Traffic Estimation for Large Urban Road Network with High Missing Data Ratio

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. 2019 Jun 24;19(12):2813. doi: 10.3390/s19122813

Algorithm 1 Particle Filter for Traffic State Estimation with Kriging Estimated Measurements [24]

Road network approximation

Use compressed sensing to select m most significant locations out of the n segments to be used for the measurement update step as defined in Section 5.3.
Initialisation

At $k = 0$ ; define all boundary conditions: number of samples, weight of samples as below,
For $l = 1, \dots N_{p}$ , $N_{p}$ number of particles;
- generate $N_{p}$ samples ${x_{0}^{(l)}}$ from the initial distribution $p (x_{0})$
- initialise the particle weights $w_{0}^{(l)} = \frac{1}{N_{p}}$ .
End for
Start the iteration for $k = 1, 2, \dots$
- (a)
  Prediction stage
  
  For $l = 1, . . ., N_{p}$ ,
  
  sample $x_{k}^{(l)} \sim p (x_{k} | x_{k - 1}^{(l)})$ according to SCM model equations
  
  End for
- (b)
  Measurement Update:
  
  This step is performed when the sampling time $t_{s}$ equals the iteration count $t_{k}$ as defined in Section 3.2
  
  i. Estimate missing measurements in the m most significant locations with Kriging using Equations (28) and (33)
  
  ii. Compute the likelihoods
  
  Based on Equation (6) compute the likelihood, $p (z_{s} | x_{s}^{(l)})$ of the particles using Equations (25)–(27)
  
  iii. Update the weights of the particles using the likelihood $p (z_{s} | x_{s}^{(i)})$ calculated from Equation (6)
  
  For $l = 1, . . ., N_{p}$
  
  $ω_{s}^{(l)} = ω_{s - 1}^{(l)} p (z_{s} | x_{s}^{(l)})$
  
  End For
  
  iv. Normalise the weights: ${\hat{ω}}_{s}^{(l)} = \frac{ω_{s}^{(l)}}{\sum_{l = 1}^{N_{p}} ω_{s}^{(l)}}$ .
- (c)
  Update the predicted states (Output): ${\hat{x}}_{s} = \sum_{l = 1}^{N_{p}} {\hat{ω}}_{s}^{(l)} x_{s}^{(l)}$
- (d)
  Re-sample the weights (Selection) only when $t_{k}$ = $t_{s}$