Monitoring long-memory air quality data using ARFIMA model

Transcription

1 ENVIRONMETRICS Environmetrics 2008; 19: Published online 9 August 2007 in Wiley InterScience ( Monitoring long-memory air quality data using ARFIMA model Jeh-Nan Pan 1 and Su-Tsu Chen 2,3 *,y 1 Department of Statistics, National Cheng-Kung University, Tainan 70101, Taiwan, R.O.C. 2 Department of Statistics, National Cheng-Kung University, Tainan 70101, Taiwan, R.O.C. 3 School of Environment and Life Sciences, Fooyin University, Kaohsiung Hsien 831, Taiwan, R.O.C. SUMMARY Statistical control chart is commonly used in the industry to help ensure stability of manufacturing process and it can also be used to monitor the environmental data, such as industrial waste or effluent of manufacturing process. However, control chart needs to be modified if the set of environmental data exhibits the property of long memory. In this paper, a control chart for autocorrelated data using autoregressive fractionally integrated moving-average (ARFIMA) model is proposed to monitor the long-memory air quality data. Finally, we use the air quality data of Taiwan as examples to compare the difference between ARFIMA and autoregressive integrated moving-average (ARIMA) models. The results show that residual control charts using ARFIMA models are more appropriate than those using ARIMA models. Copyright # 2007 John Wiley & Sons, Ltd. key words: long-memory data; control chart; ARFIMA model; ARIMA model 1. INTRODUCTION Statistical control charts are useful tools to monitor the performance of manufacturing or other environmental processes; such as the change of air quality, industrial pollution, etc. Pollutants emitted from different sources and factories are considered to be one of the environmental processes which need to be constantly monitored. Madu (1996) offers a deeper discussion of how a broad range of quality control methods, including control charts, can be applied to environmental management. Corbett and Pan (2002) showed that statistical control charts and process capability indices could be used as risk management tools for practitioners to prioritize the corrective action efforts. The application of statistical process control (SPC) techniques to the field of environmental quality management can also assist in setting up useful guidelines for evaluating actual environmental performance against the firm s environmental objectives, targets and regulatory requirements. Pan and Chen (2004) used control charts for autocorrelated data to monitor environmental performance for *Correspondence to: Su-Tsu Chen, School of Environment and Life Sciences, Fooyin University, no. 151 Chinhsueh Rd., Ta-Liao Hsiang, Kaohsiung Hsien 831, Kaohsiung, Taiwan, R.O.C. y chen8080@ms21.hinet.net Copyright # 2007 John Wiley & Sons, Ltd. Received 12 April 2006 Accepted 6 July 2007

