- Email: [email protected]
Autoregressive integrated moving averages (ARIMA) modelling of a traffic noise time series
Applied Acoustics 58 (1999) 283±294 www.elsevier.com/locate/apacoust
Autoregressive integrated moving averages (ARIMA) modelling of a trac noise time series K. Kumar*, V.K. Jain School of Environmental Sciences, Jawaharlal Nehru University, New Delhi-110 067, India Received 14 October 1997; received in revised form 25 October 1998; accepted 23 November 1998
Abstract Sound pressure levels (dBA) were measured at 10 s intervals in the vicinity of a busy road carrying vehicular trac. The resultant time series is analyzed using autoregressive integrated moving averages (ARIMA) modelling technique. The time series is found to be non stationary. After ®rst dierencing, the transformed series becomes stationary and is found to be governed by a moving average process of order 1. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: ARIMA; Trac noise; Time series
1. Introduction In the past few decades, the concern about environmental pollution in urban areas has grown appreciably. One of the important forms of pollution that aects the physiological and psychological well-being of a person is noise pollution. It is, therefore, quite important to monitor the ambient noise to which the population of an area may be exposed. This realization prompted a number of surveys of noise levels in dierent parts of the world [1±6]. A major contribution towards urban noise comes from vehicular trac. Therefore, it is not surprising that road trac noise is one of the most extensively studied areas of noise pollution. Several studies have been made on various aspects of trac noise [7±14]. Its reduction has become one of the main objectives of environmental planning in most cities all over the would. For devising appropriate strategies to ful®ll this objective, however, one needs to have a knowledge of the existing ambient noise levels. This is achieved by monitoring the noise levels in the areas of interest over a period of time. Since the nature and * Corresponding author. Department of Environmental Sciences, Guru Jambheshwar University, Hissar, India. 0003-682X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 8-4
284
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
pattern of road trac in a city may change with time, conducting such noise surveys frequently can be a very costly exercise, Moreover, it is also desirable to know the future noise levels in advance from the viewpoint of having an eective environmental plan. Therefore, it becomes essential to develop predictive models which could forecast the noise levels both spatially and temporally. To address this problem a number of mathematical models have been developed by several workers [15±23]. Baverstock et al. [19] have demonstrated that area-based prediction models using the data of land use and road trac are quite robust and reliable. Hasebe and Kaneyasu [20] have investigated the problem of prediction of trac noise from trunk roads and access roads in urban areas. Yamaguchi et al. [21] have suggested a practical method for predicting periodic non-stationary road trac noise. Zhang [22] has modelled highway noise under dierent trac ¯ow conditions. Kato et al. [23] have given a method for determining road trac noise downstream of a trac signal. Since the road trac noise is believed to be stochastic in nature, the techniques of time series analysis hold a great promise for the development of predictive models of road trac noise. However, these techniques have not been fully exploited until today, with the exception of a small number of studies [24,25]. The present study attempts to investigate the applicability of autoregressive integrated moving averages
Fig. 1. Time sequence plot.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
285
(ARIMA) modelling, a special class of time series techniques, for developing forecast models of road trac noise. 2. Site description and data collection The sampling site in the present study was located at a distance of 5 m from the kerbside of a major road (Abdul Gamel Nasir Marg) which encircles a major part of the city of Delhi. The site is about half a km away from the nearest trac intersection near the Indian Institute of Technology, Delhi. The average volume of trac passing by the sampling site is 1500 vehicles per h during 9.00 a.m. to 5.00 p.m. The trac is composed of heavy vehicles (buses and trucks), three-wheelers, twowheelers and light vehicles (cars and jeeps). The instrument used for the measurement of noise in the present study is a precision type 1 sound level meter (NL-10A, RION, Japan). The sound level meter was mounted at a height of 1.2 m above the ground. A total of 549 A-weighted instantaneous sound pressure levels of trac noise at 10 s intervals were recorded from 10.00 a.m. onwards on 27 December 1996.
Fig. 2. ACF of time series.
