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

Algorithm 2

Main steps of PRM.

S1: Calculate 𝜓^𝑟_𝑖 using (7) with initial value 𝑟_𝑖=𝑦_𝑖−med(𝑦_𝑗), calculate 𝜓^𝑥_𝑖 using (6) with 𝑡_𝑖 replaced by 𝑥_𝑖,

 and then compute 𝜓 using (11);

S2: Multiply each row of 𝑋^(𝑙) and 𝑦^(𝑙) by √𝜓, then perform

 PLS regression (see Algorithm 1) on the new PLS model. Divide each row of 𝑇 by √𝜓;

S3: Calculate 𝑟_𝑖 using (8), and 𝜓 using (6), (7), (11);

S4: If the relative difference in norm between two consecutive approximations of the regression coefficients is smaller than 10⁻³

 then continue to the next step, else go back to S2;

S5: Get 𝑇, 𝑃, 𝑞, 𝛽 in the last step of PLS at the final iteration.