Linear Exponential Research Articles

Optimal Estimation of Reliability Parameters for Modified Frechet-Exponential Distribution Using Progressive Type-II Censored Samples with Mechanical and Medical Data

The aim of this research is to estimate the parameters of the modified Frechet-exponential (MFE) distribution using different methods when applied to progressive type-II censored samples. These methods include using the maximum likelihood technique and the Bayesian approach, which were used to determine the values of parameters in addition to calculating the reliability and failure functions at time t. The approximate confidence intervals (ACIs) and credible intervals (CRIs) are derived for these parameters. Two bootstrap techniques of parametric type are provided to compute the bootstrap confidence intervals. Both symmetric loss functions such as the squared error loss (SEL) and asymmetric loss functions such as the linear-exponential (LINEX) loss are used in the Bayesian method to obtain the estimates. The Markov Chain Monte Carlo (MCMC) technique is utilized in the Metropolis–Hasting sampler approach to obtain the unknown parameters using the Bayes approach. Two actual datasets are utilized to examine the various progressive schemes and different estimation methods considered in this paper. Additionally, a simulation study is performed to compare the schemes and estimation techniques.

Forecasting short-term subway passenger flow using Wi-Fi data: comparative analysis of advanced time-series methods

Accurately monitoring passenger demand fluctuations is crucial for streamlined operations of subway systems and informed decision-making. This study presents a detailed Time Series Analysis of the Toronto subway system using Wi-Fi data connection from devices as a predictor of passenger volume. Various time series models were tested for short-term forecasting, including Linear Regression, Exponential Smoothing, ARIMA, Random Forest, N-BEATS, and T-GCN. An end-to-end modeling implementation process was carried out, and the performance of each model was evaluated. The primary objective was to assess the effectiveness of short-term prediction models for univariate time series at the system level and discuss deployment challenges. While conventional time series models are fast to implement and interpretable, they require a more in-depth data exploration phase for validation, making scaling at the system level difficult. Additionally, maintenance is more challenging with conventional models, and their exploratory analysis phases need to be repeated when the models degrade over time. Prediction difficulty varied across each subway station, indicating the need for a more thorough calibration or hybrid approach, especially for transfer stations. Despite the different uses and qualities of each model in our scenario, Random Forest and Exponential Smoothing emerged as the best performers and could be a satisfactory option for robust demand forecasting at the system level.

Statistical Inferences about Parameters of the Pseudo Lindley Distribution with Acceptance Sampling Plans

Different non-Bayesian and Bayesian techniques were used to estimate the pseudo-Lindley (PsL) distribution’s parameters in this study. To derive Bayesian estimators, one must assume appropriate priors on the parameters and use loss functions such as squared error (SE), general entropy (GE), and linear-exponential (LINEX). Since no closed-form solutions are accessible for Bayes estimates under these loss functions, the Markov Chain Monte Carlo (MCMC) approach was used. Simulation studies were conducted to evaluate the estimators’ performance under the given loss functions. Furthermore, we exhibited the adaptability and practicality of the PsL distribution through real-world data applications, which is essential for evaluating the various estimation techniques. Also, the acceptance sampling plans were developed in this work for items whose lifespans approximate the PsL distribution.

On Estimation of Shannon’s Entropy of Maxwell Distribution Based on Progressively First-Failure Censored Data

Shannon’s entropy is a fundamental concept in information theory that quantifies the uncertainty or information in a random variable or data set. This article addresses the estimation of Shannon’s entropy for the Maxwell lifetime model based on progressively first-failure-censored data from both classical and Bayesian points of view. In the classical perspective, the entropy is estimated using maximum likelihood estimation and bootstrap methods. For Bayesian estimation, two approximation techniques, including the Tierney-Kadane (T-K) approximation and the Markov Chain Monte Carlo (MCMC) method, are used to compute the Bayes estimate of Shannon’s entropy under the linear exponential (LINEX) loss function. We also obtained the highest posterior density (HPD) credible interval of Shannon’s entropy using the MCMC technique. A Monte Carlo simulation study is performed to investigate the performance of the estimation procedures and methodologies studied in this manuscript. A numerical example is used to illustrate the methodologies. This paper aims to provide practical values in applied statistics, especially in the areas of reliability and lifetime data analysis.

