Explanation of Air Quality Data Using Takagi–Sugeno Fuzzy Inference System

Abstract

The explainability of system behaviour is one of the most important concepts of modern data science. If a system is described by using rules that are clearly readable and understandable, then it is possible to model various problems arising from real life. In this paper, we present a way to create the so-called IF-THEN rules for urban air quality modelling by using the Takagi–Sugeno fuzzy inference system. The presented research study builds on previous work where such a problem was modelled by using a Takagi–Sugeno fuzzy inference system with linear membership functions. Such functions are difficult for the average person to interpret. Therefore, we replaced the output linear functions with constant functions and subsequently optimised the system to achieve the lowest approximation error. From the point of view of data analysis, this approach allows us to obtain a system with a comparatively smaller approximation error. From the point of view of model explainability, we obtain a rule base that describes the influence of individual input variables on the overall output in human terms. Finally, based on the obtained rules, we can evaluate the impact of traffic data and weather conditions on the selected air pollution parameter.

1. Introduction

As stated in Directive (EU) 2024/2881 of the European Parliament and of the Council on ambient air quality and cleaner air for Europe (see [1]), scientific evidence shows that sulphur dioxide, nitrogen dioxide and oxides of nitrogen, particulate matter ($P{M}_{10}$ and $P{M}_{2.5}$), benzene, carbon monoxide, arsenic, cadmium, lead, nickel, some polycyclic aromatic hydrocarbons and ozone are responsible for several significant negative effects on human health and are linked to several non-communicable diseases, adverse health conditions and increased mortality. The impact on human health and the environment occurs via concentrations in ambient air and via deposition. To minimise harmful effects on human health, particularly in vulnerable groups and sensitive populations, and the environment, limit values should be set for the concentrations of sulphur dioxide, nitrogen dioxide, particulate matter ($P{M}_{10}$ and $P{M}_{2.5}$), benzene, carbon monoxide, arsenic, cadmium, lead, nickel and polycyclic aromatic hydrocarbons in ambient air. The members of the council shall, in an easily understandable manner, establish and make publicly available an air quality index covering hourly updates on, at least, sulphur dioxide, nitrogen dioxide, particulate matter ($P{M}_{10}$ and $P{M}_{2.5}$) and ozone.

Particulate matter $\left(PM\right)$ represents a complex mixture of small, solid particles and liquid droplets in the air. Two basic types of $PM$ are defined—$P{M}_{10}$ and $P{M}_{2.5}$. In this article, $P{M}_{10}$ values are selected as the main indicator of air quality. $P{M}_{10}$ is composed of particles with a diameter of 10 microns or less. It is a mixture of materials that can include soot, metals, salt and dust. Major sources include vehicles, wood burning, wild or open fires, industry, dust from construction sites, gravel pits, agriculture and open landfills [2]. Exposure to high concentrations of $P{M}_{10}$ can result in a number of health impacts, ranging from coughing and wheezing to asthma attacks and bronchitis, high blood pressure, heart attacks, strokes and premature death [3].

The EU directive (see [1]) stipulates limit values for the protection of human health. A distinction is made between long-term exposure, expressed as annual average concentration values, and short-term exposure, expressed as hourly or daily average concentration values. For hourly and daily average concentrations, the maximum number of such exceedances per year is set. In the case of $P{M}_{10}$, the annual average concentration limit value is 40 $\mathsf{\mu }$g/m³. On the other hand, a limit value of 50 $\mathsf{\mu }$g/m³ per day cannot be exceeded more than 35 times per calendar year.

1.1. Description of Used Approach

In this article, we describe the way the emission data, specifically the values of $P{M}_{10}$, 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 $P{M}_{10}$ 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.

1.2. 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 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.

Let us note that in addition to external air quality, some articles are devoted to the examination of indoor air quality. In [17], an operating room air quality monitoring system based on a fuzzy decision support system is proposed to help hospital staff to guarantee a safe environment. As the authors conclude, the advantage of a fuzzy inference system is that the evaluation of air quality is based on easy-to-find input data established based on the best combination of parameters and level of alert. Even in such cases, the explainability of the system plays an important role.

All these works point to the possibility of the utilisation of fuzzy systems as explanatory elements in a decision-making process, making them relevant for the purposes of this study.

2. Description of Used Data

Several studies present the use of commercially available sensor networks (see [18,19,20]), which are commonly limited in terms of the quality of data collection. This insufficiency leads to the importance of the use of certified data and reputable data sources. Since, for the purposes of this work, we wanted to use official data, we focused on local traffic and weather conditions in the city of Prague. We used official data from certified sensors deployed by the Czech Hydrometeorological Institute (see [21]). We chose the locality Vršovická street, which has unique characteristics: it is a two-lane (per direction) road running from west to east and has one of the most significant traffic flows in the city centre area of Prague, with a speed limit of 50 km/h, and this road leads to the city centre, necessitating coordinated traffic flow management.

We used three types of data—weather, traffic and emission data. Weather conditions can have significant effect on the values of emissions; for example, a strong wind could cause an area with strong polluting elements (factories, construction of buildings, power plants, etc.) to have a similar amount of emissions as windless quiet city streets. Another important factor is the wind direction. Temperature and the amount of rainfall are also important. In the case of long-term drought, the occurrence of emissions is much higher compared with the rainy seasons of the year. Therefore, variables used to model weather in our data were Wind direction, Direction of maximum wind gust, Average wind speed, Speed of maximum wind gust, Total rainfall and Air temperature.

It is well known that transportation is related to the local production of various kinds of emissions. Transportation includes not only the movement of people but also the transport of goods. Additionally, transportation encompasses commuting, as well as travel for holidays, which must also be accounted for. As traffic data, we used the variables Intensity of vehicles and Average speed of vehicles. These variables were measured in both directions of traffic. Based on experience with traffic in large cities, where the number of cars in the city also depends on the day and hour under consideration, we decided to include the following two input variables: Day of the week and Hour of the day.

Emission data were used as the output variable. In this article, we demonstrate just the results for $P{M}_{10}$ values. As mentioned before, for $P{M}_{10}$, the limit value of 50 $\mathsf{\mu }$g/m³ per day cannot be exceeded more than 35 times per calendar year. For our location, this variable reached values in the interval $[0,108]$. Let us also state that from the 3845 measurements that we had, there were just 2 values greater than 100 $\mathsf{\mu }$g/m³, only 26 reached a value greater than 60 $\mathsf{\mu }$g/m³ and, similarly, there were just 75 values greater than 50 $\mathsf{\mu }$g/m³. Since this concerns the pollution of the urban area, the presence of a low number of high $P{M}_{10}$ values is positive, but when processing and analysing data with all permissible values, this is a rather large deficiency.

All variables were measured at hourly intervals. In the beginning, we had a matrix of the $3845\times 11$ type, specifically 3845 data measurements with 11 variables. The order and units of the used variables are given in

Table 1. Used variables, their units and order.

#	Input Variables	Units
IN1	Wind direction	Degrees
IN2	Direction of max. wind gust	Degrees
IN3	Average wind speed	Metres per second
IN4	Speed of max. wind gust	Metres per second
IN5	Total rainfall	Millimetre
IN6	Air temperature	Degree Celsius
IN7	Intensity of vehicles	Integer
IN8	Average speed of vehicles	Kilometres per hour
IN9	Day of the week	Integer
IN10	Hour of the day	Integer
	Output Variable	Units
OUT	$P{M}_{10}$ value	Microgram per cubic meter

. After a short analysis of the data, we found out that there are some measurements with incorrect values of variables—specifically, the values were equal to −5009. This value was set in the sensors to indicate a measurement error. We removed the rows of the matrix that contained these values; further, we worked with a data matrix of the $3826\times 11$ type. These data were analysed with MATLAB software, version R2018a, in the graphical user interface Fuzzy Logic Toolbox (see [22]).

3. Fuzzy Sets and Fuzzy Logic

Let us introduce the basic ideas of fuzzy sets and fuzzy logic. The first work about these mathematical structures was published in 1965 by L.A. Zadeh (see [4]). Fuzzy sets represent an extension of classical sets. They were designed to mathematically work with the concepts in the natural language. We can start our description with classical sets. We have some domain of elements $\mathbb{X}$. When using classical sets A, we know that an element x from $\mathbb{X}$ uniquely belongs to set A or uniquely does not belong to set A. Then, by using the so-called characteristic function of set A, which is denoted by ${\chi }_A$, we assign the value 1 to the elements that belong to set A and we assign the value 0 to those elements that do not belong to set A. Then, we write:

$${\chi }_A:\mathbb{X}\to \{0,1\},{\chi }_A\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x\in A,\hfill \\ 0,\hfill & x\notin A.\hfill \end{array}\right.$$

(1)

The characteristic function represents a transition from classical sets to classical logic.

Fuzzy sets were designed for working with elements which do not belong to the considered set exactly. In such a case, when we claim that some elements belong to a set to some degree, that is, not unambiguously, it is appropriate to use fuzzy sets. For example, consider high air temperature in summer. There is no specific value which divides temperature values into high and low, as this depends on the observer. It is simple to imagine other measurements with the same properties, such as the strength of wind or the intensity of vehicles.

