**2. Problem formulation**

We consider a fault detection system designed to monitor the condition of a plant, as shown in **Figure 1**, and the sequence of time-varying observations ordered in time (a time series data). Time is the independent variable, and we assume it to be discrete; thus, time-varying data are a sequence of pairs

½ � ð Þ *x*1, *t*<sup>1</sup> ;ð Þ *x*2, *t*<sup>2</sup> ; ⋯;ð Þ *xM*, *tM* with ð Þ *t*<sup>1</sup> <*t*<sup>2</sup> < ⋯ < *tM* , where each *xi* is a data point in the feature space and *ti* is the time at which *xi* is observed. The data for more than one signal are sequences of time-varying data points, so long as their sampling rates ð Þ *ti* � *ti*�<sup>1</sup> ¼ Δ*t* ¼ *η* are the same.

With this definition, we assume that:

i. The sequences of data within an historical time-varying dataset, taken by the sensor in healthy condition, are measured and are available as a memory data matrix, **X** ∈ *<sup>M</sup>*�*<sup>p</sup>*, whose elements, *xij*, are functions of the scalar parameter time, *t*, where **X** is a *p*-dimensional matrix of signals with *M* observation sequence vectors, and *xij* represents the *i*th observation of the *j*th signal.


On the basis of above descriptions and by using **X**, the objective of the present work is to develop a signal reconstruction model reproducing the plant behavior in normal conditions. Such model receives in input the observed sequence of real-time measurement matrix, **X**<sup>∗</sup> *<sup>q</sup>* , whose *r*th vector is the present measurement, **x**<sup>∗</sup> *qr* , containing the actual observations of *p* signals monitored at the present time, *t*, and produces in output, **x**^*qr* , the reconstruction values of the signals expected in normal condition. Based on this, the actual plant condition at the present time, *t* can, then, be determined by the analysis of residuals: a fault is detected if the variations between the observations and the reconstructions are large enough in, at least, one of the signals in comparison to predefined thresholds.