Universum driven cost-sensitive learning method with asymmetric loss function

Class imbalance learning (CIL) has been a critical yet challenging topic for a long time. To address the issue, this paper proposes a cost-sensitive learning method based on the linear-exponential (LINEX) loss function and the Universum (UCSSVM). On one hand, the asymmetry of LINEX loss function is leveraged to realize cost-sensitive learning, by treating each instance discriminately. On the other hand, to enhance the classification performance, a novel and effective strategy is put forward to generate Universum with the assistance of hard-to-classify samples (small margin distance). The mini-batch gradient descent (MBGD) approach is adopted to optimize UCSSVM. Meanwhile, extensive experiments demonstrate its superiority in CIL and the necessity of using Universum.

Pemodelan Pertumbuhan Penduduk Kota Kupang dengan Geogebra

This research aims to project the population of Kupang City in 2030. Data on the population of Kupang City was obtained from Badan Pusat Statistik of Kupang City. The data was used in 12 year time intervals. The process of projecting the population of Kupang City was carried out with the help of Geogebra Software. The projection of the population of Kupang City is modeled using a Linear Model, Exponential Model, Geometry Model and Logarithmic Model. The logarithmic model has the smallest error so it can be said that the logarithmic model is better than other models, in relation to projecting the population of Kupang City. The results obtained for the population of Kupang City in 2030, respectively for the Geometry Model, Logarithmic Model, Exponential Model and Linear Model are 519106.81, 535169.03, 557736.47, 535564.58. These results can be the basis for the government or parties regarding future policies.

Bayes Estimator of Exponential Distribution Parameters of Type II Censored Data with Linear Exponential Loss Function Method Based on Jeffrey Priors

Survival analysis is often used in the application of analyzing the survival of an object such as living things or objects. This analysis is identical to data censoring which is divided into three, namely: type I, II, and III censored data. Type II censored data is data censoring done by determining the number of objects to be analyzed from the total number of observation objects . Type II censored data is used when the analysis is intended to maximize the results of the analysis. Bayesian Linear Exponential (LINEX) loss function is one method that can be used to estimate parameters in survival analysis by minimizing the expected value of LINEX. The purpose of this study is to determine the Bayesian LINEX loss function parameter estimation on type II censored data using exponential distribution. This method uses the concept of posterior distribution and prior distribution. The prior distribution used is the Jeffrey prior distribution which has objective properties and is based on Fisher information theory The application of the parameter estimation results is carried out on the survival data of lung cancer patients obtained from the North Central Cancer Treatment Group. Based on the results of parameter estimation, it is concluded that the greater the value of the controller (a) can produce a smaller value of parameter estimation results (θ^) . The results of this study can be used as a reference in conducting survival tests using type II censored exponential distribution data using the LINEX loss function method based on Jeffrey priors.

Attentional ensemble model for accurate discharge and water level prediction with training data enhancement

Discharge is water flow from a higher to a lower elevation resulting from precipitation or surface runoff. It is an essential resource for human society, including drinking water and irrigation. Forecasting discharge and water level are crucial in establishing plans to secure water sources and predict floods. Deep learning has recently emerged as a potential solution for forecasting discharge and water levels. However, it faces three challenges: training data shortage, noise existence, and underestimation tendency. This research offers a novel deep learning-based method addressing the abovementioned shortcomings. To overcome data scarcity and improve prediction accuracy, we leverage the strengths of the one-dimensional Convolutional Neural Network (1D-CNN), Long short-term memory (LSTM), and Ensemble learning technique. Specifically, 1D-CNN uses a kernel moving in one direction and executing the convolution operations to extract the correlations between the features. LSTM helps to capture temporal relationships inside the data, while ensemble learning exploits multiple learning models for better predictive performance. Besides, Singular-Spectrum Analysis (SSA) technique is applied to decompose the input data into several components and eliminate noise-like ones while retaining essential parts. Besides, the attention mechanism is leveraged to assign higher weights for more essential features, further enhancing the accuracy. Finally, we utilize the Linear Exponential (LINEX) loss to prevent under- and overestimation. The experiments show that our proposal outperforms existing approaches regarding all metrics. Notably, our method improves the Nash–Sutcliffe model Efficiency coefficient (NSE) (the most critical metric) by up to 40% and 34% concerning one-step-ahead and multistep-ahead, respectively, compared to existing approaches.

