Using Fuzzy Logic to Analyse Weather Conditions

Abstract

Effective weather analysis is a very important scientific, social, and economic issue, because weather directly affects our lives and has a significant impact on various sectors, including agriculture, transport, energy, and natural disaster management. Weather analysis is therefore the basis for the operation of many decision-making support systems, especially in transport (air, sea), ensuring the continuity of supply chains for industry or the delivery of food and medicines, but also municipal economies or tourism. Its role and importance will grow with the worsening of climatic phenomena and the development of the Industry5.0 paradigm, which puts humans and their environment at the center of attention. This article presents issues related to fuzzy sets and systems and presents a weather analysis model based on them. The fuzzy system was created using Matlab, in the Fuzzy Logic Designer application, focusing on fuzzy logic. With Fuzzy Logic Designer, users can define fuzzy sets, rules, and carry out fuzzification and defuzzification processes, thereby offering great possibilities in data management.

1. Introduction

Weather is the instantaneous state of the atmosphere at a given location. It consists of a number of parameters of the troposphere that we can measure and phenomena that we can observe [1,2]. Weather is one of the phenomena over which we do not have complete control, and which crucially affects human life and activity [2]. When planning a trip, going to work, or going to a celebration, most people check the weather. Weather analysis is also one of the important factors affecting transport, industrial activity taking place on the ground, the provision of some services, the analysis of extreme phenomena, and the impact of the seasons, including energy (also: from renewable energy sources) [3,4,5]. Weather analysis is one of the key, and most challenging, tasks performed by meteorological services worldwide [6]. Weather analysis is a part of an information system that is used to predict natural conditions in the future. The most commonly used artificial intelligence technique is computer software that has a knowledge base in a particular domain and uses inference like an expert to solve problems [7]. In this case, fuzzy sets are a promising tool for modeling and inference from vague or imprecise information and offer a more flexible and realistic representation of knowledge and reasoning processes. They make it possible to model phenomena that are difficult to describe with classical mathematical methods [8]. Fuzzy methods have so far proven their usefulness in supporting the solution of many complex decision-making problems [9,10,11,12].

The aim of this paper is to introduce the concept of fuzzy logic and its applications in meteorology. Fuzzy logic has various applications in decision support systems for weather analysis [13,14,15,16], such as modeling uncertain and imprecise data, which is common in meteorology, to provide more flexible and human-like reasoning. It can be used to predict weather phenomena by incorporating qualitative data from expert opinions, which improves the decision-making process [13,14]. Fuzzy logic also facilitates the combination of different types of meteorological data, such as temperature, humidity, and wind speed, to create comprehensive analyses. In addition, it can improve short-term weather analysis by dealing with the ambiguity and nonlinearity inherent in atmospheric data. Fuzzy systems are particularly useful in regions with limited historical data, where traditional statistical models may struggle. However, there are challenges in applying fuzzy logic to weather analysis. One of the major challenges is the difficulty in defining appropriate membership functions and rules, which require expert knowledge and may be subjective. Another challenge is the potential for oversimplification, as fuzzy logic may not fully capture the complexity and dynamics of weather systems. The lack of adaptability of traditional fuzzy systems means they may not perform well when weather patterns change over time. In addition, computational efficiency can be an issue when processing the massive and complex data involved in weather analysis. Finally, integrating fuzzy logic with other advanced predictive models, such as machine learning, poses technical and conceptual challenges, particularly in ensuring consistency and reliability.

Despite the efforts of researchers and interdisciplinary cooperation [17,18,19,20], the reliability of weather analysis is still limited and atmospheric phenomena surprise the services that are responsible for both predicting the weather and changing it quickly, possibly in advance.

This article presents issues related to fuzzy sets and systems and presents a weather analysis model based on them.

The role and importance of solving weather analysis problems quickly and accurately will grow with the worsening of climate phenomena and the development of the Industry 5.0 paradigm, which puts people and their environment at the center. Effective weather analysis is a very important scientific, social, and economic issue, as weather directly affects our lives and has a significant impact on various sectors, including agriculture, transport, energy and disaster management. Weather analysis is therefore at the heart of many decision-support systems, especially in transport (air, sea), ensuring the continuity of industrial supply chains or the provision of food and medicine, but also in the municipal economy or tourism.

The use of Fuzzy logic in decision support systems for weather analysis introduces a novel approach, dealing with the inherent uncertainty and imprecision of weather data more effectively than traditional binary logic. It allows the system to process vague or incomplete input data, such as ‘partly cloudy’ or ‘slightly wet’, and generate more nuanced analyses that better reflect actual conditions in a way that is understandable to the audience (also: lay people). This method improves decision-making by providing more flexible and adaptive analysis, which can be particularly valuable in complex and dynamic weather systems. Commonly used electronic circuits and systems are sensitive to changing weather conditions and more industrial and every day solutions will incorporate weather risk assessment and analysis systems (e.g., electric cars, where battery consumption and range depend on temperature). Fuzzy logic is gaining increasing recognition in analysis and decision making systems, as it can cope not only with typical numerical data but also with imprecise or partial data, which is particularly important in weather analysis, such variable occurrences as the weather.

2. Materials and Methods

2.1. Fuzzy Sets

Fuzzy sets were introduced by Zadeh (1965) based on the assumption that a precise description of many situations in the real world is practically impossible, and that imprecisely defined ‘classes’ are important in human thinking and natural language [21]. Fuzzy set theory makes it possible to describe qualitative, fuzzy concepts and knowledge about the surrounding world, and to operate on the knowledge in this theory, creating an independent field of scientific research [22]. A fuzzy set is a class of objects with a certain degree of membership. Such a set is characterized by a membership function that assigns to each object a degree of membership ranging from zero to one [23]. One way to describe a fuzzy set A is to give its membership function μ_A: X → [0, 1] [24]. The value of μ_A(x) is thus a number in the interval [0, 1], and is called the degree of membership of an element x to the set A. The membership function has a rich choice of shapes and can be described by various mathematical methods and functions. The choice of the appropriate shape and parameters of the membership function is crucial for the effectiveness of systems using fuzzy sets. Commonly used membership functions include triangular functions, trapezoidal functions, Gaussian functions, and the singleton type of membership function [9]. This article uses triangular and trapezoidal functions:

(a)

Triangular membership function

This function is defined by three parameters a, b, c, which define the coordinates on the x-axis [25].

$${\mu }_F(x; a,b,c)=\left\{\begin{array}{c}0, x\le a\\ {\displaystyle \frac{x-a}{b-a}}, a<x\le b\\ {\displaystyle \frac{c-x}{c-b}}, b<x<c\\ 0, x\ge c\end{array}\right.$$

(b)

Trapezoidal membership function

This function is defined by four parameters a, b, c, d, which define the coordinates on the x-axis [25].

$${\mu }_F(x; a,b,c,d)=\left\{\begin{array}{c}0, x\le a\\ {\displaystyle \frac{x-a}{b-a}}, a<x<b\\ 1, b\le x\le c\\ {\displaystyle \frac{d-x}{d-c}}, c<x<d\\ 0, x\ge d\end{array}\right.$$

Membership functions are widely used in various fields such as data analysis [26], fuzzy control [27], expert systems [28], and analysis [29].

2.2. Fuzzy Systems

There are two main types of fuzzy systems, Mamdani [30] and Sugeno [31]. Mamdani and Sugeno fuzzy systems for a given system have the same number of input and output functions and the same rules. The only difference between them is the way the fuzzy output is defuzzified [32]. This paper focuses on using Mamdani inference, which is one of the most popular techniques in fuzzy logic, to design a weather analysis model.

The Mamdani system consists of IF-THEN rules that define how inputs and outputs are related to each other. Each rule can use different fuzzy sets A and B, which can be connected by AND and OR logical connectives [30,33].

Fuzzy set, IF-THEN fuzzy rule base, linguistic variables, and possibility distribution are the basic concepts of a fuzzy logic system [29]. Fuzzy systems deal with human reasoning even with a high level of abstraction. A fuzzy inference system should be able to reproduce algorithmically structured human knowledge by encoding it into a set of IF-THEN rules. Each rule represents a task to be performed [34].

2.3. Data Set

A weather analysis model was designed using 2012–2015 Seattle weather data (

Table 1. Excerpt from 2012–2015 Seattle weather data.

Date	Precipitation [mm]	Temp_max [°C]	Temp_min [°C]	Wind [m/s]
1 January 2012	0	12.8000	5	4.7000
2 January 2012	10.9000	10.6000	2.8000	4.5000
3 January 2012	0.8000	11.7000	7.2000	2.3000
4 January 2012	20.3000	12.2000	5.6000	4.7000
5 January 2012	1.3000	8.9000	2.8000	6.1000
6 January 2012	2.5000	4.4000	2.2000	2.2000
7 January 2012	0	7.2000	2.8000	2.3000
8 January 2012	0	10	2.8000	2.3000
9 January 2012	4.3000	9.4000	5	3.4000
10 January 2012	1	6.1000	0.6000	3.4000

). The database used in the research is Weather Analysis. It was downloaded on 13 May 2024 from https://www.kaggle.com/datasets/ananthr1/weather-prediction. It contains 1462 records, making it one of the more extensive and accurate databases. This database was chosen because it collects unique data on rainfall or lack thereof using certain weather conditions. The data are presented over a three-year period, where measurements were taken each day. Maximum temperature and minimum temperature are expressed in degrees Celsius, wind speed in m/s, and precipitation in mm. Due to its broad content and high-quality data, the Weather Analysis database is an invaluable resource for researchers who are looking for reliable and up-to-date data for their work and projects.

