Estimation of Urban Link Travel Time Distribution Using Markov Chains and Bayesian Approaches

Table 1

Gibbs sampling algorithm for a K component GMM combining with BIC.

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Input: N particles (i.e. ) with large weights selected by sampling strategy, K

Output: parameters ,

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Step 1. Initialization

 Determine hyperparameters:

 Set iteration

 Set , draw

 Draw

 Draw

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Step 2. Gibbs sampling

 For r = 1 to R

  Update the mixing coefficients :

  Draw , where, is the effective

  number of particles assigned to component K, and can be calculated by

  For k = 1 to K

   Update the variance :

   Draw , where

   Update the mean :

   Draw , where

  End for

 End for

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Step 3. BIC calculation

 Calculate the BIC for a K component GMM using Eq. (19)

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Note: we first execute the sampling with increasing number of components from to .