Numerical Integral Equation Research Articles

Average Run Length Computations of Autoregressive and Moving Average Process using the Extended EWMA Procedure

In the past, the control chart served as a statistical tool for detecting process changes. The Exponentially Weighted Moving Average (EWMA) control chart is highly effective for detecting small changes. This research introduces the Extended Exponentially Weighted Moving Average (Extended EWMA) control chart for the Autoregressive and Moving average process with order p = 1 and q = 1 (ARMA(1,1)) The Extended EWMA control chart incorporates two smoothing parameters ( λ1 and λ2 ) derived from the EWMA control chart. A comparative analysis of the performance of the EWMA control chart. The Average Run Length (ARL) value as determined by the explicit formulas in this research, serves as a metric for evaluating the performance of the Extended EWMA control chart and the EWMA control chart. The Numerical Integral Equation (NIE) method is used to verify the accuracy of the ARL for the explicit formulas of the two control charts which has not been before discovered. The effectiveness of control charts can also be evaluated by analyzing SDRL, ARL, MRL, RMI, AEQL, and PCI values as metrics for various design parameter values. After analyzing the results of the ARL and all five performance meters, it was determined that the Extended EWMA control chart is better than the EWMA control chart at all shift sizes of process changes. Finally, the assessment of the ARMA process is being conducted to evaluate the ARL using a dataset on PM2.5 dust levels in Bangkok, Thailand during January and February of 2024.

Efficient Monitoring of Autoregressive and Moving Average Process using HWMA Control Chart

Quality control is an essential process for manufacturing and industry because it enhances product quality, consumer satisfaction, and overall profitability. Among many other statistical process control tools, quality practitioners typically employ control charts to monitor the industrial process and detect production changes. Control charts are widely used to detect flaws in many applications, such as distributed circuits and systems, electronic devices, and systems and signals. In this study, we derived an explicit formula for Average Run Length (ARL) of the Homogenously Weighted Moving Average control chart (HWMA) under the ARMA (p,q) process. The accuracy was checked using the numerical integral equation (NIE) technique. The finding showed that the explicit formulas and numerical solutions presented an outstanding level of agreement. However, the computational time for the explicit formulas was approximately one second, which was less than that required for the NIE. Moreover, the performance efficiency of the HWMA control chart is compared with the cumulative sum control chart for ARMA (p, q) processes including ARMA (2,1), ARMA (2,3), and ARMA (1,1) processes. The results found that the HWMA control chart performance is found to be preferable to the CUSUM control chart performance. Additionally, the explicit formula of the HWMA control chart was implemented in a practical application of the count of nonconformities in printed circuit boards (PCBs).

Performance Evaluation of HWMA Control Chart based on AR(p) with Trend Model to Detect Shift Process Mean

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

Improving Sensitivity of the DEWMA Chart with Exact ARL Solution under the Trend AR(p) Model and Its Applications

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data. Doi: 10.28991/ESJ-2023-07-06-03 Full Text: PDF

Exact Average Run Length Evaluation on One-Sided and Two-Sided Extended EWMA Control Chart with Correlated Data

The Extended Exponentially Weighted Moving Average (Extended EWMA) control chart is one of the control charts that can rapidly detect a minor shift. Using the average run length (ARL), the control charts' effectiveness can be evaluated. This research aims to derive the explicit formulations for the ARL on one-sided and two-sided Extend EWMA control charts for the MA(1) model with exponential white noise, as they have not been previously presented. The analytical solution accuracy was determined and compared to the numerical integral equation (NIE) method. The results indicate that the ARL calculated using the explicit formula and the NIE method are nearly identical, demonstrating the validity of the explicit formula. In addition, this study is extended to compare the performance with the Exponentially Weighted Moving Average (EWMA) control chart. The results show that the Extended EWMA control chart has superior efficacy to the EWMA control chart. To demonstrate the efficacy of the proposed method, the analytical solution of ARL is finally applied to real-world data on Thailand's monthly fuel price.