2.4. Computational Methods

The fuzzy system was created using Matlab, in the Fuzzy Logic Designer application, focusing on fuzzy logic. Matlab, as an advanced data analysis and modelling tool, enables the creation of advanced fuzzy systems that can effectively deal with imprecision and variability in data. With Fuzzy Logic Designer, users can define fuzzy sets, rules, and carry out fuzzification and defuzzification processes, which offers great possibilities in data management.

The subsequent stages of the weather analysis system are shown in a flowchart (Figure 1) to create the weather analysis model below. This flowchart shows a classic fuzzy logic-based analysis model that applies the Mamdani method to process weather data and generate a weather analysis.

The fuzzy system was created using Matlab 2023a (MathWorks, Natick, MA, USA), in the Fuzzy Logic Designer application, which allows the design and testing of fuzzy inference systems to model complex system behavior. Fuzzy rules can be easily programmed and fuzzy models can be transparent [35].

This paper presents a weather analysis model with three input variables and one output variable. The use of fuzzy logic allows a more flexible approach to atmospheric phenomena. Instead of rigid ranges, degrees of belonging are assigned to reflect the diversity of human feelings. This approach better reflects reality and makes the modeling of atmospheric conditions more user-friendly [36]. The selection of ranges for the test data is based on a preliminary analysis of the available data. Using these ranges allows for more accurate modeling and analysis of data variability. The principle of summing to 1 is not applied due to the circumstance since the fuzzy model, in this case, operates on the data in a way that does not require normalization. The reason for this is essentially that the fuzzy model effectively captures the inherent uncertainty and variability in the data without the need for a normalized scale. Unlike traditional methods, in which data normalization is necessary to ensure consistency and comparability, the fuzzy model takes advantage of the qualitative aspects of the input data. As a result, each input data can contribute its own unique meaning without being limited to a fixed numerical range. Furthermore, this approach allows for a more detailed representation of underlying patterns and relationships in the data, enhancing the model’s ability to generate accurate results. This allows diversity in the data to be taken into account without the limitations that summing to the one would impose. It is advisable to point out that the choice of test data ranges is also based on feelings about what is considered cold, moderate, or hot. These interpretations are subjective and could be varied depending on a person’s individual preferences, experiences, and conditions. What constitutes the range of ‘hotness’ for someone, may be defined as ‘moderate’ for another person. Such a discrepancy is not an error—it is due to the fact that each person may perceive the same weather conditions differently, which is one of the advantages of the fuzzy model. It allows a flexible approach to the data, enabling an interpretation that takes into account individual differences in perception and does not impose rigid boundaries or standardized values. In this way, the model better reflects the complex nature of people’s actual feelings and preferences in the context of changing weather conditions [37,38,39].

3. Results

The output variable is precipitation and the input variables are:

(a)

Temp_max: cold, moderate, hot (Figure 2)

The variable cold represents a specific level of ‘cold’, which be calculated from the data of the lowest temperature (T_min) and the highest temperature (T_max) in the specified range. The formula for the ‘cold’ value can be represented as follows:

$$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}=(T_{min}-20\mathrm{\%}(T_{max}-T_{min}), \mathrm{where}$$

• T_min—minimum temperature value in the analysed range from the database,
• T_max—maximum temperature value in the analysed range from the database.

The calculation is presented as follows:

$$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}=(T_{min}-20\mathrm{\%}\left(T_{max}-T_{min}\right)=-1.6-20\% (35.6-\left(-1.6\right)=-9.04 \approx -9$$

Similarly, for the ‘hot’ variable, the value is determined by the following formula:

$$hot=(T_{max}-20\% (T_{min}-T_{max}), \mathrm{where}$$

• T_max—maximum temperature value in the analysed range from the database,
• T_min—minimum temperature value in the analysed range from the database.

