Robust Freeway Travel Time Prediction with State-Space Neural ...

Toward a Robust Framework for Freeway Travel time Prediction: Experiments with Simple Imputation and State-Space Neural Networks J.W.C. van Lint *, S.P. Hoogendoorn, H.J. van Zuylen Transportation Planning and Traffic Engineering Department Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1, P.O. Box 5048, 2600 GA Delft *

corresponding author: [email protected], (t) +31 15 2785061, (f) +31 15 2783179

Abstract Robustness to missing or faulty input, due to failures in the data collection system, is a key characteristic for any travel time prediction model that is to be applied in a real time environment. Previous research by van Lint et al (2002) has shown that so-called State-Space Neural Networks (SSNN) are capable of accurately predicting experienced travel times. Our paper shows that incorporating corrupt data into the training procedure does increase the robustness of these SSNN models, but at the cost of predictive performance: there is a clear trade off between robustness and model accuracy. More over, inclusion of (small) amounts of corrupted data in the training procedure makes the internal states of the SSNN model, which are closely related to the expected traffic conditions, more difficult to interpret. Application of pre-processing (imputation) strategies before feeding the input into the model does lead to a robust but still accurate model. Although there are theoretical shortcomings to simple imputation schemes, the combined framework of pre-processing and SSNN model is robust to various kinds of input failure, even though the proposed pre-processing strategies are naïve non-parameterized procedures such as exponential smoothing and spatial interpolation. We hypothesize this is due to the fact that the SSNN model is robust to the “damage” done by the pre-processor. Further research should emphasize on enhancing the proposed pre-processing procedures and applying the travel time prediction framework in real time. Words: 5388; nr. Figures 4, nr Tables: 4. INTRODUCTION There is an increasing need for systems that can provide road-users with accurate real time traffic information. Many research efforts emphasize the significance of traffic information and the potential effect of advanced traveler information systems (ATIS) on travel choice. For traffic information to have an effect on driver behavior, it should consist of clear and unambiguous messages based on accurate and reliable predictions (1). With respect to the accuracy and reliability of traffic information, the sensitivity of the quality of that information to faulty and missing input data is of particular interest. So-called State-Space Neural Networks (SSNN) are capable of accurately predicting experienced travel times, producing approximately zero mean normally distributed residuals, generally not outside a range of 10% of the real expected travel times (2). These results, however, are obtained by feeding the models with 100% accurate and reliable data. The input data to travel time prediction models, collected by a real time traffic monitoring system, will often consist of corrupted or missing values. The goal of this paper is therefore to present a travel time prediction framework, based on the SSNN model, which is robust to various kinds of input failure, as well as to noisy and possibly biased input. Robustness in this context is defined as the capability of the SSNN based framework to produce accurate predictions under all kinds and degrees of input failure. To this end we first give a brief overview of methods for dealing with datacorruption, next we present a classification of different kinds of input failure (incidental, structural and intrinsic). Subsequently, we propose various pre-processing and training strategies, and finally, we critically discuss results and present conclusions and recommendations. BRIEF OVERVIEW OF METHODS FOR DEALING WITH MISSING DATA In practice the most commonly used approach to remedy input-failure is “simple” imputation. Simple here depicts substituting missing values with sensible ad-hoc replacements, such as regression forecasts or sample mean. Schafer (3) shows that simple imputation schemes tend to change the covariance structure of the inputdata and may induce bias. Therefore, Schafer proposes EM and Markov Chain Monte Carlo based approaches that account for the missing values, and the uncertainty they inherently introduce. Examples of these and other approaches to remedy the missing data problem can be found in many fields, including neuro-computing (8) and (5), pattern recognition (4), climatology (6), and medical statistics (7), to name a few.

