The Selective Random Subspace Predictor for Traffic Flow Forecasting

Transcription

1 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE may contribute to the inefficiency in determining the corresponding ramp meters for these zone groups. Accordingly, the identification of incident impact in either the temporal domain or the spatial domain appears to be important to address the incident-responsive coordinated ramp-control issues. VI. CONCLUDING REMARKS This paper has presented a real-time coordinated ramp-control approach to address the incident-induced traffic-congestion problems on multiple mainline segments of freeways. The proposed method embeds two sequential functions in the procedure of incident-responsive coordinated ramp control: 1) identification of control-zone-based congestion patterns and 2) stochastic optimization of group-based ramp-metering control to alleviate corridor-wide incident-induced traffic congestion as much as possible. To execute the aforementioned two major functions, M-GSPRT and stochastic optimal-control approaches are used in the development of the proposed ramp-control algorithm. Our numerical results show the applicability of the proposed coordinated ramp-control method in responding to diverse incident-induced traffic-congestion conditions through appropriate identification of congestion patterns and dynamic estimation of group-based ramp meters. Results presented in this paper also suggested the relative advantages of the proposed control method compared with three specified ramp-control strategies presently used in the site. Importantly, the proposed approach finds a new and feasible solution that has never been explored in previous literature to deal with large-scale incidentinduced traffic-congestion problems. In addition, the findings observed in the numerical may stimulate more valuable research, although some limitations still remain in the current version of the proposed approach. REFERENCES [1] J.-B. Sheu, Y.-H. Chou, and L.-J. Shen, A stochastic estimation approach to real-time prediction of incident effects on freeway traffic congestion, Transp. Res. Part B, vol. 35B, no. 6, pp , 2001a. [2] J.-B. Sheu and S. G. Ritchie, Stochastic modeling and real-time prediction of vehicular lane-changing behavior, Transp. Res. Part B, vol. 35B, no. 7, pp , 2001b. [3] J. A. Wattleworth and D. S. Berry, Peak-period control of a freeway system-some theoretical investigations, Highw. Res. Rec., no. 89, pp. 1 25, [4] L. Shaw and W. R. McShane, Optimal ramp control for incident response, Transp. Res., vol. 7, pp , [5] M. Papageorgiou, H. S. Habib, and J. M. Blosseville, ALINEA: A local feedback control law for on-ramp metering, Transp. Res. Rec., no. 1320, pp , [6] H. M. Zhang and S. G. Ritchie, Freeway ramp metering using arterial neural networks, Transp. Res. Part C, vol. 5C, no. 5, pp , [7] G. L. Chang, P. K. Ho, and C. H. 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Recker, Simulation of ITS on the Irvine FOT area using the Paramics 1.5 scalable microscopic traffic simulator Phase I: Model calibration and validation, Uni. Calif., Berkeley, CA, Calif. PATH Res. Rep. UCB-ITS-PRR , The Selective Random Subspace Predictor for Traffic Flow Forecasting Shiliang Sun and Changshui Zhang, Member, IEEE Abstract Traffic flow forecasting is an important issue for the application of Intelligent Transportation Systems. Due to practical limitations, traffic flow data may be incomplete (partially missing or substantially contaminated by noises), which will aggravate the difficulties for traffic flow forecasting. In this paper, a new approach, termed the selective random subspace predictor (SRSP), is developed, which is capable of implementing traffic flow forecasting effectively whether incomplete data exist or not. It integrates the entire spatial and temporal traffic flow information in a transportation network to carry out traffic flow forecasting. To forecast the traffic flow at an object road link, the Pearson correlation coefficient is adopted to select some candidate input variables that compose the selective input space. Then, a number of subsets of the input variables in the selective input space are randomly selected to, respectively, serve as specific inputs for prediction. The multiple outputs are combined through a fusion methodology to make final decisions. Both theoretical analysis and experimental results demonstrate the effectiveness and robustness of the SRSP for traffic flow forecasting, whether for complete data or for incomplete data. Index Terms Correlation coefficient, random subspace, subset selection, traffic flow forecasting. I. INTRODUCTION Reliable analysis of historical trends of traffic flows is an important input to many of the traffic management and control systems in operation and under development. In recent years, utilizing signal processing and machine