The calculation is presented as follows:

$$hot=(T_{max}-20\% \left(T_{min}-T_{max}\right)=35.6-20\% \left(-1.6-35.6\right)=43.04 \approx 43$$

The concepts will be presented in the form of the trapezoidal function temp_max cold, moderate, hot. This function is defined in the interval [−9, 43]. For linguistic values of temp_max, the degree of membership is calculated using the formula:

$${\mathsf{\mu }}_{\mathrm{cold}}(\mathrm{x};-9,-9,5,10)=\left\{\begin{array}{c}1, x\le 5\\ {\displaystyle \frac{10-x}{5}}, 5<x<10\\ 0, x\ge 10\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{moderate}}(\mathrm{x};\mathrm{8,15,22,29})=\left\{\begin{array}{c}0, x\le 8\\ {\displaystyle \frac{x-8}{7}}, 8<x<15\\ 1, 15\le x\le 22\\ {\displaystyle \frac{29-x}{7}}, 22<x<29\\ 0, x\ge 29\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{hot}}(\mathrm{x};25,30,43,43)=\left\{\begin{array}{c}0, x\le 25\\ {\displaystyle \frac{x-25}{5}}, 25<x<30\\ 1, x\ge 30\end{array}\right..$$

(b)

Temp_min: very_cold, cold, moderate (Figure 3)

The variable very.cold represents a specific level of ‘very.cold’, which can be calculated from the data of the lowest temperature (T_min) and the highest temperature (T_max) in the specified range. The formula for the ‘very.cold’ value can be represented as follows:

$$very.cold=(T_{min}-20\mathrm{\%}(T_{max}-T_{min}), \mathrm{where}$$

• T_min—minimum temperature value in the analysed range from the database,
• T_max—maximum temperature value in the analysed range from the database.

The calculation is presented as follows:

$$very.cold=(T_{min}-20\mathrm{\%}\left(T_{max}-T_{min}\right)=-7.1-20\% \left(18.3-\left(-7.1\right)\right)=-12.18\approx -12$$

Similarly, for the ‘moderate’ variable, the value is determined by the following formula:

$$moderate=(T_{max}-20\% (T_{min}-T_{max}), \mathrm{where}$$

• T_max—maximum temperature value in the analysed range from the database,
• T_min—minimum temperature value in the analysed range from the database.

The calculation is presented as follows:

$$moderate=(T_{max}-20\% \left(T_{min}-T_{max}\right)=18.3-20\% \left(-7.1-18.3\right)=23.38 \approx 23$$

The concepts will be presented in the form of the trapezoidal function temp_minvery.cold, cold, moderate. This function is defined in the interval [−12, 23]. For linguistic values of temp_min, the degree of membership is calculated using the formula:

$${\mathsf{\mu }}_{\mathrm{very}.\mathrm{cold}}(\mathrm{x};-12,-12,0,5)=\left\{\begin{array}{c}1, x\le 0\\ {\displaystyle \frac{5-x}{5}}, 0<x<5\\ 0, x\ge 5\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{cold}}(\mathrm{x};-1.6,4,8,12)=\left\{\begin{array}{c}0, x\le -1.6\\ {\displaystyle \frac{x-(-1.6)}{5.6}},-1.6<x<4\\ 1, 4\le x\le 8\\ {\displaystyle \frac{12-x}{4}}, 8<x<12\\ 0, x\ge 12\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{moderate}}(\mathrm{x};8,15,23,23)=\left\{\begin{array}{c}0, x\le 8\\ {\displaystyle \frac{x-8}{7}}, 8<x<15\\ 1, x\ge 15\end{array}\right..$$

(c)

Wind: calm, breezy, windy (Figure 4).

The variable calm represents a specific level of ‘calm’, which can be calculated from the data of the lowest temperature (W_min) and the highest temperature (W_max) in the specified range. The formula for the ‘calm’ value can be represented as follows:

$$calm=(W_{min}-20\% (W_{max}-W_{min}), \mathrm{where}$$

• W_min—minimum wind value in the analysed range from the database,
• W_max—maximum wind value in the analysed range from the database.

The calculation is presented as follows:

$$calm=(W_{min}-20\% \left(W_{max}-W_{min}\right)=0.4-20\% \left(9.5-0.4\right)=-1.42\approx -1$$

Similarly, for the ‘windy’ variable, the value is determined by the following formula:

$$windy=(W_{max}-20\% \left(W_{min}-W_{max}\right), \mathrm{where}$$

• W_max—maximum wind value in the analysed range from the database,
• W_min—minimum wind value in the analysed range from the database.

