# An Adaptive Neuro-Fuzzy Propagation Model for LoRaWAN
## 3 Proposed Model Description
### Adaptive Neuro-Fuzzy Inference System for Propagation Estimation  
Fuzzy systems are universal approximators of nonlinear dynamic systems [26; 27]. The idea of fuzzy sets, fuzzy logics and consequently fuzzy inference systems was first proposed by [PERSON] [28]. As stated by [PERSON], fuzzy systems \"provide an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis\" [29]. The humanistic nature of the fuzzy systems allows us to define a complex system with fuzzy/linguistic variables using a human-like reasoning instead of using conventional mathematical tools or precise quantitative analysis. Fuzzy systems provide some degree of resistance to handle vague, ambiguous, imprecise, noisy, missing and uncertain information [30; 31; 32; 33]. This should provide the level of flexibility required to deal with data that is hard or rather impossible to accurately infuse into the model, such as the \(\phi\) and \(w_{s}\). This resistance also relaxes the inevitable inaccuracies in data collection. This is mainly due to the fuzzification of continuous variables. The fuzzification process transforms the crisp value of the inputs (\(x\)) to degrees of membership \(\mu(x)\) using a membership function (\(\mu\)). Next, these membership functions \(\mu\) are tuned using the gradient-descent algorithm to optimize the output. Changes in the \(\mu\) are therefore affecting the degrees of membership \(\mu(x)\) of the inputs (\(x\)).  
The proposed ANFIS architecture comprises first-order Tagaki-Sugeno (T-S)-type fuzzy systems [34], where the output membership functions are first-order polynomials. Therefore, a hybrid training allows a linear least-squares estimation to be used for the identification of the consequent parameters, and a gradient descent optimization is used to identify the premise parameters [30, 35]. Compared to most neural networks, ANFIS also has fewer parameters, many of which can be tuned with linear least-squares. These features give ANFIS the advantage of fast training and computational speed; furthermore, since there are fewer tunable parameters, the pitfall of overfitting the data would be avoided [36].  
The T-S fuzzy implication (if-then rule) is analogous to that of defining a nonlinear input-output mapping. The process can be interpreted as the decomposition of a system into a finite number of subsystems and then approximating each subsystem. The output of the T-S is determined by the aggregation of the implications. Considering a number of implications \(R^{i}\), with antecedents (premise) \(A^{i}\) and consequences \(y^{i}\), implication \(i\)th (\(i=1,\ldots,n\)) is of the format of Equation (1),  
\[\begin{split} R^{i}:\text{if }A^{i}\text{ then }y^{i}\\ A^{i}:\text{ If }x_{1}\text{ is }\mu_{1}\text{ and }\ldots\text{ and }x_{k} \text{ is }\mu_{k}{\equiv}\mu_{1}(x_{1})\Lambda\ldots\Lambda\mu_{k}(x_{k})\\ y^{i}=p_{0}+x_{1}p_{1}+\underset{}{\rightarrow}+x_{k}p_{k}\\ \overline{A}^{i}=\frac{A^{i}}{\sum\,A^{i}},\end{split} \tag{1}\]  
where \(x\) is the input vector (premise variable), \(\mu_{k}\) contains the membership functions of the \(k\)th input, \(p_{k}\) is the consequence parameters vector, and \(\overline{A}^{i}\) is the normalized firing strength, or truth value, of the implication \(R^{i}\).