# Integration of Functional Link Neural Networks into a Parameter Estimation Methodology
## 3 Proposed Dual-Response Model Based on Functional Link NNs  
### Proposed Functional Link NN  
ANNs are comprised of various processing elements known as artificial neurons or nodes, which are interconnected. Each neuron obtains input signals from the previous nodes, aggregates these signals with the related weights, and generates output signals through a transfer function (or activation function). This output signal forms the input signal for other nodes. The multilayer feed-forward back-propagation NN is the most influential model applied to various practical problems. A multilayer feed-forward back-propagation NN model comprises several layers; each layer contains several nodes. The first and last layers inside the network are considered as input and output layers, respectively, as the input and output units in the NN system are involved. Various hidden layers are located between the input and output layers. A multilayer feed-forward back-propagation NN comprises one input layer, one output layer, and various hidden layers between the input and output layer. The overall structure of a multilayer feed-forward back-propagation NN is illustrated in Figure 2.  
The universal approximation theorem for multilayer feed-forward networks was proposed by [PERSON] (2018) and [PERSON] et al. (2018). A multilayer feed-forward NN with a hidden layer can approximate a multidimensional, continuous, and arbitrary nonlinear function with any desired accuracy, as mentioned in [PERSON] (2018) and [PERSON] et al. (2018), based on the theorem stated by [PERSON] et al. (2018) and [PERSON] (2018). In the hidden area, the transfer function is used to figure out the functional formation between the input and output factors. Popular transfer functions used in ANNs include step-like, hard limit, sigmoidal, log sigmoid, hyperbolic tangent sigmoid, linear, radial basis, saturating linear, multivariate, softmax, competitive, symmetric saturating linear, universal, generalized universal, and triangular basis transfer functions ([PERSON] et al., 2018; [PERSON] et al., 2018). In RD, there are two characteristics of the output responses that are of particular interest:  
Figure 2: Multilayer feed-forward NN structure. deviation. Each output performance can be separately analyzed and computed in a single NN structure based on the dual-response estimation framework. Figure 3 illustrates the proposed functional-link-NN-based dual-response estimation approach. As shown in Figure 3, \(x_{1},\ \ x_{2},\ \ldots,\ x_{k}\) denote \(k\) control variables in the input layer. The weighted sum of the \(k\) factors with their corresponding biases \(b_{1},\ \ b_{2},\ \ldots,\ b_{h}\) can represent the input for each hidden neuron. This weighted sum is transformed by the activation function \(x+x^{2}\), also known as the transfer function. The transformed combination is the output of the hidden layer \(y_{hid}\) and refers to the input of one output layer as well. Analogously, the integration of the transformed combination of inputs with their relevant biases can represent the output neuron (\(\hat{g}\) or \(\hat{s}\)). The linear activation function \(x\) can represent the output neuron transfer function. In an \(h\)-hidden-node NN system, \(1,\ \ldots,\ j,\ \ldots,\ h\), are denoted as the hidden layer, and \(w\) and \(b\) represent the weight term and process bias, separately. In particular, the weight connection between the input factor \(x_{i}\) and hidden node \(j\) is written as \(w_{ji}\), while \(w_{j}\) is the weight connection between the hidden node \(j\) and the output. In addition, \(b_{j}^{hid}\) and \(b^{out}\) represent deviations at \(j\) and the output, respectively.