Received 17 June 2013Received in revised form 26 January 2014Accepted 13 February 2014

Short term trafc ow forecasting has received sustained attention for its ability to providethe anticipatory trafc condition required for proactive trafc control and management.

systems rely on accurate prediction of near-term trafc conditions. Considering the importance of trafc ow rate, denedt-term trafc owons.uncertainty

tication associated with level prediction, i.e., prediction interval generation (Chateld, 1993). Intuitively, short-termow rate forecasting should be informed by our understanding of the trafc ow rate dynamics that is of p

http://dx.doi.org/10.1016/j.trc.2014.02.0060968-090X/ 2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +86 25 83793131.E-mail addresses: jg2nh@yahoo.com (J. Guo), hhhwei@126.com (W. Huang), billy_williams@ncsu.edu (B.M. Williams).

1 Tel.: +86 25 83793131.2 Tel.: +1 919 5157813.

Transportation Research Part C 43 (2014) 5064

Contents lists available at ScienceDirect

Transportation Research Part C

journal homepage: www.elsevier .com/locate / t rcas the number of vehicles passing a specic road section over a predened time interval (TRB, 2000), shorrate forecasting has been identied as one of the major challenges for developing proactive ITS applicati

Short term trafc ow rate forecasting includes trafc ow rate level prediction, i.e., point forecast, and quan-trafcrimary1. Introduction

Congestion is causing serious issues for surface transportation systems around the world. Due to the increasing con-straints on new road construction or expansion, trafc management and control systems under the umbrella of intelligenttransportation systems (ITS) have become increasingly vital for improving the efciency and safety of trafc operations. Incontrast to reactive management and control systems that respond to currently observed trafc conditions, the proactiveKeywords:CongestionIntelligent transportation systemShort term trafc ow forecastingSARIMAGARCHAdaptive Kalman lterRecently, a stochastic seasonal autoregressive integrated moving average plus generalizedautoregressive conditional heteroscedasticity (SARIMA + GARCH) process has gainedincreasing notice for its ability to jointly generate trafc ow level prediction and associ-ated prediction interval. Considering the need for real time processing, Kalman lters havebeen utilized to implement this SARIMA + GARCH structure. Since conventional Kalmanlters assume constant process variances, adaptive Kalman lters that can update theprocess variances are investigated in this paper. Empirical comparisons using real worldtrafc ow data aggregated at 15-min interval showed that the adaptive Kalman lterapproach can generate workable level forecasts and prediction intervals; in particular,the adaptive Kalman lter approach demonstrates improved adaptability when trafc ishighly volatile. Sensitivity analyses show that the performance of the adaptive Kalmanlter stabilizes with the increase of its memory size. Remarks are provided on improvingthe performance of short term trafc ow forecasting.

2014 Elsevier Ltd. All rights reserved.Adaptive Kalman lter approach for stochastic short-termtrafc ow rate prediction and uncertainty quantication

Jianhua Guo a,, Wei Huang a,1, Billy M. Williams b,2a Intelligent Transportation System Research Center, Southeast University, Si Pai Lou #2, Nanjing 210096, PR ChinabDepartment of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC 27695, USA

a r t i c l e i n f o

Article history:

a b s t r a c t

ian-neural network approach (Zheng et al., 2006), hybrid fuzzy rule-based approach (Dimitriou et al., 2008), hybrid EMD-

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 51BPN (empirical mode decomposition-back propagation neural networks) approach (Wei and Chen, 2012), chaos-waveletanalysis-support vector machine approach (Wang and Shi, 2013). Intuitively, the implementation characteristics of hybridmethods are generally complex, thereby discouraging their wide-scale implementations.

In addition, ltering approaches have been widely applied, including recursive least square (RLS) (Kang et al., 1998; Yanget al., 2004), Kalman lter (Gazis and Knapp, 1971; Okutani and Stephanedes, 1984; Stathopoulos and Karlaftis, 2003; She-khar, 2004; Guo, 2005), generalized linear model (GLM) (Lan and Miaou, 1999; Lan, 2001), Bayesian dynamic linear model(Fei et al., 2011), and least mean square (LMS) lters (Lu, 1990). It is worthwhile to note that the RLS, Kalman lter, and GLMare closely related by assuming the random walk model for the forecasting algorithm state evolution (Yang et al., 2004; Lanand Miaou, 1999; Lan, 2001). These methods are promising in imparting a self-adjusting ability into the forecasting processand the Kalman lter approach is theoretically appealing for short term trafc condition forecasting. However, as pointed outpreviously, the heteroscedastic nature of trafc condition series demands a process variances adaptation mechanism in theKalman lters.

2.2. Uncertainty quantication

Compared with trafc ow level forecasting, studies on trafc uncertainty quantication or prediction interval generationare initially limited to several studies with discouraging results (Yang et al., 2004; Hugosson, 2005; Pattanamekar et al.,importance in forecasting algorithm development (Stephanedes et al., 1981). Recently, based on previous ndings in Wil-liams (1999), Williams and Hoel (2003), Shekhar and Williams (2008), etc., a stochastic structure of seasonal autoregressiveintegrated moving average plus generalized autoregressive conditional heteroscedasticity (SARIMA + GARCH) is utilized tomodel the rst and second conditional moment of trafc ow rate series, and this structure has been shown to be able togenerate desirable trafc ow rate level forecasts and prediction intervals (Guo et al., 2008).