Forecasting of Domestic Electricity Consumption in Assam, India

Though the power supply scenario in Assam is not comparable with other developed states of India, yet it is improving gradually. With increasing household electrification rate, increasing income and technological development, the share of Domestic electricity consumption in the state is experiencing a rapid growth in comparison to the other sectors like Industry, Commerce, Agriculture etc. and therefore the forecasting of domestic electricity consumption become very important to meet the power needs of the consumer. The aim of this study is to suggest the best forecasting model for domestic electricity consumption in Assam. For this purpose, three models viz. Multiple Linear Regression considering the effects of population and per capita income; ARIMA and Growth curves viz Linear, Quadratic, Cubic, Exponential, Logarithmic and Inverse are used considering the time period 1980-2018. When growth curves are compared in terms of diagnostic criteria- Adjusted R2, RMSE, AME and MAPE, then it is found that exponential model fits the data better than the other growth models. After that, comparisons are made among the models Multiple Linear Regression, ARIMA and Exponential by considering their average relative error and found the efficiency of ARIMA the highest followed by Exponential and Multiple Linear Regression models. Therefore, forecasting of domestic electricity consumption in Assam has been done with ARIMA model for the next 10 years.

Linear-exponential loss incorporated deep learning for imbalanced classification

The class imbalance issue is a pretty common and enduring topic all the time. When encountering unbalanced data distribution, conventional methods are prone to classify minority samples as majority ones, which may cause severe consequences in reality. It is crucial yet challenging to cope with such problems. In this paper, inspired by our previous work, we borrow the linear-exponential (LINEX) loss function in statistics into deep learning for the first time and extend it into a multi-class form, denoted as DLINEX. Compared with existing loss functions in class imbalance learning (e.g., the weighted cross entropy-loss and the focal loss), DLINEX has an asymmetric geometry interpretation, which can adaptively focus more on the minority and hard-to-classify samples by solely adjusting one parameter. Besides, it simultaneously achieves between and within class diversities via caring about the inherent properties of each instance. As a result, DLINEX achieves 42.08% G-means on the CIFAR-10 dataset at the imbalance ratio of 200, 79.06% G-means on the HAM10000 dataset, 82.74% F1 on the DRIVE dataset, 83.93% F1 on the CHASEDB1 dataset and 79.55% F1 on the STARE dataset The quantitative and qualitative experiments convincingly demonstrate that DLINEX can work favorably in imbalanced classifications, either at the image-level or the pixel-level.

Mathematical Proficiency of Grade 10 Students

This study aims to determine the mathematical proficiency of Grade 10 students of a Sectarian School in Bacolod City in solving functions. The data gathering instrument used was a validated teacher-made questionnaire. Based on the results and findings of the study and considering the limitations, the following conclusions were drawn: Both male and female students had average proficiency in terms of solving Linear, Polynomial and Exponential Functions. But in Logarithmic Function, the female outdid the male by achieving a high proficiency with the male only on the average. But, in terms of Quadratic Function, both male and female students scored low. Both private and public-school students had average proficiency in solving Linear Functions, Polynomial Functions, and Exponential Functions. Yet, regardless of origin of school, when it comes to proficiency in solving Quadratic Functions, all students got low. Both high- and low-income families had average level of proficiency in terms of solving Linear, Polynomial and Exponential Functions. Both also got a low level of proficiency on the topic Quadratic Function. But in terms of Logarithmic Function, those students who belonged to higher income families displayed high level of proficiency compared to those who belong to the lower income group who only had an average level of proficiency. There was a significant difference in the level of proficiency of students when grouped according to school origin. Lastly, there was no significant difference in the level of proficiency of students when grouped according to socio-economic status.

Comparison of Three Time Series Forecasting Methods on Linear Regression, Exponential Smoothing and Weighted Moving Average

The purpose of this study is to compare the 3 forecasting methods Linear Regression, Exponential Smoothing and Weighted Moving Average based on the smallest error value or close to zero. From the results of this study, the Linear Regression method was obtained as the correct method with a predicted value of 502 students, the smallest error value was MAD 27.83, MSE 1152.1 MAPE 8.1%. The Tracking Signal value moves between 1 and -1, the movement is within the control limits of the tracking signal standard deviation distribution 4 and -4, meaning that the method is correct. The Moving Range value moves between 68 and -46, this value is within the MR control limits of 117.83 and -117.83, this result shows that this means that this method has been tested for truth and can be accepted as well. Thus, indicating that the Linear Regression method as a forecasting method is appropriate and acceptable as a basis for future decision making. The level of accuracy of the error and the value in control shows that there is a time series data relationship between the x variable, namely time, and the variable y, namely actual data. In addition, it produces trending data movement patterns, meaning that data movements experience a significant increase over a long period of time or for 7 periods.