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Despite the clear theoretical shortcomings of simple imputation schemes (treated from different perspectives in (3) and (8)), the results in (9) indicate that such simple imputation schemes combined with a neural network based traffic predictor, do yield accurate traffic predictions even when up to 30% of the input data to the model is missing. One (tentative) hypothesis may be that a (properly trained) neural network is robust to the “damage” caused by the imputation scheme applied. Slight changes in the statistical properties of the input data do not cause the neural network to produce inaccurate results. Another (again tentative) hypothesis could be that the spatiotemporal patterns formed by traffic measurements have statistical properties (covariance structure, correlations through time and space) that are more invariant to simple data-repair techniques as compared to the multivariate datasets used throughout (3) which stem predominantly from medical statistics, and social sciences. Based on these findings, we will explore two different approaches in the remainder of the paper. The first one involves inclusion of corrupted data1 in the training procedure of our neural network based travel time predictor (the SSNN); the second one involves a simple imputation scheme that corrects corrupted input values before it is fed to the SSNN. But first we will classify the various kinds of input-failure we may encounter in practice. CLASSIFICATION OF INPUT FAILURE The input data to the travel time prediction models, collected by a real time traffic monitoring system, will often consist of corrupted or missing values (e.g. on average 15% of the inductive loops of the Dutch Freeway monitoring system (MONICA) may be out of operation or producing unreliable measurements2). We propose a classification of input failure as presented in Figure 1. The first type of detection failure ( Figure 1a – Incidental failures) occurs due to, for example, temporary power or communication failures in the freeway monitoring system. The second type (Figure 1b – Structural failures) occurs mainly due to physical damage or maintenance backlogs to the inductive loops or roadside equipment. Although the distinction presented here may not be as crisp in practice, the proposed distinction expresses two extreme configurations of input failure that we can expect to encounter, and is hence useful in the investigation of the robustness of travel time prediction models to input failure. A third type of failure ( Figure 1c – Intrinsic failure), measurement noise and bias, is inherent to detection devices and averaging measurements over time in general. For example: in MONICA the arithmetic time mean speed per measurement period is calculated, yielding a biased estimate of the space mean speed (the mean speed on a particular link or stretch), which is in fact the quantity of interest. Another known source of intrinsic data corruption is miscounts, double counts or false counts of vehicles, device calibration errors, round off errors, etc. THE TRAVEL TIME PREDICTION FRAMEWORK The travel time prediction framework (Figure 2) consists of two key components: a pre-processing layer, which processes and cleans the data it receives from a detection system, and the actual travel time prediction model, which is fed by the pre-processing layer and predicts travel time on the route of interest. Pre-processing Pre-processing involves all necessary operations and transformations on the input data before it is actually fed into the travel time prediction model. We argue that if the inaccuracies in the input are (partially) known, then pre-processing is always a good strategy. Based on the previous section, we propose three simple nonparameterized imputation schemes, which are discussed in the next sections. Pre-processing time series of individual inputs The first method is to consider time series of individual inputs only. Clearly, traffic measurements on fixed locations exhibit strong autocorrelation over time. Missing or corrupt measurements ud(t+1) from detector d at time instant t+1 can be replaced by a forecast f d(t+1) of some time series model, in our case by an exponentially moving average: f d (t + 1) = f d (t ) + α ⋅ [u d (t ) − fd (t )], α ∈[0,1]

(1)

with α typically set to 0.3. The quantity f d(t) denotes the exponential forecast for input d at time instant t, while ud(t) denotes the dth input value at time instant t. Obviously, more complicated AR(I)MA models or non-linear forecasting techniques can be used. Note that in case of structural detector failure, a time series forecast, such as (1), is not feasible, since it requires at least one valid measurement per input time series. 1

Corrupted or corrupt data depicts data containing missing values or values that are dubbed as unreliable by the traffic data collection system (which is the case with the Dutch Highway Data Collection System MONICA) . In this paper we assume that the actual detection of corrupt (missing or unreliable) values is done by the traffic data collection system. 2 Statistic from a week of 1 minute aggregate measureme nts of detectors on the A13 highway between Den Haag and Rotterdam, januari 2002; 9746 of 65536 measurements classified unreliable (missing or faulty) =14.9%