286
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
3. Methodology The time series of trac noise obtained as a result of sampling of instantaneous noise is analyzed using ARIMA technique [26]. A general ARIMA model of order (p,d,q) representing the time series can be written as Brd xt Bet where xt and et represent sound pressure level and random error terms at time t respectively. B is a backward shift operator de®ned by Bxt xtÿ1 , and related to r by r 1 ÿ B; rd 1 ÿ Bd ; d is the order of dierencing. B and B are autoregressive (AR) and moving averages (MA) operators of orders p and q, respectively, and are de®ned as B 1 ÿ 1 B ÿ 2 B ÿ ÿ p Bp and B 1 ÿ 1 B ÿ 2 B2 ÿ ÿq Bq
Fig. 3. Seasonal subseries plot.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
287
where 1 ; 2 ; ::::: p are the autoregressive coecients and 1 ; 2 ; :::::q are the moving average coecients. In the Box±Jenkins methodology of ARIMA modelling, the ®rst step is to determine whether the time series is stationary or non stationary. If it is non stationary it is transformed into a stationary time series by applying suitable degree of dierencing to it. This gives value of d. Then appropriate values of p,q are found by examining auto-correlation function (ACF) and partial auto-correlation function (PACF) of the time series. Having determined p,q and d the coecients of autoregressive and moving average terms are estimated using nonlinear least squares method. In the present study the time series has been analyzed using statistical computer software STATGRAPHICS. 4. Results and discussion Fig. 1 shows the horizontal time sequence plot of the time series of noise levels. The time series appears to be stationary as far as its nonseasonal behaviour is concerned. In the Box±Jenkins methodology of ARIMA modelling, it must be ®rst established that the given time series is stationary before trying to identify the
Fig. 4. Seasonal subseries plot of the seasonally dierenced time series.
288
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
orders of AR and MA processes in the model to be ®tted. The ACF of a time series is an important tool in this regard. Fig. 2 shows the ACF of time series of noise levels obtained in the present study. It can be clearly seen that the autocorrelations are signi®cantly dierent from zero even at higher time lags. It may also be noticed that the autocorrelations form a wave-like pattern. If one combines the behaviours indicated by Figs. 1 and 2, it can be inferred that the time series may have some seasonal nonstationarity. A more careful examination of Fig. 2 reveals that the autocorrelations at lags 12, 24,....... are unusually high. The neighbouring autocorrelations at lags 11, 13, 23, 25,...... are also on the higher sides. This indicates the presence of seasonality (roughly of the length 12) in the given time series, which may be associated with the switching of trac lights located about 500 m from the sampling site and corresponds to the approximate interval of 120 s between two successive green signals. This can be con®rmed by the seasonal subseries plot (season length=12) of the time series shown in Fig. 3. It is evident from the ®gure that the seasonal means are not the same for all the 12 seasons. Instead, a wave-like pattern is observed in seasonal means. Figs. 1±3 therefore point towards the presence of a seasonal nonstationarity of about the length 12 in the given time series. In view of the above indications, it becomes desirable to do the seasonal dierencing of the original time
Fig. 5. ACF of seasonally dierenced time series.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
289
series in order to remove the existing seasonal nonstationarity. Thus, a new time series is generated by dierencing the original time series at lag 12. Fig. 4 shows the seasonal subseries plot of this new time series. It becomes clear at once that the seasonal means are almost the same for all seasons now. In other words, the seasonal nonstationarity appears to have been removed in this new time series. Fig. 5 shows the ACF of this dierenced time series. It can be seen that the autocorrelation at lag 12 is signi®cantly dierent from zero. All the other autocorrelations may be considered as insigni®cant. Fig. 6 shows the PACF of the seasonally dierenced series. It can be observed that there is a pattern of exponential decay in the partial autocorrelations at lag 12, 24, 36,....... The patterns observed in Figs. 5 and 6 are quite similar to the theoretical patterns of ACF and PACF of a seasonal MA process of the order 1. This indicates the existence of a seasonal MA process of order 1 in the original time series. In view of the observations made above, a seasonal model of the order (0,1,1) [12] was identi®ed. Here p 0; q 1; d 1 and the superscript 12 represents the length of seasonality. The model ®tting results are given in Table 1. It can be seen from the table that the value of the seasonal moving average parameter 1 is less than 1 which con®rms the condition of invertibility required for moving average coecients. Further, the p-value obtained for 1 is very good, thus
Fig. 6. PACF of seasonally dierenced time series.
