A Fuzzy Implication-Based Approach for Validating Climatic Teleconnections

Abstract

Fuzzy logic, during recent decades, has evolved into one of the most influential scientific fields. To an extent, this is due to its applications which have a profound impact on the daily life of people worldwide. The goal of this paper is to focus on the applications of fuzzy logic to the study of our climate, and especially if fuzzy implications can validate the climatic teleconnections observed on climatic data. To achieve this goal, a real world case study is provided which focuses on the relationship between temperature anomalies observed at different European cities. The results of this case study are that, indeed, fuzzy implications can validate the climatic teleconnections observed on climatic data. The conclusions drawn are that fuzzy logic can assist to the definitive proof of phenomena which, till now, could only be researched experimentally.

1. Introduction

Weather forecasting plays a vital role in how every society and its people organize and live their lives. Even though in recent decades this field has made huge leaps in innovation, there are still weather phenomena which are difficult to explain or predict using conventional logic. For this reason, in this paper the possibility of using fuzzy logic instead is explored. As it can assist the effort to interpretation through maths, weather patterns are observed on climatic data. To be more specific, this study focuses on the following pattern: weather anomalies, with respect to the average monthly climatic means, between different parts of Europe are often at opposite signs. A real world example of this phenomenon, is the fact that cold and dry winters in Northern Europe are accompanied by warm and wetter than average winters in Southern European regions. Moreover, warm and dry Northern European summers are accompanied by cooler and wetter than average Southern European summers. These contrasting anomalies between distant geographic regions is a type of teleconnection [1,2,3,4,5] driven by changes in the upper tropospheric circulation (the jet-stream) which governs the movement of mid-latitude weather systems near the surface. Such teleconnections become pronounced in periods when the NAO (North Atlantic Oscillation) index (broadly defined as a fluctuation in the mean sea level pressure difference between the Azores high pressure region and the Icelandic low pressure region) becomes positive or negative. The larger the absolute value of the NAO index is, the larger the observed anomalies. In order to interpret this pattern, the fuzzy implication theory is introduced as a methodology to check the validity of teleconnection hypotheses (which emerge from the observation of the pattern) that can be formulated as propositional logic statements of the form: “Weather anomaly of type A in a given location/area imply weather anomaly type B in another remote location/area”. An attractive feature of this approach, which is complementary to standard statistical approaches, is that it can give straightforward, easy to interpret measures on the validity of a broad number of hypotheses that emerge in the complex Earth climate system.

The published research concerning this study can be divided into two parts

Table 1. Published research.

Category	Published Research
Implications	“Fuzzy Implications” [6]
	“On the characterizations of implications” [7]
	“Automorphisms, negations and
	implication operators” [8]
	“Actions of Automorphisms on Some Classes
	of Fuzzy Bi-implications” [9]
	“Generalization of Fuzzy Connectives” [10]
	“A New Approach. Modern Discrete
	Mathematics and Analysis” [11]
	“Parametric Fuzzy Implications Produced via
	Fuzzy Negations with a Case Study
	in Environmental Variables” [12]
	“Application of Algorithmic Fuzzy Implications on Climatic Data” [13]
Teleconnections	“Teleconnections” [1]
	“Teleconnections in the Geopotential Height Field
	during the Northern Hemisphere Winter” [2]
	“Progress during TOGA in understanding and
	modeling global teleconnections associated with
	tropical sea surface temperatures” [3]
	“Barotropic Wave Propagation and Instability,
	and Atmospheric Teleconnection Patterns” [4]
	“Inter-annual temperature and precipitation variations
	over the Litani Basin in response to
	atmospheric circulation patterns” [5]

, one concerning the implications and one about the teleconnections.

Implications: the book [6] and the paper [7] provided the definitions, properties, and theorems of fuzzy implications. Furthermore, the papers [8,9,10] defined the fuzzy implications via automorphism functions, which became the theoretical basis of this paper. Finally, the book [11] and the papers [12,13] have demonstrated a fuzzy implication in table form, which has been constructed with real climatic data.

Teleconnections: the book [1] researched the teleconnections between Southeastern Europe–Iberian Peninsula and Northwestern Europe–Scandinavia during the winter using statistic methods. Moreover, the paper [2] focused on the teleconnections between the North Atlantic and the Pacific (North America) and created new teleconnection patterns. Furthermore, the papers [3,4] are centered around atmospheric teleconnection patterns which they study with statistic methods. Additionally, the paper [5] investigates the implication of the North Atlantic Oscillation (NAO), El Niño Southern Oscillation (ENSO), Indian Monsoon, and 10 other teleconnection patterns of the Northern Hemisphere.

