# Forecasting Economy-Related Data Utilizing Weight-Constrained Recurrent Neural Networks
## 1 Introduction  
The rapid advances in digital technologies, as well as the vigorous development of the Internet and the significant storage capabilities of electronic media have enabled economic research centers to accumulate and store large repositories of data. This enormous amount of valuable data yields information of various economic activities such as credit history of bank customers, stock market prices and movement, companies and business funds' sales records, and other statistics. It is a fundamental need for business, companies, and banks to extract useful knowledge from these data. This knowledge constitutes a key element for better decision-making in the increasing market volatility and competition. Economic data mining constitutes an essential process where intelligent methods are applied to extract data patterns from economic databases. This research area has rapidly grown and gained popularity in the modern economic era due to its potential to assist managerial decision-making. During the last two decades, researchers and financial managers have begun to analyze economic data utilizing machine learning and data mining techniques for supporting hard policy decisions (see [1; 2; 3; 4] and the references therein). As a result, the area of economic analysis has been dramatically changed from a rather qualitative science to a more quantitative science, which is also based on knowledge extraction from databases. Nevertheless, economic data are usually imbalanced and are usually characterized by complex dimensionality and enormous noise, hindering their analysis and modeling. Therefore, the process of leveraging these data constitutes an attractive and challenging task for many experts, which often requires huge efforts. Artificial Neural Networks (ANN) have been established as some of the most dominant machine learning algorithms for extracting useful knowledge; thus their value has been demonstrated across an impressive spectrum of applications [5; 6; 7; 8]. Due to their excellent capability of self-learning and self-adapting, they are very appealing to deal with economic and financial problems withpoorly-defined system models, noisy data, and strong presence of nonlinear effects. In the literature, several neural network architectures have been proposed [9]. Recurrent Neural Networks (RNNs) constitute a class of neural networks that are well known for their power to memorize time dependencies and model nonlinear systems.In contrast to the classical feed-forward neural networks in which all the inputs and outputs are independent of each other, RNNs allow previous outputs to be used as inputs while having hidden states. Their main advantages are their ability to deal with time-varying input and output through their own natural temporal operation [10; 11; 12]. Mathematically, the problem of efficiently training an RNN can be formulated as the minimization of an error function \(E(w)\), which depends on the connection weights \(w\) of the network. Recently, [PERSON] [13] proposed a new approach for improving the generalization ability of neural networks, by applying additional conditions on the weights in the form of bound constraints, during the training process. The motivation behind this approach focuses on defining the weights of the trained network in a more uniform way in order for all inputs and neurons of the network to be efficiently explored and exploited. Therefore, the problem of training a neural network is reformulated as a constrained optimization problem, namely:  
\[\min\{E(w)\,:\,w\in\mathcal{B}\}, \tag{1}\]  
with:  
\[\mathcal{B}=\{w\in\mathbb{R}^{n}\,:\,l\leq w\leq u\}, \tag{2}\]  
where \(l\in\mathbb{R}^{n}\) and \(u\in\mathbb{R}^{n}\) denote the lower and upper bounds on the weights, respectively. Moreover, to evaluate the efficacy and the efficiency of this approach, [PERSON] proposed a new weight-constrained neural network training algorithm.