290
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
indicating that the value of the coecient is signi®cantly dierent from zero. The p-value obtained for the mean on the other hand indicates that mean cannot be taken as signi®cantly dierent from zero. This, of course, is expected since the original time series has been dierenced once and the model, therefore, gives the mean for the seasonally dierenced time series. The non-signi®cant value of mean in the model further indicates the absence of any deterministic trend in the original time series. In other words, it can be said that time series is stochastic in nature and that it lacks any deterministic component. Once an ARIMA model is ®tted, it is important to investigate how well the model ®ts the given time series. This comprises the step of diagnostic checking in model building. Investigation of the behaviour of residuals is a useful tool in this regard. Fig. 7 shows the ACF of residuals obtained from the model ®tted in Table 1. It is evident from the ®gure that all autocorrelations in the ACF plot of residuals are insigni®cant. This implies that the residuals are not autocorrelated and are statistically independent. This means that the model is adequate. This is further con®rmed by the 2 test statistic computed for the ®rst 20 residual autocorrelations. The computed value of 2 test statistic 23.3834 is less than the tabulated value 28.8693 of 2 with 18 d.f. at 0.05 level of signi®cance. Thus the joint null hypothesis, that the
Fig. 7. Estimated residual ACF.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
291
residual autocorrelations are independent, cannot be rejected. Hence, it may be inferred that the residual autocorrelations are not signi®cantly dierent from zero. The model equation on the basis of results shown in Table 1 can be written as 1 ÿ B12 xt 1 ÿ 0:85439B12 et
1
An alternative model found worth considering for ®tting to time series was the one with an additional non-seasonal autoregressive term. The model ®tting results are presented in Table 2 which are quite similar to those presented in Table 1 except that in the above model one nonseasonal AR(1) parameter 1 has been included. The signi®cance level of 1 is just satisfactory. It can be seen that the value of 1 is less than 1. This satis®es the stationarity condition for an ARIMA model. Fig. 8 shows the residual ACF for the alternative model. After comparing Fig. 7 with Fig. 8, it can easily be seen that the behaviour of residuals in both the models is quite similar. The alternative model may be represented by the following equation. 1 ÿ 0:08314B 1 ÿ B12 xt 1 ÿ 0:8573B12 et
2
Table 1 Summary of ®tted model for the observed time series of noise levels Parameter
Estimate
Standard error
t-value
p-value
SMA (12) Mean Constant
0.85439 0.00017 0.00017
0.0213 0.02414 ±
42.44129 0.00704 ±
0.0000 0.99439 ±
Model ®tted to seasonal dierences of order 1 with seasonal length=12. Estimated white noise variance=13.5842 with 534 d.f. Estimated white noise standard deviation (SE)=3.68567. Chi-square ( 2 ) test statistic on ®rst 20 residual autocorrelations=23.3834 with probability of a larger value given white noise=0.220875.
Table 2 Summary of the alternative model ®tted to the observed time series Parameter
Estimate
Standard error
t-value
p-value
AR (1) SMA (12) Mean Constant
0.08314 0.85730 ÿ0.00005 ÿ0.00004
0.04318 0.02014 0.02563 ±
1.92537 42.57278 ÿ0.00186 ±
0.05471 0.00000 0.99852 ±
Model ®tted to seasonal dierences of order 1 with seasonal length=12. Estimated white noise variance=13.5191 with 533 d.f. Estimated white noise standard deviation (SE)=3.67683. Chi-square ( 2 ) test statistic on ®rst 20 residual autocorrelations=19.3563 with probability of a larger value given white noise=0.3702.
292
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
One of the basic principles of ARIMA model building is that the model should be parsimonious (i.e. the model which adequately describes the time series with a relatively less number of parameters). Keeping in view the above principle, it can be concluded that the model represented by Eq. (1) is the most appropriate for the observed time series. The model Eq. (1) can be used for predictive purposes by expanding it in the following manner. xt xtÿ12 et ÿ 0:85439etÿ12
3
where t is assumed to be the current time period (i.e. the 549th, the last observation). In order to forecast xt1 , all the subscripts in Eq. (3) are increased by 1. It gives xt1 xtÿ11 et1 ÿ 0:85439etÿ11
4
Since the term et1 i.e. the forecast error one period ahead is not known, it is assigned the value zero (the expected value of feature random errors). The values of other terms on the right-hand side of Eq. (4) are known. Thus the forecast one period ahead is determined. Forecasts for future periods can similarly be obtained. To summarize, the road trac noise variations in the present study have been shown to be governed by a seasonal moving average (0,1,1) [12] process. It would be