The main goals of this paper, which are realized by its findings, are:

• To provide a new implementation of fuzzy logic that affects the daily life;
• To propose a new view point on the way we study weather phenomena by confirming the hypothesis that weather patterns observed on climatic data can be proved through fuzzy logic.

Last but not least, the motivation behind this paper was the observation of weather teleconnections present in Europe and the possibility of studying them using fuzzy logic instead of pure statistical methods.

The paper follows the following structure:

The Section 2 “Preliminaries” presents the theoretical background of the paper, such as the definitions and properties of the Fuzzy Negations, Triangular norms, Fuzzy Implications, and automorphism functions.

The Section 3 “Materials and Methods” is the main body of the paper, in which the construction of parametric implications as well as the processing of the Matlab code takes place. Moreover, the empiristic and implication table are generated and compared. Finally, the python code exports the anomalies and their graphs.

The Section 4 “Results” provides a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn. Furthermore, the results and how they can be interpreted in perspective of previous studies and of the working hypotheses is discussed. The findings and their implications are being discussed in the broadest context possible and limitations of the work highlighted. Future research directions are also mentioned.

The Section 5 “Conclusions” highlights the major contribution of this study and the limitations of the proposed methodology. The description of future research directions is also explored.

2. Preliminaries

In this section, the definitions and basic properties of the negation, conjunction, and implication operators in fuzzy logic are provided. The concepts of automorphism is used throughout the whole paper.

2.1. Fuzzy Negations

Some definitions retrieved from the literature can be found in the following references: (Baczyński M., 1.4.1–1.4.2 Definitions, pp. 13–14, [6]), (Bedregal B.C., p. 1126, [14]), (Fodor J., 1.1–1.2 Definitions, p. 3, [15]), (Gottwald S., 5.2.1 Definition, p. 85, [16]), (Weber S., 3.1 Definition, p. 121, [17]) and (Trillas E., p. 49, [18]).

Definition 1.

A function $N:(0,1)\to [0,1]$ is called a Fuzzy negation if

• $\left(N1\right)N\left(0\right)=1,N\left(1\right)=0;$
• $\left(N2\right)Nisdecreasing.$
• A fuzzy negation N is called strict if, in addition to the former properties, the following apply:
• (N3) N is strictly decreasing;
• (N4) N is continuous.
• A fuzzy negation N is called strong if the following property is satisfied:
• $\left(N5\right)N\left(N\right(x\left)\right)=x,x\in [0,1].$

The following

Table 2. Basic fuzzy negations classes.

Designation	Equation
Sugeno class	$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda \in (-1,+\infty )$
Yager class	$N^W\left(x\right)={\left(1-x^w\right)}^{\frac{1}{w}},w\in (0,+\infty )$

presents two well-known families of fuzzy negations. Those fuzzy negations can be found in the work by Baczyński M., p. 15, [6].

2.2. Triangular Norms (Conjunctions)

The following definition can be found in: (Klement E.P et al., 1.1 Definition, pp. 4–10, [19]), (Baczyński M., 2.1.1, 2.1.2 Definitions, pp. 41–42, [6]) and (Weber S., 2.1 Definition, pp. 116–117, [17]).

Definition 2.

A function $T:{[0,1]}^2\to [0,1]$ is called a triangular norm, shortly, t-norm, if it satisfies, for all $x,y\in [0,1]$, the following conditions:

• $\left(T1\right)T(x,y)=T(y,x),\left(commutativity\right);$
• $\left(T2\right)T(x,T(y,z\left)\right)=T\left(T\right(x,y),z),\left(associativity\right);$
• $\left(T3\right)ify\le z,thenT(x,y)\le T(x,z),\left(monotonicity\right);$
• $\left(T4\right)T(x,1)=x,\left(boundarycondition\right).$

In the following

Table 3. Basic t-norms.

Designation	Equation
Minimum	$T_M(x,y)=min\{x,y\}$
Algebraic product	$T_p(x,y)=x·y$
Lukasiewicz	$T_{LK}(x,y)=max(x+y-1,0)$

, three well-known t-norms are presented. Those t-norms can be found in: (Baczyński M., p. 42, [6]).

2.3. Fuzzy Implications

The fuzzy implication functions are probably some of the main functions in fuzzy logic. They play a similar role to that played by classical implications in crisp logic. The fuzzy implication functions are used to execute any fuzzy “if–then” rule on fuzzy systems. The following definition can be found: (Baczyński M., p. 2, [6]) and (Fodor J., p. 299, [20]).

Definition 3.

A binary operator $I:{[0,1]}^2\to [0,1]$ is said to be an implication function, or an implication, if, for all $x,y\in [0,1]$, it satisfies:

$\begin{array}{c}\left(I1\right):I\left(x,z\right)\ge I\left(y,z\right)whenx\le y,thefirstplaceantitonicity;\hfill \\ \left(I2\right):I\left(x,y\right)\le I\left(x,z\right)wheny\le z,thesecondplaceisotonicity;\hfill \\ \left(I3\right):I\left(0,0\right)=1,boundarycondition;\hfill \\ \left(I4\right):I\left(1,1\right)=1,boundarycondition;\hfill \\ \left(I5\right):I\left(1,0\right)=0,boundarycondition.\hfill \end{array}$

A function $I:{[0,1]}^2\to [0,1]$ is called a fuzzy implication only if it satisfies $\left(I1\right)$–$\left(I5\right)$. The set of all these fuzzy implications will be denoted by $FI$.

2.4. Automorphism Functions

Automorphism functions play an instrumental role in fuzzy connectives. This is the case because they are necessary for their generalization.

The following definition can be found in: (Bedregal B., p. 1127, [14]) and (Bustince H, B., p. 211, [8]).

Definition 4.

A mapping $\phi :[a,b]\to [a,b]\left(\right[a,b]\subset \mathbb{R})$ is an automorphism of the interval [a, b] if it is continuous and strictly increasing and satisfies the boundary conditions: $\phi \left(a\right)=aand$ $\phi \left(b\right)=b$. If φ is an automorphism of the unit interval, then ${\phi }^{-1}$ is also an automorphism of the unit interval.

In the following

Table 4. Basic automorphism functions.

Designation	Equation
polynomial	$\phi \left(x\right)=x^n$
implicit	$\phi \left(x\right)=\sqrt[n]{x}$

, two well-known automorphism functions are presented.

3. Materials and Methods

Based on weather changes, Europe can be dissected into two regions, North-West Europe (Region A) and South-East Europe (Region B). Especially, it has been observed that if the summer months of Region A are cold and wet, then the corresponding months of Region B would be hot and dry. This observation motivated the creation of this paper, which studies in depth the weather data (temperature, precipitation) of those regions. The data mentioned were retrieved from the European Union’s Earth Observation Programme Copernicus and cover the last 72 years (1950–2021). The steps followed in order to experimentally validate this observation are evident below.

• Step 1: Fuzzy Implication Construction;
• Step 2: Defining the linguistic rule;
• Step 3: Defining Region A and Region B;
• Step 4: Retrieving the average daily temperature values per six hours of the summer months of both Region A and Region B;
• Step 5: Calculating the average monthly temperature values of both Region A and Region B for the last 72 years;
• Step 6: Calculating the long-time average monthly temperature values of both Region A and Region B;
• Step 7: Locating the temperature anomalies of both Region A and Region B;
• Step 8: Plotting the graphs of the temperature anomalies for Region A and Region B;
• Step 9: Validation of the linguistic rule.

Each step is explained in depth in the following paragraphs:

Step 1: Based on the results of our recently published paper “Generalization of Fuzzy Connectives” (see [10]) twelve new fuzzy implications were constructed.

The following theorems prove on an axiomatic basis the above mentioned stages with the help of Theorem 6 of [10].

Theorem 1.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_M:[0,1]\to [0,1]$ function, which is defined $T_M(x,y)=min\{x,y\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^1}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^1}_{\phi }(x,y)=\sqrt[n]{\frac{1-min\left\{x^n,\frac{1-y^n}{1+\lambda ·y^n}\right\}}{1+\lambda ·min\left\{x^n,\frac{1-y^n}{1+\lambda ·y^n}\right\}}}$$

(1)

Proof. 

$$\begin{array}{cc}& {I^1}_{\phi }(x,y)=\sqrt[n]{N^{\lambda }\left(T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n]{\frac{1-T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}}=\\ & \sqrt[n]{\frac{1-min\left\{x^n,\frac{1-y^n}{1+\lambda ·y^n}\right\}}{1+\lambda ·min\left\{x^n,\frac{1-y^n}{1+\lambda ·y^n}\right\}}}\end{array}$$

□

By applying the climatic data to the first fuzzy implication, the graph Figure 1 is generated.