Pre-processing spatial patterns The second pre-processing method considers spatial traffic patterns on the route of interest. Suppose a route is composed with D measurement devices, each measuring mean speeds and flows. Then consider flows and mean speeds measured at time instant t as spatial patterns {…, qd, …} and {…, ud, …} respectively. From traffic theory we know that traffic properties over adjacent links are strongly correlated. If measurements on intermediate locations on the route are fraud we can use cubic spline interpolation to fill in the gaps (Figure 3 gives an example in which 5 out of 13 data-points are corrupt). If D* ∈{3, ..., D} out of D measurements ud on measurement locations xd are available in such a spatial pattern, the cubic interpolation procedure fills in the gaps by solving the following polynomial between each two adjacent data-points: uˆ d ( x) = u d + a d ( x − xd ) + bd ( x − xd ) 2 + cd ( x − xd )3 , x ∈ [ xd , xd +1 ],

(2)

d = {1,.., D − 1}, *

constraint by the D* known data-points ud, D*-1 known gradients (first differences) and the D*- 1 known concavities (second differences) between the data-points. Equation (2) has 3D* unknowns so two more conditions are required. The easiest evolve when we set the concavity zero at the first and last known datapoints, which is the case in the matlab implementation of this algorithm (“interp1” function) that we used in our tests. If less then 3 valid data-point in a spatial pattern were available, we use linear interpolation to fill in the gaps. Finally, if measurements are fraud at the edges of our route we estimate these values with the aid of (1) (smoothing over space instead of over time). Pre-processing spatio-temporal patterns Since traffic flow yields patterns in both space and time, a combination of time series modeling and spatial interpolation might prove a useful preprocessing method. A missing value could then be replaced by a linear combination of a time series forecast and a value resulting from spatial interpolation. f d (t ) = βMA f dMA (t) + β interp fdinterp (t )

(3)

where ƒMA and ƒinterp denote the imputed values for a missing datum by means of (1) and (2) respectively, and β MA and β interp depict weighting parameters.with in the simplest case, both β values set to 0.5. More sophisticated approaches are possible; in this study, however, we will focus on non-parameterized pre-processing strategies only. Travel time prediction model: State-Space Neural Network The second key component in the robust travel time prediction framework is the actual travel time prediction model itself. In (2) it is argued that accurate and reliable travel time prediction either requires highly sophisticated traffic flow models, or data driven models that are capable of learning the non-linearity’s of congested traffic from data. The State Space Neural Network (SSNN) model proposed by (2) is an example of a model of the second category (Figure 4). One of the strategies we will investigate to increase the robustness of the travel time prediction framework in Figure 2, is to include corrupted data in the data with which the SSNN model will be trained. The idea behind it is that the allow the SSNN to develop an internal model, which takes data corruption into account automatically. We might expect that the SSNN will be able to learn from corrupted data, but that an increase in robustness is at the expense of its predictive accuracy, since we have deliberately increased the complexity and non-linearity of the problem at hand. Finally, a clear advantage of using a neural network model for travel time prediction is that it is generally not very sensitive to small disturbances (noise) and some bias in its inputs. In this sense, a neural network approach intrinsically satisfies a degree of robustness to noisy and biased input data (Figure 1c), given that it is designed and trained properly (10). EXPERIMENTAL SETUP Data used for experiments We will use synthetic data obtained from a micro-simulation to test the robustness of the framework above. To this end we have set up a network in the microscopic simulation model FOSIM (developed at the Delft University of Technology, see e.g. (13)) for the southbound stretch of the A13 highway between two of the major Dutch cities in the western part of the Netherlands (The Hague and Rotterdam), similar to the one used in (2). The road stretch measures 7,3 kilometers and yields travel times of 4 up to15 minutes in free flow and congested conditions respectively. The input to FOSIM (Dynamic OD matrices) was scaled to fit collected data