learning techniques for traffic flow forecasting has drawn more and more attention [1] [5]. These innovative methods activate and enlighten Intelligent Transportation Systems to a large extent. In this paper, we concentrate on using the ideology of random subspace to deal with the issue of short-term traffic flow forecasting and propose a new predictor. The problem of short-term traffic flow forecasting is to determine the traffic flow data in the next time interval, usually in the range of 5 min to half an hour. Up to the present, some approaches ranging from simple to complex are proposed for traffic flow forecasting. Simple ones, such as random walk, historical average, and informed historical average can only work well in specific situations [1]. One class of the complex approaches (Urban Traffic Control System prediction methods) are based on forecasting philosophies similar to the above methods. There are also other elaborate methods proposed including Manuscript received October 31, 2005; revised July 23, 2006, October 11, 2006, October 23, 2006, November 3, 2006, and November 4, This work was supported by the National Basic Research Program of China under Project 2006CB The Associate Editor for this paper was M. Papageorgiou. The authors are with the Department of Automation, Tsinghua University, Beijing , China ( shiliangsun@gmail.com; sunsl02@mails. tsinghua.edu.cn; zcs@mail.tsinghua.edu.cn). Digital Object Identifier /TITS /$ IEEE

2 368 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE 2007 approaches based on time series models (including autoregressive integrated moving average (ARIMA), seasonal ARIMA), Kalman filter theory, neural network approaches, nonparametric methods, simulation models, local regression models, layered models known as the ATHENA model and the KARIMA model, the fuzzy-neural approach, and the Markov chain model [1], [6] [19]. Although the methods mentioned above have alleviated difficulties in traffic modeling and forecasting to some extent, from a careful review, we can still find a problem, that is, most of them have not made good use of spatial information from the viewpoint of a network to analyze the traffic flow trends of object sites. Although Chang et al. utilized the data from other roadways to make judgmental adjustments, the information was still not used to its full potential [20]. Yin et al. developed a fuzzy-neural model to predict traffic flows in an urban street network whereas it only utilized the upstream flows in the current time interval to forecast the selected downstream flow in the next interval [17]. The total information in a transportation network was also not used fully. Most importantly, the existing methods hardly work when the data used for forecasting is incomplete, i.e., partially missing or substantially contaminated by noises, while this situation often occurs in practice. The incomplete data can be caused by malfunctions or measurement errors in data collection and recording systems, such as failed loop detectors, faulty loop amplifiers, or signal communication and processing errors. Although the historical average method (filling up the incomplete data with their historical averages) could be applied to deal with this matter, the forecasting performance is quite limited. For the topic of traffic flow forecasting with incomplete data, we have reported results in the papers [3] and [19]. In [19], we proposed the sampling Markov chain method to carry out prediction using the idea of Monte Carlo integration. The performance of the sampling Markov chain method is a little inferior to the Markov chain method for the case of complete data. However, this method would be time consuming when the number of incomplete variables is more than two because of the inherent complexity of Monte Carlo sampling. The Bayesian network method proposed in [3] integrating the casual relationship between adjacent traffic flows is a very prominent approach for incomplete traffic flow forecasting, but when the amount of incomplete data is a little high, it would become complicated for calculation since a lot of nodes need to be involved to compensate for the influence of incomplete data. The development of a method which is effective and robust for incomplete traffic flow forecasting is in a great demand. The main contribution of this paper is that we present the selective random subspace predictor (SRSP), which combines the available spatial information with temporal information in a transportation network to implement traffic flow modeling and forecasting and is effective and robust enough to carry out traffic flow forecasting for both complete data and incomplete data. To forecast an object traffic flow at a given road link, the SRSP tries to find correlated links over the entire network and utilizes the traffic flows on the correlated links as input to complete