Theorem 2.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_P:[0,1]\to [0,1]$ function, which is defined $T_P(x,y)=x·y$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^2}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^2}_{\phi }(x,y)=\sqrt[n]{\frac{1-x^n·\frac{1-y^n}{1+\lambda ·y^n}}{1+\lambda ·x^n·\frac{1-y^n}{1+\lambda ·y^n}}}$$

(2)

Proof. 

$${I^2}_{\phi }(x,y)=\sqrt[n]{N^{\lambda }\left(T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n]{\frac{1-T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}}=\sqrt[n]{\frac{1-x^n·\frac{1-y^n}{1+\lambda ·y^n}}{1+\lambda ·x^n·\frac{1-y^n}{1+\lambda ·y^n}}}$$

□

By applying the climatic data to the second fuzzy implication, the graph Figure 2 is generated.

Theorem 3.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_{LK}:[0,1]\to [0,1]$ function, which is defined $T_{LK}(x,y)=max\{x+y-1,0\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^3}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^3}_{\phi }(x,y)=\sqrt[n]{\frac{1-max\left\{x^n+\frac{1-y^n}{1+\lambda ·y^n}-1,0\right\}}{1+\lambda ·max\left\{x^n+\frac{1-y^n}{1+\lambda ·y^n}-1,0\right\}}}$$

(3)

Proof. 

$$\begin{array}{cc}& {I^3}_{\phi }(x,y)=\sqrt[n]{N^{\lambda }\left(T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n]{\frac{1-T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}}=\\ & \sqrt[n]{\frac{1-max\left\{x^n+\frac{1-y^n}{1+\lambda ·y^n}-1,0\right\}}{1+\lambda ·max\left\{x^n+\frac{1-y^n}{1+\lambda ·y^n}-1,0\right\}}}\end{array}$$

□

By applying the climatic data to the third fuzzy implication, the graph Figure 3 is generated.

In the following concept map, Figure 4, each stage of the construction of the first three fuzzy implications is explained visually:

Theorem 4.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_M:[0,1]\to [0,1]$ function, which is defined $T_M(x,y)=min\{x,y\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^4}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^4}_{\phi }(x,y)=\sqrt[n.\omega ]{1-{\left(min\left\{x^n,\sqrt[\omega ]{1-y^{n·\omega }}\right\}\right)}^{\omega }}$$

(4)

Proof. 

$$\begin{array}{cc}& {I^4}_{\phi }(x,y)=\sqrt[n]{N^{\omega }\left(T_M\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n·\omega ]{1-{\left(T_M(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right))\right)}^{\omega }}=\\ & \sqrt[n.\omega ]{1-{\left(min\left\{x^n,\sqrt[\omega ]{1-y^{n·\omega }}\right\}\right)}^{\omega }}\end{array}$$

□

By applying the climatic data to the fourth fuzzy implication, the graph Figure 5 is generated.

Theorem 5.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_P:[0,1]\to [0,1]$ function, which is defined $T_P(x,y)=x·y$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^5}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^5}_{\phi }(x,y)=\sqrt[n·\omega ]{1-{\left(x^n·\sqrt[\omega ]{1-y^{n·\omega }}\right)}^{\omega }}$$

(5)

Proof. 

$$\begin{array}{cc}& {I^5}_{\phi }(x,y)=\sqrt[n]{N^{\omega }\left(T_P\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n·\omega ]{1-{\left(T_P\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}^{\omega }}=\\ & \sqrt[n·\omega ]{1-{\left(x^n·\sqrt[\omega ]{1-y^{n·\omega }}\right)}^{\omega }}\end{array}$$

□

By applying the climatic data to the fifth fuzzy implication, the graph Figure 6 is generated.

Theorem 6.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_{LK}:[0,1]\to [0,1]$ function, which is defined $T_{LK}(x,y)=max\{x+y-1,0\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=x^n$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=\sqrt[n]{x}$.