Due to the requirement of real time processing for many ITS-related transportation applications, the SARIMA + GARCHstructure needs to be handled in an online fashion. In this regard, Kalman lter, due to its recursive nature, is one of the mostwidely adopted tools of achieving this purpose. In doing so, the time series models are converted into state space models,including a state transition equation and an observation equation for the hidden state process and the observation process,respectively. In the conventional Kalman lter, the process variances are important seeding parameters calibrated usuallyusing historical trafc ow rate series. However, as shown in Guo (2005) and Guo et al. (2012), trafc condition is heteros-cedastic in nature, and adaptive Kalman lters with time-varying process variances adaptation are expected to be theoret-ically more appealing than the conventional Kalman lters. In this paper, the performance of the adaptive Kalman lter forshort term trafc ow rate forecasting is investigated. Following a review of related literatures, the methodology is pre-sented, including the denition of the stochastic structure, state space representation, and adaptive Kalman recursion.The paper then presents the empirical results from application of the adaptive Kalman lter to real world data. The paperconcludes with summaries and remarks.

2. Literature review

Over the decades, there has been a variety of approaches published on short term trafc ow forecasting. In this section,we summarize the studies of level forecasting and uncertainty quantication, i.e., point forecast and prediction intervalgeneration.

2.1. Trafc condition level forecasting

Trafc ow level forecasting, i.e., point prediction, has been widely investigated and many studies have been published inthe literature. First, linear time series model has been widely applied, including exponential smoothing (Ross, 1982), auto-regressive integrated moving average (ARIMA) model (Ahmed and Cook, 1979; Levin and Tsao, 1980; Nihan and Holersland,1980; Hamed et al., 1995), SARIMA (Williams, 1999;Williams and Hoel, 2003; Guo, 2005; Guo et al., 2008), multivariate timeseries models (Williams, 2001; Kamarianakis and Prastacos, 2003; Min and Wynter, 2011), spectral analysis (Nicholson andSwann, 1974; Tchrakian et al., 2012). In these models, the trafc dynamics are inherently assumed to be linear. It is worth-while to note that SARIMA has been shown to generate promising performances (Smith et al., 2003; Guo et al., 2008; Lippiet al., 2013).

Second, non-parametric approaches have been applied, including neural network and its variations (Clark et al., 1992;Smith and Demetsky, 1994; Park et al., 1998; Zhang, 2000; Dia, 2001; Chen and Grant-Muller, 2001; Yin et al., 2002; Jiangand Adeli, 2005; Vlahogianni et al., 2005; Dunne and Ghosh, 2012), k-nearest neighbor approach (Davis and Nihan, 1991;Smith and Demetsky, 1996, 1997), kernel smoothing (El Faouzi, 1996), and local linear regression (Sun et al., 2003). Thesemethods are in general automatic and do not make strong assumptions on the underlying model form. They have a notableadvantage of adaptive learning of the underlying trafc dynamics through a historical trafc condition database.

Hybrid methods have also been exploited to enhance the performances of single forecasting approaches, including com-bined Kohonen maps with ARIMA (KARIMA) model (Der Voort et al., 1996), ATHENA (Danech-Pajouh and Aron, 1991), Bayes-

sented; nally the adaptive Kalman recursion is provided.

3.1. SA

Asdidatei.e., thics ofdene

52 J. Guo et al. / Transportation Research Part C 43 (2014) 5064/BUB 1 B 1 B xt hBHB et 1

et ht

pet 2

ht a0 Xvi1aie2ti

Xui1

bihti 3

et IN0;1 4where

t: time index;p the order of the short-term autoregressive polynomial;q: order of the short-term moving average polynomial;d: order of the short-term differencing;P: order of the seasonal autoregressive polynomial;Q: order of the seasonal moving average polynomial;D: order of the seasonal differencing;B: backshift operator such that Bxt = xt1;(1 BS)D: seasonal differencing;(1 B)d: short term differencing;/(B) = 1 /1B /2B2 /pBp: short-term autoregressive polynomial;h(B) = 1 h1B h2B2 hqBq: short-term moving average polynomial;U(BS) = 1 U1(BS) U2(BS)2 UP(BS)P: seasonal autoregressive polynomial;H(BS) = 1 H1(BS) H2(BS)2 HQ(BS)Q: seasonal moving average polynomial;ht: conditional variance at t, i.e., et|Wt1 N(0, ht) with Wt1 as the information up to t 1;u: autoregressive order of GARCH process with uP 0;v: moving average order of GARCH process with v > 0;a0: positive constant coefcient;ai,i=1,. . .,v: non-negative coefcients of the lagged sample variance e2ti;bi,i=1,. . .,u: non-negative coefcients of the lagged conditional variance hti.RIMA + GARCH structure

mentioned previously, the stochastic SARIMA + GARCH structure is gradually emerging as one of the promising can-s for modeling trafc ow series. In this structure, the SARIMA component captures the rst conditional moment,e dynamics of trafc ow levels, and the GARCH component captures the second conditional moment, i.e., the dynam-trafc ow variances. Formally, for a discrete trafc ow series xt, the SARIMA(p,d,q)(P,D,Q)S + GARCH(u,v) structure isd as Eqs. (1)(4).