from inductive loop-detectors on a number of typical afternoon peaks of January 2000. The scaling was done heuristically. All data used for training and testing the neural networks (both inputs and outputs) are linearly scaled to the interval [+0.1, +0.9], based on the rule-of-thumb that this leads to faster and more stable learning. Representation of input failure All input failures result in input values of –1, allowing the pre-processing layer and/or travel time prediction model to detect these values as “outliers”. If a measurement from detector d at time period p is dubbed corrupt then all values (i.e. both spe ed and flow) measured at {d, p} are replaced with –1. Incidental input failure is generated with a random generator ϑ, producing numbers from a uniform distribution on [0, 1], such that each measurement {d, p} has an equal probability to be labeled corrupt. If the required level of corruption is set to 20% then a measurement is labeled corrupt if ϑ(d,p) ≤ 0.2. The maximum amount of incidental input failure considered is set at 40%. In case of structural detection failure ALL measurements {dk, p} from a specific detector dk are labeled corrupt. Considering all possible combinations of failing detectors leads to a very large amount of test data, which is why we will only consider cases where 1 or 2 detectors (out of 13) are structurally down, yielding a total of 13! 13! + = 13 + 78 = 91 test-sets 1!12! ⋅ 2!11! ⋅

We will not include structural failure of detectors located on- or off-ramps here. Obviously, spatial interpolation does not apply to structural detector failure at onramps, since no adjacent detectors are available. Training and Testing Strategies In this section we will propose a number of strategies in handling faults or incompleteness in the input data based on the framework in Figure 2. Each strategy (listed in Table 1) consists of pre-processing method and a neural network training scheme, and is consequently tested against a dataset consisting of (a) 100% clean data, (b) a dataset with increasing amount of incidental input failure (detector failure is randomly drawn from a uniform probability distribution), and – in three of the five strategies - (c) a dataset with structural detector failure (one or two detectors on the main carriage way structurally breaks down). Note that the SSNN’s are developed and trained with Levenberg-Marquardt backpropagation and Bayesian regularization (12) in all cases. Note that we do not need to test the performance of S2, S3 and S4 on clean data, since pre-processing has no effect if no corruption is present. Performance indicators Let T p and Tpest denote the experienced and the predicted mean travel time for a vehicle departing in time period p on the route described above, and let P the total number of periods. The prediction error for departure time p est then equals ∆ est p = T p − Tp . We will make use of the following performance measures to assess the various strategies: RMSE

(Root of Mean Squared Error)

2 1 ∑ ( ∆ estp ) P p

MRE

(Mean Relative Error)

100 ∆ p ∑ P p Tp

est

∑( SRE

(Standard Deviation of Relative Error)

p

100∆ est p − MRE )2 Tp P −1

RESULTS Testing on Incidental Failure Table 2 (a, b, and c) show the MRE, SRE and RMSE respectively on one of the two test datasets for all proposed strategies. SSNN models trained with an increasing percentage of (incidental) failure in their training datasets. The best results (in terms of overall RMSE) are obtained with strategy S2, which performs well at all degrees of incidental input failure corruptness. This supports the idea that a neural network is robust to the “damage” caused by a simple imputation scheme (the MA procedure in this case). We can also conclude that incorporating corrupt values into the training datasets (strategies S1), does increase the robustness of the model to incidental input failure but at the expense of accuracy. Clearly, the SSNN trained with 40% corrupt input data, produces a larger bias and larger variance when tested against clean test-data, than a SSNN model trained with 100% clean



