4 Numerical example
4.3 Flow Prediction
4.3 Flow Prediction
After using the specified model to predict the O-D flow, we denote
and the prediction of traditional and the revised model, respectively. Since the O-D flow matrix is unobservable, we can check the appropriateness only through comparing the predicted with the observation.
simple
In the following there are fourteen figures, each of which describes the patterns of the two predicted flows of both of the models. As we can see that the ups and downs of the observations are in general followed pretty well by both predicted series and one series performs better than the other for different cases, but neither of them is better for all the nodes. For example, in Figure 3 for the first origin, the patterns of the predicted series are similar as that of the observations, although the simple model gives closer predicted values. But for the other origins, the more sophisticated model
outperforms the simple one in both the predicted values and the patterns. As to destination nodes, both models show more variation, in some figures the traditional model even shows better patterns like Figure 14 and Figure 16. Note that in the Figure 16, there has an outlier in the observation exceeding 800 vehicles in a five minutes interval, the traditional model faithfully predicts the outlier value and the revised model shows the similar pattern but not the similar value of observation pattern at the price of underestimate at all other time intervals. So it seems that the revised model performs better on the average and the simple model performs better when the outlier happens.
10 30 50 70 90
Time (5 minutes) 100
150 200 250
Flow (vehicles)
Mainstream (96 kilometer)
O1
observation simple updated
Figure 3: Observed and predicted flow entering the
1
th origin.10 30 50 70 90 Time (5 minutes)
20 40 60 80 100
Flow (vehicles)
Jhulin (90 kilometer)
O2
observation simple updated
Figure 4: Observed and predicted flow entering the
2
th origin.10 30 50 70 90
Time (5 minutes) 0
20 40 60 80
Flow (vehicles)
Guansi (79 kilometer)
O3
observation simple updated
Figure 5: Observed and predicted flow entering the 3th origin.
10 30 50 70 90 Time (5 minutes)
20 40 60 80 100
Flow (vehicles)
Longtan (68 kilometer)
O4
observation simple updated
Figure 6: Observed and predicted flow entering the
4
th origin.10 30 50 70 90
Time (5 minutes) 20
40 60 80 100
Flow (vehicles)
Dasi (63 kilometer)
O5
observation simple updated
Figure 7: Observed and predicted flow entering the 5th origin.
10 30 50 70 90 Time (5 minutes)
0 20 40 60 80
Flow (vehicles)
Yingge (54 kilometer)
O6
observation simple updated
Figure 8: Observed and predicted flow entering the 6th origin.
10 30 50 70 90
Time (5 minutes) 0
100 200 300
Flow (vehicles)
Tuchen (42 kilometer)
O7
observation simple updated
Figure 9: Observed and predicted flow entering the 7th origin.
10 30 50 70 90 Time (5 minutes)
0 10 20 30 40
Flow (vehicles)
Jhulin (90 kilometer)
D1
observation simple updated
Figure 10: Observed and predicted flow exiting the
1
th destination.10 30 50 70 90
Time (5 minutes) -10
0 10 20 30 40
Flow (vehicles)
Guansi (79 kilometer)
D2
observation simple updated
Figure 11: Observed and predicted flow exiting the
2
th destination.10 30 50 70 90 Time (5 minutes)
-40 -20 0 20 40
Flow (vehicles)
Longtan (68 kilometer)
D3
observation simple updated
Figure 12: Observed and predicted flow exiting the 3th destination.
10 30 50 70 90
Time (5 minutes) 0
20 40 60
Flow (vehicles)
Dasi (63 kilometer)
D4
observation simple updated
Figure 13: Observed and predicted flow exiting the
4
th destination.10 30 50 70 90 Time (5 minutes)
20 40 60 80 100 120 140
Flow (vehicles)
Yingge (54 kilometer)
D5
observation simple updated
Figure 14: Observed and predicted flow exiting the 5th destination.
10 30 50 70 90
Time (5 minutes) 0
20 40 60 80
Flow (vehicles)
Tuchen (42 kilometer)
D6
observation simple updated
Figure 15: Observed and predicted flow exiting the 6th destination.
10 30 50 70 90 Time (5 minutes)
0 200 400 600 800
Flow (vehicles)
Mainstream (37 kilometer)
D7
observation simple updated
Figure 16: Observed and predicted flow exiting the 7th destination.
Let
(
ijn)
t ij ij ij
ij
T T T T
T =
1,
2, " , " ,
be the flow entering the origin andexiting destination during each time period. We will compare the two patterns
i
thj
th) ,
,
(
1, ,,
n simple ij simple
ij simple
ij
T T
T = "
andT
ij,updated= ( T
ij1,updated, " , T
ijn,updated)
which are the same row of the and . There are totally twenty-eight O-D pairs. Here we choose some more interesting figures meaningful to discuss.
simple
Xˆ Xˆ
updatedMost figures show that both patterns behave similarly as in Figures 17 and 18.
But when the flows are low, less than 20, say, the simple method behaves badly as to give large negative numbers, and the updated method seldom to give unreasonable results of this kind, see Figure 19 and 20. In the first and the last , Figure 21 and Figure 22 show the large difference between the two models. The simple model gives big spikes without any signs in the data, the only reason we can give is that the
O
1D
7nodes for these two figures are the boundaries of the study area and all the vehicles entering or exiting the system were assigned to
O
1 orD
7, respectively.Comparing from the figures of and we find that most figures have
similar pattern for these three series. But from the figures of , the simple pattern have too many negative points. Since the flow is always positive, we suggest that the model considering the travel time is better than the simple model.
j
i
D
O , T
ijT
ij10 30 50 70 90
time(5 minutes) 0
10 20 30
flows(vehicle)
96km~79km
O1D2
updated simple
Figure 17: Predicted flow between the upstream and Gaunsi.
10 30 50 70 90 time(5 minutes)
1 11 21 31
flows(vehicle)
96km~63km
O1D4
updated simple
Figure 18: Predicted flow between the upstream and Dasi.
10 30 50 70 90
time(5 minutes) -15
-10 -5 0 5 10
flows(vehicle)
90km~79km
O2D2
updated simple
Figure 19: Predicted flow between Jhulin and Gaunsi.
10 30 50 70 90 time(5 minutes)
-12 -7 -2 3 8 13
flows(vehicle)
90km~68km
O2D3
updated simple
Figure 20: Predicted flow between Jhulin and Dasi destination.
10 30 50 70 90
time(5 minutes) 0
40 80 120
flows(vehicle)
96km~37km
O1D7
updated simple
Figure 21: Predicted flow between the upstream and downsteram.
10 30 50 70 90 time(5 minutes)
0 20 40 60 80 100
flows(vehicle)
90km~37km
O2D7
updated simple
Figure 22: Predicted flow between Gaunsi and downstream.
5 Conclusion
In this thesis we propose to incorporate the effect of travel time into the Gaussian state space model and developed an algorithm to estimate the unknown transition matrix and forecast the O-D flow matrix simultaneously. By doing so the performance of the model improved in different aspects. In the most figures of flow prediction, we find that the trends of flows have much different behavior in the different time interval. This means that we should assume that the matrix varies with time. In other words, when using this model to estimator O-D flow, the total studying time period should not be too long in order not to let the pattern of O-D flow be unreasonable. In the future work, we could extend matrix to vary with time or depend on some exogenous factors to improve the ability in describing the reality of the model. Non-Gaussian distribution is also an alternative for the distribution of the error terms.
F
F
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