S S D d S2003; Sun et al., 2004). Recently, models are borrowed from nancial analysis eld for uncertainty quantication with prom-ising results, including primarily GARCH approach (Guo, 2005; Kamarianakis et al., 2005; Tsekeris and Stathopoulos, 2006;Guo et al., 2008; Sohn and Kim, 2009; Karlaftis and Vlahogianni, 2009;Yang et al., 2010; Guo and Williams, 2010; Chen et al.,2011; Zhang et al., 2013), and stochastic volatility approach (Tsekeris and Stathopoulos, 2010). Note that GARCH model canbe processed using Kalman lter (Guo, 2005; Guo et al., 2008; Guo and Williams, 2010); therefore, adaptive mechanism canalso be used for improving the performance. In addition, as a conventional prediction model, neural network has been en-hanced to provide the ability of generating prediction intervals (van Hinsbergen et al., 2009; Khosravi et al., 2011; Mazloumiet al., 2011).

2.3. Summary

In summary, many studies have been proposed for short term trafc ow level forecasting and prediction interval gen-eration. Specically, Kalman lter approach developed based on time series models, in particular, the SARIMA + GARCHstructure, has been recognized as promising for short term trafc ow forecasting. However, conventional Kalman lter can-not update its process variances in real time. Therefore, this paper will investigate and test an adaptive Kalman lter ap-proach using real world trafc ow data collected around the world.

3. Methodology

In this section, rst, the stochastic structure of SARIMA + GARCH is dened; then its state space representation is pre-

whereinterp

Woldary tra

IMA(0movin

Th

1 B x 1HB w 6where w is the output of the seasonal operator. A further manipulation gives Eqs. (7) and (8) as

whichseason

For

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 53rion, a parsimonious model is preferable for eld implementations. In this paper, the GARCH(1,1) model is selected for itssimplicity. For this model, Guo (2005) showed that the information criteria of the best model and the GARCH(1,1) modelare close to each other for multiple trafc ow series collected around the world, indicating the validity of choosing thisGARCH(1,1) model.

In summary, for the 15-min trafc ow data, the SARIMA(1,0,1)(0,1,1)672 + GARCH(1,1) structure can be utilized for shortterm trafc ow level prediction and uncertainty quantication. Note that this structure can be processed in two steps, i.e.,SARIMA processing and GARCH processing using outputs from SARIMA processing, without efciency lost (Engle, 1982; Bol-lerslev, 1986). This is essential for constructing the online short term trafc ow forecasting system.

3.2. State space representation

Recall that the SARIMAmodel can be interpreted as a cascade of a seasonal operator and a short-term operator; therefore,the SARIMA (1,0,1)(0,1,1)672 + GARCH(1,1) structure can be processed as IMA(0,1,1)672 + ARMA (1,1) + GARCH(1,1), whichcan be converted into a seasonal exponential smoothing operator plus two state space representations as follows.models local variations after the seasonal variation is removed from the original trafc ow series through theal operator.the GARCH(u,v) process, although an exact best model can be identied by minimizing the sample information crite-t

1HB672 1HB6721HB672 xt wt 7

and

xt 1HB672 HB6722 H2B6723 xt wt 8indicating that the seasonal operator is exactly a seasonal exponential smoothing with smoothing parameter a = 1 H. Infact, this seasonal operator will retrieve the seasonal pattern in trafc ow series.

The short-term operator is dened as Eq. (9), i.e.,

1 /Bwt 1 hBet 9t t

672 672,1,1)672, i.e., the integrated moving average operator, and a short-term operator of ARMA(1,1), i.e., the autoregressiveg average operator.e seasonal operator is dened as Eq. (6), i.e.,process using real world trafc ow data collected from the United States and the United Kingdom. By investigating into thisSARIMA model, we can see that SARIMA(1,0,1)(0,1,1)672 model can be processed as a cascade of a seasonal operator ofDecomposition Theorem and the assertion that a one-week lagged rst seasonal difference will yield a weakly station-nsformation, Williams (1999) and Williams and Hoel (2003) showed that trafc ow series can be modeled a SARIMAGiven the specication of the SARIMA + GARCH structure, the orders of SARIMA and GARCH can be determined, respec-tively. For the SARIMA structure, previous studies have shown that the SARIMA(1,0,1)(0,1,1)672 model can be used to de-scribe trafc ow rate series aggregated at 15-min interval (Williams, 1999; Williams and Hoel, 2003). Based on thegt et ht that is serially uncorrelated with mean zero and n = max (u, v) (Bollerslev, 1986). Note that this ARMAretation provides a handy treatment of converting the GARCH model to a state space representation.In above representation, Eq. (1) denes the SARIMA structure, and Eqs. (2)(4) dene the GARCH structure. For the SAR-IMA structure, it is assumed that the roots of /(B), h(B),U(BS), andH(BS) are outside of unit circle to ensure the causality andinvertibility of the model, and also these polynomials have no common factors. Note that the causality property ensures theprocess can be expressed in terms of past innovations and the invertibility property ensures the process can be expressed aspast observations, which is important for model estimation and prediction (Fuller, 1996; Box et al., 1994, 2008). et is theresidual series, conventionally assumed to be Gaussian white noise with mean zero and constant variance; however, for het-eroscedastic trafc ow series as shown in (Guo et al., 2012), the addition of the GARCH component is needed. In Guo et al.(2012), through comprehensive transformation analyses and heteroscedasticity tests with different testing powers, hetero-scedasticity is shown to be intrinsic to trafc ow series, which validates the necessity of further investigations using ap-proaches such as the GARCH model.