For each element from the domain, fuzzy sets assign a certain degree of membership to the set. The function which assigns the degree of membership is called the membership function, it is denoted by $\mu$, and it maps to the interval $[0,1]$. The utilisation of fuzzy sets is simple, since each fuzzy set is uniquely determined by its membership function. In such a case, when we use fuzzy sets, we assign the value 1 to the points that clearly belong to the considered set. We assign the value 0 to the points that clearly do not belong to the given set, and to the remaining elements, we assign a value from the interval $(0,1)$, for example by connecting the previous endpoints with a straight line.

As an example, Figure 2 displays the membership function modelling the high temperature in summer problem mentioned above. From this figure, we can see that the considered domain for temperature in summer is a set $\mathbb{X}=[0,50]$. The temperatures from the interval $[30,35]$ surely belong to the set high temperature in summer (they were assigned a value of 1). The temperatures from the intervals $[0,20]$ and $[40,50]$ surely do not belong to the set high temperature in summer (they were assigned a value of 0). The temperature values from the interval [20, 30] were assigned a degree of membership to the set high temperature in summer by simply connecting the points $A\left[200\right]$ and $B\left[301\right]$ with a straight line. Similarly, the temperature values from the interval [35, 40] were assigned a degree of membership to the set high temperature in summer by simply connecting the points $C\left[351\right]$ and $D\left[400\right]$ with a straight line.

Mathematically, we write it as follows:

$${\mu }_H:\mathbb{X}\to [0,1],{\mu }_H\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{x-20}{10}},\hfill & x\in [20,30],\hfill \\ 1,\hfill & x\in [30,35],\hfill \\ {\displaystyle \frac{40-x}{5}},\hfill & x\in [35,40],\hfill \\ 0,\hfill & x\in [0,20]\&[40,50].\hfill \end{array}\right.$$

(2)

This type of membership function is called the trapezoidal membership function. Note that we can uniquely determine such functions by using four parameters $\left[abcd\right]$. For the function displayed in Figure 2, these parameters are $\left[20303540\right]$, i.e., the first coordinates of points A, B, C and D. Notation using four coordinates is also used in MATLAB software. As we can see, we chose these values as integers. Since the values of these parameters significantly affect the result, even a small change in the parameter values can lead to a significant improvement in the overall result. Various optimisation methods are used to find more accurate values. They are often incorporated into software for working with fuzzy sets.

There is a large number of different types of membership functions. When we want to capture expert knowledge, we usually use the mentioned trapezoidal membership function (for examples of such uses, see [23,24]). When we have a sufficient number of data, we can obtain the membership functions by approximating the considered values with the chosen function. In our work, we chose the so-called Gaussian membership function to approximate the data. The Gaussian function has a general prescription:

$$f\left(x\right)=a{\mathrm{e}}^{-{\displaystyle \frac{{(x-b)}^2}{2c^2}}}$$

(3)

where $\mathrm{e}$ is Euler’s constant, a is the height of the curve’s peak, b is the position of the centre of the peak and c is the standard deviation.

Figure 3 displays the membership function of the fuzzy set high temperature in summer using the Gaussian membership function. Note that this function has just two parameters $\left[uv\right]$. In MATLAB software, parameter u represents the standard deviation of the curve, and parameter v represents the position of the centre of the curve peak. When using the fuzzy set approach, the height of this curve is always equal to one.

Figure 3 displays the Gaussian membership function with $u=9$ and $v=32.5$, i.e.,

$${\mu }_H\left(x\right)={\mathrm{e}}^{-{\displaystyle \frac{{(x-32.5)}^2}{2{\left(9\right)}^2}}}.$$

(4)

Similarly to the trapezoidal membership functions, when using these functions in real applications, it is necessary to optimise the function parameters.

When analysing data by using fuzzy sets, each input variable needs to be described with several membership functions. For example, the input variable Air temperature can obtain the values low air temperature, middle air temperature and high air temperature. This allows us to create a set of human-friendly rules which consists of the statements like If the temperature is high and the wind speed is low, then there is a lot of particular matter in the air.

Several fuzzy inference systems have been developed to work with such rules. The most used of these are the Mamdani fuzzy inference system (see [25]) and the Takagi–Sugeno fuzzy inference system (see [6]). For the Mamdani FIS, it is typical that the variables are determined by using fuzzy sets in both the assumptions and conclusions of the rule. To determine the output of the system, a process of defuzzification is needed, in which the fuzzy set is transformed into a specific number. This transformation is commonly computationally demanding and time-consuming. Since, in our work, we consider a sizable Smart City system, which needs to be able to analyse data in close-to-real time, we are not able to use the Mamdani FIS.

The input variables of the Takagi–Sugeno FIS are defined by using fuzzy sets as in the case of the Mamdani FIS. However, the output variables of this system are determined by using simple functions (linear or constant functions). The overall output of the system is determined by the function that represents the weighted arithmetic average of the individual output functions. The weights of the individual functions are determined by using the fuzzy set values for specific input values. Since this work focuses on the application of the Takagi–Sugeno FIS as an explanatory element in air quality prediction, we present the use of such a system in the next section in detail.

4. Takagi–Sugeno Fuzzy Inference System

The Takagi–Sugeno fuzzy inference system was created to analyse data that are describable by linear functions in certain parts of the domain of definition. This leads to the fact that the data need to be approximated in the remaining parts of the definition domain (see [6]). To work with such data, IF-THEN rules are created. These rules consist of assumptions (the IF part of the rules), in which input linguistic variables are represented by fuzzy sets, and consequences (the THEN part of the rules), in which output linguistic variables are represented by linear functions. In specific problems, there is a possibility to use constant functions as the consequence functions. Here is an example of the rules used in the Takagi–Sugeno fuzzy inference system:

$$R_k:IF{X}_{1}is{A}_{1k}AND{X}_{2}is{A}_{2k}AND\dots AND{X}_{n}is{A}_{nk},THENYisf(x_1,x_2,\dots x_n).$$

(5)

Above, $R_k(k=1,2,\dots ,K)$ are the rules, $X_i(i=1,2,\dots ,n)$ are input linguistic variables, $A_{ik}$ are the values of the input variables expressed by fuzzy sets, Y is an output linguistic variable, f is the linear or constant function which represents the value of the output variable, and $x_i(i=1,2,\dots ,n)$ are specific, measured values of input variables. The rules can be created by experts from the considered area or can be obtained by analysing a large number of known data. Of course, these two approaches can be combined.

Let us mention some methods usable when analysing a large number of data. These methods are based on the design and subsequent optimisation of the rules. In the scope of MATLAB software, there are two possible methods for the creation of the Takagi–Sugeno fuzzy inference system. The first one, the so-called Grid Partition method, is based on the principle that the definition scope of each input variable is divided into the required number of parts. In this way, the definition scope of an n-dimensional cuboid of inputs is divided into smaller parts (n-dimensional cuboids), and to each part (cuboid), the output value is assigned. The number of rules is equal to the number of cuboids the domain is divided into. Such an approach is suitable if a small number of input variables is used.

The second one, the Subtractive Clustering method, is based on a specific clustering method where the points with the largest number of neighbours are chosen as cluster centres (see [26]). To each cluster centre, some output value is assigned. Then, the rules are created in such a way that the values of the input variables are computed from the values of cluster centres and the output values are the same as the values assigned to the output values of the cluster centres. Depending on the parameters set by the user, the method determines the number of cluster centres. With the change in the method parameters, it is possible to achieve a smaller/larger number of cluster centres and thus also a smaller/larger number of rules, which leads to the possibility of the optimisation of the number of rules in the system.

In the next step, the parameters of input and output variables need to be optimised (see Section 3). Several techniques have been developed for the parameter optimisation of Takagi–Sugeno FISs. For example, evolutionary algorithms (see [27]) or artificial neural networks (see [28]) can be used. We worked with an adaptive network-based fuzzy inference system (ANFIS), which is an optimisation method using artificial neural network principles (see [28]). The ANFIS consists of a neural network of five layers, where two of the layers are adaptive (one of them optimises the parameters of the input variables, and the other optimises the parameters of the output variables) and the rest of the layers are computational. As an input of the ANFIS, some Takagi–Sugeno FIS is given. Then, as a result, a new Takagi–Sugeno FIS with the same number of rules but with new variable parameters is obtained.

