Industry 4.1. Группа авторов
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Название: Industry 4.1
Автор: Группа авторов
Издательство: John Wiley & Sons Limited
Жанр: Техническая литература
isbn: 9781119739913
isbn:
In this section, WPT serves as the major time‐frequency analysis method to extract useful SFs for various machinery applications. WPT is a generalization of DWT to provide a richer information and it can be implemented by DWT‐based MRA as introduced in Section 2.3.2.2.
As illustrated in Figure 2.16, although DWT provides flexible time‐frequency resolution, it suffers from a relatively low resolution in the high‐frequency region since only the approximation coefficients
Figure 2.21 illustrates a WPT‐based fully binary decomposition tree. In WPT, the decomposition occurs in both approximation and detail coefficients. Then the same signal as illustrated in Figure 2.16 can be successively decomposed into different levels using a series low‐pass g[k] (scaling function) and high‐pass h[k] (wavelet function) filters that divide spectrums into one low‐frequency band and one high‐frequency band, which can be represented by approximation
Figure 2.21 WPT decomposition binary tree.
Note that, even detail coefficients in the high‐frequency region can be decomposed into higher level with a better resolution. Finally, a three‐level WPT produces a total of eight frequency sub‐bands in the third level, with each frequency sub‐band covering one‐eighth of the signal frequency spectrum.
Thus, for a discrete signal with length N, and given WPT coefficients
where
u uth wavelet packet node at level L, u= 1, 2, …, L;
v subband length for each wavelet packet node at level L, v = N/2L.
The signal’s energy distribution contained in a specific frequency band is calculated based on all cL[n] in each wavelet packet node using (2.15) and can be used as a SF [16], which provides more useful information than directly using cL[n].
In this way, the WPT technique precisely localizes information behind the non‐stationary signals in both time and frequency domains and thus it is widely applied to mechanical fault diagnosis.
2.3.3.4 Autoencoder
Recently, AEN becomes an important and popular technique to efficiently reduce the dimensionality and generate the abstract of large volumes of data [11, 12]. AEN is an unsupervised backpropagation neural‐network consisting of three fully‐connected layers of encoder (input), code (middle), and decoder (output).
The encoder layer encodes and compresses the data to the code layer, and then decoder layer reconstructs the compressed internal representation of input data from the code layer into output data as closer to the original input as possible. As depicted in Figure 2.22, the architecture of the encoder, code, and decoder can be designed to constitute at least one layer each.
Figure 2.22 Architecture of the AEN.
Let x be one variable of the input set, then the mathematical relationships between layers can be defined as (2.16) and (2.17), and its output
where
h compressed code of the middle layer;
output reconstructed from c in the middle layer;
fEN encoder layer;
fDE decoder layer;
fa activation function;
WEN network weight for node in the encoder;
WDE network weight for node in the decoder;
bEN bias for node in the encoder layer;
bDE bias for node in the decoder layer.
The number of input and output nodes depends on the size of raw data, while the number of nodes in the code layer is a hyperparameter that varies according to the AEN architecture and input data format as other hyperparameters do.
All weights and biases are usually initialized randomly, and then the learning procedure starts to iteratively update weights through back‐propagation algorithm, which minimizes the reconstruction errors between x and
Instead of adopting the entire AEN, the compressed code h is widely СКАЧАТЬ