Autoregressive integrated moving averages (ARIMA) modelling of a traffic noise time series

Autoregressive integrated moving averages(ARIMA) modelling of a tra�c noise time series

K. Kumar*, V.K. JainSchool 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 roadcarrying vehicular tra�c. The resultant time series is analyzed using autoregressive integratedmoving averages (ARIMA) modelling technique. The time series is found to be non stationary.

After ®rst di�erencing, the transformed series becomes stationary and is found to be governedby a moving average process of order 1. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: ARIMA; Tra�c noise; Time series

1. Introduction

In the past few decades, the concern about environmental pollution in urban areashas grown appreciably. One of the important forms of pollution that a�ects thephysiological and psychological well-being of a person is noise pollution. It is,therefore, quite important to monitor the ambient noise to which the population ofan area may be exposed. This realization prompted a number of surveys of noiselevels in di�erent parts of the world [1±6]. A major contribution towards urban noisecomes from vehicular tra�c. Therefore, it is not surprising that road tra�c noise isone of the most extensively studied areas of noise pollution. Several studies havebeen made on various aspects of tra�c noise [7±14]. Its reduction has become one ofthe main objectives of environmental planning in most cities all over the would. Fordevising appropriate strategies to ful®ll this objective, however, one needs to have aknowledge of the existing ambient noise levels. This is achieved by monitoringthe noise levels in the areas of interest over a period of time. Since the nature and

Applied Acoustics 58 (1999) 283±294

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* Corresponding author. Department of Environmental Sciences, Guru Jambheshwar University,

Hissar, India.

pattern of road tra�c in a city may change with time, conducting such noise surveysfrequently can be a very costly exercise, Moreover, it is also desirable to know thefuture noise levels in advance from the viewpoint of having an e�ective environ-mental plan. Therefore, it becomes essential to develop predictive models whichcould forecast the noise levels both spatially and temporally. To address this prob-lem a number of mathematical models have been developed by several workers[15±23]. Baverstock et al. [19] have demonstrated that area-based prediction modelsusing the data of land use and road tra�c are quite robust and reliable. Hasebe andKaneyasu [20] have investigated the problem of prediction of tra�c noise fromtrunk roads and access roads in urban areas. Yamaguchi et al. [21] have suggested apractical method for predicting periodic non-stationary road tra�c noise. Zhang [22]has modelled highway noise under di�erent tra�c ¯ow conditions. Kato et al. [23]have given a method for determining road tra�c noise downstream of a tra�c signal.Since the road tra�c noise is believed to be stochastic in nature, the techniques of

time series analysis hold a great promise for the development of predictive models ofroad tra�c 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 attemptsto investigate the applicability of autoregressive integrated moving averages

Fig. 1. Time sequence plot.

284 K. Kumar, V.K. Jain /Applied Acoustics 58 (1999) 283±294

(ARIMA) modelling, a special class of time series techniques, for developing fore-cast models of road tra�c noise.

2. Site description and data collection

The sampling site in the present study was located at a distance of 5m from thekerbside of a major road (Abdul Gamel Nasir Marg) which encircles a major part ofthe city of Delhi. The site is about half a km away from the nearest tra�c intersec-tion near the Indian Institute of Technology, Delhi. The average volume of tra�cpassing by the sampling site is �1500 vehicles per h during 9.00 a.m. to 5.00 p.m.The tra�c is composed of heavy vehicles (buses and trucks), three-wheelers, two-wheelers and light vehicles (cars and jeeps).The instrument used for the measurement of noise in the present study is a preci-

sion type 1 sound level meter (NL-10A, RION, Japan). The sound level meter wasmounted at a height of 1.2m above the ground. A total of 549 A-weighted instan-taneous sound pressure levels of tra�c noise at 10 s intervals were recorded from10.00 a.m. onwards on 27 December 1996.

Fig. 2. ACF of time series.