Performance Measurement of a DMEWMA Control Chart on an AR(p) Model with Exponential White Noise

The double-modified exponentially weighted moving average (DMEWMA) control chart running an autoregressive (AR) process is proposed to detect unusual events. The AR equation and the DMEWMA statistic are combined to evaluate the control limit of the exponential residual term to obtain the explicit formula for the average run length (ARL). The ARLs computed using the explicit formula approach and the well-established numerical integral equation method were compared to validate the former. The efficiencies of the original EWMA, MEWMA, and DMEWMA control charts running AR processes based on simulation and real data were compared by using the results of ARL and relative mean index calculations. The results indicate that the explicit formula for the ARL of an AR process running on a double-modified EWMA control chart detected changes more quickly than on either of the other two control charts for small and moderate changes. Finally, real data on COVID-19 is provided to demonstrate the application of this explicit formula.

Statistical Design for Monitoring Process Mean on Modified EWMA Control Chart based on Autocorrelated Data

This research endeavor is focused on establishing explicit formulas for the computation of the average run length (ARL) within the context of a moving average process characterized by exogenous variables, denoted as MAX(q,r), and subjected to exponential white noise. Additionally, we aim to conduct a comparative analysis of their performance against the exponentially weighted moving average (EWMA) and the modified exponentially weighted moving average (modified EWMA) methodologies. The evaluation of their performance will be based on metrics such as the absolute percentage relative error (APRE) and the relative mean index (RMI). Furthermore, we undertake a rigorous assessment of the accuracy of these explicit formulas in relation to ARL by considering CPU time, utilizing the numerical integral equation (NIE) method derived through the application of the Gauss-Legendre quadrature rule. This comparative evaluation is carried out for both control chart methodologies. To ascertain the efficacy of our explicit formulas approach, we apply it to two distinct datasets. The first dataset pertains to the closing price of natural gas, with the crude oil WTI price serving as the exogenous variable. The second dataset encompasses the closing stock price of KTB Public Company Limited, with daily foreign exchange rates for USD/JPY and EUR/USD as the exogenous variables. The results of applying the ARL based on the explicit formulas to these two datasets demonstrate that, under these conditions, the modified EWMA control chart outperforms the EWMA control chart.

ARL Evaluation of a DEWMA Control Chart for Autocorrelated Data: A Case Study on Prices of Major Industrial Commodities

The double exponentially weighted moving average (DEWMA) control chart, an extension of the EWMA control chart, is a useful statistical process control tool for detecting small shift sizes in the mean of processes with either independent or autocorrelated observations. In this study, we derived explicit formulas to compute the average run length (ARL) for a moving average of order q (MA(q)) process with exponential white noise running on a DEWMA control chart and verified their accuracy by comparison with the numerical integral equation (NIE) method. The results for both were in good agreement with the actual ARL. To investigate the efficiency of the proposed procedure on the DEWMA control chart, a performance comparison between it and the standard and modified EWMA control charts was also conducted to determine which provided the smallest out-of-control ARL value for several scenarios involving MA(q) processes. It was found that the DEWMA control chart provided the lowest out-of-control ARL for all cases of varying the exponential smoothing parameter and shift size values. To illustrate the efficacy of the proposed methodology, the presented approach was applied to datasets of the prices of several major industrial commodities in Thailand. The findings show that the DEWMA procedure performed well in almost all of the scenarios tested. Doi: 10.28991/ESJ-2023-07-05-020 Full Text: PDF

Accurate Average Run Length Analysis for Detecting Changes in a Long-Memory Fractionally Integrated MAX Process Running on EWMA Control Chart

Numerical evaluation of the average run length (ARL) when detecting changes in the mean of an autocorrelated process running on an exponentially weighted moving average (EWMA) control chart has received considerable attention. However, accurate computation of the ARL of changes in the mean of a long-memory model with an exogenous (X) variable, which often occurs in practice, is challenging. Herein, we provide an accurate determination of the ARL for long-memory models such as the fractionally integrated MAX processes (FIMAX) with exponential white noise running on an EWMA control chart by using an analytical formula based on an integral equation. From a computational perspective, the analytical formula approach is accomplished by solving the solution for the integral equation obtained via the Fredholm integral equation of the second kind. Moreover, the existence and uniqueness of the solution for the analytical formula were confirmed via Banach’s fixed-point theorem. Its efficacy was compared with that of the ARL derived by using the well-known numerical integral equation (NIE) technique under the same circumstances in terms of the ARL percentage accuracy and computational processing time. The percentage accuracy was 100%, which indicates excellent agreement between the two methods, and the analytical formula also required much less computational processing time. An example to illustrate the effectiveness of the proposed approach with a process involving real data running on an EWMA control chart is also provided herein. The explicit formula method offers an accurate determination of the ARL and a new approach for validating its computation, especially for long-memory scenarios running on EWMA control charts.