data. Nonetheless, the inclusion of small amounts (10 - 20%) of corruptness in the training datasets offers a good balance between accuracy and robustness. Testing on structural detector failure As noted before, structural detector failure may require a neural network to learn the complex dynamics of traffic flow in a different way, by “looking” at different detectors to detect congestion and thus delays. In other words: structural detector failure might require an entirely different SSNN model, than a situation with clean or randomly corrupted data. Consequently, we a-priori expect that a pre-processing strategy might prove more useful than a strategy where SSNN’s are trained with various degrees of data corruption. Table 3 shows the performance for a SSNN model trained with clean data, four different SSNN models, trained with 10 – 40% (incidental) corruption, and a SSNN model trained with clean data fed with pre-processed data using spatial interpolation on test sets in which one out of the 13 detectors of the main carriage way is structurally down. Again we see that a simple pre-processing strategy (S3 – spatial interpolation) outperforms the models resulting from strategy S1 on all performance indicators. Again this indicates the robustness of the neural network to the “damage” caused by a simple imputation scheme (spatial interpolation), although structural input failure at some of the significant detectors still causes an increase in bias. This increase is never larger than 6% and significantly less than in any of the “S1 models”. Nonetheless, all “S1 models” still perform substantially better than the model trained with clean data, but no clear picture emerges as to how much corruption in the training datasets leads to the best overall result. If we look at the RMSE cost for failure of detectors connected to so-called “significant” links3, {2,3,5,6,7,8,10,11}, then the SSNN trained with 20% corruption can be considered the best model for strategy S1. A couple of remarks need to be made. First, it is important to note that structural failure of the significant detectors leads to a larger bias (MRE), rather than larger variance (SRE), for all strategies. This is a result from the fact that these detectors are associated with those links where congestion sets in first (bottlenecks), and thus provide crucial information on the progression of congestion and hence the expected travel time. Secondly, the test sets are produced by including input failure in only one of the available original test sets. Finally, a remarkable result is that structural failure of detectors connected to these significant links result in the largest errors in the SSNN trained with clean data, but do not necessarily yield the largest errors in SSNN models trained with more than 10% corruption. We have no clear explanation for this phenomenon. Finally, let us consider structural failure of two detectors on the main carriageway, for which Table 4 presents the total performance on all 78 test-sets. Again we see that the inclusion of small amounts of corruption in the training datasets leads to an improvement of the performance and thus robustness of the model to corrupted input. But, as could be expected, the preprocessing strategy (S3 – spatial interpolation) outperforms the S1 strategies. Overall comparison and discussion In the previous sections we tested two general approaches to increase the robustness of the SSNN travel time prediction model to both incidental and structural input failure. The first involved incorporating corrupted data in the SSNN training procedure, the second pre-processing strategies with naïve non-parameterized procedures such as an exponentially moving average and spatial interpolation. It appears that both approaches do improve the robustness to both incidental and structural input failure considerably, albeit that the pre-processing strategies performed best. This is remarkable since the proposed preprocessing algorithms are very simple naïve non-parameterized procedures. This supports both hypotheses posed in the second section on the robustness of a neural network to the “damage” caused by simple imputation schemes, and the invariant nature of traffic data to imputation schemes (to a degree). There is yet another, more qualitative, argument that favours the pre-processing approach to the training approach. The internal states of SSNN travel time prediction model (Figure 4) are strongly correlated to the actual traffic processes and can be interpreted as metrics representative for the expected traffic conditions on the route of interest (van Lint et al, 2001). Incorporating corrupt data in the training procedure makes these internal states more difficult to interpret, and makes the model less useful for analytical purposes. As stated earlier, the distinction between incidental and structural detector failure may not be as crisp as presented here. The results of the previous section show that a time series approach works best for incidental failure, but is useless for structural failure, where spatial interpolation is the preferable alternative. Thus, in real life a more sophisticated pre-processing strategy is required. In this strategy, we need to keep track of detector failure, classify it on-line as incidental or structural, and consequently apply the appropriate pre-processing

3

For example: detector 10 and 11 are connected to link 10, which is considered the principle bottleneck on our test network, where congestion sets in. Another “significant” link is number 5, to which detectors 5 and 6 are connected. For more details we refer to (2).