forecasting. Apparently, the traffic flows from adjacent links are among the most informative ones for the purpose of forecasting. We have studied the problem of using adjacent links to forecast the traffic flow of the current link in our previous paper [3]. Besides, there are many distant links which are also informative and perhaps should be taken account in forecasting the traffic flow of the current link. The motivation is very intuitive. Although many links may be located at different even distant parts of a transportation network, there may exist some common kind of hidden sources influencing their own traffic flows. Some of the possible sources include shopping centers, residential communities, car parks, etc. People s activities around the same kind of sources usually obey some consistent laws in a long time period, such as the usually common time range for working hours. These hidden sources imply some related information that is useful for traffic flow forecasting in different links, and this correlation could be partially reflected by the traffic flows on these links. To be specific, the SRSP adopts the measurement of Pearson correlation coefficient to integrate spatial and temporal relevant information in a transportation network to construct a set of strongly related traffic flows. The set forms a space called the selective input variable space, and each strongly related traffic flow serves as one dimension in the space. The dimensionality of this space may be very large, and it is of no need for us to use all the elements therein to implement forecasting. Instead, a small subset of input dimensions (input traffic flows) from this space are randomly selected to form a random subspace. Also, sequentially, we use our previous method the [Gaussian mixture model (GMM)] to approximate the statistical relationship between each input subspace and the output (object traffic flow) [3]. Based on this relationship, traffic flow forecasting is performed under the criterion of minimum mean square error (mmse). In view of the factor of noises usually existing in recorded data and the problem of forecasting in case of incomplete data, the above procedure is repeated for multiple times. That is, we randomly select multiple subspaces, use GMMs to model the statistical relationship between each subspace and the output, and then carry out forecasting. At last, the outputs from multiple subspace inputs are combined to make the final prediction. II. SPATIO TEMPORAL INFORMATION COMBINATION USING THE SRSP In a transportation network, almost all the other road links would directly or indirectly relate to the traffic flow forecasting task of the current link as a result of vehicles movement. However, taking traffic flows from all the links as input variables would involve much redundancy and be prohibitive for computation. Consequently, a variable selection procedure to seek the most informative traffic flows from both adjacent and distant links as input is of great demand. The potential benefits of variable selection for traffic flow forecasting are evident: reducing the storage requirement, reducing training and utilization time, and enabling traffic flow forecasting more compactly and precisely. Up to date, many variable selection algorithms adopt variable ranking as a principal or auxiliary selection mechanism because of their simplicity, scalability, and good empirical success [21] [24]. In this paper, we also adopt the variable ranking mechanism, and the Pearson correlation coefficient is used as the specific ranking criterion defined for individual variables. In other words, we use the Pearson correlation coefficient to rank traffic flows from different links according to their respective correlations with the object traffic flow. By the ranking mechanism, a number of traffic flows could be found to form a selective input space whose dimensionality might be high. Since there is usually a lot of redundant information encoded in the selective input space, not all the input variables are required for traffic flow forecasting. Each time, we only need one subset from the selective input space to serve as the forecasting input, and this would also simplify the computation. The subsets are selected uniformly and randomly in this paper. Furthermore, the fact that in the real world traffic flows are often partially missing or contaminated by noises is another reason that why we adopt the random subspace idea to carry out traffic flow forecasting instead of using the whole selective input space as input. The SRSP randomly selects multiple variable subspaces, each of which includes several traffic flows from the selective input space, and then constructs a predictor with each subspace as an input. For the selected input subspace, a prediction formulation is derived from the GMM approximation of the statistical

3 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE relationship between the input traffic flows and the output traffic flow. The final forecasting result is obtained by averaging the outputs of all the individual predictors. TABLE I FLOW CHART OF THE SRSP A. SRSP We first carry out variable ranking to rank traffic flows from different links in a transportation network before selecting input variable subsets. Variable ranking can be regarded as a filter method: It is a preprocessing step independent of the choice of the sequent predictor [25]. Under certain independence or orthogonality assumptions, variable ranking may be optimal with respect to a given predictor [21]. Even when variable ranking is not optimal, it may be preferable to other variable subset selection methods because of its computational and statistical scalability: Computationally, it is efficient since for n variables it requires only the computation of n scores and sorting the scores; Statistically, it is robust against overfitting because it introduces bias but it may have considerably less variance [26]. Consider a set of m samples {x k,y k }(k =1,...,m) consisting of n input variables x k,i (i =1,...,n) and one output variable y k. Γ(i) is a correlation scoring function which is computed from x k,i and y k (k =1,...,m). Assume a higher score indicates a more valuable variable, and the variables are sorted in decreasing order of the corresponding Γ(i) values. Furthermore, let X i denote the random variable corresponding to the ith component of input vector x, and let Y denote the random variable with the instantiation y. The Pearson correlation coefficient between X i and Y is defined as cov(x i,y) R(i) = (1) var(xi )var(y ) where cov denotes the covariance and var the variance. When there are m samples, the estimate of R(i) is given by [21] m k=1 R(i) = (x k,i x i )(y k y) m (x k=1 k,i x i ) 2 m (y (2) k=1 k y) 2 where x i =1/m m x k=1 k,i,andy =1/m m y k=1 k. Since highly positive and negative correlation coefficients both indicate strong correlation, the norm R(i) is used as our traffic flow ranking criterion, i.e., Γ(i) = R(i). After the variable ranking stage, we can obtain M input traffic flows making up of the selective input space S, which will be very informative for the prediction of the object flow according to the correlation criterion. In a city, there might be many related traffic flows for the current link. Therefore, M is probably very large. Using the entire high-dimensional input space as input to forecast traffic flow directly would arouse intensive computation and overadaptation because comparatively, it is hard to obtain enough training data to estimate the prediction relationship. In addition, it would introduce the problem of redundant dimensions. Alternatively, we adopt the random subspace ideology proposed in [27] to generate K random subspaces {S i } K i=1 and every time randomly select a subspace of m traffic flows with replacement from S and use them as an input to forecast the object traffic flow. The dimension m of random subspace can be determined by the training set to get good training results or be determined empirically. Given the object traffic flow and the selected input traffic flows, we utilize GMMs to approximate their joint probability distribution, whose parameters are estimated through the competitive expectation maximization (CEM) algorithm. Then, we can obtain the optimum prediction formulation as an analytic formulation under the mmse criterion. The benefits of GMMs involve at least three aspects, as listed in [3]. One prominent advantage of the CEM algorithm is that it could automatically determine the number of Gaussian components. For details about the GMM, CEM algorithm, and the prediction formulation, see [3], [18], and [28]. In succession, we can repeat the above operation K times and would obtain K forecasting results {F (S k )} K k=1 for the current object flow. The outputs are combined using the fusion methodology of taking average, and the average is regarded as the final forecasting outcome F ( S) of the SRSP method, i.e., F ( S) = 1 K K F (S k ). (3) k=1 B. Flow Chart for Traffic Flow Forecasting In this section, we describe the flow chart of the SRSP for traffic flow forecasting. The data set is divided into two parts: one serving as training set for input traffic flow selection and prediction parameter estimation and the other test set for evaluating the proposed method. The flow chart of our forecasting procedure is described in Table I. It should be noticed that the flow chart can be largely reduced and completely meet the requirement of real-time utility when forecasting a new traffic flow, because the prediction formulation need only be derived once based on the historical traffic flows (training data set) and thus can be obtained in advance. C. Traffic Flow Forecasting With Incomplete Data The SRSP