Let us note that when working with an ANFIS, similar to using any other neural network, it is necessary to divide data randomly into the training, test, and checking datasets. In our work, to divide data into the mentioned sets, the ratio of 70%:15%:15% was used. Since neural networks represent iterative algorithms, a suitably chosen termination condition is also important. In MATLAB software, we can choose from two options for the termination condition—Error Tolerance and Number of Epochs. We set Error Tolerance equal to 0. We gradually changed the Number of Epochs value. We noticed an interesting phenomenon. In the beginning, we set this parameter to a high value. In some cases, the system error stopped changing after a certain Number of Epochs. However, after the next training run, the error decreased. This is an interesting observation, because in classical neural networks, if the error does not change once, the next training run does not reduce the error. This phenomenon occurs because the ANFIS is a structurally specific type of neural network. While in classical neural networks, the weights are determined based on the connections between the neurons of individual layers, in ANFISs, the weights are determined based on the nodes of the network. Therefore, we recommend running the ANFIS several times. First, a higher value should be chosen for the Number of Epochs (for example, 1000), and when the system error stops changing during training, it is advisable to choose a very low value for the number of training epochs (for example, from 5 to 10). If the error does not change, we can consider the training to be completed. If the error decreases, we continue training by using a small value for the Number of Epochs. However, it may happen that the error starts to grow, so we recommend saving the obtained FIS after each training run.

Since MATLAB software computes the error for training and checking samples separately, it is necessary to evaluate the obtained FIS with respect to all used data together (i.e., training, testing and checking). We decided to use RMSE (Root Mean Square Error; see [29]) to evaluate the system we created because it is also used by other models implemented in MATLAB, with which we compared our results. The formula for the computation of RMSE is as follows:

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^j}{(y_i-{\widehat{y}}_i)}^2}{j}}},$$

(6)

where $y_i$ is the actual i-th value of variable y, ${\widehat{y}}_i$ is the predicted i-th value of the variable and j is the number of observations in dataset.

5. Data Analysis

The main objective of the presented work is to find a function that approximates the relationship between input and output values as accurately as possible. Part of this research study was performed and mentioned in [8]. In this work, we continue the research study with an emphasis on the explainability of the system.

As mentioned before, we consider ten input variables and one output variable. As an approximation method, we used the Takagi–Sugeno FIS. For this system, an IF-THEN rule base needs to be created. Let us give an example of such a rule for our task:

$R_k$: IF The wind speed is high AND … AND The number of vehicles is low,

THEN The value of $P{M}_{10}$ is low.

The values low, high, etc., in the assumptions are expressed by using fuzzy sets. Note that fuzzy sets are uniquely determined by using their membership functions, which can be of several types (triangular, trapezoidal, Gaussian, bell-shaped, etc.). In this research study, we chose Gaussian membership functions, as we assume a normal data distribution. These functions need to be specified by using two parameters $\left[uv\right]$. In MATLAB software, parameter u represents the standard deviation of the curve, and parameter v represents the position of the centre of the curve peak. The height of this curve is always equal to one. The values in the consequences could be expressed by using constant or linear functions. If we want to design a system that is explainable, it is more appropriate to use a constant function for an output.

In the first step, we wanted to create the rules ourselves. Based on the knowledge from experts, we knew some relationships between variables, for example, the aforementioned rule that strong wind can cause an area with strong polluting elements to have a similar amount of emissions as windless quiet city streets. We wanted to determine relationships among multiple inputs and find further dependencies among them. Therefore, we proceeded as follows: We sequentially sorted the input data based on the first input, second input, etc., and we tried to evaluate the effect of specific input values on the overall output. Since we had ten input variables and this problem is quite complex, we were not able to state rules that would cover the entire input data space. Therefore, we decided to use another approach—analyse the known data by using methods the described in Section 4.

The first approach, using the Grid Partition method, allows the definition domain of each input to be divided into several parts (cuboids). From them, n-dimensional cuboids, containing the input values, are subsequently created. The output value is assigned to each cuboid. Then, the processed values of the cuboid together with the output value represent one IF-THEN rule. Since we have ten input variables, as an example, we divide each definition domain into three parts, and we obtain $3^{10}$ = 59,049 rules. MATLAB software is unable to generate such a large rule base. Even if we could generate the entire rule base, it would be problematic to explain the behaviour of the system. For this reason, we need a lower number of the rules.

This lower number of the rules is given as a result of another method for Takagi–Sugeno FIS creation—Subtractive Clustering. This method consists of two steps: In the first step, the ideal number of clusters is found by using a clustering algorithm called Subtractive Clustering (see [26]). In the second step, each cluster centre is identified with the variables of one IF-THEN rule. Note that the Subtractive Clustering method, which is implemented in MATLAB software, always assigns Gaussian membership functions for input variables and linear functions for output variables.

