A Partial Robust M-Regression-Based Prediction and Fault Detection Method

Algorithm 1

SIMPLS algorithm.

S1: Normalize 𝑋 and 𝑦 into zero mean and unit variance.

S2: Set 𝐴₀=𝑋^𝑇𝑦, 𝑀₀=𝑋^𝑇𝑋, 𝐶₀=𝐼, for 𝑖=1,…,ℎ.

 2.1 compute 𝑞_𝑖, which is the dominant eigenvector of 𝐴^𝑇_𝑖𝐴_𝑖;

 2.2 𝑤_𝑖=𝐴_𝑖𝑞_𝑖,𝜏_𝑖=𝑤^𝑇_𝑖𝑀_𝑖𝑤_𝑖,𝑤_𝑖=𝑤_𝑖/√𝜏_𝑖;

 2.3 𝑝_𝑖=𝑀_𝑖𝑤_𝑖;

 2.4 𝑞_𝑖=𝐴^𝑇_𝑖𝑤_𝑖;

 2.5 𝑣_𝑖=𝐶_𝑖𝑝_𝑖,𝑣_𝑖=𝑣_𝑖/𝑣_𝑖;

 2.6 𝐶_𝑖+1=𝐶_𝑖−𝑣_𝑖𝑣^𝑇_𝑖;

 2.7 𝑀_𝑖+1=𝑀_𝑖−𝑝_𝑖𝑝^𝑇_𝑖;

 2.8 𝐴_𝑖+1=𝐶_𝑖𝐴_𝑖.

S3: 𝑃=𝑝₁⋯𝑝_ℎ, 𝑊=𝑤₁⋯𝑤_ℎ,

 𝑞 = 𝑞₁⋯𝑞_ℎ, 𝑇=𝑋𝑊, 𝛽=𝑊𝑞^𝑇