# An e-Learning Toolbox Based on Rule-Based Fuzzy Approaches
## 2 Materials and Methods  
The data mining methodologies designed and used in this research are mainly based on fuzzy inductive reasoning (FIR) methodology. Therefore, a brief description of this approach and its main developments are presented in this section. FIR is a modeling and simulation methodology derived from [PERSON]'s General Systems Problem Solver (GSPS) [31]. It is a tool for general system analysis that allows studying the conceptual modes of behavior of dynamical systems. FIR models are non-parametric models based on fuzzy logic. Unlike other widely used soft computing techniques, such as neural networks (NN), which perform a network training process from a data set in order to obtain a model, FIR performs a model synthesis process from the data set, which is less costly in time. Moreover, in this process FIR extracts the behavioral patterns from the data and represents them in the form of fuzzy rules. Therefore, the rules represent the knowledge associated with the data. In contrast, NN use the data to train a model by fitting a set of weights so that it approximates a function. Once the model is obtained, the data is discarded and not used in the prediction process. This implies that a NN will always predict something while a FIR model will only predict if it is possible to validate its prediction on the available data set. This ensures that the model never predicts if the available data are not sufficient to support the prediction. FIR methodology consists of four main processes, namely: fuzzification, qualitative model identification, fuzzy forecast, and defuzzification. The conversion of the quantitative data available from the system under study to qualitative data is performed within the fuzzification module. The fuzzification function converts a single quantitative value into a qualitative triple consisting of class, membership, and side values for each quantitative element. The qualitative model identification process is responsible to obtain the best model that represents the behavior of the system. To this end, it is necessary to find the temporal and causal relationships between the input and output variables. An example of input-output relationship is presented in Equation (1). In this multivariate time series example, we have two input variables, i.e., _Inp1_ and _Inp2_, and an output variable, i.e., _Out_. Previous values in time (\(t\)-\(\propto\)\(\delta t\)), of the input and output variables may be relevant to predict the output value at current time (\(t\)). \[\left[ \begin{array}{ccc}\textit{Inp1}&\textit{Inp2}&\textit{Out}\\ t-2\delta t&0&t-2\delta t\\ 0&t-\delta t&0\\ 0&0&t\end{array} \right] \tag{1}\]  
Equation (1) shows the causal and temporal relation between the inputs and the output, i.e., it defines the model structure, which, in the FIR methodology, is called mask. In the example, the value of the output variable at time \(t\), is obtained taking into account the values of the _Inp1_ and _Out_ variables two time steps back into the past and the value of the _Inp2_ variable one time step back into the past. A FIR model is composed of a mask and a pattern rule base (set of rules). The mask (see example in Equation (1)), represents the structure of the model, i.e., it selects those variables that best capture the behavior of the system. The optimal mask is obtained performing a feature selection process by means of the Shannon entropy measure, using either a mechanism of exhaustive search of exponential complexity, or by a suboptimal search strategy of polynomial complexity. The pattern rule base is the set of rules obtained from the training data when the mask is used to extract the set of patterns that describe the behavior of the system. The next process is the prediction process, which, in the FIR methodology, is called fuzzy forecast. This inference process is based on the k-nearest neighbor technique.