K. Kumar, V.K. Jain /Applied Acoustics 58 (1999) 283±294 285

3. Methodology

The time series of tra�c noise obtained as a result of sampling of instantaneousnoise is analyzed using ARIMA technique [26]. A general ARIMA model of order(p,d,q) representing the time series can be written as

��B�rdxt � ��B�et

where xt and et represent sound pressure level and random error terms at time trespectively. B is a backward shift operator de®ned by Bxt � xtÿ1, and related to rby r � 1ÿ B;rd � �1ÿ B�d; d is the order of di�erencing. ��B� and ��B� areautoregressive (AR) and moving averages (MA) operators of orders p and q,respectively, and are de®ned as

��B� � 1ÿ �1Bÿ �2Bÿ � � � � � � � � ��ÿ�pBp

and

��B� � 1ÿ �1Bÿ �2B2 ÿ � � � � � � � � ��ÿ�qBq

Fig. 3. Seasonal subseries plot.


where �1; �2; :::::�p are the autoregressive coe�cients and �1; �2; :::::�q are the mov-ing average coe�cients.In the Box±Jenkins methodology of ARIMA modelling, the ®rst step is to deter-

mine whether the time series is stationary or non stationary. If it is non stationary itis transformed into a stationary time series by applying suitable degree of di�eren-cing to it. This gives value of d. Then appropriate values of p,q are found by exam-ining auto-correlation function (ACF) and partial auto-correlation function (PACF)of the time series. Having determined p,q and d the coe�cients of autoregressive andmoving average terms are estimated using nonlinear least squares method. In thepresent study the time series has been analyzed using statistical computer softwareSTATGRAPHICS.

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 con-cerned. In the Box±Jenkins methodology of ARIMA modelling, it must be ®rstestablished that the given time series is stationary before trying to identify the

Fig. 4. Seasonal subseries plot of the seasonally di�erenced time series.


orders of AR and MA processes in the model to be ®tted. The ACF of a time seriesis an important tool in this regard. Fig. 2 shows the ACF of time series of noiselevels obtained in the present study. It can be clearly seen that the autocorrelationsare signi®cantly di�erent from zero even at higher time lags. It may also be noticedthat the autocorrelations form a wave-like pattern. If one combines the behavioursindicated by Figs. 1 and 2, it can be inferred that the time series may have someseasonal nonstationarity. A more careful examination of Fig. 2 reveals that theautocorrelations at lags 12, 24,....... are unusually high. The neighbouring auto-correlations at lags 11, 13, 23, 25,...... are also on the higher sides. This indicates thepresence of seasonality (roughly of the length 12) in the given time series, which maybe associated with the switching of tra�c lights located about 500m from the samplingsite and corresponds to the approximate interval of 120 s between two successive greensignals. This can be con®rmed by the seasonal subseries plot (season length=12) ofthe time series shown in Fig. 3. It is evident from the ®gure that the seasonal meansare not the same for all the 12 seasons. Instead, a wave-like pattern is observed inseasonal means. Figs. 1±3 therefore point towards the presence of a seasonal non-stationarity of about the length 12 in the given time series. In view of the aboveindications, it becomes desirable to do the seasonal di�erencing of the original time

Fig. 5. ACF of seasonally di�erenced time series.


series in order to remove the existing seasonal nonstationarity. Thus, a new timeseries is generated by di�erencing 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 otherwords, the seasonal nonstationarity appears to have been removed in this new timeseries. Fig. 5 shows the ACF of this di�erenced time series. It can be seen that theautocorrelation at lag 12 is signi®cantly di�erent from zero. All the other auto-correlations may be considered as insigni®cant. Fig. 6 shows the PACF of the sea-sonally di�erenced series. It can be observed that there is a pattern of exponentialdecay in the partial autocorrelations at lag 12, 24, 36,....... The patterns observed inFigs. 5 and 6 are quite similar to the theoretical patterns of ACF and PACF of aseasonal MA process of the order 1. This indicates the existence of a seasonal MAprocess 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 lengthof 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 para-

meter �1 is less than 1 which con®rms the condition of invertibility required formoving average coe�cients. Further, the p-value obtained for �1 is very good, thus

Fig. 6. PACF of seasonally di�erenced time series.