Modification of ARL for detecting changes on the double EWMA chart in time series data with the autoregressive model

This research aims to derive the average run length (ARL) evaluation of the double exponentially weighted moving average (double EWMA) control chart for observation data that follows exponential white noise in a time series model with an autoregressive model. Since most real-world data is automatically correlated, autoregressive models are available. Comparisons were made between the ARLs obtained using the explicit formula and the numerical integral equation (NIE) approach. The results showed that the explicit formula's use of the ARL outperformed the NIE approach in terms of computation time. After that, the efficacy of the exponentially weighted moving average (EWMA) and double EWMA charts is then compared using the suggested explicit ARL formula. The ARL of the double EWMA chart was found to perform better than the ARL of the EWMA chart in all situations. It also uses natural gas and diesel prices on stock exchanges around the world as cases studies. The results show that the double EWMA chart has better detection sensitivity than the EWMA chart, and the results are consistent with the experimental results. As a result, the sensitivity of the double EWMA chart in detecting changes makes it a good alternative for monitoring processes with real-world data.

Alternative to Detecting Changes in the Mean of an Autoregressive Fractionally Integrated Process with Exponential White Noise Running on the Modified EWMA Control Chart

The modified exponentially weighted moving-average (modified EWMA) control chart is an improvement on the traditional EWMA control chart. Herein, we provide more details about the modified EWMA control chart using various values of an additional design parameter for detecting small-to-moderate shifts in the process mean of an autoregressive fractionally integrated (ARFI(p, d)) process with exponential white noise running thereon. The statistical performances of the two charts were evaluated in terms of the average run length (ARL) obtained by solving integral equations (IEs). This provides an exact formula with proven existence and uniqueness verified by applying Banach’s fixed-point theorem. The accuracy of the proposed formula for the ARL was compared with the ARL derived by using the numerical IE technique for the out-of-control state. Although their accuracies were identical for various out-of-control situations and long-term memory processes, the exact formula method required less than 0.01 s to compute the ARL whereas the numerical IE method took 3–4 s. The strengths of using the exact formula are that it is simple to calculate and the computational time is significantly reduced. Comparing their standard deviations of the run length and median run lengths yielded the same results. Finally, practical examples with real-life data corresponding to ARFI(p, d) processes with exponential white noise are provided to demonstrate the applicability of the proposed exact formula.

On the Performance of the Extended EWMA Control Chart for Monitoring Process Mean Based on Autocorrelated Data

The extended exponentially weighted moving average (EEWMA) control chart is an instrument for detection. It can quickly identify small shifts in the process. The benchmark for the control chart's performance is the average run length (ARL). In this paper, we present the efficiency of the EEWMA control chart to detect tiny shifts when the observations are autocorrelated with exponential residuals through the explicit formulas of the ARL. The accuracy of the solution was verified with the numerical integral equation (NIE) method. After that, the ARL effectiveness of the ARL on the EEWMA control chart was expanded to compare with the traditional EWMA control chart. Finally, using two real datasets that indicate the percentages of internet users using Windows 7 and iOS, the applicability of the offered method is shown. Our findings support the notion that the EEWMA control chart performs better when using autocorrelated data to track tiny changes.

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average (extended EWMA) control chart is one of the control charts and can be used to quickly detect a small shift. The performance of control charts can be evaluated with the average run length (ARL). Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p) model has not been reported previously. The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA control chart for the trend AR(p) model as well as the trend AR(1) and trend AR(2) models with exponential white noise. The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numerical integral equation (NIE) method. The results show that the ARL obtained by the explicit formula and the NIE method is hardly different, but the explicit formula can help decrease the computational (CPU) time. Furthermore, this is also expanded to comparative performance with the Exponentially Weighted Moving Average (EWMA) control chart. The performance of the extended EWMA control chart is better than the EWMA control chart for all situations, both the trend AR(1) and trend AR(2) models. Finally, the analytical solution of ARL is applied to real-world data in the health field, such as COVID-19 data in the United Kingdom and Sweden, to demonstrate the efficacy of the proposed method.

