# Explanation of Air Quality Data Using Takagi-Sugeno Fuzzy Inference System
## 1 Introduction
### Related Work  
Air quality modelling represents one of the main topics in many studies. The authors usually analyse data from the selected area (a city or even part of a city), trying to determine the dependence between the input variables and the monitored output. Subsequently, they try to predict the values of the output variable, as well as determining the influence of individual input variables on air quality itself. This is usually closely related to the interpretation of the created models. In [11], the authors analyse pollution data from a government source in three metropolitan cities in India (Bangalore, Delhi and Mumbai). In the study, Fuzzy type-1 and type-2 systems are used to model and predict the level of carbon monoxide. In [12], the air quality index of Kampala, a city in East Africa, is used to build a prediction model based on the fuzzy logic inference system. This system is used for the determination of the air quality in Kampala city, according to the air pollutant. The result of the study is that fuzzy logic algorithms can determine the air quality index and can therefore be used to predict and estimate the air quality index in real time, based on the given air pollutant concentrations. In [13], three approaches, i.e., hybrid data-driven artificial neural networks, the so-called nonlinear autoregressive method with external input and neural networks and adaptive neuro-fuzzy inference approaches, were used for estimating the air quality in an urban area. The findings show notable performance of used models for high-dimensional data assessment. Traditional time-series forecasting models assume a linear relationship between variables, while there are nonlinear and complex components in air pollution modelling. The authors of study [14] aimed to address to this limitation by improving the accuracy of the daily prediction of pollutants by using adaptive neuro-fuzzy inference system modelling. A nonlinear multivariate regression model was developed and experimentally refined to obtain the lowest error possible. Data on pollutants were collected in the city of Tehran, Iran. According to the performance indicators of the created models, the adaptive neuro-fuzzy inference system was more accurate in predicting time-series data than the regression models. Of course, even deep neural networks (DNNs) are used to model air quality. However, the \"black box\" nature of these models often limits their interpretability, which is crucial to informed decision making. Therefore, the effort to propose deep neural networks along  
Figure 1: Flowchart of data analysis (T-S FIS denotes Takagi–Sugeno fuzzy inference system). with some explanatory model becomes more relevant. For example, in [15], the authors introduce a temporal selection layer technique within deep learning models for time-series forecasting to tackle this issue. This technique not only improves prediction accuracy by embedding feature selection directly into the neural network but also enhances the interpretability of the created model and reduces computational costs. The application of explainability techniques to evaluate the impact of weather and time-related factors on air pollution helped assess feature importance. The results show that the mentioned approach improved both prediction accuracy and model interpretability, leading to more effective pollution management strategies. In general, many articles are devoted to explaining DNN models. Let us mention an article [16] where the authors state that systems implemented by using deep neural networks achieve better results than linear models. However, when using nonlinear systems, analysts may need help to understand the outcome of the decision-making process. Therefore, they must have an architecture that allows them to interpret the findings. In the mentioned article, a post hoc explanation model was developed in parallel with the black-box classifier to provide explainable classifications consistent with the black-box classifier. These explanatory classifiers contained fuzzy logical functions.