**3.1 Mathematical framework of AABKR**

In AABKR, the validation of the signals at present time, *t* is based on the analysis of the on-line query pattern, **X**<sup>∗</sup> *<sup>q</sup>* , to reconstruct the current vector, **x**<sup>∗</sup> *qr* , at time, *t*. The basic idea behind AABKR is to capture both the spatial and temporal correlations in the time-series data (see **Figure 2**), for effective signal reconstruction in the transient operation of industrial systems. The historical memory matrix **X** is reorganized into **A** ∈ *<sup>N</sup>*�*r*�*<sup>p</sup>* sequences of array of *N* matrices of length *r*, containing the measurement vectors having ð Þ *r* � 1 overlapping between the two

**Figure 2.** *Graphical representation of the bilateral directions for a time-series.*

consecutive time windows, with the *r* sequence vectors in each matrix of **A** array represented as A*<sup>k</sup> <sup>r</sup>* <sup>∈</sup> *<sup>r</sup>*�*p*, where *<sup>k</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>*, and ð Þ *<sup>N</sup>* <sup>¼</sup> *<sup>M</sup>* � *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> .

The AABKR is expressed in such a way that each neighboring value is weighted on its proximity in space and time. Hence, the mathematical framework of the AABKR is summarized as follows [54, 55]:
