An Advanced Broyden–Fletcher–Goldfarb–Shanno Algorithm for Prediction and Output-Related Fault Monitoring in Case of Outliers

Algorithm 2

ABFGS algorithm.

(1)	Calculate the robust starting value of weight . The residual weight is obtained by and (12). The leverage weight is calculated by (11), and the score vector is replaced by .
(2)	The th rows of and are multiplied by to obtain weighted data matrices and . BFGS regression analysis is performed on the new weighted data matrices and . The updated score matrix is obtained from the regression analysis results. Then, each row of the new matrix is corrected by dividing . The vector of the original problem is calculated by the modified score matrix .
(3)	Update residual by formula . Then, use to correct the weight of .
(4)	Repeat steps 2 and 3 until converges. As long as the norm of the difference between two successive approximations of is less than the specified threshold (e.g., 0.01), the iteration will stop. Go to Step 5.
(5)	The final regression coefficient vector is obtained through the last ABFGS regression step. The final prediction function is . Here, is the new sample point and is the corresponding output prediction. Establish an online output prediction scheme by using the prediction function.