# Explanation of Air Quality Data Using Takagi-Sugeno Fuzzy Inference System
## 1 Introduction
### Description of Used Approach  
In this article, we describe the way the emission data, specifically the values of \(PM_{10}\), can be computed based on the known values of traffic data and weather conditions by using fuzzy systems (see [4; 5; 6; 7]). A similar process could be used in many other cases. Despite the fact that nowadays, deep neural networks are used in most similar data-based computations, fuzzy systems could reach comparable results. Moreover, while deep neural networks need a large number of data for computations, fuzzy systems can be used with smaller sets of data.  
Various types of fuzzy systems have been proposed for the task of data analysis. We will mention just one of these systems--the Takagi-Sugeno fuzzy inference system (Takagi-Sugeno FIS). This system was designed to analyse data describable by linear functions in certain parts of the domain of definition, while there is a problem of data approximation in the remaining parts of said domain (see [6]). To work with such data, IF-THEN rules are created. These rules consist of assumptions, in which variables are represented by fuzzy sets, and consequences, in which variables are represented by linear functions. In specific problems, there is a possibility to use constant functions as the consequence functions. The rules can be created by experts from the considered area or can be obtained by analysing a large number of known data.  
Since the research study presented in this work follows up our ongoing research, we used the same dataset as in [8; 9; 10]. In ongoing research, we approximate the values of \(PM_{10}\) based on the known values of traffic data and weather conditions in Prague, Czech Republic. We want to prove the existence of a strong correlation between traffic data and emission data. The results can be applied in the implementation of the Smart City concept in the city of Prague. Identifying the specific inputs that have an impact on obtaining high output values, we can help city officials implement measures to reduce emissions in the city.  
In the previous works, the Takagi-Sugeno fuzzy inference system with linear output was implemented to approximate data values in the selected problem. This system demonstrably reaches one of the best results when compared with other conventionally used approximation methods. In this paper, we present the modification of the Takagi-Sugeno fuzzy inference system with linear output into the Takagi-Sugeno fuzzy inference system with constant output in such a way that the error rate of the approximation inflates minimally. Subsequently, we focus on the overall explainability of the created rule base.  
The entire data analysis process is shown in a flowchart (see Figure 1).  
In the rest of this section, we offer a review of selected works related to the use of fuzzy systems as explainable predictors. The body of the article then consists of six sections. In Section 1, there is a short introduction to the problem and related work. In Section 2, the used data are presented. In Section 3, the basics of fuzzy sets and fuzzy logic are given. In Section 4, the basics of Takagi-Sugeno fuzzy inference systems are given. We also mention the methods of system optimisation while the focus is put on one of these methods--adaptive network-based fuzzy inference system. The obtained results are presented in Section 5 and a discussion in Section 6. In Section 7, the conclusions and future work are mentioned.