Fig. 8. Estimated residual ACF of the alternative model.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
293
appropriate here to mention that the above result cannot be generalized to other trac sites. In fact the identi®cation of the stochastic process (whether autoregressive or moving average or a combination of both) governing any time series will have to be done afresh in order to develop a predictive model of trac noise for a given site. The utility of the time series modelling approach in the present study is limited insomuch as the forecasting of long-term noise levels is concerned. However, the study does reveal that even on short time scales the ¯uctuations in noise levels are governed by stochastic processes which can be identi®ed with the help of the Box±Jenkins methodology. Certainly, it would have been better to consider a time series over much longer time scales (e.g. months or years) but it was not feasible in the present study due to the lack of the facility of a permanent automatic noise monitoring station. Nevertheless, the adequacy of ARIMA modelling technique demonstrated in the present study does provide the basis to further investigate the applicability of these techniques to forecast noise levels over longer time scales. References [1] Mochizuki T, Imaizumi N. City noise in Tokyo. Journal of Acoustical Society of Japan 1967;23:146±7. [2] Price AJ. A community noise survey of great Vancouver. Journal of Acoustical Society of America 1972;52:488±92. [3] Safeer HB. Community noise levelsÐa statistical phenomenon. Journal of Sound and Vibration 1973;26:489±502. [4] Webster JE. Community noise survey in Medford, Massachussetts. Journal of Acoustical Society of America 1973;54:985±95. [5] Attenborough K, Clark S, Utley WA. Background noise levels in the United Kingdom. Journal of Sound and Vibration 1976;48:359±75. [6] Singh BB, Jain VK. A comparative study of noise levels in some residential, industrial and commercial areas of Delhi. Environmental Monitoring and Assessment 1995;35:1±11. [7] Cannelli GB. Trac noise pollution in Rome. Applied Acoustics 1974;7(2):103±16. [8] Rylander R, Sorenson S, Kajland A. Trac noise exposure and annoyance reactions. Journal of Sound and Vibration 1976;47:237±42. [9] Ko NWM. Trac noise in a high rise city. Applied Acoustics 1978;11:225±39. [10] Bjorkman M. Maximum noise levels in road trac noise. Journal of Sound and Vibration 1988;127(3):583±7. [11] Carter NL, Ingham P, Tran K. Overnight trac noise measurements in bedrooms and outdoors, Pennant Hills Road, SydneyÐcomparisons with criteria for sleep. Acoust. Aust. 1992;20(2):49±55. [12] Brown AL. Exposure of Australian population to road trac noise. Applied Acoustics 1994;43(2):169±76. [13] Hofman WF, Kumar A, Tulen JHM. Cardiac reactivity to trac noise during sleep on man. Journal of Sound and Vibration 1995;179(4):577±89. [14] Ohrstrom E. Eects of low levels of road trac noise during the night: a laboratory study on number of events, maximum noise levels and noise sensitivity. Journal of Sound and Vibration 1995;179(4):603±15. [15] Johnson DR, Saunders EG. The evaluation of noise from freely ¯owing road trac. Journal of Sound and Vibration 1968;7:287±309. [16] Burgess MA. Noise prediction for urban trac conditionsÐrelated to measurements in the Sydney metropolitan area. Applied Acoustics 1977;10:1±7. [17] Delany ME, Harland DG, Hood RA, Scholes WE. The prediction of noise levels L10 due to road trac. Journal of Sound and Vibration 1976;48(3):305±25.
294
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
[18] Favre BM. Noise emission of road vehicles: evaluation of some simple models. Journal of Sound and Vibration 1983;91(4):571±82. [19] Baverstock SJ, Pocock RL, Attenborough K. Development of area-based methods for predicting ambient noise. Applied Acoustics 1991;33:303±12. [20] Hasebe M, Kaneyasu K. Prediction of trac noise from trunk roads and access roads in urban areas. Noise Control Engineering Journal 1991;37(2):71±6. [21] Yamaguchi S, Ishihara S, Kato Y. A practical method of predicting periodic nonstationary road trac noise based on the time rate of average number of ¯owing vehicles. Acoustical Letters 1992;16(6):123±8. [22] Zhang J. A study on the highway noise prediction model applicable to dierent trac ¯ow. Noise Control Engineering Journal 1993;41(3):371±5. [23] Kato Y, Yamaguchi S, Takagi K. A practical prediction method of road trac noise around trac signals and its experimental studies in actual roads. Prediction at the side of the road, downstream from the trac signal. Journal of Acoustical Society of Japan 1994;50(2):91±102. [24] DeVor, RE, Schomer, PD, Kline, WA, Neathamer, RD. Development of temporal sampling strategies for monitoring noise. Journal of Acoustical Society of America 1979;66(3):763±71. [25] Schomer PD, De Vor RE, Kline WA. Sampling strategies for monitoring noise in the vicinity of airports. Journal of Acoustical Society of America 1983;73(6):2041±50. [26] Box GEP, Jenkins, GM. Time series analysis forecasting and control. San Francisco: Holden-Day, 1976.
