A Batch Rival Penalized Expectation-Maximization Algorithm for Gaussian Mixture Clustering with Automatic Model Selection

Algorithm 2

Batch RPEM algorithm.

Initialization: Given a specific 𝑘(𝑘≥𝑘^∗,𝑘^∗isthetruenumberofclusters),

        initialize the parameter Θ.

Step  1: Given Θ^(𝑛), we compute ℎ(𝑗∣𝐱_𝑡,Θ^(𝑛))’s and 𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))’s for

   all 𝐱_𝑡’s via (2) and (7), respectively.

Step  2: Fixing ℎ(𝑗∣𝐱_𝑡,Θ^(𝑛))’s and 𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))’s, we update Θ by

𝛼_𝑗^(𝑛+1)=𝜙_𝑗^(𝑛)/∑^𝑘_𝑗=1𝜙_𝑗^(𝑛),

𝝁_𝑗^(𝑛+1)=(1/𝜙_𝑗^(𝑛))∑^𝑁_𝑡=1𝐱_𝑡𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛)),

𝐂_𝑗^(𝑛+1)=(1/𝜙_𝑗^(𝑛))∑^𝑁_𝑡=1𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))(𝐱_𝑡−𝝁_𝑗^(𝑛))(𝐱_𝑡−𝝁_𝑗^(𝑛))^𝑇,

      where 𝜙_𝑗^(𝑛)=∑^𝑁_𝑡=1𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))

Step  3: Let 𝑛=𝑛+1, and go to Step  1 for the next iteration until Θ

   is converged.