For the GARCH structure, the error series et is assumed to be normal for facilitating the construction of the predictionintervals. An alternative representation establishes that GARCH process can be interpreted as autoregressive moving average(ARMA) process in e2t as presented in Eq. (5).

e2t a0 Xni1

ai bie2ti Xui1

bigti gt 5

2

at:Yt:Xt:

CoFor

Combined, Eqs. (15) and (16) constitute the state space representation for GARCH(1,1) model, where the reparameterized

Th

ances

54 J. Guo et al. / Transportation Research Part C 43 (2014) 5064and Tapley (1976) is selected that uses a memory of observation errors and state estimation errors to ne-tune the processvariances. Using the state space representation in Eqs. (12) and (13) as an example, the steps of the adaptive Kalman recur-sion are described as Eqs. (17)(27).is preferred, termed as the so called adaptive Kalman lter. In this paper, the adaptation mechanism proposed in Myers3.3. Adaptive Kalman recursion

Though the two state space models described above can be readily solved using the well-known Kalman recursions (Kal-man, 1960), it is necessary to point out that the process variances required for seeding the Kalman recursion could be eitherconstant or time varying. For heteroscedastic trafc ow rate series, an adaptation mechanism for updating the process vari-nential smoothing operator, i.e., Eq. (10), and two state space representations, i.e., Eqs. (12) and (13) and Eqs. (15) and(16). Since Eq. (10) is recursive in nature and the two state space models can be solved recursively, an online prediction sys-tem can then be constructed.erefore, the SARIMA(1,0,1)(0,1,1)672 + GARCH(1,1) structure can be processed consecutively using the seasonal expo-parameter vector a0 a b T is treated as the hidden state and the squared residual series from ARMA(1,1) model is treatedas the driving observation process.e2t 1 e2t1 gt1

ab

B@ CA gt 15and the state transition equation as Eq. (16), i.e.,

a0ab

0B@

1CA

t

diag k12n o a0

ab

0B@

1CA

t1

zt 16

wheregt: observation noise series with observation noise covariance matrix CovgtgTt Rt;zt: state noise series with state noise covariance matrix CovztzTt Qt .a00 1where b = b1 and a = a1 + b1. Then, we dene the observation equation as Eq. (15), i.e.,mbined, Eqs. (12) and (13) constitute the state space representation for the ARMA(1,1) operator.the GARCH(1,1) component, according to Eq. (5) and through a reparameterization, we have Eq. (14) as

e2t a0 ae2t1 bgt1 gt 14et: observation noise series with observation noise covariance matrix Rt Covetet .state noise series with state noise covariance matrix Qt CovataTt ;current observation dened as xt;time varying observation matrix dened as xt1 et1 T;

TFor IMA(1,0,1)672, recognizing its equivalence with seasonal exponential smoothing shown previously, the operator canbe processed as Eq. (10), i.e.,

x^t axt672 1 ax^t672 10where a is the smoothing parameter with a = 1 H and x^t is the predicted value at time t. In this paper, we assume x^t xtfor t = 1,2,. . .,672 to start the equation.

For ARMA(1,1) + GARCH(1,1), Guo and Williams (2010) presented two state space representations for modeling and pre-dicting trafc speed series. For the ARMA component, we reorganize the ARMA(1,1) model into Eq. (11), i.e.,

xt /xt1 het1 et 11then by treating the ARMA model parameters as hidden state to be estimated and using the random walk model with a for-getting factor as its evolution dynamics, we dene the state transition equation as Eq. (12), i.e.,

wt Uwt1 at: 12In addition, by treating the trafc ow rate series as the driving observation process, we dene the observation equation

as Eq. (13), i.e.,

Yt XTt wt et 13where

wt: state variable dened as / h T;U: state transition matrix dened as diagfk12g, with k dened as a forgetting factor;

bPtjt1 UbPt1jt1U Qt 18

Ste

Ste

bPtjtbPtjtKt:

e^t:at:a^t:N:

3.4. Su

this st

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 55man recursion mechanism will be applied onto the two state space representations so that two adaptive Kalman lters canbe developed to process sequentially the trafc ow series in real time.

4. Empirical study

In this section, we present the real world data used in this study, the experimental design for carrying out the investiga-tion, the empirical results on level forecasting and uncertainty quantication, and a sensitivity analysis on the memory sizeof the adaptive Kalman lters.ructure can be converted into a seasonal operator and two state space representations. In this paper, an adaptive Kal-

In summary, the SARIMA(1,0,1)(0,1,1)672 + GARCH(1,1) structure can be used to model the 15-min trafc ow data, andaverage observation errors;system state estimation errors;average system state estimation errors;prescribed memory size of the adaptive Kalman lter.

mmaryet: observation errors;1: prior state estimation error covariance;: posterior state estimation error covariance;Kalman gain at time t;wherej1