2 210 J.-N. PAN AND S.-T. CHEN air pollution in Taiwan and selected the most appropriate one, which will give a warning signal in advance. Unlike the traditional control charts in which observations are assumed to be independent, observations of air quality and other environmental processes usually have autocorrelations. For example, according to Pan and Chen (2004), PM10 (particles between 2.5 and 10 micrometers) and O 3 of air quality in the Taipei city follow autoregressive integrated moving-average (ARIMA) models. Furthermore, Chan and Hwang (1996) stated that PM10 is the most important pollutant that deteriorates the air quality in Taiwan. Thus, PM10 is the main pollutant discussed in this paper and its time series model is studied. When applying control chart to autocorrelated data, it is commonly assumed that statistical models and white noises could fit the data. After fitting an appropriate model to the data, the residuals can be calculated. If the model is suitable, residuals should be independently and identically normally distributed and residual control charts can be applied then. If the control chart gives a signal, the process will be intervened and necessary corrective actions need to be taken. Autocorrelated data usually have significant autocorrelation functions (ACFs) just for small lags. However, the air pollution data have significant ACFs even at a very large lag due to the property of long-memory processes (e.g., Cope and Hess, 2005; Varotsos et al., 2005; Iglesias et al., 2006; Varotsos and Kirk-Davidoff, 2006). Therefore, the objective of this study is to develop a procedure of applying autoregressive fractionally integrated moving-average (ARFIMA) models to monitor long-memory air quality data. We show that natural logarithm of PM10 in southern Taiwan follows ARFIMA models instead of ARIMA models. Comparison of the suitability of applying ARIMA and ARFIMA models to air quality data is conducted. 2. DEVELOPMENT OF STATISTICAL CONTROL CHARTS FOR LONG-MEMORY DATA A fundamental assumption in the typical application of control charts is that the observations are uncorrelated. Unfortunately, this assumption is often not valid in many manufacturing or environmental processes. To deal with such kind of data, various control charts for autocorrelated data have been discussed and developed (see Lu and Reynolds, 1999). As to the development of control charts, it is clear that the monitored processes are initially assumed to be independent. Then control charts are extended to autocorrelated processes such as ARMA models. When time series data have autocorrelations that persist for a long time, that is, the so-called long-memory processes, statistical control charts for ARFIMA models are needed. Caballero et al. (2002) analyzed daily time series of mid-latitude near-surface air temperature at central England, Chicago, and Los Angeles. The spectral densities of these time series were plotted to show their long-range dependence. They also used a number of tests to detect the presence of long memory and concluded that long memory is indeed present in all the time series. There are several possible definitions of the property long memory (Baillie, 1996). For any discrete time series P process with ACF r(h) at lag h, the process possesses long memory if the quantity lim n n!1 h¼ n jrðhþj is nonfinite. Equivalently, the spectral density f(v) will be unbounded at low frequencies. Besides, according to Brockwell and Davis (1991), a weakly stationary process has long memory if its ACF has a hyperbolic decay, rðhþ~ch 2d 1 as h!1, where C 6¼ 0 and d < 0.5. In contrast to a long-memory process with hyperbolic decaying ACF, a weakly stationary process has a short memory when its ACF is geometrically bounded rðhþ Cr h for some C > 0, 0 < r < 1. ARIMA( p, d, f) model can be extended to non-integer values of d as the following definition of ARFIMA model proposed by Granger and Joyeux (1980) and Hosking (1981).

3 ARFIMA MODEL 211 Definition 2.1: Let Z t be a stationary process such that F p ðbþð1 BÞ d Z t ¼ u 0 þ Q q ðbþa t (1) for some 0.5 < d < 0.5. Then Z t is called a fractional ARIMA( p, d, q) or an ARFIMA( p, d, f) process. An ARFIMA process has long memory or long-range dependence if 0 < d < 0.5. For 0.5 < d < 0, the process is called intermediate memory or overdifferenced (Brockwell and Davis, 1991). When data have long memory and r(h) is significant even for a very large h, then the plot of ACF is very helpful to decide whether a process is long memory or not. Besides, the spectral density of a process is also valuable for detecting the property of long memory because that it has a pole at frequency zero, that is, the spectral density f(v) tends to infinity as v approaches zero. By examining both ACF and spectral density, a process with long memory can be determined. Recently, much work has been done in the area of model identification and estimation of long-memory models. Diagnostic checking of the adequacy of fitted models has also received much attention. In short, the study of long-memory data still attracts many scholars today. Wang (2002) considered the problem of detecting changes in an I(d) process, that is, ARFIMA(0, d, 0) process, and an ARFIMA( p, d, q) process by statistical control charts. To monitor an ARFIMA( p, d, q) process, it is transformed into an I(d) process, then control charts are applied to this I(d) process. The control limits of these control charts are calculated with the mean and the asymptotic variance of statistics based on the I(d) process. Instead of transforming the processes into an I(d) process like Wang s work, we propose the use of residual charts of ARFIMA models to monitor long-memory air quality data. The collected data are fitted to ARFIMA model first and an exponentially weighted moving-average (EWMA) control chart is applied to the residuals then. There are two phases of constructing the residual control chart for ARFIMA model. Phase I is to establish control limits using a historical data set. Phase II is the period of using these limits to monitor the process. If control charts signal, operators should step in to bring processes back to in-control state. Unlike manufacturing processes, control charts for air quality data may help the public in another way. If air quality is suspected to deteriorate and the news could reach the public as soon as possible, for example, posted in a website, people especially those with respiratory problems may stay at home instead of going out to avoid harm from it. Although a regular weather forecast does the job, it cannot release detailed information to any specific place. Listed below are the proposed procedures for constructing control limits for the control chart using ARFIMA models in Phase I. 1. Collect historical air quality or environmental data. 2. Fit the data collected in step 1 into an appropriate model and perform estimation of parameters. Check the suitability of the model. After a proper model is selected, residuals can be calculated. 3. Establish control limits for the residuals. 4. Delete any residuals fallen beyond the control limits and estimate parameters of control charts. 5. Re-establish control limits for the residuals. 6. Repeat steps 4 and 5 until there are no outliers/out-of-control signals. If models and parameters of processes are known in advance, then the control limits could be calculated and one can bypass the Phase I. The control limits established in Phase I are used to monitor processes in Phase II. The above procedures help in constructing an appropriate chart using ARFIMA models. Most control charts for autocorrelated processes do not follow the procedures mentioned

