Simultaneous-Fault Diagnosis of Automotive Engine Ignition Systems Using Prior Domain Knowledge and Relevance Vector Machine

Algorithm 1

Algorithm of the proposed framework for simultaneous-fault diagnosis of time-dependent ignition patterns.

Given a training dataset TRAIN_F of single-fault patterns only, a validation dataset VALID_F and a test dataset TEST_F of single-

fault and simultaneous-fault patterns (all datasets have been preprocessed by the combination of DK and WPT and PCA, as

presented in Figure 7(a)):

 (i) Train the probabilistic classifier 𝑓_class

  𝑓_class includes 𝑑(𝑑−1)/2 pairwise classifiers 𝐶_𝑖𝑗 as shown in Figure 6(b).

 (ii) ∀ f  ∈  VALID_F, prepare the probability vector 𝝆

   Calculate 𝑓_class (f) = 𝝆 = [𝜌₁,𝜌₂,…,𝜌_𝑑] (Figure 7(c)).

 (iii) For k = 1 to 𝑀// Run a direct search technique, such as GA or PSO, M times

   Produce an initial population 𝜖for the decision threshold 𝜀

   (a) ∀𝜀 ∈ 𝜖, find the classification vector y(f) = y = [𝑦₁,𝑦₂,…,𝑦_𝑑]= [𝜀(𝜌₁),𝜀(𝜌₂),…,𝜀(𝜌_𝑑)] according to (1).

   (b) Calculate the 𝐹-measure 𝐹_me with y(f) and l(f) using (12), that is, find 𝐹_me over VALID_F,

     where l(f) = [𝑙₁,𝑙₂,…,𝑙_𝑑] is the true classification vector for input f provided from VALID_F.

   (c) Produce next generation of 𝜀

   Until convergence or matching stopping criteria, return the best solution ε as the 𝜀_𝑘.

 (iv) Among all 𝜀_𝑘, k = 1 to M, choose the one producing the highest F-measure 𝐹_me as the optimal decision threshold 𝜀_opt.

 (v) Return the trained probabilistic classifier 𝑓_class and the optimized decision threshold 𝜀_opt as the main components of the

  intelligent diagnostic system.

 (vi) The performance of 𝑓_class and 𝜀_opt can be evaluated with TEST_F and 𝐹_me as illustrated in Figure 7(d).