indicating that the value of the coe�cient is signi®cantly di�erent from zero. Thep-value obtained for the mean on the other hand indicates that mean cannot be takenas signi®cantly di�erent from zero. This, of course, is expected since the originaltime series has been di�erenced once and the model, therefore, gives the mean forthe seasonally di�erenced time series. The non-signi®cant value of mean in themodel further indicates the absence of any deterministic trend in the original timeseries. In other words, it can be said that time series is stochastic in nature and that itlacks 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 modelbuilding. 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 isevident from the ®gure that all autocorrelations in the ACF plot of residuals areinsigni®cant. This implies that the residuals are not autocorrelated and are statisti-cally independent. This means that the model is adequate. This is further con®rmedby the �2 test statistic computed for the ®rst 20 residual autocorrelations. Thecomputed 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.


residual autocorrelations are independent, cannot be rejected. Hence, it may be inferredthat the residual autocorrelations are not signi®cantly di�erent from zero.The model equation on the basis of results shown in Table 1 can be written as

�1ÿ B12�xt � �1ÿ 0:85439B12�et �1�

An alternative model found worth considering for ®tting to time series was the onewith an additional non-seasonal autoregressive term. The model ®tting results arepresented in Table 2 which are quite similar to those presented in Table 1 except thatin the above model one nonseasonal AR(1) parameter �1 has been included. The sig-ni®cance level of �1 is just satisfactory. It can be seen that the value of �1 is less than1. This satis®es the stationarity condition for an ARIMA model. Fig. 8 shows theresidual ACF for the alternative model. After comparing Fig. 7 with Fig. 8, it caneasily be seen that the behaviour of residuals in both the models is quite similar. Thealternative model may be represented by the following equation.

�1ÿ 0:08314B��1ÿ B12�xt � �1ÿ 0:8573B12�et �2�

Table 2

Summary of the alternative model ®tted to the observed time series

Parameter Estimate Standard error t-value p-value

AR (1) 0.08314 0.04318 1.92537 0.05471

SMA (12) 0.85730 0.02014 42.57278 0.00000

Mean ÿ0.00005 0.02563 ÿ0.00186 0.99852

Constant ÿ0.00004 ± ± ±

Model ®tted to seasonal di�erences 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.

Table 1

Summary of ®tted model for the observed time series of noise levels

Parameter Estimate Standard error t-value p-value

SMA (12) 0.85439 0.0213 42.44129 0.0000

Mean 0.00017 0.02414 0.00704 0.99439

Constant 0.00017 ± ± ±

Model ®tted to seasonal di�erences 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.


One of the basic principles of ARIMA model building is that the model should beparsimonious (i.e. the model which adequately describes the time series with a rela-tively less number of parameters). Keeping in view the above principle, it can beconcluded that the model represented by Eq. (1) is the most appropriate for theobserved time series. The model Eq. (1) can be used for predictive purposes byexpanding 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 xt�1, all the subscripts in Eq. (3) are increased by 1. It gives

xt�1 � xtÿ11 � et�1 ÿ 0:85439etÿ11 �4�

Since the term et�1 i.e. the forecast error one period ahead is not known, it isassigned the value zero (the expected value of feature random errors). The values ofother terms on the right-hand side of Eq. (4) are known. Thus the forecast one per-iod ahead is determined. Forecasts for future periods can similarly be obtained.To summarize, the road tra�c 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.


appropriate here to mention that the above result cannot be generalized to othertra�c sites. In fact the identi®cation of the stochastic process (whether auto-regressive or moving average or a combination of both) governing any time serieswill have to be done afresh in order to develop a predictive model of tra�c noise fora given site. The utility of the time series modelling approach in the present study islimited insomuch as the forecasting of long-term noise levels is concerned. How-ever, the study does reveal that even on short time scales the ¯uctuations in noiselevels are governed by stochastic processes which can be identi®ed with the help ofthe Box±Jenkins methodology. Certainly, it would have been better to consider atime series over much longer time scales (e.g. months or years) but it was not feasiblein the present study due to the lack of the facility of a permanent automatic noisemonitoring station. Nevertheless, the adequacy of ARIMA modelling techniquedemonstrated in the present study does provide the basis to further investigate theapplicability of these techniques to forecast noise levels over longer time scales.

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Autoregressive integrated moving averages (ARIMA) modelling of a traffic noise time series

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