Qk 1N

XNj1

atj1 a^

atj1 a^ T N 1

NUtj1bPtjjtjUTtj1 bPtj1jtj1h i

27a^ 1N

atj1 26p 7: Update state process covariance matrix QtXN

t tjt t1jt1w^tjt w^tjt1 Kt Yt XTt w^tjt1 23

bPtjt I KtXTt bPtjt1 24p 6: State estimation errors computationa w^ Uw^ 25Step 2: Observation errors computation

et Yt XTt w^tjt1 19Step 3: Update observation process covariance matrix Rt

e^ 1N

XNj1

etj1 20

Rt 1NXNj1

etj1 e^etj1 e^T N 1N Xtj1bPtj1jtjXTtj1

21

Step 4: Kalman gain computation

Kt bPtjt1Xt

XTt bPtjt1Xt Rt 22Step 5: Posterior state estimation and posterior state estimation error covariance estimationStep 1: State propagation and prior state estimation error covariance estimation

w^tjt1 bUw^t1jt1 17 T

4.1. Data

Real world trafc data collected from four highway systems are used in this paper, including the motorway system in theUnited Kingdom, and the metropolitan freeway systems in Maryland, Minnesota, and Washington State of the United States.Though originally archived at different time intervals, these data are aggregated into 15-min data according to Edie (1963),i.e., the 15-min trafc ow data were computed as the sum of trafc ow data for all the original time intervals within this15-min interval. In performing the aggregation, missing values were propagated upwards, and simple screening procedures,i.e., threshold test and hang-on test, were applied to eliminate the obvious erroneous data. In addition, missing values wereimputed using the SARIMA(1,0,1)(0,1,1)672 model through applying the back forecasting technique iteratively. In total, 36trafc ow series were selected and a summary of the data is provided in Table 1.

4.2. Experimental design

In order to investigate the performances of the proposed adaptive Kalman lter, four approaches are compared as belowin Table 2.

Note that in Table 2, for BATCH processing, the SARIMA model outperforms HoltWinters method, three heuristic models(i.e., random walk, historical average, and derivation from historical average), neural network, and k-nearest neighborregression in terms of level forecasting in these studies (Williams, 1999; Smith et al., 2003; Williams and Hoel, 2003),and the GARCH model is the primary approach that has been investigated in terms of uncertainty quantication. Therefore,the comparison of the four selected approaches will provide a comprehensive picture on short term trafc ow rateforecasting.

For each approach, three measures of level forecasting are used, i.e., mean absolute error (MAE), mean absolute percent-age error (MAPE), and root mean square error (RMSE). Given xt as real trafc ow rate observations, x^t as the forecasted

Table 1Data overview.

Region Highway Station Number of lanes Start End Sample size

UK M25 4762a 4 9/1/1996 11/30/1996 8736UK M25 4762b 4 9/1/1996 11/30/1996 8736UK M25 4822a 4 9/1/1996 11/30/1996 8736UK M25 4826a 4 9/1/1996 11/30/1996 8736UK M25 4868a 4 9/1/1996 11/30/1996 8736UK M25 4868b 4 9/1/1996 11/30/1996 8736UK M25 4565a 4 1/1/2002 12/31/2002 35,040UK M25 4680b 4 1/1/2002 12/31/2002 35,040

MN I35W-NB 60 4 1/1/2004 12/31/2004 35,136

56 J. Guo et al. / Transportation Research Part C 43 (2014) 5064MN I35W-SB 578 3 1/1/2004 12/31/2004 35,136MN I35E-NB 882 3 1/1/2004 12/31/2004 35,136MN I35E-SB 890 3 1/1/2004 12/31/2004 35,136MN 169-NB 442 2 1/1/2004 12/31/2004 35,136MN 169-SB 737 2 1/1/2004 12/31/2004 35,136WA I5 ES-179D_MN_Stn 4 1/1/2004 6/29/2004 17,298WA I5 ES-179D_MS_Stn 3 1/1/2004 6/29/2004 17,298WA I5 ES-130D_MN_Stn 4 4/1/2004 9/30/2004 17,516WA I5 ES-130D_MS_Stn 4 4/1/2004 9/30/2004 17,516WA I405 ES-738D_MN_Stn 3 7/1/2004 12/29/2004 17,406WA I405 ES-738D_MS_Stn 3 7/1/2004 12/29/2004 17,406

Note: UK: the United Kingdoms; MD: Maryland; WA: Washington State; MN: Minnesota. See Guo et al. (2012) for more information.UK M1 2737a 3 2/13/2002 12/31/2002 30,912UK M1 2808b 3 2/13/2002 12/31/2002 30,912UK M1 4897a 3 2/13/2002 12/31/2002 30,912UK M6 6951a 3 1/1/2002 12/31/2002 35,040MD I270 2a 3 1/1/2004 5/5/2004 12,096MD I95 4b 4 6/1/2004 11/15/2004 16,128MD I795 7a 2 1/1/2004 5/5/2004 12,096MD I795 7b 2 1/1/2004 5/5/2004 12,096MD I695 9a 4 1/1/2004 5/5/2004 12,096MD I695 9b 4 1/1/2004 5/5/2004 12,091MN I35W-NB 60 4 1/1/2000 12/31/2000 35,136MN I35W-SB 578 3 1/1/2000 12/31/2000 35,136MN I35E-NB 882 3 1/1/2000 12/31/2000 35,136MN I35E-SB 890 3 1/1/2000 12/31/2000 35,136MN 169-NB 442 2 1/1/2000 12/31/2000 35,136MN 169-SB 737 2 1/1/2000 12/31/2000 35,136

trafcdictioratio i

Intrafctionssure a

4.3. Re

Table 2Approaches under comparison.

