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## یک روش اثربخش برای تجزیه و تحلیل نقطه تغییر داده های پانلی | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

پژوهش در مدیریت تولید و عملیات | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

دوره 14، شماره 4 - شماره پیاپی 35، بهمن 1402، صفحه 49-59 اصل مقاله (1.03 M) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

نوع مقاله: مقاله پژوهشی | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

شناسه دیجیتال (DOI): 10.22108/pom.2024.138146.1513 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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ناصر رفیعی؛ کریم آتشگر^{*} ؛ مهرداد فضل علی
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^{}گروه مهندسی صنایع، دانشکده مدیریت و مهندسی صنایع، دانشگاه صنعتی مالک اشتر، تهران ایران | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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**Introduction**
Advances in data analysis have allowed engineers and practitioners to model a large number of process measurements to reconstruct panel data to evaluate its stability over time. In other words, there is in many applications, multi-dimensional data that includes time series observations of a large number of cross-sectional units. This structure of data is referred to as the panel data where one considers both sectional data and time series. The panel data approach provides the opportunity to investigate the individual effects of the cross-sectional data over time, separately. Changepoint refers to the time when a process shifts from an in-control condition to an unexpected condition. Identification of the change point of panel data leads one to an effective root cause analysis of the process. Literature addresses that Bai (2010) used the least square error (LSE) method to estimate the common change point in means of panel data. Bai (2010) also used the quasi-maximum likelihood (QML) method to estimate the change point in mean, variance, and both. Horváth and Hušková (2012) investigated the statistics of the change point estimator proposed by Bai (2010) and proposed a test based on the likelihood approach. Horváth & Hušková (2012) used their proposed method to identify the change point of the Gini coefficient for the panel data of 33 countries including European countries, Australia, the United States, South America, China and Taiwan. Li et al. (2015) developed the statistic of Horváth and Huşková (2012) and proposed a new statistic based on the cumulative sum (CUSUM) method to identify the common change point of the variance in panel data. Chen and Hu (2017) focused on using CUSUM to estimate the mean change point of panel data. The estimator proposed by Chen and Hu (2017) is less complex as it is more accurate compared to the LSE method proposed by Bai (2010). Pestova and Pesta (2015) proposed a test to identify a common change point in means of panel data using Bootstrap term. Their proposed statistic is based on the CUSUM method. Pestova and Pesta (2017) also proposed a common change point estimator using panel data based on the LSE method. Maciak et al. (2018) proposed a method using CUSUM statistics. The three recent studies used claim amounts paid by insurance companies to evaluate the capability of the proposed methods. De Wachter and Tzavalis (2012) proposed a test to detect the change point in the structure of a dynamic linear panel data model. De Wachter and Tzavalis (2012) approached the GMM framework. Zhu et al. (2013) proposed a method for the detection of the change point in the case that unbalanced panel data follows a dependence structure. In this approach, the copula method is used to describe the dependency structure of the panel data. Zhu et al. (2013) used their proposed method to identify the start and the end of the credit crisis in Chinese banking. Enomoto and Nagata (2016) developed the Mahalanobis – Taguchi (MT) method proposed by Taguchi (2002) using Bayes inference to identify the change point in panel data. They used an annual beverage consumption case in Japan to evaluate the performance of the proposed method. Cho (2016) proposed the Double CUSUM (DC) statistic to identify the change point in panel data. Cho (2016) used financial data sets from stock prices of S & P 100 index components as a real case study to evaluate the capability of the proposed method. Atashgar and Rafiee (2019) using the exponentially weighted moving average (EWMA) and double CUSUM statistic proposed Double CUSUM-EWMA (DCE) statistic to detection of the change point in the panel data means. They showed the superiority of their proposed method compared to the Cho (2016) method. Atashgar et al. (2022) proposed a method for detecting the change point in panel data approaching a hybrid statistic called Double CUSUM-Modified EWMA (DCME). Atashgar et al. (2022) showed numerically higher sensitivity to their proposed method compared to the methods proposed by Cho (2016) and Atashgar and Rafiee (2019). Recently also, Atashgar and Rafiee (2020) investigated the annual car production in the world using change point analysis in the panel data. They expressed that the change point analysis has the capability of evaluating strategic issues, effectively. This study attempts to propose a new method with high sensitivity compared to the methods proposed in the literature for detecting the common change point location in panel data. The remainder of this paper is structured as follows; the next section introduces the concept of the change point issue and the importance of its identification. The third section is allocated to describe the research methodology and the proposed method for detecting the location of the common change point in the panel data. The fourth section provides the capability of the proposed method by investigating two real case studies. The fifth section is allocated to the findings of this research. Section 6 compares the capability of the proposed method numerically considering several different cases. Finally, the last section is dedicated to conclusions.