Autoregressive integrated moving averages (ARIMA) modelling of a trac noise time series K. Kumar*, V.K. Jain School of Environmental Sciences, Jawaharlal Nehru University, New Delhi-110 067, India Received 14 October 1997; received in revised form 25 October 1998; accepted 23 November 1998
Abstract Sound pressure levels (dBA) were measured at 10 s intervals in the vicinity of a busy road carrying vehicular trac. The resultant time series is analyzed using autoregressive integrated moving averages (ARIMA) modelling technique. The time series is found to be non stationary. After ®rst dierencing, the transformed series becomes stationary and is found to be governed by a moving average process of order 1. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: ARIMA; Trac noise; Time series
1. Introduction In the past few decades, the concern about environmental pollution in urban areas has grown appreciably. One of the important forms of pollution that aects the physiological and psychological well-being of a person is noise pollution. It is, therefore, quite important to monitor the ambient noise to which the population of an area may be exposed. This realization prompted a number of surveys of noise levels in dierent parts of the world [1±6]. A major contribution towards urban noise comes from vehicular trac. Therefore, it is not surprising that road trac noise is one of the most extensively studied areas of noise pollution. Several studies have been made on various aspects of trac noise [7±14]. Its reduction has become one of the main objectives of environmental planning in most cities all over the would. For devising appropriate strategies to ful®ll this objective, however, one needs to have a knowledge of the existing ambient noise levels. This is achieved by monitoring the noise levels in the areas of interest over a period of time. Since the nature and * Corresponding author. Department of Environmental Sciences, Guru Jambheshwar University, Hissar, India. 0003-682X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 8-4
284
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
pattern of road trac in a city may change with time, conducting such noise surveys frequently can be a very costly exercise, Moreover, it is also desirable to know the future noise levels in advance from the viewpoint of having an eective environmental plan. Therefore, it becomes essential to develop predictive models which could forecast the noise levels both spatially and temporally. To address this problem a number of mathematical models have been developed by several workers [15±23]. Baverstock et al. [19] have demonstrated that area-based prediction models using the data of land use and road trac are quite robust and reliable. Hasebe and Kaneyasu [20] have investigated the problem of prediction of trac noise from trunk roads and access roads in urban areas. Yamaguchi et al. [21] have suggested a practical method for predicting periodic non-stationary road trac noise. Zhang [22] has modelled highway noise under dierent trac ¯ow conditions. Kato et al. [23] have given a method for determining road trac noise downstream of a trac signal. Since the road trac noise is believed to be stochastic in nature, the techniques of time series analysis hold a great promise for the development of predictive models of road trac noise. However, these techniques have not been fully exploited until today, with the exception of a small number of studies [24,25]. The present study attempts to investigate the applicability of autoregressive integrated moving averages
Fig. 1. Time sequence plot.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
285
(ARIMA) modelling, a special class of time series techniques, for developing forecast models of road trac noise. 2. Site description and data collection The sampling site in the present study was located at a distance of 5 m from the kerbside of a major road (Abdul Gamel Nasir Marg) which encircles a major part of the city of Delhi. The site is about half a km away from the nearest trac intersection near the Indian Institute of Technology, Delhi. The average volume of trac passing by the sampling site is 1500 vehicles per h during 9.00 a.m. to 5.00 p.m. The trac is composed of heavy vehicles (buses and trucks), three-wheelers, twowheelers and light vehicles (cars and jeeps). The instrument used for the measurement of noise in the present study is a precision type 1 sound level meter (NL-10A, RION, Japan). The sound level meter was mounted at a height of 1.2 m above the ground. A total of 549 A-weighted instantaneous sound pressure levels of trac noise at 10 s intervals were recorded from 10.00 a.m. onwards on 27 December 1996.
Fig. 2. ACF of time series.
