# Data Fusion-Based Structural Damage Identification Approach Integrating Fractal and RCPN
## 2 Counter-Propagation Network (CPN)  
CPN is developed on the basis of the Kohonen feature mapping network and Grossberg basic competitive network, and is a combination of both. Recently, CPN has been widely used in data compression, statistical analysis and pattern recognition. Because CPN has been described in great detail in relevant literature [35; 36], only a brief introduction about how CPN works was given in this paper. The structure of CPN network model can be shown in Figure 1. Obviously, the CPN network consists of three layers: input layer, competitive layer and output layer. Here, the input and competitive layers constitute self-organising maps (SOMs) networks, while the competitive and output layers constitute a competitive networks model. CPN as a whole is a teacher-type network, but a SOM network is a typical teacher-less network. Therefore, the CPN network model combines the advantages of two kinds of network models, and can give play to the strengths of each network model. It is an organic combination of these two types of network. Assuming there are \(N\) neurons, the input mode for \(P\) consecutive values is \(A_{k}=(a_{1}^{k},a_{2}^{k},\dots a_{N}^{k})\); there are \(Q\) neurons in the competition layer, and the corresponding binary output vector is \(B_{k}=(b_{1}^{k},b_{2}^{k},\dots b_{N}^{k})\); There are \(M\) neurons in the output layer, and the output of its continuous value is \(C_{k}^{f}=(c_{1}^{k},c_{2}^{k},\dots c_{M}^{k})\), the target output vector is \(C_{k}=(c_{1}^{k},c_{2}^{k},\dots c_{M}^{k})\), where, \(k=1,2,\dots,p\). The connection weight vector from the input layer to the competitive layer is \(W_{j}=(w_{j1},w_{j2},\dots w_{jN})\), (\(j=1,2,\dots,Q\)); The connection weight vector from the competitive layer to the output layer is, \((v_{i1},v_{i2},\dots,v_{iQ})\) where, (\(i=1,2,\dots,M\)); The learning and working rules of the CPN model can be summarized as follows:  
1. Initialize. 2. The \(K\)th input mode \(A_{k}\) is supplied to the input layer of the network, the connection weight vectors \(W_{j}\) and \(V_{l}\) are assigned in the interval [0,1], and all input modes \(a_{i}^{k}\) are normalized: \[a_{i}^{k}=\frac{a_{i}^{k}}{\|A_{k}\|},\|A_{k}\|=\sqrt{\sum_{i=1}^{N}{(a_{i}^{ k})}^{2}}\,\ i=1,2,\dots,N\] (1) At the same time, each neuron in the competition layer is assigned a variable \(t\) (the initial value of \(t\) is set to 0), which is used to record the number of wins of this neuron, and the maximum number of wins is set as \(T\), and the error limit is set as \(e\). 3. The connection weight vector \(W_{j}\) is normalized as follows: \[w_{ji}=\frac{w_{ji}}{\|w_{ji}\|},\,\|w_{ji}\|=\sqrt{\sum_{i=1}^{N}w_{ji}^{2}} \ \ \ \ i=1,2,\dots,N\] (2)
4. The input activation value of each neuron in the competitive layer is obtained, namely the sum of weighted input: \[s_{j}=\sum_{i=1}^{N}a_{i}^{k}w_{ji}\ \ j=1,2,\dots,Q\] (3)
5. Find the vector \(W_{g}\), which is closest to \(A_{k}\) in the connection weight vector \(W_{j}\): \[W_{g}=\max_{j=1,2,\dots,Q}\sum_{i=1}^{N}a_{j}^{k}w_{ji}=\max_{j=1,2,\dots,Q}s_ {j}\] (4) In addition, the output of neuron \(g\) is set to 1, and the output of other competing layer neurons is set to 0: \[b_{j}=\left\{ \begin{array}{ll}1&\ \ \ j=g\\ 0&\ \ \ j\
eq g\end{array} \right.\] (5)  
Figure 1: CPN model structure. (5) Adjust the connection weight vector \(W_{g}\): \[w_{gi}(t+1)=w_{gi}(t)+\alpha(\alpha_{i}^{k}-w_{gi}(t))\quad i=1,2,\ldots,N\] (6) where, \(\alpha\) is the learning rate, \(0<\zeta<1\). (6) The connection weight vector \(W_{g}\) is re-normalized as above, step (3). (7) Adjusting the connection weight vector \(V_{l}\) from the winning neuron in the competition layer to the neuron in the output layer: \[v_{li}(t+1)=v_{li}(t)+\beta b_{j}(c_{j}-c_{l}^{\prime})\quad l=1,2,\ldots,M, \quad j=1,2,\ldots,Q\] (7) where, \(\beta\) is learning rate and \(0<\beta<1\).