The calculation is presented as follows:

$$windy=(W_{max}-20\% \left(W_{min}-W_{max}\right)=9.5-20\% \left(0.4-9.5\right)=11.32\approx 11$$

The concepts will be presented in the form of the trapezoidal function wind calm, breezy, windy. This function is defined in the interval [−1, 11]. For linguistic values of wind, the degree of membership is calculated using the formula:

$${\mathsf{\mu }}_{\mathrm{calm}}(\mathrm{x};-1,-1,1,3)=\left\{\begin{array}{c}1, x\le 1\\ {\displaystyle \frac{3-x}{2}}, 1<x<3\\ 0, x\ge 3\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{breezy}}(\mathrm{x};2,3,6,7.5)=\left\{\begin{array}{c}0, x\le 2\\ {\displaystyle \frac{x-2}{1}}, 2<x<3\\ 1, 3\le x\le 6\\ {\displaystyle \frac{7.5-x}{1.5}}, 6<x<7.5\\ 0, x\ge 7.5\end{array}\right.,$$

$${\mathsf{\mu }}_{\mathrm{windy}}(\mathrm{x};6,8,11,11)=\left\{\begin{array}{c}0, x\le 6\\ {\displaystyle \frac{x-6}{2}}, 6<x<8\\ 1, x\ge 8\end{array}\right..$$

The input and output variables were prepared and assigned a degree of membership ranging from zero to one. The precipitation after blurring is assigned to five sets (Figure 5):

• Very low;
• Low;
• Normal;
• High;
• Very high.

The concepts will be presented in the form of the triangular function precipitation very.low, low, normal, high, very.high. This function is defined in the interval [0, 56]. The range of values from 0 to 56 is divided into five equal ranges. For linguistic values of precipitation, the degree of membership is calculated using the formula:

$${\mu }_{very.low}(x; 0, 7, 14)=\left\{\begin{array}{c}0, x\le 0\\ {\displaystyle \frac{x}{7}}, 0<x<7\\ {\displaystyle \frac{14-x}{7}}, 7<x<14\\ 0, x\ge 14\end{array}\right.,$$

$${\mu }_{low}(x; 10.5, 17.5, 24.5)=\left\{\begin{array}{c}0, x\le 10.5\\ {\displaystyle \frac{x-10.5}{7}}, 10.5<x<17.5\\ {\displaystyle \frac{24.5-x}{7}}, 17.5<x<24.5\\ 0, x\ge 24.5\end{array}\right.,$$

$${\mu }_{normal}(x; 21, 28, 35)=\left\{\begin{array}{c}0, x\le 21\\ {\displaystyle \frac{x-21}{7}}, 21<x<28\\ {\displaystyle \frac{35-x}{7}}, 28<x<35\\ 0, x\ge 35\end{array}\right.,$$

$${\mu }_{high}(x; 31.5, 38.5, 45.5)=\left\{\begin{array}{c}0, x\le 31.5\\ {\displaystyle \frac{x-31.5}{7}}, 31.5<x<38.5\\ {\displaystyle \frac{45.5-x}{7}}, 38.5<x<45.5\\ 0, x\ge 45.5\end{array}\right.,$$

$${\mu }_{very.high}(x; 42, 49, 56)=\left\{\begin{array}{c}0, x\le 42\\ {\displaystyle \frac{x-42}{7}}, 42<x<49\\ {\displaystyle \frac{56-x}{7}}, 49<x<56\\ 0, x\ge 56\end{array}\right..$$

The basis of the fuzzy system is a fuzzy rule base [40]. To obtain the predicted output in the fuzzy rule base, Mamdani-type inference based on the logical function IF-THEN usingAND as a conjunction. The input value comes from a fuzzy process and is then fed into a rule that was created to be used as a fuzzy output. Based on the input data values of wind, minimum temperature, and maximum temperature and the output data of precipitation, thirty-five rules were created, according to the project assumptions and the 2012–2015 Seattle weather data. The system used the Mamdani inference method, which involved drawing conclusions from the rules used. In the process, fuzzy logical operations such as AND (using the MIN operator) were used to combine different rules and produce a fuzzy output. The rules are as follows:

• If (temp.max is moderate) and (temp.min is moderate) and (wind is windy) then (precipitation is very.high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is breezy) then (precipitation is very.high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is windy) then (precipitation is very.high)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is breezy) then (precipitation is very.high)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is breezy) then (precipitation is high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is breezy) then (precipitation is high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is windy) then (precipitation is high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is calm) then (precipitation is high)
• If (temp.max is moderate) and (temp.min is cold) and (wind is breezy) then (precipitation is normal)
• If (temp.max is cold) and (temp.min is cold) and (wind is breezy) then (precipitation is normal)
• If (temp.max is cold) and (temp.min is cold) and (wind is windy) then (precipitation is normal)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is breezy) then (precipitation is normal)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is windy) then (precipitation is normal)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is calm) then (precipitation is normal)
• If (temp.max is moderate) and (temp.min is cold) and (wind is calm) then (precipitation is normal)
• If (temp.max is moderate) and (temp.min is cold) and (wind is windy) then (precipitation is normal)
• If (temp.max is cold) and (temp.min is cold) and (wind is breezy) then (precipitation is low)
• If (temp.max is cold) and (temp.min is cold) and (wind is calm) then (precipitation is low)
• If (temp.max is moderate) and (temp.min is cold) and (wind is windy) then (precipitation is low)
• If (temp.max is cold) and (temp.min is cold) and (wind is windy) then (precipitation is low)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is windy) then (precipitation is low)
• If (temp.max is moderate) and (temp.min is cold) and (wind is breezy) then (precipitation is low)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is calm) then (precipitation is low)
• If (temp.max is moderate) and (temp.min is cold) and (wind is calm) then (precipitation is low)
• If (temp.max is cold) and (temp.min is very.cold) and (wind is calm) then (precipitation is very.low)
• If (temp.max is cold) and (temp.min is very.cold) and (wind is breezy) then (precipitation is very.low)
• If (temp.max is cold) and (temp.min is cold) and (wind is breezy) then (precipitation is very.low)
• If (temp.max is cold) and (temp.min is cold) and (wind is calm) then (precipitation is very.low)
• If (temp.max is moderate) and (temp.min is cold) and (wind is breezy) then (precipitation is very.low)
• If (temp.max is moderate) and (temp.min is very.cold) and (wind is calm) then (precipitation is very.low)
• If (temp.max is moderate) and (temp.min is cold) and (wind is calm) then (precipitation is very.low)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is calm) then (precipitation is very.low)
• If (temp.max is moderate) and (temp.min is moderate) and (wind is breezy) then (precipitation is very.low)
• If (temp.max is hot) and (temp.min is moderate) and (wind is breezy) then (precipitation is very.low)
• If (temp.max is hot) and (temp.min is moderate) and (wind is windy) then (precipitation is very.low)

The final step was defuzzification, in which the fuzzy result (precipitation) was transformed into a concrete numerical value using one of the available defuzzification methods, in this case centroid. All these elements, including the use of the MIN operator for implication, worked together to create a fuzzy inference system that made it possible to predict precipitation based on input variables.

Figure 6 shows a 3D plot showing the analysis of precipitation based on the input variables temp.max and wind.

The results of weather analysis can be analyzed using 3D surface plots, as shown in Figure 6 and Figure 7. In Figure 6, it may be seen that as the wind and maximum temperature increase, the amount of precipitation also increases. On the other hand, in Figure 7 we can see that precipitation increases rapidly with a rising wind, especially at low minimum temperature values. It can be clearly seen that at the lowest minimum temperature and wind, precipitation is close to zero. A similar pattern is seen in Figure 6, where at the lowest values of maximum temperature and wind, precipitation is also minimal. Based on the analysis of both graphs, it can be concluded that the key factor affecting the amount of precipitation is the wind, because as the wind increases, precipitation becomes more intense, and Figure 8 on the input variables temp.min and wind.

Once the fuzzy system was designed, it was loaded in Matlab using the readfis function. Then 30 different test data were collected, from which a matrix was created, which contains a set of test data consisting of three columns, namely temp.max, temp.min, and wind. These data are selected from different seasons and months to best reflect the weather in Seattle. The test data were not chosen randomly. After defining the membership functions, the rule base was developed based on real-world weather patterns and logical relationships between the input variables. The data ranges were selected to reflect typical weather conditions, with the focus on how different combinations of max. temperature, min. temperature, wind, and precipitation are perceived in practice. The number of rows in test_data determines the number of datasets that are analyzed by the fuzzy system. The next step is to initialize the results. The test_outputs variable is initialized as a null vector with a length equal to the number of test data sets. A simulation based on the input data is then made on the fuzzy model using the evalfis function. The results are displayed sequentially. Next, a matrix is created that contains the actual rainfall data, which is compared with the analysis results obtained from the fuzzy system. These values are used to evaluate the accuracy of the model. Then we calculate the mean squares of the differences between the analysis and actual precipitation data (MSE) and the coefficient of determination ($R^2$), which help us understand how well the model predicts actual results and allow us to further improve the model. Mean Squared Error (MSE) is the average squared difference between the observed values in a statistical study and the values predicted by a model. When comparing observations with predicted values, it is necessary to square the differences, as some data values will be greater than the analysis, while others will be less. Since observations are equally likely to be greater or less than the predicted values, the differences would otherwise sum to zero. Squaring these differences prevents this from happening [41]. Coefficient of determination ($R^2$) is a measure used to evaluate the ability of a model to predict or explain an outcome in the linear regression setting. R² represents the proportion of the variance in the dependent variable (Y) that is predicted or explained by linear regression and the predictor variable (X, also known as the independent variable) [42].