We used the above-mentioned procedure to analyse our data. In the first step, we used the predefined parameters of Subtractive Clustering. The algorithm found 28 clusters and generated 28 rules. The number of obtained rules was acceptable, but we wanted to reach an even lower number of rules. Therefore, we gradually increased the value of the parameter Range of influence of the cluster centre from the value $0.5$ to the value $0.8$. Using the value 0.65, the algorithm determined 12 clusters and generated 12 rules. Such a number of rules is suitable for the explainability of the created system. In the next step, we used an ANFIS (as mentioned at the end of Section 4) to optimise the parameters of the created rules.

When we optimised the rule base with 28 rules, we achieved a total RMSE equal to 9.0016. After the optimisation of the rule base with 12 rules using ANFIS, we achieved a new FIS (see Figure 4) with a total RMSE equal to 9.2817.

To compare the obtained models with other models, we applied all regressors offered by MATLAB software to the examined data. MATLAB version R2018a offers the Regression Learner app, which includes four linear regression models, three types of regression trees, six methods using SVM, four types of Gaussian progress regression (GPR) models, and two types of ensemble tree models. We used all of them to determine which method produces the lowest error on our data (see

Table 2. Used models.

#	Regression Methods Implemented in MATLAB Software	RMSE
1	Fine Tree	11.400
2	Medium Tree	10.717
3	Coarse Tree	10.365
4	Linear Regression	10.821
5	Interaction Linear Regression	10.509
6	Robust Linear Regression	10.891
7	Stepwise Linear Regression	10.446
8	Linear SVM	10.930
9	Quadratic SVM	10.691
10	Cubic SVM	69.599
11	Fine Gaussian SVM	10.058
12	Medium Gaussian SVM	9.9058
13	Coarse Gaussian SVM	10.444
14	Rational Quadratic GPR	9.2061
15	Squared Exponential GPR	9.6948
16	Matem 5/2 GPR	9.3543
17	Exponential GPR	9.3088
18	Ensemble Boosted Trees	9.9197
19	Ensemble Bagged Trees	9.6739
	Proposed FIS Methods with Linear Outputs	RMSE
20	FIS with 12 rules	9.2817
21	FIS with 28 rules	9.0016

). The lowest value of RMSE with the use of MATLAB-implemented models was reached by using Rational Quadratic Gaussian Progress Regression, where the error was equal to 9.2061. The second-best result was reached using Exponential Gaussian Progress Regression, where the RMSE was equal to 9.3088.

Based on the comparison with the methods used by us, it is obvious that our model with 28 rules achieved the lowest error. Similarly, our model with 12 rules achieved the third-best result. All these results, together with some ideas to improve them, are mentioned in [8]. Now, we pay attention to the explainability of the created system.

The problem with the obtained models is that the Subtractive Clustering method, which is implemented in MATLAB software, always assigns linear functions to the output variables. If we want to create an explainable system, it is appropriate for the output variables to be defined by constant functions. We tried to answer the question of whether it is possible to change the output variable function from linear to constant in such a way that the error rate inflates minimally. If, for a certain combination of our inputs, we assumed an output value of, for example, 45, then the rule with a constant output would look like this:

$R_k$: IF The wind speed is high AND … AND The number of vehicles is low,

THEN The value of $P{M}_{10}$ is 45.

We decided to use an already-created model with 12 rules. The research question was how to create constant output values if the input values are specified by using Gaussian membership functions (see Figure 4). The natural answer was that for each rule, we take the values of the position of the centres of the input Gaussian functions (position of their peaks) and insert them into the prescription of the output linear function. As mentioned before, in MATLAB software, we have been working on a graphical user interface called Fuzzy Logic Toolbox. Now, it is the time to look at the code of the created Takagi–Sugeno FIS. Figure 5 presents function parameters of some individual input and output variables created by MATLAB software.

In Figure 5, the value of the position of the centre of the Gaussian function for the first input variable in the first rule is marked in yellow. Of course, we considered the values of the position of the centres of all the Gaussian functions that occurred in the first rule. We have ten values (let us label them $x_{1,1},x_{1,2},\dots ,x_{1,10}$). On the other hand, to each rule, an output in the form of 11 values was assigned (they are marked in green in Figure 5). The first 10 values represent the linear terms (let us label them $a_{1,1},a_{1,2},\dots ,a_{1,10}$), and the 11th value represents the absolute term (let us label it $b_1$) of the linear function. Then, the constant output for the first rule can be computed as

$$c_1=a_{1,1}{x}_{1,1}+a_{1,2}{x}_{1,2}+\dots +a_{1,10}{x}_{1,10}+b_1.$$

(7)

We determined the output values for all 12 rules by using the same procedure. Then, based on the obtained values, we created a new Takagi–Sugeno FIS with constant outputs. We evaluated the generated FIS against our data and obtained an RMSE of 14.7281. The error of the created system was quite large, so we optimised the system again by using the ANFIS method. Through optimisation, we were able to reduce the error to a value of 9.9313, which is still a usable result compared with all the mentioned regression models (see Table 2).

From a mathematical point of view, as the output of the mentioned FIS, we obtained a nonlinear function that covers the entire input space. We are now able to assign to each tuple of inputs, as an output, one specific number that represents the value of $P{M}_{10}$.

From a scientific point of view, we have obtained a fuzzy rule base with which it is possible to present relationships between input and output variables by using human speech. We can create these rules as follows: We take the first rule and determine for which values of the individual input variables all Gaussian membership functions take the value 1 (parameter v of each Gaussian function from the first rule). Then, we assign an output value that is identical to the constant output value of the first FIS rule and is easily readable in MATLAB software. By using a similar principle with all rules, we obtain the rule base which has values as displayed in

Table 3. Values of input and output variables in created rule base with 12 rules.

#	IN1	IN2	IN3	IN4	IN5	IN6	IN7	IN8	IN9	IN10	OUT
R1	224.229	231.127	1.152	2.795	0.077	10.157	1412.038	43.316	4.172	7.716	19.411
R2	220.996	207.963	1.555	3.901	0.015	9.972	98.000	31.562	2.828	22.001	19.893
R3	187.044	180.090	2.858	5.705	0.100	9.776	801.994	41.860	4.879	8.254	8.959
R4	167.882	162.783	−2.970	2.405	−1.000	3.632	491.931	37.351	7.519	17.812	47.971
R5	222.906	207.862	−0.307	3.184	−0.050	13.320	822.013	43.239	5.152	5.242	28.331
R6	73.016	90.221	0.509	2.028	0.294	22.259	100.003	33.187	6.084	19.873	29.211
R7	211.959	236.062	0.330	2.543	0.052	5.598	70.008	27.661	3.761	2.918	16.469
R8	142.039	135.051	0.296	3.479	−0.219	−0.292	1491.981	44.787	4.806	10.125	36.897
R9	246.249	225.135	0.540	5.885	0.459	17.695	292.003	36.410	6.945	16.965	13.941
R10	130.887	123.929	−0.818	2.123	0.053	2.209	80.035	33.252	4.362	3.031	40.331
R11	323.971	292.967	1.988	4.425	−0.052	15.781	1267.966	43.460	1.150	8.888	11.050
R12	48.938	61.896	3.057	2.875	−0.130	21.712	1056.000	41.600	4.484	8.029	20.969

.

From the first row of Table 3, it follows that the first rule has the following form:

IF Wind direction is approximately 224.229 AND Direction of max. wind gust is approximately 231.127 AND Average wind speed is approximately 1.152 AND Speed of max. wind gust is approximately 2.795 AND Total rainfall is approximately 0.077 AND Air temperature is approximately 10.157 AND Intensity of vehicles is approximately 1412.038 AND Average speed of vehicles is approximately 43.316 AND the Day of the week is approximately 4.172 AND Hour of the day is approximately 7.716 THEN Value of $P{M}_{10}$ is approximately 19.411.

Moreover, from Table 3, it follows that the greatest output value of the created rule base is approximately 48 $\mathsf{\mu }$g/m³ (see Rule 4 in Table 3). Let us remind the reader that the data analysed by us had the specificity that the resulting values were from the interval $[0,108]$, but out of 3826 data measurements, only 26 reached a value greater than 60 $\mathsf{\mu }$g/m³. We were interested in the quality of the approximated data whose output value was greater than 60 $\mathsf{\mu }$g/m³. Therefore, we sorted the data according to the output value and plotted them in a graph together with the approximated data (see Figure 6).

We found that the biggest error occurs precisely for data whose input value is greater than 60 $\mathsf{\mu }$g/m³. We looked at the input data again, this time only at those whose output value was greater than 60 $\mathsf{\mu }$g/m³, and tried to add two to three more rules to the already existing rule base that would apply to those output values. Given the nature of the data, we tried a combination of multiple rule parameters. Only when using one combination of the two new rules (and subsequent optimisation by using the ANFIS), we managed to achieve a lower error, an RMSE of 9.8095 (see

Table 4. Used models 2.

	Proposed FIS Methods with Constant Outputs	RMSE
22	FIS with 12 rules	9.9313
23	FIS with 14 rules	9.8095

, and compare it with Table 2).

The values of the variables of the created rule base with 14 rules are displayed in

Table 5. Values of input and output variables in created rule base with 14 rules.