Pengurangan Bullwhip Effect pada Rantai Pasok Air Minum Dalam Kemasan Menggunakan Metode Collaborative Planning, Forecasting, and Replenishment (CPFR): Studi Kasus

CV Tirta Alam Dodu is a privately owned company engaged in the beverage industry in the form of AMDK. The problem faced by the company is that there is a difference between demand and sales, or better known as the bullwhip effect problem. The purpose of this study is to reduce the bullwhip effect on the supply chain and optimize the amount of product inventory according to the amount of demand. The method used is the Collaborative Planning, Forecasting, and Replenishment (CPFR) method. Based on the results of the research conducted, it can be concluded that the analysis of demand forecasting calculations on AMDK products at CV Tirta Alam Dodu was carried out using historical data on product demand through five analytical methods, including Linear Trend, Moving Average, Exponential Smoothing, Decomposition, and Winter's Model. The best forecasting resulted in Winter's Model which showed the smallest error value compared to the other four models. The way to minimize so that it does not become a bullwhip effect is to use the CPFR method, where the bullwhip effect value is obtained on glass mineral water products and gallons of mineral water of 0,985 and 0,898, respectively. This result shows that there is no amplification of demand and has reached the company's target, which is to fall below number one. The way to optimize inventory so that there is no excess stock or shortage of stock in the product is to make the amount of inventory based on the amount of demand from the forecasting results plus the safety stock.

Bayesian Estimation of the Scale Parameter of Weibull Distribution using Ranked Set Samples

This research article is primarily focused on the investigation of Bayes estimators for the scale parameter of the Weibull distribution. Specifically, we focus on a sampling method known as Ranked Set Sample with Unequal Samples (RSSU), as introduced and studied by Bhoj (2001). Our objective is to derive Bayes estimates for the scale parameter utilizing two distinct loss functions: the squared error loss (SEL) function and the linear exponential (LINEX) loss function. We consider the scenario where the scale parameter follows either a gamma prior distribution or a Jeffreys prior distribution. To evaluate the performance of these estimators, we conduct simulations and analyse their bias and mean squared error (MSE). Our findings show that RSSU-based estimators outperform those based on Simple Random Sampling (SRS) and Ranked Set Sampling (RSS), when either Jeffreys prior or gamma prior distribution is used.

A Bayesian estimation of the Gini index and the Bonferroni index for the Dagum distribution with the application of different priors

Abstract Bayesian estimators and highest posterior density credible intervals are obtained for two popular inequality measures, viz. the Gini index and the Bonferroni index in the case of the Dagum distribution. The study considers informative and non-informative priors, i.e. the Mukherjee-Islam prior and the extension of Jeffrey’s prior, respectively, under the presumption of the Linear Exponential (LINEX) loss function. A Monte Carlo simulation study is carried out in order to obtain the relative efficiency of both the Gini and Bonferroni indices while taking into consideration different priors and loss functions. The estimated loss proves lower when using the Mukherjee-Islam prior in comparison to the extension of Jeffrey’s prior and the LINEX loss function outperforms the squared error loss function (SELF) in terms of the estimated loss. Highest posterior density credible intervals are also obtained for both these measures. The study used real-life data sets for illustration purposes.

Bayesian Estimation Using Expected LINEX Loss Function: A Novel Approach with Applications

The loss function plays an important role in Bayesian analysis and decision theory. In this paper, a new Bayesian approach is introduced for parameter estimation under the asymmetric linear-exponential (LINEX) loss function. In order to provide a robust estimation and avoid making subjective choices, the proposed method assumes that the parameter of the LINEX loss function has a probability distribution. The Bayesian estimator is then obtained by taking the expectation of the common LINEX-based Bayesian estimator over the probability distribution. This alternative proposed method is applied to estimate the exponential parameter by considering three different distributions of the LINEX parameter, and the associated Bayes risks are also obtained in consequence. Extensive simulation studies are conducted in order to compare the performance of the proposed new estimators. In addition, three real data sets are analyzed to investigate the applicability of the proposed results. The results of the simulation and real data analysis show that the proposed estimation works satisfactorily and performs better than the conventional standard Bayesian approach in terms of minimum mean square error and Bayes risk.

