PSI - Issue 64

Marco Pirrò et al. / Procedia Structural Integrity 64 (2024) 661–668 Author name / Structural Integrity Procedia 00 (2019) 000–000 3 characteristics ) contained in the input data by compressing the input set; subsequently, the decoder layer uses the hidden characteristics, stored in the hidden layer, to reconstruct the input data from the output nodes. If x n ∈ Â I is the n-th realization of the measurement from a single sensor – with I equal to the number of samples – the encoding operation can be mathematically formalized by an activation function f expressed in: h n = f ( W x n + b 1 ) (1) where h n ∈ Â J is the n-th vector of hidden characteristics (with J < I), W ∈ Â J ´ I is the encoding weight matrix and b 1 ∈ Â J is the bias term for the input layer. Similarly, the decoding operation performs a mapping of the learned patterns h n through another function g, as expressed in: y n = g ( W ' h n + b 2 ) (2) where, y n ∈ Â I represents the n-th reconstructed output, W ' ∈ Â I ´ J is the decoding weight matrix and b 2 ∈ Â I is the bias term for the hidden layer. Typical activation functions f and g are the sigmoid function, the hyperbolic tangent function or the linear function (Giglioni et al., 2022) . The training of the AE model permits to estimate the parameters = ( W, W ' , b 1 , b 2 ) , by minimizing a reconstruction loss function Z( x n , y n ) that penalizes the differences between x n and y n . 663

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Fig. 1. Schematic of an autoencoder network.

3. The adopted anomaly detection technique

The AE-based anomaly detection procedure herein described consists of three main steps: (a) organization of the signals database, (b) training of the network and (c) anomaly detection by data reconstruction. 3.1. Organization of the input database The paper proposes a training procedure of a single AE model based on the simultaneous use of several measurements (ideally collected by all the sensors belonging to the same monitoring system). Consequently, referring to Fig. 1, in which the illustrated network refers to a single channel, x n represents the n-th realization of all the channels, meaning that x n is a M ´ I matrix – with M equal to the number of channels and I equal to the number of samples – that must be consistent throughout the dataset. Furthermore, the signal portions contained in each x n should be long enough to capture the dynamics of the investigated structure. 3.2. Training of the autoencoder As usual in unsupervised anomaly detection, during the training of the model the structure is supposed to be in its healthy state under typical EOV. The time series collected within this period – organized in M ´ I matrices – are