4 212 J.-N. PAN AND S.-T. CHEN above. A literature survey would undoubtedly reveal that distinction between these two phases is lacking in most papers (Faltin et al., 1997). There seems to be a tendency to focus on Phase I, although this is usually not explicitly stated. Under the traditional assumption of in-control process data, that is, independent and identically distributed (i.i.d.), control charts constructed in Phase I are used for monitoring processes in Phase II. If a control chart does not signal, then the process is in-control, and it may have the same distribution function as the historical data in Phase I. However, the independence assumption sometimes is not reasonable for processes in many applications because the dynamics of the process either produce autocorrelated observations or the underlying processes are correlated, for example, the ozone content in the air. If the autocorrelated process in Phase I is in-control, then the process in Phase II should be fitted with the same time series model in Phase I. Generally speaking, if the larger mean shift is concerned, X chart would be a better choice. If the smaller mean shift is to be detected, then either EWMA or CUSUM chart can do the job well. The application of the above procedures will be demonstrated by two empirical examples of long-memory air quality data of southern Taiwan. 3. COMPARISON OF ARFIMA AND ARIMA MODELS 3.1. Example of using the PM10 data of Nantsz There are 58 surveillance stations established by EPA of Taiwan to monitor the air quality in Taiwan. The largest industrial city, Kaohsiung, located in the southern Taiwan, has serious air pollution problems. Nantsz is an administrative district in Kaohsiung city, where a big refinery plant of Chinese Petroleum Corporation and many other industries are located. It is also well known for a long history of public protest for pollution. The air quality data of PM10 collected by Nantsz station are discussed in this paper since PM10 is the important index to health as mentioned in Section 2. The hourly data were collected between 1999 and 2002 by Nantsz station. In this example, daily average is used and we drop some missing data due to measurement failure or unbelievable measurement. Thus, a total 725 observations were recorded during and 712 observations were recorded between 2001 and There are two phases for the construction of control charts. In Phase I, a set of historical data is chosen and parameters of the fitted model are estimated. In this stage, the 725 observations gathered during 1999 and 2000 shown in Figure 1 are treated as historical data. Obviously, Figure 1 indicates that there is seasonal influence and the variance of PM10 is not constant. Since the variance of PM10 is not constant, a transformation of PM10 is performed to stabilize variance as suggested by Wei (1990). A natural logarithm of PM10, denoted has successfully achieved this goal without deseasonalizing and detrending the raw data. Moreover, the ACFs of ln(pm10) data decay at a very slow rate as shown in Figure 2, which further confirms that it does exhibit long memory, so ARFIMA model would be a better one. In order to find a suitable ARFIMA model for ln(pm10), several models are selected as candidates. By the Akaike Information Criterion, it is found that ARFIMA(0, d, 1) is suitable for ln(pm10). After model fitting, diagnostic of residuals are performed to check the suitability of residual control chart. The results indicate that residuals follow normal distribution. To monitor the change of residuals, it is suggested by Montgomery (2004) that EWMA chart be used since it is known for its sensitivity to detect small-sustained shift of process and its robustness to non-normal data. Assume residual at tth time is r t. The control statistic of EWMA residual control chart can be written as Equation (2). Y t ¼ð1 lþy t 1 þ lr t (2)