**The concept and the importance of the change point**
Assume is panel data including independent observations, where . In this case, N and T denote the number of cross-sectional units and the length of the time series for each cross-sectional unit, respectively. When a panel data of a process works under common causes, the values of the panel vary over time under a known normal distribution. Equation 1 indicates the case in the panel of the process is produced without affecting any special cause. This case is referred to as the in-control condition of the process panel data.
In Equation 1, denotes the mean of the cross-section
In Equation 2, τ indicates, the time of possible change, and addresses the change value in mean after an unknown time of τ Î[1, T). Time
**Research methodology and the proposed method**
Let for t = 1, …, T denotes a random variable of time series. Assume a change point occurs at an unknown time τ. In this case, follows a common distribution function for t = 1,..., τ , and it follows a common distribution function for t = τ+1, ..., T, where . Let be defined for t = 1, …, T as follows:
where the function sgn(x) is defined as the follow:
Pettitt (1979) proposed a nonparametric test to identify the change point in the mean of time series. In this approach, to identify the presence of the change point
Based on Atashgar and Rafiee (2020) it is possible to use Monte Carlo simulation to access the test criterion. Once the test criterion is equal to and , the null hypothesis (i.e. no change has occurred) is rejected and the location of the change point can be estimated as Equation 7:
Based on the definition of Equation 2, the condition of Pettitt (1979) model is developed for the process panel data case. Assuming is defined for i = 1,…, N and t = 1,…, T, as well as is defined for i = 1,…, N and t = 1,…, T the following equation can be written:
In this case, to detect the location of the common change point in the panel data Equation 9 is proposed.
where is the test statistic of Equation (5) and is the test criterion for time series observations of cross-section
**Case studies**
In this section to analyze the performance of the proposed method, two real cases are considered. The reports in this section show the capability of the proposed method to identify the change point of the process panel data.
This section analyzes the change point identification for a real panel case to investigate the strategic terms of a holding company. The results of the real case of this section indicate that the proposed model of this paper is capable of performing an effective strategic analysis. The data in this section correspond to 4 important variables of 5 industries related to an Iranian holding company during the years 2010 to 2019. The terms of this case study are 1) total sales ( The analysis using DCE and DCME methods addresses the presence of a common change point for the panel data. The analysis indicates the existence of change points for the variables except for
The analysis of the change point for the real panel data using the proposed model of this paper addresses = 5. It means that the common change point for the three variables was triggered in the year 2014. Table 3 shows the results of the change point analysis.
To perform a strategic root cause analysis, the change point status for each industry of the holding company is considered. In this consideration, the retrospective time series data corresponding to the industries are analyzed separately to identify the change point for each industry. In this step, the change point detection method in the time series proposed by Pettitt (1979) is used. Table 4 shows the analysis results for
The change point analysis of the above case study leads one to the following conclusions: - As shown in Figure 2, the total sale (TS) term affecting the responsible factor(s) increases after the identified change point of the industries.
- After the identified mean change point corresponding to SCC, the mean observations of all 5 companies follow a different trend. Figure 3 addresses graphically the values after the detected change point.
- Figure 4 indicates that the mean value of TIP for all 5 industries decreases, after the change point.
The investigation of the performance reports of the industries from 2006 to 2014 considering the change point analysis, indicates that executive policies of the strategic plan have been changed affecting some important factors such as turnover of management. 4.2 Automotive manufacturing change point identification In this section, a real case corresponding to car manufacturing studied by Atashgar and Rafiee (2020) is considered. In this case, the number of cars manufactured by different companies around the world in 19 years (in the time interval of 2000 to 2018) is analyzed based on the change point concept. The countries considered in this evaluation are listed in Table 5. Based on the analysis of the proposed method of this paper, the location of the common change point for the automotive manufacturing case is estimated in 2008. Figure 5 indicates the result of the evaluation in obtaining the change point location in the case of panel data of the automotive industry. As shown in Figure 5 in the region of the maximum value the cure is relatively flat and it is not very peaked at the maximum value. This evaluation addresses the location of the change point and it is invaluable information for economic analysis. The automotive industry crisis in 2008-2010 was a part of the financial crisis of 2007-2008 and the great recession has been reported [11]. This crisis was reported for American industries and then it affected European, Canadian, and Asian industries.
**Findings**
Identification of the change point in a panel data case is an important step for practitioners, in the case that the process works under an out-of-control condition. An identified change point allows one to analyze and identify the source(s) that affected the process at the time when the process has been shifted to an unnatural condition. In this paper, a new statistic (as defined in Equation 10) is proposed to lead practitioners to find the change point statistically.