The results are presented in

Table 2. Results of the test implemented.

Temp.max	Temp.min	Wind	Actual Precipitation	Results
35	17.2	3.3	0	7.0
33.3	17.8	3.4	0	7.0
18.3	15.0	5.2	30.5	30.3721
5.6	2.8	4.3	27.4	17.5052
8.9	4.4	5.1	18.5	24.8854
6.7	3.9	6.0	21.8	17.5064
13.3	9.4	6.5	33.5	28.0
8.9	2.2	4.1	22.4	24.8854
13.3	6.7	8.0	29.5	33.2407
21.1	13.3	4.7	28.7	30.3559
1.1	−3.3	3.2	5.3	7.0
1.7	−2.8	5.0	2.5	7.0
11.1	10.0	7.2	30.0	30.3941
31.1	14.4	2.5	0	7.0
8.3	5.0	3.9	39.1	19.5958
−1.1	−2.8	1.6	15.2	7.0
6.7	−0.6	4.2	3.6	13.3740
8.3	0.6	6.2	19.3	19.5958
13.9	11.7	1.9	15.7	18.6014
33.9	16.7	3.7	0	7.0
30.6	15.0	3.0	0	7.0
30.6	15.0	2.8	0	7.0
31.1	16.7	4.7	0	7.0
32.2	13.3	3.1	0	7.0
17.8	6.7	2.0	20.8	22.7593
15.0	12.2	2.8	34.5	29.8978
9.4	6.1	2.4	32	28.0
8.3	7.2	6.2	19.6	19.5958
7.2	0.6	3.7	13.2	16.5565
6.7	−1.7	3.0	4.1	7.0

and Figure 8. The MSE is 41.2659 and R² is 0.75673, which indicate that the model is useful, although there is potential for further optimization to improve the quality of the weather analysis. This result means that the model can recognize the weather with around 75% accuracy, which is quite good, but not ideal. The model has particular difficulties when analysing heavier rainfall, which may be related to the fact that heavy rainfall is uncommon in the study area. Such phenomena are more difficult to predict due to their irregularity and less availability of historical data. To improve the model, consideration could be given to including additional variables or expanding the dataset, especially to include those for more extreme weather conditions. Such measures could increase the accuracy of the analysis, especially for the more unusual phenomena.

4. Discussion

The inspiration for this article is the growing desire to predict the weather. The subject literature confirms that weather analysis is an essential and important element, especially in security, planning daily activities, agriculture and economy, transportation and logistics, public health, tourism and recreation, and energy management. For these reasons, using three input variables, minimum temperature, maximum temperature, and wind, it was found that it was possible to predict the occurrence of precipitation [43,44,45,46,47,48,49]. The goal of the study was to create a model to predict precipitation occurrence and intensity based on historical meteorological data from the Seattle area. According to the Fuzzy Logic Model Description study [6,29], the number of inputs has been increased, which makes the model even more accurate. In addition, it was found that, as in Conclusions, “a fuzzy inference system using the Mamdani method can be used as a reference in weather analysis” [7]. The linguistic parameters must be properly prepared beforehand [43]. Referring to others [9,29,35,40], studies are frequently based on trapezoidal or triangular membership functions, so, in this paper, we also decided to use such functions. The creation of rule bases is the most important factor in creating a fuzzy logic model [40].

The main objective was to evaluate the effectiveness of the fuzzy system in analysing selected weather parameters. Taking into account real-time dynamics and changes undoubtedly plays a key role in forecasting weather conditions. Therefore, in the next stages of the research, we plan to focus on developing this analysis by integrating mechanisms that allow real-time data processing. This evolution of the system will enable more accurate and timely forecasts, which is essential for further progress in this field. We see this article as an introduction to our research, which will continue with a focus on the dynamic aspects of data processing.

Following on from previous studies [6,7,17], the results confirm as well as complement the information available on the effectiveness of fuzzy logic in weather analysis. Studies such as [6,22,35] have shown that fuzzy logic systems are more beneficial than traditional methods.

The application of fuzzy logic in a decision support system (DSS) for weather analysis can help to predict the weather. Fuzzy logic, with its ability to process imprecise and ambiguous information, represents a new approach to analysing weather conditions that traditional systems can struggle with. The results presented in this study are intended to explore the potential of applying a fuzzy inference system using the Mamdani method. To reliably assess the effectiveness of this approach, it is necessary to conduct further comparative studies with existing precipitation analysis methods, preferably using the same empirical data, in order to accurately determine the improvement of weather analysis through fuzzy logic.