5 ARFIMA MODEL PM Figure 1. The run chart of Nantsz s PM10 data ACF Figure 2. The ACF of Nantsz s ln(pm10) data

6 214 J.-N. PAN AND S.-T. CHEN When EWMA chart is used, the parameter l and the in-control ARL should be decided first. A different size of mean shift needs different l. If a smaller mean shift is concerned, a smaller l needs to be used. The parameter l of the above residual EWMA chart is set to be 0.1 and control limits are set to have in-control ARLs according to Montgomery (2004). The initial EWMA statistics showed that there were eight points out of control. After deleting out-of-control points, the EWMA control chart is applied to the remainder of residuals again. These procedures are repeated until all statistics are in-control. The ARFIMA model is then fitted again for the left data from 2001 to The appropriate model can be written as Equation (3): ð1 BÞ 0:47 ðlnðpm10 t Þ 4:34Þ ¼ð1 þ 0:16BÞ" t (3) In Phase II, we first obtain the residuals of the 712 data from 2001 to 2002 by fitting Equation (3). The EWMA residual control limits constructed in Phase I is applied to these residuals as shown in Figure 3. Figure 3 indicates that no residual is out of control, which implies that ln(pm10) data in Phase II are likely to follow the similar pattern of Phase I. Thus, we may conclude that there is no evidence that the air quality of PM10 at Nantsz in Phase II is different from Phase I. This means that the air quality in Nantsz area has not been improved from 1999 to 2002 period. Further corrective actions need to be done. If the long-term autocorrelations were ignored, then the most commonly used models to fit time series are ARIMA models. In contrast with the ARFIMA model, ARIMA model is compared for assessing the suitability of model selection. It is found that ARIMA(0, 1, 2) model could fit the air quality data of Nantsz from 1999 to However, the residuals are not normally distributed, thus, with a proper Box-Cox transformation (l ¼ 2) the residuals are normally distributed. Similarly, in Phase I, EWMA control chart is applied to these transformed residuals. EWMA statistics time Figure 3. EWMA chart for the residuals of Nantsz s ln(pm10) data in Phase II using ARFIMA model

7 ARFIMA MODEL 215 ewma statistics time Figure 4. EWMA chart for the residuals of Nantsz s ln(pm10) data in Phase II using ARIMA model After deleting the only out-of-control point, the air quality data are modeled again. The appropriate model can be written as Equation (4). ð1 BÞ lnðpm10 t Þ¼ 0:0003 þð1 0:4031B 0:2883B 2 Þ" t (4) In Phase II, we use Equation (4) to fit ln(pm10) of Nantsz collected between 2001 and Despite the fact that residuals could not been transformed to be normally distributed with Box-Cox method in this phase, EWMA control charts are applied to monitor the residuals without transformation. Figure 4 indicates that there are two points out of control and its pattern is different from Figure 3. This Nantsz example demonstrates that the ARFIMA model is more appropriate than ARIMA model. False alarms would occur if one selects a wrong ARIMA model instead of using ARFIMA model. In addition, if the diagnoses of residuals have been performed cautiously, it will be found that the ARIMA model does not fit the data because of the residuals have non-constant variance and the ACF of ARIMA model does not resemble the one of ln(pm10) data. The ACF is noticeably important and should be calculated to a very large lag to help us identify the long-memory property. In short, the model fitting in the Phase I should be performed very carefully to avoid the incorrect selection of models, which may lead to false alarms in Phase II Example of using the PM10 data of Tsoying To compare with Nantsz s data, the practical application of PM10 data collected by Tsoying station is discussed too. In Phase I, the 725 observations gathered during 1999 and 2000(as shown in Figure 5)

8 216 J.-N. PAN AND S.-T. CHEN PM are treated as historical data. Obviously, it indicates that there is seasonal influence and the variance of PM10 is not constant, then natural logarithm of PM10 is taken to stabilize the variance without deseasonalizing and detrending the raw data. As to the property of long memory of the data, the ACF shown in Figure 6 further confirms that Tsoying s ln(pm10) data do exhibit long memory. Therefore, ARFIMA model would be a better one. By the Akaike Information Criterion, it is found that ARFIMA(2, d, 0) is suitable for ln(pm10) data. After model fitting, a diagnostic check of residuals is 1.00 Figure 5. The run chart of Tsoying s PM10 data ACF Figure 6. The ACF plot of Tsoying s ln(pm10) data