Developing Average Run Length for Monitoring Changes in the Mean on the Presence of Long Memory under Seasonal Fractionally Integrated MAX Model

The cumulative sum (CUSUM) control chart can sensitively detect small-to-moderate shifts in the process mean. The average run length (ARL) is a popular technique used to determine the performance of a control chart. Recently, several researchers investigated the performance of processes on a CUSUM control chart by evaluating the ARL using either Monte Carlo simulation or Markov chain. As these methods only yield approximate results, we developed solutions for the exact ARL by using explicit formulas based on an integral equation (IE) for studying the performance of a CUSUM control chart running a long-memory process with exponential white noise. The long-memory process observations are derived from a seasonal fractionally integrated MAX model while focusing on X. The existence and uniqueness of the solution for calculating the ARL via explicit formulas were proved by using Banach's fixed-point theorem. The accuracy percentage of the explicit formulas against the approximate ARL obtained via the numerical IE method was greater than 99%, which indicates excellent agreement between the two methods. An important conclusion of this study is that the proposed solution for the ARL using explicit formulas could sensitively detect changes in the process mean on a CUSUM control chart in this situation. Finally, an illustrative case study is provided to show the efficacy of the proposed explicit formulas with processes involving real data.

Approximation of average run length on extended EWMA control chart for autoregressive process with explanatory variables using numerical integral equation method

The objective of this research is to study numerical integral equation method (NIE) using five quadrature rules namely, composite midpoint rule, composite trapezoidal rule, composite Simpson’s rule, composite Boole’s rule and Gauss-Legendre rule for approximating the average run length on extended exponentially weighted moving average (extended EWMA) control chart for autoregressive process with explanatory variables when white noise follows an exponential distribution. Furthermore, the comparison on performance of classical EWMA and extended EWMA control chart had been conducted. The results show that extended EWMA control chart can detect changes quicklier than EWMA control chart for all conditions.

Detection Sensitivity of a Modified EWMA Control Chart with a Time Series Model with Fractionality and Integration

Among the many statistical process control charts, the modified exponentially weighted moving average (EWMA) control chart has been designed to swiftly detect a small shift in a process parameter. Herein, we propose two explicit formulas for the average run length (ARL) for integrated moving average (IMA) and fractional integrated moving average (FIMA) models combined with the modified EWMA control chart for time series prediction. The application of the suggested control chart procedures depends on the residuals of the IMA and FIMA models. The performance of the control chart with both models is evaluated by using the ARL. Explicit formulas for the ARL for the two models with the modified EWMA statistic are derived and their precision is compared with the numerical integral equation method. The explicit formulas could accurately predict the true ARL while markedly decreasing the computational processing time compared to the numerical integration method. The capabilities of the IMA and FIMA models with the modified EWMA control chart were studied by varying g times the last term and exponential smoothing parameter λ, with the relative mean index being used to evaluate these situations. The results show that the modified EWMA control chart with either model performed better than the original EWMA control chart. Furthermore, the modified EWMA control chart with either model was highly efficient when g increased and λ was small. Two applications involving energy commodity prices are used to illustrate the efficacies of the proposed approaches, the results of which were in accordance with the experimental study. Doi: 10.28991/ESJ-2022-06-05-015 Full Text: PDF

Performance of the CUSUM Control Chart Using Approximation to ARL for Long- Memory Fractionally Integrated Autoregressive Process with Exogenous Variable