algorithm, which could be a time series model, a spatial interpolation model or even more sophisticated (spatiotemporal) models such as Kalman filters, macroscopic traffic models or even neural networks. Finally, one should note that this is a first exploratory study into the applicability of the various strategies to enhance robustness of the SSNN based travel time prediction framework. More extensive experiments with a wider range of test data - preferably real data - are required before we can draw sound conclusions on the robustness and accuracy of the resulting models. Clear insight in how the hypothesized robustness of the SSNN to the “damage” caused by simple non-parameterized imputation schemes is transferable to other locations/networks must be assessed by testing the model in different settings. Since the pre-processing procedures are not dependent on the traffic measurements, the portability of the travel time prediction framework proposed here, depends largely on the transferability and the limitations of the SSNN model, which is, at least in terms of its parameters, location specific. We hypothesize, though, that the SSNN model can be applied on any freeway stretch, given that this stretch equipped with enough detectors on relevant locations, and that the stretch is not too long (in the range of 5 to 15 kilometers). Both these constraints on the portability and applicability of the SSNN model need to be addressed and specifically quantified in future studies. CONCLUSIONS AND FURTHER RESEARCH TOPICS Robustness to missing or faulty input, due to failures in the data collection system, is a key characteristic for any travel time prediction model that is to be applied in a real time environment. We have tested two strands of approaches to deal with different kinds of input failure (incidental, structural) in conjunction with a neural network based travel time prediction model (the SSNN model). The main findings are: 1. Both naïve non-parameterized imputation schemes, and incorporation of corrupt data in the training data of the SSNN, improve the robustness of the travel time prediction model to incidental and structural input-failure considerably. 2. The pre-processing strategies give the best overall results. Although there are clear theoretical shortcomings to simple imputation schemes, their use may be justified in this particular application: the results indicate that the SSNN is robust to the “damage” done by naïve imputation schemes. 3. Nonetheless, in real life more sophisticated pre-processing procedures need to be developed since the distinction between incidental and structural detector failure will not be as crisp as presented here. 4. Inclusion of corruption in the training procedure of the SSNN model also improves robustness, but at the cost of predictive accuracy. Further research should focus real time application of the robust travel time prediction framework proposed here, and enhancing the two components (SSNN model and pre-process layer) to meet real-time requirements.

REFERENCES 1. Berkum, P. van, Mede, P. van der, (1993), The impact of Traffic Information: Dynamics in route and departure time choice, Delft University Press 2. Lint, J.W.C., van, Hoogendoorn, S.P., Zuylen, H.J., van (2002). Freeway travel time prediction with State-Space Neural Networks, Proceedings of the Transportation Research Board Annual Meeting (CD Rom), 14-17 January 2002, Washington D.C., USA 3. Schafer, J.L., 1997, Analysis of Incomplete Multivariate Data, Chapman & Hall, London, UK 4. Gabrys, B., 2002, Neuro-fuzzy approach to processing inputs with missing values in pattern recognition problems, International Journal of Approximate Reasoning, Volume 30, Issue 3, September 2002, Pages 149-179 5. Meert, K., 1996, A Real-Time Recurrent Learning Network Structure for Dealing with Missing Sensor Data, Proceedings of the 1996 IEEE Conference on Neural Networks part 3 pp1600-1605, Washington D.C., USA 6. Jeffrey, S., J., Carter, J., O., Moodie, K., B., Beswick, A., R., 2001, Using spatial interpolation to construct a comprehensive archive of Australian climate data, Environmental Modeling & Software, Volume 16, Issue 4, June 2001, Pages 309-330 7. Faris, P., D., Ghali, W., A., Brant, R., Norris, C., M., Galbraith, P., D., Knudtson, M., L., 2001, Multiple imputation versus data enhancement for dealing with missing data in observational health care outcome analyses, Journal of Clinical Epidemiology, Volume 55, Issue 2, February 2002, Pages 184-191 8. Armitage, W.D., 1994, Enhancing the robustness of a Feedforward Neural Network in the Presence of Missing Data, Proceedings of the 1994 IEEE Conference on Neural Networks part 2 pp836-839, Orlando, Florida, USA 9. Chen, H., Grant-Muller, S., Mussone, L., Montgomery, F., 2001, A Study of Hybrid Neural Network Approaches and the Effects of Missing Data on Traffic Forecasting, Neural Computing and Applications, (2001) vol. 10, pp277-286, Springer-Verlag London.