#	IN1	IN2	IN3	IN4	IN5	IN6	IN7	IN8	IN9	IN10	OUT
R1	224.229	231.122	1.488	2.792	0.084	10.339	1412.043	43.372	4.586	7.733	19.351
R2	220.998	207.966	1.536	3.935	0.018	9.954	98.000	31.521	2.764	22.122	19.868
R3	187.046	180.100	3.039	5.886	0.116	9.821	803.564	41.851	5.090	8.165	10.022
R4	167.874	162.780	−3.103	2.810	−1.011	3.605	491.931	37.313	7.351	17.970	44.771
R5	222.908	207.861	−0.354	3.145	−0.052	13.261	822.013	43.208	5.087	5.322	26.302
R6	73.036	90.234	0.636	1.993	0.279	22.540	100.002	33.170	6.009	19.871	28.828
R7	211.952	236.067	0.546	2.527	0.053	5.534	70.009	27.676	3.838	3.020	17.015
R8	142.026	135.052	0.275	3.432	−0.251	−0.470	1491.977	44.752	4.909	10.172	35.544
R9	246.255	225.139	0.630	6.053	0.466	17.632	292.004	36.516	7.023	16.929	13.544
R10	130.890	123.925	−0.865	2.145	0.053	2.173	80.035	33.232	4.312	3.020	40.310
R11	323.960	292.952	1.803	4.425	−0.056	15.367	1267.964	43.438	1.188	8.884	11.621
R12	48.934	61.885	3.137	2.942	−0.118	21.707	1056.002	41.629	4.569	7.976	20.754
R13	73.036	135.052	1.488	1.000	−	5.534	100.015	35.068	4.909	7.013	2397.486
R14	73.036	135.052	1.488	1.000	−	5.534	100.015	35.068	4.909	19.640	63.097

. Note that there are two cells that do not contain any value. This means that the output value does not depend on that specific input variable. During the creation of the 13th and 14th rules, we used only the values from the interval $[60,108]$, which represent the real values of $P{M}_{10}$ obtained in Prague. It is also interesting that after training, we obtained a value of one output variable equal to 2397.486.

6. Discussion

There are many real situations in which it is necessary to determine or predict the behaviour of data. In this paper, we created an explainable Takagi–Sugeno FIS with constant output functions for real data. It was created from the Takagi–Sugeno FIS with linear output functions. In this way, we have achieved the objective that we can interpret each rule in the following form:

IF the first input has a value around $x_{k,1}$

AND the second has a value around $x_{k,2}$

AND … AND the tenth input has a value around $x_{k,10}$

THEN output has a value around $c_k$.

Such a rule base helps us interpret the system and allows us to find input values that contribute to individual outputs, which is of great benefit to searching for the possibility of system correction. The accuracy of the created system decreased only minimally, and (as mentioned at the end of the previous section) there is a possibility to increase it by adding a few rules to existing rule base.

The second interesting information presented in this article is the method of training the ANFIS. Let us remind the reader that sometimes, it happens that after a certain Number of Epochs, the system error stops changing. However, after the next training run, the error decreases. This phenomenon occurs because the ANFIS is a specific type of neural network. In the ANFIS, the weights are determined based on the nodes of the network. Therefore, we recommend running the ANFIS several times. First, a higher value should be chosen for the Number of Epochs (for example, 1000), and when the system error stops changing during training, it is advisable to choose a very low value for the number of training epochs (for example, from 5 to 10). If the error does not change, we can consider the training to be completed. If the error decreases, we continue training by using a small value for the Number of Epochs. However, it may happen that the error starts to increase, so we recommend saving the obtained FIS after each training run.

7. Conclusions

In this paper, we presented a method of knowledge extraction from measurement datasets of air quality data. We pointed out a new way of gaining knowledge about air quality modelling—using Takagi–Sugeno fuzzy inference systems. We created a fuzzy rule base with which it is possible to present relationships between input and output variables by using human speech. We chose meteorological and traffic data in the city of Prague as input data. We used them to model emission values, specifically $P{M}_{10}$ values. The results can be applied in the implementation of the Smart Cities concept in the city of Prague. By identifying 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 future, we want to use fuzzy inference systems to predict emission values. Such models can help predict, for example, what $P{M}_{10}$ values will be like at some future time. Predictions for the next few minutes to hours can, for example, help create alerts for the elderly and people suffering from allergies when they should not leave their homes. On the other hand, longer-term predictions can help in planning the so-called smart parking lots located at the edge of a city for cars to stop at at some time and thus prevent unwanted increases in the values considered.