286
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
3. Methodology The time series of trac noise obtained as a result of sampling of instantaneous noise is analyzed using ARIMA technique [26]. A general ARIMA model of order (p,d,q) representing the time series can be written as Brd xt Bet where xt and et represent sound pressure level and random error terms at time t respectively. B is a backward shift operator de®ned by Bxt xtÿ1 , and related to r by r 1 ÿ B; rd 1 ÿ Bd ; d is the order of dierencing. B and B are autoregressive (AR) and moving averages (MA) operators of orders p and q, respectively, and are de®ned as B 1 ÿ 1 B ÿ 2 B ÿ ÿ p Bp and B 1 ÿ 1 B ÿ 2 B2 ÿ ÿq Bq
Fig. 3. Seasonal subseries plot.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
287
where 1 ; 2 ; ::::: p are the autoregressive coecients and 1 ; 2 ; :::::q are the moving average coecients. In the Box±Jenkins methodology of ARIMA modelling, the ®rst step is to determine whether the time series is stationary or non stationary. If it is non stationary it is transformed into a stationary time series by applying suitable degree of dierencing to it. This gives value of d. Then appropriate values of p,q are found by examining auto-correlation function (ACF) and partial auto-correlation function (PACF) of the time series. Having determined p,q and d the coecients of autoregressive and moving average terms are estimated using nonlinear least squares method. In the present study the time series has been analyzed using statistical computer software STATGRAPHICS. 4. Results and discussion Fig. 1 shows the horizontal time sequence plot of the time series of noise levels. The time series appears to be stationary as far as its nonseasonal behaviour is concerned. In the Box±Jenkins methodology of ARIMA modelling, it must be ®rst established that the given time series is stationary before trying to identify the
Fig. 4. Seasonal subseries plot of the seasonally dierenced time series.
288
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
orders of AR and MA processes in the model to be ®tted. The ACF of a time series is an important tool in this regard. Fig. 2 shows the ACF of time series of noise levels obtained in the present study. It can be clearly seen that the autocorrelations are signi®cantly dierent from zero even at higher time lags. It may also be noticed that the autocorrelations form a wave-like pattern. If one combines the behaviours indicated by Figs. 1 and 2, it can be inferred that the time series may have some seasonal nonstationarity. A more careful examination of Fig. 2 reveals that the autocorrelations at lags 12, 24,....... are unusually high. The neighbouring autocorrelations at lags 11, 13, 23, 25,...... are also on the higher sides. This indicates the presence of seasonality (roughly of the length 12) in the given time series, which may be associated with the switching of trac lights located about 500 m from the sampling site and corresponds to the approximate interval of 120 s between two successive green signals. This can be con®rmed by the seasonal subseries plot (season length=12) of the time series shown in Fig. 3. It is evident from the ®gure that the seasonal means are not the same for all the 12 seasons. Instead, a wave-like pattern is observed in seasonal means. Figs. 1±3 therefore point towards the presence of a seasonal nonstationarity of about the length 12 in the given time series. In view of the above indications, it becomes desirable to do the seasonal dierencing of the original time
Fig. 5. ACF of seasonally dierenced time series.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
289
series in order to remove the existing seasonal nonstationarity. Thus, a new time series is generated by dierencing the original time series at lag 12. Fig. 4 shows the seasonal subseries plot of this new time series. It becomes clear at once that the seasonal means are almost the same for all seasons now. In other words, the seasonal nonstationarity appears to have been removed in this new time series. Fig. 5 shows the ACF of this dierenced time series. It can be seen that the autocorrelation at lag 12 is signi®cantly dierent from zero. All the other autocorrelations may be considered as insigni®cant. Fig. 6 shows the PACF of the seasonally dierenced series. It can be observed that there is a pattern of exponential decay in the partial autocorrelations at lag 12, 24, 36,....... The patterns observed in Figs. 5 and 6 are quite similar to the theoretical patterns of ACF and PACF of a seasonal MA process of the order 1. This indicates the existence of a seasonal MA process of order 1 in the original time series. In view of the observations made above, a seasonal model of the order (0,1,1) [12] was identi®ed. Here p 0; q 1; d 1 and the superscript 12 represents the length of seasonality. The model ®tting results are given in Table 1. It can be seen from the table that the value of the seasonal moving average parameter 1 is less than 1 which con®rms the condition of invertibility required for moving average coecients. Further, the p-value obtained for 1 is very good, thus
Fig. 6. PACF of seasonally dierenced time series.