Approach Description

EXPRW Uses seasonal exponential smoothing for capturing the historical pattern, and uses a random walk model to capture the local variations;considering the established robustness of exponential smoothing, the seasonal exponential smoothing parameter a is selected as 0.15,and so is for the following three approaches (Williams, 1999; Williams and Hoel, 2003; Shekhar, 2004; Guo, 2005). This method does notsupport the construction of time varying prediction intervals

BATCH Uses SAS PROC ARIMA to process the SARIMA(1,0,1)(0,1,1)672 model according to the standard Box-Cox approach, and uses SAS PROCAUTOREG to process the GARCH(1,1) model. Square root transformation is applied in the data series according to Guo (2005) and Guoet al. (2012). Time varying prediction intervals are computed at 95% signicant level

KF Uses seasonal exponential smoothing and two standard Kalman lters for processing the SARIMA(1,0,1)(0,1,1)672 + GARCH(1,1) structure;square root transformation is applied in the data series according to Guo (2005) and Guo et al. (2012). Time varying prediction intervals

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 57The results on level forecasting are presented in Table 5 for all the trafc ow observations. To compare these approachespair-wisely, a protected repeated t-test is used on the performance measures of MAE, MAPE, and RMSE over the 36 stations,

respecInstituferencmanceseasonow rate observations, and the width to ow ratio is calculated as the average of width to ow ratios for all the pre-n intervals. Ideally, the kickoff percentage is expected to be close to 5% for 95% signicant level, and the width to ows expected to be small.order to show the detailed performance of each approach, these measures will be investigated by groups according toow levels and time of day as dened in Tables 3 and 4, respectively. Note that groups with fewer than 10 observa-are ignored due to limited sample size. In addition, the mean performance measures will be computed for each mea-cross these trafc ow series.

sults on level forecastingtrafc ow rates, and n as the total number of trafc ow rate observations processed, the three measures are dened in Eqs.(28)(30), respectively.

MAE 1n

Xnt1

jxt x^t j 28

MAPE 100n

Xnt1

xt x^txt

29

RMSE 1n

nXn

t1xt x^t2

q30

For each approach, except for EXPRW that cannot generate time varying prediction intervals, twomeasures of uncertaintyquantication performance are calculated, namely kickoff percentage and width to ow ratio. The kickoff percentage is cal-culated as the total number of trafc ow observations lying outside of prediction intervals divided by the total number of

are computed at 95% signicant levelAKF Uses seasonal exponential smoothing and two adaptive Kalman lters for processing the SARIMA(1,0,1)(0,1,1)672 + GARCH(1,1) structure;

square root transformation is applied in the data series according to Guo (2005) and Guo et al. (2012). Time varying prediction intervalsare computed at 95% signicant level. The memory of each adaptive Kalman lter is selected as 672, i.e., one weektively, with same group code in Table 5 indicating no signicant statistical difference in the mean performances (SASte Inc., 1999). On observing Table 5, rst, for all the three measures, the statistically signicant mean performance dif-e between BATCH, AKF, and KF cannot be established through the repeated t-test, showing comparable mean perfor-s of the three methods in level forecasting. In addition, the three methods outperformed EXPRW in that though theal exponential smoothing part of EXPRW has the theoretical foundation to capture the historical trafc ow rate pat-

Table 3Groups by trafc level.

Group Group description

TTL For all trafc observationsG1 P0 and

T15 [22:0024:00)

58 J. Guo et al. / Transportation Research Part C 43 (2014) 5064tern, the random walk model part of EXPRW is overly simplistic for capturing the local variations in general. Second, AKF

Table 5Comparison of level forecasting.

Method Groupa Mean MAE (veh/h/ln) Group Mean MAPE (%) Group Mean RMSE (veh/h/ln)

BATCH B 43.5 B 7.15 B 65.91AKF B 44.98 B 7.31 B A 68.78KF B A 46.44 B 7.69 B A 69.63EXPRW A 50 A 8.4 A 74.78

a Same code in the group column indicates no statistically signicant difference can be found in the mean performances of corresponding approaches.Table 4Groups by time of day.

Group Group description

T1 [00:0004:00)T2 [04:0006:00)T3 [06:0007:00)T4 [07:0008:00)T5 [08:0009:00)T6 [09:0010:00)T7 [10:0012:00)T8 [12:0014:00)T9 [14:0016:00)T10 [16:0017:00)T11 [17:0018:00)T12 [18:0019:00)T13 [19:0020:00)T14 [20:0022:00)outperforms KF consistently, indicating an increased performance of AKF over KF in terms of level forecasting. This demon-strates that improved performance can be expected through updating in real time the process variances for the Kalmanlters.

In addition to above overall performances, detailed level forecasting performances are shown in Figs. 1 and 2 with respectto trafc levels and time of day, respectively. From these two gures, rst, for almost all trafc level and time of day groups,the BARCH approach generates the best level forecasting performances and KF and AKF generate comparable performanceswith AKF outperforms KF slightly. In addition, the EXPRW approach in general yields the worst performance except for trafclevel L5 when EXPRW shows the best performance. Note that for Level 5, trafc is near capacity with ow rate over 2000 veh/h/ln and this high level trafc might not leave much room for the trafc oscillating. Therefore, simple forecasting approachessuch as EXPRW might be able to extrapolate trafc with desirable performance.

