## 5 Methodology of IPAM Based on FCM and ELM
Also, small variations in weights allow the system to become more stable and noise-resistant as input errors will not be amplified at the output of a linear system with little correlation between input and output weights. Thus, the random ranking of weights, which produces loosely coupled characteristics at the hidden layer, allows for a satisfactory solution and good generalization performance to be achieved. Corresponding to a linear output level, the Generalized Least Squares Approximation [32] approach is used, where such a solution is also linear and very fast to calculate. Thus, the random weighting, which produces weakly correlated features at the hidden layer, allows for a satisfactory solution and a good generalization performance. Respectively, for a linear output layer, the Generalized Least Squares Approximation approach is used, where such a solution is also linear and very fast to calculate. Specifically, in an ELM using SHLFFNN and random representation of the hidden layer neurons, the input data are mapped to a random \(L\)-dimensional space with a discrete set of training \(N\), where \((x_{i},t_{i})\), \(i\in\llbracket 1,N\rrbracket\)\(\mu\epsilon\)\(x_{i}\in R^{d}\) and \(t_{i}\in\ R^{c}\). The output of the network can be represented as follows [29]:  
\[f_{L}(x)=\sum_{i=1}^{L}\beta_{i}h_{i}(x)=h(x)\beta\ i\in\llbracket 1,N\rrbracket \tag{6}\]  
where \(\beta=[\beta_{1},\ldots,\beta_{L}]^{T}\) is the output of the weights matrix between the hidden nodes and the output nodes, \(h(x)=[g_{1}(x),\ldots,g_{L}(x)]\) are the exits of hidden nodes (random hidden attributes) for input x, and \(g_{1}(x)\) is the output of the hidden node \(i\). Based on dataset \(\mathrm{N}[(x_{i},t_{i})]_{i=1}^{N}\), an ELM can solve the learning problem \(H\beta=T\), where \(T=[t_{1},\ldots,t_{N}]^{T}\) are the target tags and \(H\) the output panel of the hidden layer shown below:  
\[H\!\left(\omega_{j},b_{j},x_{i}\right)\!=\!\left[ \begin{array}{ccc}g(\omega _{1}x_{1}+b_{1})&\cdots&g(\omega_{l}x_{1}+b_{l})\\ \vdots&\ddots&\vdots\\ g(\omega_{1}x_{N}+b_{1})&\cdots&g(\omega_{l}x_{N}+b_{l})\end{array} \right]_{N\! \times\!l} \tag{7}\]  
Prior to the training, the input weights table \(\omega\) and the biases \(\beta\) are randomly generated in the space \([-1,1]\), with \(\omega_{j}=\left[\omega_{j1},\ \omega_{j2},\ldots,\omega_{jm}\right]^{T}\) and \(\beta_{j}=\left[\beta_{j1},\ \beta_{j2},\ldots,\beta_{jm}\right]^{T}\). The output weights table of the hidden level \(H\) is calculated by the activation function and the training data based on the following function:  
\[H=g(\omega x+b) \tag{8}\]  
The output weights \(B\) can be calculated from the relation:  
\[\beta=\left(\frac{\mathcal{C}}{C}+H^{T}H\right)^{-1}\!H^{T}X \tag{9}\]  
where \(H=[h_{1},\ldots,h_{N}]\) are the outputs of the hidden layer and \(X=[x_{1},\ldots,x_{N}]\) are the input data. B can also be calculated from the general relation:  
\[\beta=H^{+}T \tag{10}\]  
where \(H^{+}\) is the Moore-Penrose generalized inverse matrix for matrix \(H\). The hidden layer is responsible for transforming the input data into a different representation. The transformation is a two-stage process:Input data enters the hidden layer, using weights and corresponding biases of the input layer. 2. The data is transformed based on a non-linear transformation function. The hidden layer, and in particular, the second stage of input data transformation, is the only point where a non-linear process is used throughout the method, which significantly increases the learning abilities of ELMs. After the transformation, the data in the hidden layer are used to find the weights of the output layer. The construction of hidden-layer neurons in a random manner is an ingenious architectural technique for fast learning, which essentially eliminates the problem of overfitting. At the hidden layer there is no explicit constraint and various transformation functions can be used, such as sigmoidal, tangential, and threshold.