Estimation of Some Lifetime Parameters of Flexible Reduced Logarithmic‐Inverse Lomax Distribution under Progressive Type‐II Censored Data

In this study, the estimation of parameters of a three‐parameter flexible reduced logarithmic‐inverse Lomax (FRL‐IL) distribution based on progressive type‐II right censored sample is studied. These methods include maximum likelihood estimations (MLEs) and Bayesian estimators. Approximate confidence intervals (ACIs) for the reliability and hazard functions are estimated based on the asymptotic distribution of maximum likelihood estimates (MLEs). In addition, two bootstrap CIs are also proposed. Bayesian estimates are obtained for symmetric and asymmetric loss functions such as squared error loss (SEL) and linear‐exponential (LINEX) loss functions. The Gibbs within Metropolis‐Hasting sampler procedure is applied using the Markov Chain Monte Carlo (MCMC) technique to get the Bayes estimates of the unknown parameters and their credible intervals (CRIs). Finally, a real‐life dataset that represents a group of patients with bladder cancer is considered an application of the proposed methods.

Indonesian Trade Deficit with China: Background and Grey Forecasting

Indonesia's trade balance with China has remained negative since 2010. The current study forecasts Indonesia's trade deficit with China for five years using the Even Grey Forecasting model EGM (1,1,α,θ). The sample was conducted by collecting the data of traded deficits for the past ten years. Data were collected from the official websites of Indonesia's Central Bureau of Statistics of (BPS), Ministry of Trade, among others. By building upon the literature, the study argues that trade deficits might have occurred from internal and external factors, such as the lack of infrastructure, the depreciation of the Rupiah (Indonesian currency) against the U.S. dollar, and the ASEAN-China Free Trade Agreement. Comparative analysis with Linear Regression (LR), Exponential Regression (ER), and Exponential Triple Smoothing (ETS) revealed the superiority of the grey forecasting model for trade deficit prediction. The study found that the trade deficit was minimum during the COVID-19 pandemic. It also showed an increasing trade deficit in the post-COVID period. The study concludes with some recommendations for Indonesia to minimize the trade deficit.

Bayesian Estimations of Exponential Distribution Based on Interval-Censored Data with a Cure Fraction

Censored data are considered to be of the interval type where the upper and lower bounds of an event’s failure time cannot be directly observed but only determined between interval inspection times. The analyses of interval-censored data have attracted attention because they are common in the fields of reliability and medicine. A proportion of patients enrolled in clinical trials can sometimes be cured. In some instances, their symptoms mostly disappear without any recurrence of the disease. In this study, the proportion of such patients who are cured is estimated. Furthermore, the Bayesian approach under the gamma prior and maximum likelihood estimation (MLE) is used to estimate the cure fraction depending on the bounded cumulative hazard (BCH) model based on interval-censored data with an exponential distribution. The Bayesian approach uses three loss functions: squared error, linear exponential, and general entropy. These functions are compared with the MLE and used between estimators. Moreover, they are obtained using the mean squared error, which locates the best option to estimate the parameter of an exponential distribution. The results show that the BCH model and lambda parameter of the exponential distribution based on the interval-censored data can be best estimated using the Bayesian gamma prior with a positive loss function of the linear exponential.

Demand Forecasting of Toyota Avanza Cars in Indonesia: Grey Systems Approach

Toyota Avanza car is a popular four-wheeler among Indonesia middle-class customers. The current study aims to forecast the demand for Toyota Avanza cars in Indonesia in the next six years using the grey forecasting model EGM (1,1, α, θ). The comparative analysis of the results obtained from the grey model with those of Linear Regression, Exponential Regression, and Exponential Triple Smoothing techniques revealed the superiority of the grey model as it produced most accurate forecasts. The accuracy was measured through the Mean Absolute Percentage Error. The results revealed, the car sales are likely to decline in the future. Although forecasts are never completely accurate, forecasting can provide a reference for developing strategy to meet future demand. The results are important for Toyota Avanza car manufacturers in Indonesia.