Application of Wavelet in Linear Prediction of Traffic Volume

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Application of Wavelet in Linear Prediction of
Traffic Volume
Ping Yi, Li Sheng University of
Akron
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Introduction

• Short-Term Prediction of traffic volume is one of
  the most important components in traffic control.

• Random fluctuations in Traffic Volume affect the
  accuracy of prediction, but are very difficult to
  address.

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• Current methods to predict traffic volume
• Empirical-based methodsstandard statistical
  methodology, which includes the non-linear
  regression, Box-Jenkins type ARIMA, neural
  network and PID control.
• Traffic process-based physical modeling of
  vehicle passage variables on the supply side and
  the behavioral modeling of the trip and OD flows
  for the demand side.
• Studies showed that there is a large degree of
  variations between the predicted data and the
  field measurements. Further research is needed to
  improve the accuracy of prediction.

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Objective

• Find out the inherent features of traffic volume
  fluctuation by decomposing the original traffic
  volume data.
• Take advantage of the features to improve the
  accuracy of prediction.

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Linear Predictor

• One of the Empirical-Based methods.
• An optimal filter applied to the random process.

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Structure of Linear Predictor
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Model of Linear Predictor
where is the predicted data from the
preceding data. defines the M-dimensional
space spanned by the previous samples,
, ,.
--coefficients for this filter
--order of the filter
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Traffic Volume Data Characteristics

• Random Fluctuation
• the Monte Carlo simulation method is applied in
  this study.

• Components
• it can be considered as a general waveform, which
  includes sets of sub-components.

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Fundamental of Wavelet Transform

• Mathematical definition of wavelet
• wavelet

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• The vector space representation
• represents a sequence of real numbers

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Discrete Wavelet Transform(DWT)

• Data series sk,
• Orthonormal wavelet bases
• Scaling function
• Wavelet function
• Coefficients in wavelet domain
• Trend component (low frequency filter)
  
• Fluctuation component (high frequency filter)

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Discrete Wavelet Transform(DWT)

• Any data series can be decomposed into a set of
  sub-series in wavelet domain and reconstructed by
  these sub-series in time domain through the
  wavelet bases.

sd1d2 d3 dkak
where d1 is the first level fluctuation
sub-signal, dk is the kth level fluctuation
sub-signal and ak the kth level approximation
signal.
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Discrete Wavelet Transform(DWT)
akak1 ak2 akN/2k
dkdk1 dk2 dkN/2k
N total number of data points
k number of level for decomposition
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Simulation Data Generation

• The Monte Carlo simulation is applied here
• Only consider the situation in urban network.

Stop Bar
Detector
Figure 2 Detector Configuration
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Simulation Data Generation(cont)
where q represents the flow rate, t represents
the inter-arrival time between vehicles, h is the
headway, and P defines the probability of the
occurrence of a certain headway value.
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Architecture for Predictor Incorporated with
Wavelet Method
Linear Predictor
ak
Linear Predictor
dk
Wavelet Decomposition
Wavelet Reconstruction
d1
Linear Predictor
Figure 3
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Characteristics of traffic volume

• Fast fluctuations exist.

Figure 4
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Methods

• The results are based on the WLSE algorithm for
  the 5-order linear prediction.

• The DB2 wavelet is applied to the simulated data

• 3 level decomposition is performed

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DB2 Wavelet
Figure 5
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DB2 3-level Decomposition of Original Data
Figure 6
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Reconstructed data after Linear Predictor
Figure 7
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Comparison of Original Data and Predicted data
w/o wavelet
Figure 8
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Comparison of Original Data and Predicted data w/
wavelet
Figure 9
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Results

• Mean Absolute Percentage Error (MAPE)
• Mean of Error (ME)
• Standard Deviation of Error(SDE)
• x(i) simulated traffic volume for ith time
  interval
• s(i) predicted traffic volume for ith time
  interval
• N sample size

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Comparative Statistics Before and After Wavelet
Transform
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Results (cont)

• Linear Prediction with wavelet transform compares
  much better to the original data from these
  figures.
• The MAPE describes the average deviation from the
  original data, whereas a larger MSPE index
  specifies that there are more larger-sized
  differences in the data than in those of the
  counterpart for comparison. The average
  improvement for this study is 56.8 after
  wavelet.
• The ME describes the mean of difference between
  original data and predicted data. The bigger it
  is, the bigger the difference. The average
  improvement is 56.6.
• The SDE describes the concentration of the
  error. The average improvement is 52.6.

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Conclusion

• Using the wavelet incorporated linear predictor,
  the predicted traffic volume compares
  overwhelmingly better with the original data. The
  proposed approach is feasible.
• Ability of wavelet decomposition in data feature
  extraction is fully demonstrated.
• Further research is needed to apply this
  algorithm to different situation to test the
  reliability and tolerance of this algorithm.

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Thank you!
Any Question?