10. Bishop, C.M., 1995, Neural Networks for pattern recognition, Oxford University Press, United Kingdom 11. Elman, J., 1990, Finding structure in Time, Cognitive Science 14, pp. 179-211 (1990) 12. Foresee, F.D., Hagan, M.T., 1997, Gauss-Newton approximation to Bayesian learning, International Conference on Neural Networks, Vol 3 , pp. 1930 -1935 vol.3 (1997) 13. Vermijs, R.G.M.M., Schuurman, H., (1994), Evaluating capacity of freeway weaving sections and onramps using the microscopic simulation model FOSIM, Proceedings of the second international symposium on highway capacity, Vol 2., pp 651-670



LIST OF FIGURES missing values

space

missing values

measurements are biased and noisy

space

space

time

time

time

(a) Incidental failures

(c) Intrinsic failures

(b) Structural failures

Figure 1: Incidental, Structural and Intrinsic failures in detection system

Travel time prediction framework Pre-processing

Detection System(s)

Travel Time prediction Model

Travel Time Information Service Figure 2: Robust travel time prediction framework

u (mean speed measured at detectors in [m/s])

Example spatial interpolation procedure

actual measurements corrupted measurements repaired measurements cubic spline curve

120

100

80

60

40

20

0

-20 0

1000

2000

3000

4000

5000

6000

7000

8000

x (detector locations in [m])

Figure 3: Example of spatial interpolation pre-processing procedure when 5 out of 13 detectors produce corrupted values. In this case the pre-processing procedure makes substantial errors.



outputlayer Consists of one neuron, which calculates the expected travel time on a route of K Links

y(t+1)

x1(t)

x1(t+1 )

u1(t)

…..…

xi(t+1)

ui(t)

…..…

xK (t+1 )

xi(t)

State Space Neural Network for Travel Time Prediction, based on the NN topology proposed by Elman (1990) Context Layer stores the previous states of the hidden neurons xK (t)

Hidden layer: each hidden neuron represents a link along the route. On a route of K links, there are K hidden neurons

uK(t)

Input: Each input vector u i(t), consists of averaged speeds and flows measured at detector stations connected to link i, during time period [t-1… t]. If a link is connected to an on-ramp (off-ramp), u i(t) also contains the inflow (outflow) measured at the onramp (offramp), during time period [t-1… t]. Thus, the size of u i(t-1) may vary for each hidden neuron.

Figure 4: Fully Connected State-Space Neural Network (SSNN) for freeway travel time prediction, based on a state-space formulation of the travel time prediction problem and the recurrent neural network topology proposed in (11)



LIST OF TABLES Table 1: Combined strategies of pre-processing and training

Strategy

pre-processing

training

S0 S1 S2 S3 S4

none none MA Interpolation MA+Interpolation

100% clean 10-40% incidental 100% clean 100% clean 100% clean

testing 100% clean x x

incidental x x x x x

structural x x x

Table 2: Performance of travel time prediction framework on increasing amount of incidental input-failure a) Mean Relative prediction Error of strategies S0 (top row), S1 (2nd to 5th row), S2 (6 th row), S3 (7th row) and S4 (8th row) to incidental input failure at increasing amounts of corrupted data in the test datasets (total nr records in test data: 5 x 396) % of incidental failure in test-set

pre processing

training with corrupted data

MRE (%)

MRE on ALL test-sets

0%

10%

20%

30%

40%

S0 S1 (10%)

0 -1

26 0

51 4

71 6

81 12

46 4

S1 (20%) S1 (30%) S1 (40%) S2 (ma) S3 (interp) S4 (int/ma)