is very robust and can still work well when the input data are incomplete (missing or substantially contaminated by noises). If some incomplete data appear in a certain subspace, we can just remove this subspace and generate a new one until there are no incomplete data involved in the used subspace, and this almost does not influence the final fusion result as we just do random selection for subspaces. Therefore, this is a very straightforward and natural way to deal with incomplete data. We will discuss this matter in the following section with a specific instance. III. EXPERIMENTS The problem addressed in this paper is to forecast the future traffic flow rates at given roadway locations from the historical data on a transportation network. The field data analyzed are the vehicle flow rates of discrete time series recorded during every interval of 15 min along many road links by the UTC/SCOOT system of the Traffic Management Bureau of Beijing, whose unit is vehicles per hour (veh/h). For the current short-term traffic flow forecasting problem, we carry out a one-step prediction, and the prediction time horizon is taken as 15 min. That is, we use historical data to forecast the traffic flow rates

4 370 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE 2007 TABLE II TWENTY MOST CORRELATED TRAFFIC FLOWS WITH Ka(t) Fig. 1. Representative transportation network taken from the urban traffic map of Beijing. on the next interval of 15 min. From the real urban traffic map, we select a representative patch to verify the proposed approach, which is given in Fig. 1. Each circle node in the sketch map denotes a road link. An arrow shows the direction of traffic flow, which reaches the corresponding downstream link from its upstream link. Paths without arrows are of no traffic flow records. The raw data for use are of 25 days (24 h/day) and 2400 sample points in total taken from March The starting time for data recording on a day is 00:00 at midnight. To validate our approach objectively, the first 2112 points (training set) of them are employed to estimate parameters of forecasting formulation, and the rest (test set) are employed to test the forecasting performance. To compare with the presented SRSP, two other approaches (the random walk method and the Markov chain method) are utilized as base lines. Random walk is a classical method for traffic flow forecasting, whose core idea is to forecast the current value using its last value [1]. It can be formulated as ˆx(t +1)=x(t). For our forecasting problem, random walk uses the traffic flow rates of the last 15 min to determine the flow rates of the current 15 min. The Markov chain method models traffic flows at different times as a high-order Markov chain. It has shown great merits over several approaches such as historical average, informed historical average, K nearest neighborhood, and sampling Markov chain model [18], [19]. In this paper, the joint probability distribution for the Markov chain method is also approximated by the GMM, whose parameters are estimated through the CEM algorithm. The dimension of subspace in the SRSP is taken as 4 (the same with the Markov chain method in [18]) for each object link. This entire configuration makes an equitable comparison as much as possible. Resultantly, the only difference between our proposed method and the Markov chain method is that the SRSP utilizes the whole spatial and temporal information to forecast, while the latter only uses the temporal information of the object site. A. Input Selection We take road link Ka as an instance to show our approach. Ka represents the vehicle flow from the upstream link H to the downstream link K. All the available traffic flows which may be informative to forecast Ka in the analyzed transportation network include {Ba, Bb, Bc,Ce, Cf,Cg, Ch,Da, Db,Dc, Dd, Eb, Ed, Fe,Ff,Fg,Fh,Gb, Gd, Hi, Hk, Hl, Ia, Ib, Id, Jf, Jh, Ka, Kb, Kc, Kd}. Considering the time factor, in order to forecast the traffic flow Ka(t), we might need to judge its relevancy from the above sites with different time indexes, such as {Ba(t 1), Ba(t 2),...,Ba(t d)}, {Bb(t 1), Fig. 2. Observed flow versus forecasted flow for road link Ka. TABLE III PERFORMANCE COMPARISON OF THREE METHODS FOR TRAFFIC FLOW FORECASTING AT FIVE DIFFERENT ROAD LINKS Bb(t 2),...,Bb(t d)}, etc. The time unit of the time step is 15 min. In this paper, d is taken as 100 empirically. Table II lists M (M =20in this paper) most correlated traffic flows that construct the entire selective input space. The M traffic flows are selected with the correlation variable ranking criterion, and their corresponding correlation coefficients with traffic flow Ka(t) are also given in Table II. B. Experimental Results With the selected input space, the SRSP generates K (K =10 in this paper) random subspaces. The joint probability distribution between the input (the chosen random subspace) and the output Ka(t) are approximated with GMMs, and thus, we can derive the prediction formulation, and carry out traffic flow forecasting on the test data set. Finally, we combine the K outputs to form one forecasting output. The forecasted flow versus observed flow on the test set for Ka is shown in Fig. 2, where the dotted star line shows the forecasted data, while the solid line shows the observed data. In addition, we also conducted experiments on four other traffic flows Ch, Dd, Fe, and Gd. Table III gives the forecasting results of all the five road links through

