introduce two competitive algorithms, namely Ordering-based Expansion Algorithm (OEA) and Followers-based Peeling Algorithm (FPA), wherein they greedily expand or peel vertices, respectively, to identify the maximal one in a breadth-first manner, utilizing a novel vertex ordering and followers concept. To further accelerate the computation, we propose an advanced Expand-and-Filter Algorithm (EFA++) that first acquires a slightly larger community and subsequently executes a filtering phase employing a dynamic programming technique. Furthermore, we define the “Affected Area” for edge insertion or deletion that caters for dynamic scenarios. Finally, the theoretical analysis and extensive experimental evaluation on several real-life datasets demonstrate the effectiveness and efficiency of the proposed algorithms.

Self-supervised Bipartite Graph Representation Learning: A Dirichlet Max-margin Matrix Factorization Approach

Bipartite graph representation learning aims to obtain node embeddings by compressing sparse vectorized representations of interactions between two types of nodes, e.g., users and items. Incorporating structural attributes among homogeneous nodes, such as user communities, improves the identification of similar interaction preferences, namely, user/item embeddings, for downstream tasks. However, existing methods often fail to proactively discover and fully utilize these latent structural attributes. Moreover, the manual collection and labeling of structural attributes is always costly. In this article, we propose a novel approach called Dirichlet Max-margin Matrix Factorization (DM3F), which adopts a self-supervised strategy to discover latent structural attributes and model discriminative node representations. Specifically, in self-supervised learning, our approach generates pseudo group labels (i.e., structural attributes) as a supervised signal using the Dirichlet process without relying on manual collection and labeling, and employs them in a max-margin classification. Additionally, we introduce a Variational Markov Chain Monte Carlo algorithm (Variational MCMC) to effectively update the parameters. The experimental results on six real datasets demonstrate that, in the majority of cases, the proposed method outperforms existing approaches based on matrix factorization and neural networks. Furthermore, the modularity analysis confirms the effectiveness of our model in capturing structural attributes to produce high-quality user embeddings.

It's not all about residual confounding: a plea for QBA for epidemiologic researchers and educators.

We spend a great deal of time on confounding in our teaching, in our methods development and in our assessment of study results. This may give the impression that uncontrolled confounding is the biggest problem that observational epidemiology faces, when in fact, other sources of bias such as selection bias, measurement error, missing data, and misalignment of zero time may often (especially if they are all present in a single study) lead to a stronger deviation from the truth. Compared to the amount of time we spend teaching how to address confounding in a data analysis, we spend relatively little time teaching methods for simulating confounding (and other sources of bias) to learn their impact and develop plans to mitigate or quantify the bias. We review a paper by Desai et al that uses simulation methods to quantify the impact of an unmeasured confounder when it is completely missing or when a proxy of the confounder is measured. We use this article to discuss how we can use simulations of sources of bias to ensure we generate better and more valid study estimates, and we discuss the importance of simulating realistic datasets with plausible bias structures to guide data collection. If an advanced life form exists outside of our current universe and they came to earth with the goal of scouring the published epidemiologic literature to understand what the biggest problem epidemiologists have, they would quickly discover that the limitations section of publications would provide them with all the information they needed. And most likely what they would conclude is that the biggest problem that we face is uncontrolled confounding. It seems to be an obsession of ours.

Proportion Estimation Using Enhanced Class of Estimators Under Simple Random Sampling: Application with Real Data Sets and Simulation

ABSTRACT This study aims to suggest a generalized class of estimators for population proportion under simple random sampling, which uses auxiliary attributes. The bias and MSEs are considered derived to the first degree approximation. The validity of the suggested and existing estimators is assessed via an empirical investigation. The performance of estimators has been evaluated using real data sets and a simulation study. We found that the suggested estimators have minimum mean squared error (MSE) and a higher percentage relative efficiency when compared to existing estimators based on real data sets and simulation studies. The suggested estimator performs better than the existing ones, according to the theoretical and empirical findings.