-3 -3 1 0 0 0

-1 4 1 1 -2 2

4 8 5 0 -3 2

7 20 10 1 -6 2

11 31 11 1 -8 1

4 12 6 1 -4 1

b) Standard deviation of Relative prediction Error of strategies S0 (top row), S1 (2nd to 5th row), S2 (6th row), S3 (7th row) and S4 (8t h row) to incidental input failure at increasing amounts of corrupted data in the test datasets (total nr records in test data: 5 x 396) % of incidental failure in test-set

pre processing


SRE (%) S0 S1 (10%) S1 (20%) S1 (30%) S1 (40%) S2 (ma) S3 (interp) S4 (int/ma)

0% 5 6 7 8 13 5 5 5

10% 27 8 9 10 12 5 7 6

20% 43 12 12 10 12 5 10 7

30% 53 11 13 17 17 6 12 7

40% 59 18 16 22 16 7 12 9

SRE on ALL test-sets 52 13 13 19 15 6 10 7

c) Root of Mean of Squared prediction Errors of strategies S0 (top row), S1 (2nd to 5th row), S2 (6 th row), S3 (7 th row) and S4 (8th row) to incidental input failure at increasing amounts of corrupted data in the test datasets (total nr records in test data: 5 x 396)

0%

10%

20%

30%

40%

RMSE on ALL test-sets

S0

17

115

192

249

280

195

S1 (10%)

22

29

42

44

64

43

S1 (20%) S1 (30%) S1 (40%) S2 (ma)

28 33 53 17

33 46 48 17

39 53 56 18

50 91 67 23

60 124 68 25

44 77 59 20

S3 (interp)

17

30

40

58

62

45

S4 (int/ma)

17

25

29

33

35

28

% of incidental failure in test-set

pre processing


RMSE (seconds)



Table 3: Performance of strategies S0, S1, and S3 to structural input failure at 1 out of 13 detectors on the main carriage way (total nr records in test data: 13 x 396). detector structurally down

12

13

Performance on ALL test-data

-13 -24 17 40 37 64 30 12 6 7 8 20 17 35 20 10 56 124 71 140 132 220 104 44

24 44 62 8 16 28 39 6 81 150 213 33

5 5 26

24 32 123

-1 6 22

1 6 25

23 10 92

21 13 76

-5 9 44

10 7 47

-1 6 22

2 13 47

-2 7 27

1 -16 3 7 7 9 27 82 40

-1 6 24

-3 -10 -20 9 6 8 8 8 25 49 86 41

24 16 87

-4 6 29

6 8 44

-2 7 28

-1 13 51

-2 8 30

3 -11 0 15 9 10 65 49 38

4 11 54

2 9 39

7 9 47

11 8 53

11 10 59

5 8 35

6 10 52

-4 8 33

2 11 46

1 13 53

10 18 90

-6 11 41

5 11 55

-1 9 35

2 -10 -4 13 9 13 53 48 48

13 12 72

13 -19 27 -1 12 11 17 12 68 94 132 48

2 17 69

0 5 17

6 6 33

0 5 17

1 5 17

-1 5 19

4 7 33

1 5 18

-3 6 25

1 6 23

1 S0

MRE (%) SRE (%)

S3 (interp)

S1 (40%)

S1 (30%)

S1 (20%)

S1 (10%)

RMSE (s) MRE (%) SRE (%) RMSE (s) MRE (%) SRE (%) RMSE (s) MRE (%) SRE (%) RMSE (s) MRE (%) SRE (%) RMSE (s) MRE (%) SRE (%) RMSE (s)

2

3

4

5

0 -10 -1 6 6 5 22 49 21

6

7

8

-6 -12 2 5 8 10 35 55 41

-4 7 30

-1 5 19

0 5 18

9

10

11

6 6 34

0 5 17

1 5 17

Table 4: Total Performance on structural detector failure at 2 out of 13 detectors on the main carriageway, indicators calculated on 78 test-sets of 396 records each

S0 S1 (10%) S1 (20%) S1 (30%) S1 (40%) S3

MRE (%) 43 4 0 8 4 2


SRE (%) 46 18 18 16 22 7

RMSE (seconds) 185 64 65 66 83 29