In summary, for level forecasting, though BATCH works best that agrees with previous ndings (e.g., Williams and Hoel,2003; Smith et al., 2003), both KF and AKF perform reasonably well and AKF slightly outperforms KF. Considering the onlineprocessing nature of AKF, it is fair to claim that AKF is promising for short term trafc ow level forecasting.

4.4. Results on uncertainty quantication

The results in terms of kickoff percentage with respect to trafc levels and time of day are presented in Figs. 3 and 4,respectively. Note that these gures present the difference between the computed kickoff percentage and 5% for clarity pur-pose. First, it can be seen that in terms of all observations (for TTL in Fig. 3), though both KF and AKF are conservative com-pared to BATCH for generating prediction intervals at 95% signicant level, AKF outperforms KF in yielding kickoff percentagecloser to 5%; in addition, by further looking into the performance of AKF and KF for different trafc levels and time of daygroups, we can see that for all groups (G1 to G5) in Fig. 3 and most groups except for T3 and T4 in Fig. 4, AKF outperformsKF, indicating the added adaptive ability integrated into AKF through estimating and updating recursively its process vari-ances. Second, when investigating into the kickoff percentages of AKF and BATCH, we nd that for volatile trafc condition athigher trafc levels (G3, G4, and G5 in Fig. 3) or for morning/afternoon rush hours (T4, T5, T6, T10, T11,T12, and T13 in Fig. 4),AKF outperforms BATCH by yielding kickoff percentage closer to 5%. On reection, this nding might not be surprising in thata xed set of parameters for BATCH is used for all trafc conditions without addressing trafc pattern variations, while AKFcan adjust the parameters with respect to trafc variations. This is desirable since most trafc management and controloperations are targeting volatile trafc conditions.

12) AKF

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 59(b)02468

10

Traffic Level

Mea

n M

APE

(% BATCHEXPRWKF

120140

G1 G2 G3 G4 G5

hr/ln)

(a)0

20

40

60

80

100

Traffic Level

Mea

n M

AE

(veh/h

r/ln)

AKFBATCHEXPRWKF

14

G1 G2 G3 G4 G5The results of width to ow ratio with respect to trafc levels and time of day are presented in Figs. 5 and 6, respectively.On observing the results, rst, we nd BATCH consistently generates smaller width to ow ratio than KF and AKF for all traf-c levels and time of day; however, considering that BATCH has higher kickoff percentages for volatile trafc conditions, i.e.,G3, G4, G5 in Fig. 3 and T4, T5, T6, T10, T11, T12, T13 in Fig. 4, the added value of narrower prediction intervals is offset by theadded kickoff percentages for these volatile trafc conditions. In addition, by comparing KF and AKF, it can be found that KFoutperforms AKF for all trafc levels and time of day groups; however, for volatile trafc conditions, the mean values ofwidth to ow ratios for AKF varies from 0.34 to 0.59, which is workable for these trafc conditions.

In summary, unlike the performances for level forecasting, the uncertainty quantication performances are mixed forthese approaches. However, considering the requirements of trafc management and control systems on volatile trafc con-ditions, it is safe to conclude that the online approach of AKF can generate workable prediction intervals for these volatiletrafc conditions.

4.5. Sensitivity analysis on AKF memory

In the AKF approach, the memory size N is critical for estimating the process variances that are to be adjusted into theadaptive Kalman lter structure. For investigating the performances of AKF with respect to memory size, a sensitivity anal-ysis of AKF memory size is conducted with results illustrated in Figs. 7 and 8 for MAPE and kickoff percentage, respectively.For clarity, eight of the total 36 trafc ow series are selected. On observing the gures, we nd that AKF performances dem-onstrate a converging pattern when the memory size increases. For MAPE, the performance stabilizes when the memory sizeis around 3 days, and for kickoff percentage, the performance stabilizes when the memory size is around 5 days. Therefore,for simplicity and uniformity, the memory size is selected to 7 days (a week) in this paper, translating into 672 data point for15-min interval. Considering this requirement, the eld implementation of AKF will need an adjusting time of about oneweek to function normally. Taking advantage of the continuous nature of trafc data archiving systems, AKF can be initial-ized using historical trafc ow data for seeding the adaptive Kalman recursion.

(c)0

20406080

100

G1 G2 G3 G4 G5Traffic Level

Mea

n R

MSE

(veh/ AKF

BATCHEXPRWKF

Fig. 1. Level prediction performance comparison by trafc level.

(a)

0

20

40

60

80

100

Time of Day

Mea

n M

AE

(veh/h

r/ln)

AKFBATCHEXPRWKF

(b)

468

1012141618

Time of Day

Mea

n M

APE

(%) AKF

BATCHEXPRWKF

(c)

0

20

406080

100

120

Time of Day

Mea

n R

MSE

(veh

/hr/ln

)

AKFBATCHEXPRWKF

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

Fig. 2. Level prediction performance comparison by time of day.

0123456789

10

TTL G1 G2 G3 G4 G5Traffic Level

%

BATCHAKFKF

Fig. 3. Kickoff percentage comparison by trafc level.

0123456789

10

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

Time of Day

%

BATCHAKFKF

Fig. 4. Kickoff percentage comparison by time of day.