5 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE the random walk method, Markov chain method, and our proposed approach, respectively, with performances evaluated by root mean square error (rmse) [3]. For the same traffic flow in Table III, the smaller rmse corresponds to the better forecasting accuracy. From the experimental results given in Table III, we can find the effectiveness of the SRSP, which integrates both spatial and temporal information for forecasting. The performance of the random walk method is inferior to those of the Markov chain method and the SRSP to a large extent. One important reason is that if used the least the information as input. Besides, the SRSP has obtained significant improvement on forecasting accuracy compared to the Markov chain method. This is due to the fact that the SRSP integrates the whole spatial and temporal information to forecast, while the Markov chain method only utilizes the temporal information of the object site. Since the traffic flows from distant links can have high correlation to the object traffic flow as shown in Table II for Ka, they are very helpful in forecasting the object traffic flow. In addition, the SRSP can still work in the case of incomplete data. For example, in forecasting Ka(t), the Markov chain method would use Ka(t 1), Ka(t 2), Ka(t 3), and Ka(t 4) as input. However, if the input is incomplete, such as Ka(t 1) being missing or contaminated, then Markov chain method would lose its applicability, while the SRSP can still work since it can use the other M 1 traffic flows in the selective input space to generate subspaces, and the final performance would not change much. Furthermore, if multiple input flows, such as Ka(t 1), Ka(t 2) are missing or contaminated, the SRSP can still work stably, while the Markov chain method cannot. IV. ANALYSIS ON THE EFFECTIVENESS AND ROBUSTNESS OF THE SRSP It is reported that subset selection methods for input variables are unstable procedures for prediction, such as best subsets, forward variable selection, backward variable selection, or variants thereof [29]. Therefore, if one specific subset is used to implement prediction, we will take the risk of a large forecasting bias. The original idea of random subspace for classification which adopts an ensemble of decision trees to make a final robust output gives inspiration to us [27]. Based on the theoretical analyses for bagging methods and random forests in [30] [32], we provide the following justification of the original random subspace predictor. Consider a training set T of m examples {x k,y k }(k =1,...,m) consisting of n input variable x k,i (i =1,...,n) and one output variable y k. Each (x, y) case in T is independently drawn from the underlying probability distribution P. Define random vector Θ as the feature subset selection vector, which consists of a number of independent random integral indexes uniformly drawn between 1 and n. The predictor derived from the training set T and the subset selection vector Θ using a fixed forecasting model (e.g., the GMM) is defined as f(x, T, Θ). Thus, the random subspace predictor is f RSP (x, T )=E Θ f(x, T, Θ).TakeX, Y to be random vectors having the distribution P and independent of the training set T. The mean prediction error e of f(x, T, Θ) is e = E T E X,Y E Θ (Y f(x, T, Θ)) 2. (4) Also, the mean prediction error of the random subspace predictor can be given as e RSP = E T E X,Y (Y f RSP (X, T)) 2 = E T E X,Y (Y E Θ f(x, T, Θ)) 2. (5) Applying Jensen s inequality EZ 2 (EZ) 2 gives E Θ (Y f(x, T, Θ)) 2 (Y E Θ f(x, T, Θ)) 2. (6) Therefore, we can draw the conclusion that e RSP e, and the instability of f(x, T, Θ) with respect to Θ would make e RSP strictly lower than e. Supposing Θ Ω, the SRSP considers how to select a different random vector Θ Ω such that e SRSP = E T E X,Y (Y E Θ f(x, T, Θ )) 2 <e RSP (7) where e SRSP is the mean prediction error of the SRSP. If E X,Y (Y E Θ Ω f(x, T, Θ ) 2 <E X,Y (Y E Θ Ω f(x, T, Θ)) 2, we can replace the original field Ω with the new field Ω and obtain a more precise predictor with error e SRSP <e RSP. It can be shown that this is feasible for traffic flow forecasting, and the algorithm we present in this paper is an instantiation which successfully implements this process. As given