4.1. Limitations of Current Studies

Fuzzy logic, while useful for dealing with uncertainty in decision support systems, has several limitations in predicting weather conditions. The accuracy of analysis can be compromised by the subjective nature of defining fuzzy sets and membership functions. Fuzzy logic cannot cope with large-scale data and dynamic, non-linear interactions typical of weather systems, potentially leading to oversimplified models [50]. Fuzzy logic is unable to learn from new data because it relies on predefined rules that do not adapt over time. Interpretability of fuzzy systems can be a challenge when dealing with complex rules, making it difficult to understand or refine the model [51]. Fuzzy logic may not adequately account for extreme weather events as it tends to smooth the data, potentially underestimating risk. The computational efficiency of fuzzy logic can be an issue when processing high-dimensional data, often related to weather analysis. Integrating fuzzy logic with other predictive models, such as machine learning, can be complex and require significant tuning [52]. Fuzzy logic may not work well in highly chaotic systems such as weather, where small changes can lead to very different results. The lack of standardised methods for constructing fuzzy systems can lead to inconsistencies in different implementations [53]. Fuzzy logic can struggle with scalability, especially when more variables and rules are added to the system [54]. While the proposed fuzzy model proves effective in analysing precipitation, it is important to note that it cannot achieve an exact value of 0, which may present a limitation, especially when predicting the absence of precipitation.

4.2. Directions for Further Research

Further research on fuzzy logic in weather analysis decision support systems could focus on several promising directions. Improving the design of fuzzy functions and membership sets using adaptive methods such as machine learning could increase accuracy by better capturing the complexity of weather data. Integrating fuzzy logic with deep learning models could enable automatic rule generation and adaptation, making the system more responsive to new data [55]. Hybrid systems combining fuzzy logic with probabilistic models such as Bayesian networks could offer a more robust approach to dealing with uncertainty and variability in weather analysis [56]. Developing techniques to better model and incorporate extreme weather events in fuzzy systems would improve risk assessment and response strategies. Research on computational efficiency, perhaps through exploration of parallel processing or optimized algorithms, could make fuzzy logic more suitable for large-scale, real-time weather analysis [57]. Improving the interpretability of fuzzy systems by simplifying complex rule sets while maintaining accuracy could make models more accessible to meteorologists [58]. The use of fuzzy logic in combination with chaos theory could provide new insights into managing the inherent unpredictability of weather systems. Exploring the potential of quantum computing to support fuzzy logic computations could provide breakthroughs in processing power and speed [59]. A unified framework for fuzzy logic systems in weather analysis could be developed to ensure consistency and reliability across implementations. Incorporating user feedback mechanisms into fuzzy logic systems could enable continuous improvement and adjustment based on actual performance and needs [60]. In the next phases of the work, it is planned to use fuzzy systems to enable more flexible data processing and trend modeling, a key step toward increasing the accuracy of the analysis. Ultimately, it is planned to extend the developed fuzzy system with the ability to process trends using Ordered Fuzzy Numbers (OFN) [9]. The use of OFNs in weather data analysis will enable better prediction of climate change and forecasting of extreme weather events, which could have important implications for risk management and planning in various sectors.

5. Conclusions

The fuzzy logic weather analysis model uses 3 input variables in the form of maximum temperature, minimum temperature, and wind, as well as 35 rules that generate the weather forecast, namely precipitation, which can be very low, low, normal, high, and very high. There are many factors that need to be taken into account when developing a weather analysis system so that the analyses produced are increasingly accurate. Weather analysis, even with advanced technology and complex mathematical models, is a hard task. The developed weather analysis model provides reliable analysis, but it is important to bear in mind that the climate is dynamic and constantly changing. Consequently, it is extremely difficult to obtain a model that is close to 100% accurate. This topic encourages further work on developing weather analysis models that can provide increasingly better results and motivates further research and improvement of analysis tools that can provide even better analysis relevant to many aspects of daily life in the future. One of the next steps in the development of such models could be more advanced weather forecasting, which would take into account additional factors such as climate change patterns, extreme weather events, or seasonal anomalies. Integrating such data into models could not only improve the accuracy of short-term forecasts, but also increase their effectiveness in long-term forecasting, such as in analyzing the impact of climate change on future weather conditions. As weather analysis models develop, it will be possible to create more precise tools that not only improve weather prediction, but also impact other areas such as urban planning, energy management, or natural disaster prevention.