A RULE-BASED APPROACH USING THE ROUGH SET ON COVID-19 DATA

The COVID-19 pandemic has not only caused loss of life but also significantly affected people's emotional state. These emotional impacts have had serious consequences on societies and economies around the world. In order to repair these devastations in society, it is important to analyse these emotional effects in depth. In this study, the effects of the pandemic on human emotions are analysed using soft computing techniques. A rule-based approach is proposed for the analysis with the help of a rough set. The proposed method is based on two main components. The first one is the process of selecting the optimal subset (OFS) from the whole feature set with the help of k best known feature selection approaches. The second component involves the use of rough clustering methods to generate rules on the selected feature subset OFS. In the study, the first real data set called " Real World Concern Dataset", which is obtained from emotional responses to COVID-19, was used. The dataset consists of 5,000 items (2,500 short + 2,500 long). In the experimental studies, the proposed approach was tested with both labelled and unlabelled data, and it was observed that effective results were obtained with an accuracy rate of over 85%. It was also found that people were highly concerned about the future due to the pandemic.

Goodness–of–fit tests based on the min–characteristic function

Tests of fit for classes of distributions that include the Weibull, the Pareto and the Fréchet families are proposed. The new tests employ the novel tool of the min–characteristic function and are based on an L2–type weighted distance between this function and its empirical counterpart applied on suitably standardized data. If data–standardization is performed using the MLE of the distributional parameters then the method reduces to testing for the standard member of the family, with parameter values known and set equal to one. Asymptotic properties of the tests are investigated. A Monte Carlo study is presented that includes the new procedure as well as competitors for the purpose of specification testing with three extreme value distributions. The new tests are also applied on a few real–data sets.

Simulation study to evaluate when Plasmode simulation is superior to parametric simulation in estimating the mean squared error of the least squares estimator in linear regression.

Simulation is a crucial tool for the evaluation and comparison of statistical methods. How to design fair and neutral simulation studies is therefore of great interest for both researchers developing new methods and practitioners confronted with the choice of the most suitable method. The term simulation usually refers to parametric simulation, that is, computer experiments using artificial data made up of pseudo-random numbers. Plasmode simulation, that is, computer experiments using the combination of resampling feature data from a real-life dataset and generating the target variable with a known user-selected outcome-generating model, is an alternative that is often claimed to produce more realistic data. We compare parametric and Plasmode simulation for the example of estimating the mean squared error (MSE) of the least squares estimator (LSE) in linear regression. If the true underlying data-generating process (DGP) and the outcome-generating model (OGM) were known, parametric simulation would obviously be the best choice in terms of estimating the MSE well. However, in reality, both are usually unknown, so researchers have to make assumptions: in Plasmode simulation studies for the OGM, in parametric simulation for both DGP and OGM. Most likely, these assumptions do not exactly reflect the truth. Here, we aim to find out how assumptions deviating from the true DGP and the true OGM affect the performance of parametric and Plasmode simulations in the context of MSE estimation for the LSE and in which situations which simulation type is preferable. Our results suggest that the preferable simulation method depends on many factors, including the number of features, and on how and to what extent the assumptions of a parametric simulation differ from the true DGP. Also, the resampling strategy used for Plasmode influences the results. In particular, subsampling with a small sampling proportion can be recommended.

Statistical analysis of the inverse power Zeghdoudi model: Estimation, simulation and modeling to engineering and environmental data

Abstract In this research, we investigate a brand-new two-parameter distribution as a modification of the power Zeghdoudi distribution (PZD). Using the inverse transformation technique on the PZD, the produced distribution is called the inverted PZD (IPZD). Its usefulness in producing symmetric and asymmetric probability density functions makes it the perfect choice for lifetime phenomenon modeling. It is also appropriate for a range of real data since the relevant hazard rate function has one of the following shapes: increasing, decreasing, reverse j-shape or upside-down shape. Mode, quantiles, moments, geometric mean, inverse moments, incomplete moments, distribution of order statistics, Lorenz, Bonferroni, and Zenga curves are a few of the significant characteristics and aspects explored in our study along with some graphical representations. Twelve effective estimating techniques are used to determine the distribution parameters of the IPZD. These include the Kolmogorov, least squares (LS), a maximum product of spacing, Anderson-Darling (AD), maximum likelihood, minimum absolute spacing distance, right-tail AD, minimum absolute spacing-log distance, weighted LS, left-tailed AD, Cramér-von Mises, AD left-tail second-order. A Monte Carlo simulation is used to examine the effectiveness of the obtained estimates. The visual representation and numerical results show that the maximum likelihood estimation strategy regularly beats the other methods in terms of accuracy when estimating the relevant parameters. The usefulness of the recommended distribution for modelling data is illustrated and displayed visually using two real data sets through comparisons with other distributions.