60 J. Guo et al. / Transportation Research Part C 43 (2014) 5064

J. Guo et al. / Transportation Research Part C 43 (2014) 5064 610.00.20.40.60.81.01.21.41.61.82.0

TTL G1 G2 G3 G4 G5Traffic Level

Rat

io

BATCHAKFKF

Fig. 5. Width to ow ratio comparison by trafc level.

22.22.4

AKF5. Summaries and discussions

Over the decades, short term trafc ow forecasting has received a sustained attention from many transportation engi-neers and researchers for imparting the anticipatory ability required for developing proactive trafc control and manage-ment systems. Short term trafc ow forecasting includes predicting the level of trafc ow series, i.e., to predict howmany vehicles will be arriving within the next time interval, and constructing the prediction interval, i.e., to decide the dis-persion (uncertainty) of the prediction. Recently, studies shows that the conditional mean and the conditional variance oftrafc ow series can be modeled as the SARIMA + GARCH structure. Based on this structure, online algorithm can be devel-oped for processing this structure consecutively through a seasonal exponential smoothing operator and two state space rep-resentation based Kalman ltering models. However, for conventional Kalman lters, the seeding process variances areconstants that are estimated from historical trafc ow data. Therefore, in this paper, the adaptive Kalman recursion is inves-tigated to estimate and update the process variances in real time.

The proposed adaptive Kalman lters are applied for level forecasting and prediction interval construction, respectively.Using real world data from four regions around the world, the adaptive Kalman lters are compared with three other ap-proaches. For level forecasting, the empirical results show the added performance of adaptive Kalman lter over conven-

00.20.40.60.8

11.21.41.61.8

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15

Time of Day

Rat

io

KFBATCH

Fig. 6. Width to ow ratio comparison by time of day.

4

8

12

16

20

1 2 3 4 5 6 7AKF Memory size (day)

MA

PE (%

)

UK 1996 4762aUK 2002 6951aMD 2004 9aMD 2004 4bMN 2000 442MN 2004 737WA 2004 ES_130D_MN_StnWA 2004 ES_738D_MN_Stn

Fig. 7. Sensitivity of MAPE with respect to N.

62 J. Guo et al. / Transportation Research Part C 43 (2014) 5064tional Kalman lter through updating the process variances. For uncertainty quantication, the results are mixed while theempirical investigation shows that the adaptive Kalman lter can generate workable prediction intervals; in particular, theadaptive Kalman lter approach showed improved adaptability when trafc is highly volatile. Sensitivity analysis is also con-ducted, showing that the performance of the adaptive Kalman lter stabilizes with the increase of its memory size.

Some remarks are provided as follows. First, it should be noted that the performances of AKF are affected fundamentallyby the stochastic structure from which the state space models are developed. Therefore, compared with level forecastingwhere SARIMA has been shown in many studies to be able to capture the rst conditional moment of trafc ow series,GARCH model is only adopted for uncertainty quantication in the past few years. Consequently, it is not surprising tosee the mixed results in uncertainty quantication performances. In fact, future studies should be conducted to advancethe investigation into the second order moment of trafc condition series.

Second, uniform performance measures should be investigated to evaluate the prediction intervals. In this paper, kickoffpercentage and width to ow level are used to gauge the coverage and length of the prediction intervals, respectively.However, a uniform measure is still lacking to determine the overall performance. Therefore, more work should be con-ducted to evaluate the performance of prediction intervals, in particular, for cases when the coverage measure conicts withthe length measure as found in this paper.

In the end, more adaptation mechanism should be investigated. As discussed in Noriega and Pasupathy (1997), the pro-cess noise adaptation mechanism adopted in this study is a suboptimal sequential estimator which can handle time variantmodels with relatively modest computational cost. However, for more improvements, further investigations are recom-mended on the variations of adaptation mechanisms.

Acknowledgements

0

5

10

15

1 2 3 4 5 6 7AKF memory size (day)

Pred

ictio

n in

terv

al k

icko

ff pe

rcen

tage

(%)

UK 1996 4762aUK 2002 6951aMD 2004 9aMD 2004 4bMN 2000 442MN 2004 737WA 2004 ES_130D_MN_StnWA 2004 ES_738D_MN_Stn

Fig. 8. Sensitivity of kickoff percentage with respect to N.The authors would like to thank the Minnesota Department of Transportation, the Washington State Department ofTransportation, the Maryland Department of Transportation, and the United Kingdoms Highways Agency for providingthe data used in this study. This work was supported in part by the Natural Science Foundation of China under GrantNos. 71101025 and 60904026, the National Key Technology R&D Program under grant No. 2011BAK21B01, and the Funda-mental Research Funds for the Central Universities.

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64 J. Guo et al. / Transportation Research Part C 43 (2014) 5064

Adaptive Kalman filter approach for stochastic short-term traffic flow rate prediction and uncertainty quantification1 Introduction2 Literature review2.1 Traffic condition level forecasting2.2 Uncertainty quantification2.3 Summary

3 Methodology3.1 SARIMA+GARCH structure3.2 State space representation3.3 Adaptive Kalman recursion3.4 Summary

4 Empirical study4.1 Data4.2 Experimental design4.3 Results on level forecasting4.4 Results on uncertainty quantification4.5 Sensitivity analysis on AKF memory

5 Summaries and discussionsAcknowledgementsReferences