Metric Learning Method Aided Data-Driven Design of Fault Detection Systems

Algorithm 1

Learning Mahalanobis matrix by iterative algorithm.

Input: 𝑥_𝑘: a given data sets of 𝑛 points, 𝑠: set of pairs of data in same

categories, 𝐷: set of pairs of data in different categories, 𝐻₀: a given

matrix, 𝑠: a given upper bound, 𝑏: a given lower bound, 𝛽: the slack

variable

Output: 𝐻: the target Mahalanobis matrix

(1) 𝐻 = 𝐻₀, 𝜆𝑖,𝑗=0

(2) 𝑚𝑖,𝑗=𝑠 when 𝑖,𝑗∈𝑆, 𝑚𝑖,𝑗=𝑏 when 𝑖,𝑗∈𝐷

(3) Repeat until convergence:

 (3.1) Pick a pairs of data 𝑥_𝑖,𝑥_𝑗𝑡∈𝑥_𝑘

 (3.2) 𝑙𝑖,𝑗_𝑡=1 when 𝑖,𝑗_𝑡∈𝑆, 𝑙𝑖,𝑗_𝑡=−1 when 𝑖,𝑗_𝑡∈𝐷

 (3.3) 𝑔_𝑡=𝑥_𝑖−𝑥_𝑗^𝑇𝐻_𝑡𝑥_𝑖−𝑥_𝑗

 (3.4) 𝑘_𝑡=𝑙𝑖,𝑗_𝑡⋅min𝜆𝑖,𝑗_𝑡,𝑙𝑖,𝑗_𝑡21𝑔_𝑡−𝛽𝑚𝑖,𝑗_𝑡

 (3.5) 𝜇_𝑡=𝑘_𝑡1−𝑘_𝑡𝑔_𝑡

 (3.6) 𝑚𝑖,𝑗_𝑡+1=𝛽𝑚𝑖,𝑗_𝑡𝛽+𝑘_𝑡𝑚𝑖,𝑗_𝑡

 (3.7) 𝜆𝑖,𝑗_𝑡+1=𝜆𝑖,𝑗_𝑡−min𝜆𝑖,𝑗_𝑡,𝑙𝑖,𝑗_𝑡21𝑔_𝑡−𝛽𝑚𝑖,𝑗_𝑡

 (3.8) 𝐻_𝑡+1=𝐻_𝑡+𝜇_𝑡𝐻_𝑡𝑥_𝑖−𝑥_𝑗𝑥_𝑖−𝑥_𝑗^𝑇𝐻_𝑡

Return: 𝐻