290
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
indicating that the value of the coecient is signi®cantly dierent from zero. The p-value obtained for the mean on the other hand indicates that mean cannot be taken as signi®cantly dierent from zero. This, of course, is expected since the original time series has been dierenced once and the model, therefore, gives the mean for the seasonally dierenced time series. The non-signi®cant value of mean in the model further indicates the absence of any deterministic trend in the original time series. In other words, it can be said that time series is stochastic in nature and that it lacks any deterministic component. Once an ARIMA model is ®tted, it is important to investigate how well the model ®ts the given time series. This comprises the step of diagnostic checking in model building. Investigation of the behaviour of residuals is a useful tool in this regard. Fig. 7 shows the ACF of residuals obtained from the model ®tted in Table 1. It is evident from the ®gure that all autocorrelations in the ACF plot of residuals are insigni®cant. This implies that the residuals are not autocorrelated and are statistically independent. This means that the model is adequate. This is further con®rmed by the 2 test statistic computed for the ®rst 20 residual autocorrelations. The computed value of 2 test statistic 23.3834 is less than the tabulated value 28.8693 of 2 with 18 d.f. at 0.05 level of signi®cance. Thus the joint null hypothesis, that the
Fig. 7. Estimated residual ACF.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
291
residual autocorrelations are independent, cannot be rejected. Hence, it may be inferred that the residual autocorrelations are not signi®cantly dierent from zero. The model equation on the basis of results shown in Table 1 can be written as 1 ÿ B12 xt 1 ÿ 0:85439B12 et
1
An alternative model found worth considering for ®tting to time series was the one with an additional non-seasonal autoregressive term. The model ®tting results are presented in Table 2 which are quite similar to those presented in Table 1 except that in the above model one nonseasonal AR(1) parameter 1 has been included. The signi®cance level of 1 is just satisfactory. It can be seen that the value of 1 is less than 1. This satis®es the stationarity condition for an ARIMA model. Fig. 8 shows the residual ACF for the alternative model. After comparing Fig. 7 with Fig. 8, it can easily be seen that the behaviour of residuals in both the models is quite similar. The alternative model may be represented by the following equation. 1 ÿ 0:08314B 1 ÿ B12 xt 1 ÿ 0:8573B12 et
2
Table 1 Summary of ®tted model for the observed time series of noise levels Parameter
Estimate
Standard error
t-value
p-value
SMA (12) Mean Constant
0.85439 0.00017 0.00017
0.0213 0.02414 ±
42.44129 0.00704 ±
0.0000 0.99439 ±
Model ®tted to seasonal dierences of order 1 with seasonal length=12. Estimated white noise variance=13.5842 with 534 d.f. Estimated white noise standard deviation (SE)=3.68567. Chi-square ( 2 ) test statistic on ®rst 20 residual autocorrelations=23.3834 with probability of a larger value given white noise=0.220875.
Table 2 Summary of the alternative model ®tted to the observed time series Parameter
Estimate
Standard error
t-value
p-value
AR (1) SMA (12) Mean Constant
0.08314 0.85730 ÿ0.00005 ÿ0.00004
0.04318 0.02014 0.02563 ±
1.92537 42.57278 ÿ0.00186 ±
0.05471 0.00000 0.99852 ±
Model ®tted to seasonal dierences of order 1 with seasonal length=12. Estimated white noise variance=13.5191 with 533 d.f. Estimated white noise standard deviation (SE)=3.67683. Chi-square ( 2 ) test statistic on ®rst 20 residual autocorrelations=19.3563 with probability of a larger value given white noise=0.3702.
292
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
One of the basic principles of ARIMA model building is that the model should be parsimonious (i.e. the model which adequately describes the time series with a relatively less number of parameters). Keeping in view the above principle, it can be concluded that the model represented by Eq. (1) is the most appropriate for the observed time series. The model Eq. (1) can be used for predictive purposes by expanding it in the following manner. xt xtÿ12 et ÿ 0:85439etÿ12
3
where t is assumed to be the current time period (i.e. the 549th, the last observation). In order to forecast xt1 , all the subscripts in Eq. (3) are increased by 1. It gives xt1 xtÿ11 et1 ÿ 0:85439etÿ11
4
Since the term et1 i.e. the forecast error one period ahead is not known, it is assigned the value zero (the expected value of feature random errors). The values of other terms on the right-hand side of Eq. (4) are known. Thus the forecast one period ahead is determined. Forecasts for future periods can similarly be obtained. To summarize, the road trac noise variations in the present study have been shown to be governed by a seasonal moving average (0,1,1) [12] process. It would be
Fig. 8. Estimated residual ACF of the alternative model.