The cumulative sum (CUSUM) control chart is a well-known process monitoring tool that is sensitive to small-to-moderate changes in process parameters. In this paper, we propose an approximated average run length (ARL) method based on the numerical integral equation (NIE) method for monitoring the mean of a long-memory autoregressive fractionally integrated process with an exogenous variable (ARFIX) running on a CUSUM control chart. The approximated ARL based on the NIE method is realized by solving a system of linear equations and integration based on the partitioning and summation of the area under the curve of a function derived by using the Gauss-Legendre quadrature. In a comparative study with the ARL based on analytical formulas, the proposed approximated ARL method could detect shifts of various sizes in the process mean of an ARFIX process running on a CUSUM control chart. In addition, the proposed method was compared with their analytical formulas in terms of the relative percentage change (r%) to verify the accuracy of the ARL results. The results revealed that the ARL results obtained from the NIE method are an approach to analytical formulas with r% of less than 0.25. Hence, the NIE method is very accurate and in excellent agreement with the analytical formulas approach. Apparently, the NIE method is an alternative as efficiently as the analytical method for this situation. It also performed well in comparison with the approximated ARL for the same process running on an exponentially weighted moving average control chart. In addition, real datasets are also used to demonstrate the efficacy of the proposed method.

On the Average Run Lengths of Quality Control Schemes Using a Numerical Integral Equation Approach

This research presents the approach of estimating the average run length (ARL) by using the numerical integral equation (NIE) approach, such as the Gaussian, Midpoint, Trapezoidal, and Simpson’s rules for the extended exponentially weighted moving average (EEWMA) control chart, when observations are continuous distributions namely exponential, Weibull and Gamma distributions. In addition, the performance of the extended exponentially weighted moving average (EEWMA) control chart is compared with the modified exponentially weighted moving average (modified EWMA) and exponentially weighted moving average (EWMA) control charts. The performance metric is the out-of-control average run length (ARL1). The results show that the EEWMA control chart performs the best among the modified EWMA and EWMA control charts. Furthermore, the efficacies of the control charts using the approximated ARL solutions were also applied to real-world applications.

Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables

The exponentially weighted moving average (EWMA) control chart is a popular tool used to monitor and identify slight unnatural variations in the manufacturing, industrial, and service processes. In general, control charts operate under the assumption of normality observation of the attention quality feature, but it is not easy to maintain this assumption in practice. In such situations, the data of random processes are correlated data, such as stock price in the economic field or air pollution data in the environment field. The characteristics and performance of the control chart are measured by the average run length (ARL). In this article, we present the new explicit formula of ARL for EWMA control chart based on MAX(q,r) process. The proposed explicit formula of ARL for the MAX(q,r) process is proved using the Fredholm integral equation technique. Moreover, ARL values are also assessed using the numerical integral equations method based on Gaussian, midpoint, and trapezoidal rules. Banach's fixed point theorem guarantees the existence and uniqueness of the solution. Furthermore, the accuracy of the proposed explicit formula is assessed in absolute percentage relative error compared with the numerical integral equations method. The results found that the explicit formula's ARL values are similar to those obtained using the numerical integral equation method; the absolute percentage relative errors are less than 0.0001 percent. As a result, the essential conclusion is that the explicit formula outperforms the numerical method in computational time. Consequently, the proposed explicit formula and the numerical integral equation have been the alternative approaches for computing ARL values of the EWMA control chart. They would be applied in various fields, including economics, environment, biology, engineering, and others.

Integral equation solutions for the average run length for monitoring shifts in the mean of a generalized seasonal ARFIMAX(P, D, Q, r)s process running on a CUSUM control chart.

The CUSUM control chart is suitable for detecting small to moderate parameter shifts for processes involving autocorrelated data. The average run length (ARL) can be used to assess the ability of a CUSUM control chart to detect changes in a long-memory seasonal autoregressive fractionally integrated moving average with exogenous variable (SARFIMAX) process with underlying exponential white noise. Herein, new ARLs via an analytical integral equation (IE) solution as an analytical IE and a numerical IE method to test a CUSUM control chart's ability to detect a wide range of shifts in the mean of a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise are presented. The analytical IE formulas were derived by using the Fredholm integral equation of the second type while the numerical IE method for the approximate ARL is based on quadrature rules. After applying Banach's fixed-point theorem to guarantee its existence and uniqueness, the precision of the proposed analytical IE ARL was the same as the numerical IE method. The sensitivity and accuracy of the ARLs based on both methods were assessed on a CUSUM control chart running a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise. The results of an extensive numerical study comprising the examination of a wide variety of out-of-control situations and computational schemes reveal that none of the methods outperformed the IE. Specifically, the computational scheme is easier and can be completed in one step. Hence, it is recommended for use in this situation. An illustrative example based on real data is also provided, the results of which were found to be in accordance with the research results.