**Discussion**
The above analysis indicated that the proposed method is capable of detecting the change point effectively. To analyze the performance of the proposed method using a comparative approach, a low-dimension (10 × 10) process is simulated. In this analysis, it is assumed that the panel data is affected by a step shift after the change point. In this section, three models of the literature including Cho (2016), Atashgar and Rafiee (2019) and Atashgar et al. (2022) are compared with the performance of the proposed model of this paper. The simulated data follow the ARMA (2, 2) model, as Equation 11:
and , , and ϱÎ{0.2, 0.5}. It is noted that ϱ adjusts the correlation degree of the cross-section. Assume that all the cross-sections of the panel are affected by a step change at time τ. The change size for each cross-section follows the uniform function U (0.75, 1.25) with ratio 𝛿 Î{0.1, 0.2, 0.3}. In this evaluation, according to Atashgar et al. (2022), λ = 0.6 and k = λ / 2 values are considered. In this analysis, the simulation is iterated 1000 times for each change value. To evaluate the location accuracy indicator of the change point, is considered, where indicates the estimated change point τ. Table 6 shows the results of the analysis. Table 1 compares numerically the accuracy of the change point location detection for the four methods, i.e. DC, DCE, DCME, and PN (the proposed method of this paper). The analysis of Table 6 leads one to conclude the following results: - When the common change point manifests itself into the panel near the beginning and the end time points of the time interval [1, T], the location estimation accuracy of the change point by the proposed PN statistic of this paper is much better compared to the other three models.
- When the real common change point occurs in the middle of the time interval [1, T], the location accuracy of the change point estimated by all four models improves as the size of the change is increased.
- The reports indicate that the capability of improving the proposed model PN is superior compared to the other three models.
**Conclusions**
A precise change point identification allows practitioners to conduct an effective root cause analysis and remove the unnatural cause that affected the process. Change point analysis is started after the process panel data has been affected by a special cause and has shifted to an unnatural condition. Hence identifying the change point of a panel data is evaluated as an important issue in process management and data analysis issues. In this research, a new statistic is proposed for detecting the location of the change point in the panel data case. The proposed method allows practitioners to estimate the change point effectively. The comparative numerical performance analysis indicated that the capability of the proposed method is superior compared to the existing models of the literature. The investigation of two real panel case studies in this paper indicated that the proposed method is an effective approach to analysing different panel cases. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Atashgar, K., and Rafiee, N. (2019). Identifying the change point of panel data using simultaneously EWMA and CUSUM methods. Atashgar, K., Rafiee, N. and Karbasian, M. (2022). A new hybrid approach to panel data change point detection. Atashgar, K., and Rafiee, N. (2020). Identification of the Automotive Manufacturing Change Point Approaching Panel Data. International Conference on Industrial Engineering and Operations Management Dubai, UAE. Bai, J. (2010). Common breaks in means and variances for panel data. Chen, Z., and Hu, Y. (2017). Cumulative sum estimator for change-point in panel data. Statistical Papers, 58 (3), 707–728. https://doi.org/10.1007/s00362-015-0722-y Cho, H. (2016). Change-point detection in panel data via double CUSUM statistic. Electronic Journal of Statistics, 10(2), 2000–2038. https://doi.org/10.48550/arXiv.1611.08631 De Wachter, S., and Tzavalis, E. (2012). Detection of structural breaks in linear dynamic panel data models. Computational Statistics & Data Analysis, 56(11), 3020–3034. https://doi.org/10.1016/j.csda.2012.02.025 Enomoto, T., and Nagata, Y. (2016). Detection of change points in panel data based on the Bayesian MT method. Total Quality Science, 2(1), 36–47. https://doi.org/10.17929/tqs.2.36 Horváth L., Hušková, M. (2012). Change-point detection in panel data. Journal of Time Series Analysis, 33(4), 631–648. https://doi.org/j.1467-9892.2012.00796.x Li, F., Tian, Z., Xiao, Y., and Chen, Z. (2015). Variance change-point detection in panel data models. Economics Letters, 126, 140–143. https://doi.org/10.1016/j.econlet.2014.12.005 Maciak, M., Pestova, B., and Pesta, M. (2018). Structural breaks in dependent, heteroscedastic, and external panel data. Kybernetika, 54, (6), 1106–1121. https://doi.org/10.14736/kyb-2018-6-1106 Pestova, B., and Pesta, M. (2015). Testing structural changes in panel data with small fixed panel size and bootstrap. Metrika, 78(6), 665–689. https://doi.org/10.48550/arXiv.1509.01291 Pestova, B., and Pesta, M. (2017). “Change point estimation in panel data without boundary issue”. Risks, 5(1), 1-22. https://doi.org/10.3390/risks5010007 Pettitt, A.N. (1979). A non-parametric approach to the change-point problem. Applied Statistics, 28(2), 126-135. https://doi.org/10.2307/2346729 Taguchi, G. (2002), Technological development in the MT system. Japan Standards Association. (In Japanese) Zhu, X., Li, Y., Liang, C., Chen, J. and Wu, D. (2013). Copula-based change point detection for financial contagion in Chinese banking. Information Technology and Quantitative Management, 17, 619–626. https://doi.org/10.1016/j.procs.2013.05.080 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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