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
293
appropriate here to mention that the above result cannot be generalized to other trac sites. In fact the identi®cation of the stochastic process (whether autoregressive or moving average or a combination of both) governing any time series will have to be done afresh in order to develop a predictive model of trac noise for a given site. The utility of the time series modelling approach in the present study is limited insomuch as the forecasting of long-term noise levels is concerned. However, the study does reveal that even on short time scales the ¯uctuations in noise levels are governed by stochastic processes which can be identi®ed with the help of the Box±Jenkins methodology. Certainly, it would have been better to consider a time series over much longer time scales (e.g. months or years) but it was not feasible in the present study due to the lack of the facility of a permanent automatic noise monitoring station. Nevertheless, the adequacy of ARIMA modelling technique demonstrated in the present study does provide the basis to further investigate the applicability of these techniques to forecast noise levels over longer time scales. References [1] Mochizuki T, Imaizumi N. City noise in Tokyo. Journal of Acoustical Society of Japan 1967;23:146±7. [2] Price AJ. A community noise survey of great Vancouver. Journal of Acoustical Society of America 1972;52:488±92. [3] Safeer HB. Community noise levelsÐa statistical phenomenon. Journal of Sound and Vibration 1973;26:489±502. [4] Webster JE. Community noise survey in Medford, Massachussetts. Journal of Acoustical Society of America 1973;54:985±95. [5] Attenborough K, Clark S, Utley WA. Background noise levels in the United Kingdom. Journal of Sound and Vibration 1976;48:359±75. [6] Singh BB, Jain VK. A comparative study of noise levels in some residential, industrial and commercial areas of Delhi. Environmental Monitoring and Assessment 1995;35:1±11. [7] Cannelli GB. Trac noise pollution in Rome. Applied Acoustics 1974;7(2):103±16. [8] Rylander R, Sorenson S, Kajland A. Trac noise exposure and annoyance reactions. Journal of Sound and Vibration 1976;47:237±42. [9] Ko NWM. Trac noise in a high rise city. Applied Acoustics 1978;11:225±39. [10] Bjorkman M. Maximum noise levels in road trac noise. Journal of Sound and Vibration 1988;127(3):583±7. [11] Carter NL, Ingham P, Tran K. Overnight trac noise measurements in bedrooms and outdoors, Pennant Hills Road, SydneyÐcomparisons with criteria for sleep. Acoust. Aust. 1992;20(2):49±55. [12] Brown AL. Exposure of Australian population to road trac noise. Applied Acoustics 1994;43(2):169±76. [13] Hofman WF, Kumar A, Tulen JHM. Cardiac reactivity to trac noise during sleep on man. Journal of Sound and Vibration 1995;179(4):577±89. [14] Ohrstrom E. Eects of low levels of road trac noise during the night: a laboratory study on number of events, maximum noise levels and noise sensitivity. Journal of Sound and Vibration 1995;179(4):603±15. [15] Johnson DR, Saunders EG. The evaluation of noise from freely ¯owing road trac. Journal of Sound and Vibration 1968;7:287±309. [16] Burgess MA. Noise prediction for urban trac conditionsÐrelated to measurements in the Sydney metropolitan area. Applied Acoustics 1977;10:1±7. [17] Delany ME, Harland DG, Hood RA, Scholes WE. The prediction of noise levels L10 due to road trac. Journal of Sound and Vibration 1976;48(3):305±25.
294
K. Kumar, V.K. Jain / Applied Acoustics 58 (1999) 283±294
[18] Favre BM. Noise emission of road vehicles: evaluation of some simple models. Journal of Sound and Vibration 1983;91(4):571±82. [19] Baverstock SJ, Pocock RL, Attenborough K. Development of area-based methods for predicting ambient noise. Applied Acoustics 1991;33:303±12. [20] Hasebe M, Kaneyasu K. Prediction of trac noise from trunk roads and access roads in urban areas. Noise Control Engineering Journal 1991;37(2):71±6. [21] Yamaguchi S, Ishihara S, Kato Y. A practical method of predicting periodic nonstationary road trac noise based on the time rate of average number of ¯owing vehicles. Acoustical Letters 1992;16(6):123±8. [22] Zhang J. A study on the highway noise prediction model applicable to dierent trac ¯ow. Noise Control Engineering Journal 1993;41(3):371±5. [23] Kato Y, Yamaguchi S, Takagi K. A practical prediction method of road trac noise around trac signals and its experimental studies in actual roads. Prediction at the side of the road, downstream from the trac signal. Journal of Acoustical Society of Japan 1994;50(2):91±102. [24] DeVor, RE, Schomer, PD, Kline, WA, Neathamer, RD. Development of temporal sampling strategies for monitoring noise. Journal of Acoustical Society of America 1979;66(3):763±71. [25] Schomer PD, De Vor RE, Kline WA. Sampling strategies for monitoring noise in the vicinity of airports. Journal of Acoustical Society of America 1983;73(6):2041±50. [26] Box GEP, Jenkins, GM. Time series analysis forecasting and control. San Francisco: Holden-Day, 1976.