9 ARFIMA MODEL 217 performed to assess the suitability of the residuals. The results indicate that residuals follow normal distribution. The initial EWMA chart of Tsoying s data showed that there were 12 points out of control. After deleting out-of-control points, the EWMA control chart is applied to the remainder of residuals again. These procedures are repeated until all statistics are in-control. The ARFIMA model is then fitted again for the remaining data from 1999 to The appropriate model can be written as: ð1 0:17B þ 0:17B 2 Þð1 BÞ 0:49 ðlnðpm10 t Þ 4:27Þ ¼" t (5) In Phase II, Equation (5) is used to fit the 717 data from 2001 to 2002 and then EWMA residual control charts constructed in Phase I are applied to monitor the residuals as shown in Figure 7. Notice that there is one out-of-control point in Figure 7. It indicates that 1n (PM10) data in Phase II is likely to follow the similar pattern of Phase I except one day. If the long-term autocorrelations were ignored, then the most commonly used models to fit time series are ARIMA models. Similar to Nantsz s example, ARIMA model is compared with ARFIMA model for assessing the suitability of model selection. It is found that ARIMA(0, 1, 2) model could fit the air quality data of Tsoying from 1999 to Since the residuals cannot be converted into normally distributed data by Box-Cox transformation, EWMA control chart is used to monitor the untransformed residuals because of its robustness to non-normal data. Since there is no out-of-control point in this EWMA chart, the appropriate model can be written as: ð1 BÞ lnðpm10 t Þ¼ 0:0002 þð1 0:6855B þ 0:1171B 2 Þ" t (6) In Phase II, Equation (6) is used to fit ln(pm10) data of Tsoying collected between 2001 and 2002 and then EWMA control charts are applied to monitor the residuals without transformation. Figure 8 shows that there are two out-of-control points and its pattern is different from Figure 7. This Tsoying s EWMA statistics time Figure 7. EWMA control chart for the residuals of Tsoying s data in Phase II using ARFIMA model

10 218 J.-N. PAN AND S.-T. CHEN EWMA chart of residuals of EWMA statistics time Figure 8. EWMA control chart for the residuals of Tsoying s data in Phase II using ARIMA model example demonstrates that the ARFIMA model is more appropriate than ARIMA model. Similar to Nantsz s example, if the diagnosis of residuals is performed cautiously, it can be found that the ARIMA model is not appropriate since the residuals of Tsoying s data have non-constant variance. Therefore, it should be noted that the model fitting in the Phase I should be performed very cautiously to avoid incorrect selection of models, which may lead to false alarms in Phase II. 4. CONCLUSIONS In this paper, control charts using ARFIMA model are proposed to monitor long-memory air quality data. A proper use of control charts can help us understand whether the underlying long-memory air quality model has changed. The proposed procedures of applying ARFIMA models to monitor air quality data can also be used for monitoring other long-memory environmental data. Through two empirical examples of air quality of southern Taiwan, one set of ln(pm10) data follows ARFIMA(0, 0.47, 1) model while another one follows ARFIMA(2, 0.49, 0) model, we have demonstrated that control charts using ARFIMA models are more appropriate than control charts using ARIMA models in monitoring long-memory air quality data. When monitoring data with autocorrelation, the meaning of out-of-control indicates not only the residuals of processes may deviate from what are assumed, but also the underlying model of the process might be changed. Due to the complexity of the ARFIMA model, procedures for constructing an appropriate control chart especially in Phase II are more difficult than the traditional control charts for autocorrelated data. To ensure the efficiency and effectiveness of monitoring environmental data with long memory, it is suggested that an on-line analyzer of the residual control chart using ARFIMA model be developed to help practitioners understand the quality change of the environment, so a timely corrective action can be made.

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