in (3), when carrying out forecasting, we use the average of results from different subspaces to approximate the mathematic expectation involved in f RSP (X, T), since the computation of mathematic expectation is intractable for the SRSP. To assess the influence of correlation upon the forecasting performance, we show the curve of forecasting errors with respect to input combinations of different correlation values for traffic flow forecasting of two road links K a and F e in Fig. 3. For each link, we first find its 100 most correlated input traffic flows and then calculate forecasting errors using as input the combination of every four consecutive input traffic flows in the order of decreasing correlation. Thus, we can depict the underlying relationship between correlation and forecasting results on 100/4 =25combinations. The resultant tendency is that higher correlated input would basically give better forecasting results (smaller errors and smaller fluctuations). Therefore, the new field Ω can be selected to include many high correlated flows with respect to the object traffic flow (the original field Ω can be regarded as including all the related traffic flows in a transportation network). In the meantime, the fluctuation of forecasting errors depicted in Fig. 3 also implies the instability of subset selection and supports the idea of forecasting using an ensemble predictor. These are the theoretical justification of why the SRSP works. To evaluate the robustness of the SRSP, now we try to give a general idea about the proportion of road links whose data are allowed to be completely missing. Also, road link Ka is taken as an example. As one can see, there are totally 31 road links with traffic flow recordings in the transportation network shown in Fig. 1. Also, the 20 most correlated traffic flows in Table II come from 11 road links. Since the used dimension of subspace is 4, theoretically, the SRSP can still work as long as there are no more than 11 4=7missing road links in the selective input space. Of course, the precondition is that the remaining four road links can give a fine forecasting result. The average value and standard deviation of the forecasting errors for the K (K =10) individual predictors in the SRSP can be obtained as ± 2.2, which are basically comparable to the forecasting error of the SRSP for road link, Ka. Therefore, the precondition is in this case met. Also, we can draw a rough conclusion that the SRSP can work well for road link Ka, even in case that there are 7/31 23% road links in a transportation network which are completely missing. The proportion can still be enlarged if the SRSP employs some optimized parameters, such as the size of the selective input space. In a city, there are usually hundreds of road links, and thus, in forecasting the traffic flow on an object link, there are rather many candidate links available to be included in the selective input space. If we fix the dimensionality of

6 372 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE 2007 With regard to the fusion strategy for multiple subspace forecasting results, we adopt the strategy of taking average, as shown in (3). We have also considered weighting different subspace outputs according to their performance. However, the difference of performance between different subspace predictors is small in our experiments, and weighting has not brought much improvement. Therefore, we use the strategy of taking average. There are still some valuable directions to be discussed and improved upon in the future, such as exploring how to determine the optimum dimensionality of the random subspaces and how to carry out traffic flow forecasting in the case of congestion and exploring the relationship between the forecasting accuracy and the number of the individual predictors in an ensemble, etc. Fig. 3. Relationship between the forecasting error and the combination of different inputs with decreasing correlations to the object traffic flow. (a) Object traffic flow Ka. (b) Object traffic flow Fe. the selective input space (e.g., 20 as in this paper), then only a small proportion of road links would be included in the space. Consequently, we can allow the data from many road links to be completely missing. Generally speaking, the robustness of the SRSP will increase with the number of the road links in a transportation network. V. C ONCLUSION In this paper, we propose an SRSP integrating the whole spatial and temporal information available in a transportation network to carry out short-term traffic flow forecasting. It is effective, robust, and still operates well when encountering missing or contaminated data. Theoretical analysis and experimental results with the urban vehicular flow of Beijing and comparisons with other methods demonstrate the applicability of the SRSP. 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