Then, there is a fuzzy implication ${I^6}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^6}_{\phi }(x,y)=\sqrt[n·\omega ]{1-{\left(max\left\{x^n+\sqrt[\omega ]{1-y^{n·\omega }}-1,0\right\}\right)}^{\omega }}$$

(6)

Proof. 

$$\begin{array}{cc}& {I^6}_{\phi }(x,y)=\sqrt[n]{N^{\omega }\left(T_{LK}\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}=\sqrt[n·\omega ]{1-{\left(T_{LK}\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}^{\omega }}=\\ & \sqrt[n·\omega ]{1-{\left(max\left\{x^n+\sqrt[\omega ]{1-y^{n·\omega }}-1,0\right\}\right)}^{\omega }}\end{array}$$

□

By applying the climatic data to the sixth fuzzy implication, the graph Figure 7 is generated.

In the following concept map Figure 8 each stage of the construction of the fuzzy implications 4, 5, and 6 is explained visually:

Theorem 7.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_M:[0,1]\to [0,1]$ function, which is defined $T_M(x,y)=min\{x,y\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^7}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^7}_{\phi }(x,y)={\left(\frac{1-min\left\{\sqrt[n]{x},\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}\right\}}{1+\lambda ·min\left\{\sqrt[n]{x},\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}\right\}}\right)}^n$$

(7)

Proof. 

$$\begin{array}{cc}& {I^7}_{\phi }(x,y)={\left(N^{\lambda }\left(T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)\right)}^n={\left(\frac{1-T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_M\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}\right)}^n=\\ & {\left(\frac{1-min\left\{\sqrt[n]{x},\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}\right\}}{1+\lambda ·min\left\{\sqrt[n]{x},\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}\right\}}\right)}^n\end{array}$$

□

By applying the climatic data to the seventh fuzzy implication, the graph Figure 9 is generated.

Theorem 8.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_P:[0,1]\to [0,1]$ function, which is defined $T_P(x,y)=x·y$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^8}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^8}_{\phi }(x,y)={\left(\frac{1-\sqrt[n]{x}·\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}}{1+\lambda ·\sqrt[n]{x}·\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}}\right)}^n$$

(8)

Proof. 

$$\begin{array}{cc}& {I^8}_{\phi }(x,y)={\left(N^{\lambda }\left(T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)\right)}^n={\left(\frac{1-T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_P\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}\right)}^n=\\ & {\left(\frac{1-\sqrt[n]{x}·\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}}{1+\lambda ·\sqrt[n]{x}·\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}}\right)}^n\end{array}$$

□

By applying the climatic data to the eighth fuzzy implication, the graph Figure 10 is generated.

Theorem 9.

We assume the following:

1. 

A strong negation $N^{\lambda }:[0,1]\to [0,1]$ function, which is defined

$N^{\lambda }\left(x\right)=\frac{1-x}{1+\lambda x},\lambda >-1$;

2. 

A continuous Archemedean and strictly t-norm $T_{LK}:[0,1]\to [0,1]$ function, which is defined $T_{LK}(x,y)=max\{x+y-1,0\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^9}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^9}_{\phi }(x,y)={\left(\frac{1-max\left\{\sqrt[n]{x}+\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}-1,0\right\}}{1+\lambda ·max\left\{\sqrt[n]{x}+\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}-1,0\right\}}\right)}^n$$

(9)

Proof. 

$$\begin{array}{cc}& {I^9}_{\phi }(x,y)={\left(N^{\lambda }\left(T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)\right)\right)}^n={\left(\frac{1-T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}{1+\lambda ·T_{LK}\left(\phi \left(x\right),N^{\lambda }\left(\phi \left(y\right)\right)\right)}\right)}^n=\\ & {\left(\frac{1-max\left\{\sqrt[n]{x}+\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}-1,0\right\}}{1+\lambda ·max\left\{\sqrt[n]{x}+\frac{1-\sqrt[n]{y}}{1+\lambda ·\sqrt[n]{y}}-1,0\right\}}\right)}^n\end{array}$$

□

By applying the climatic data to the ninth fuzzy implication, the graph Figure 11 is generated.

In the following concept map Figure 12 each stage of the construction of the fuzzy implications 7, 8, and 9 is explained visually:

Theorem 10.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_M:[0,1]\to [0,1]$ function, which is defined $T_M(x,y)=min\{x,y\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^{10}}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^{10}}_{\phi }(x,y)={\left(\sqrt[\omega ]{1-{\left(min\left\{\sqrt[n]{x},\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}\right\}\right)}^{\omega }}\right)}^n$$

(10)

Proof. 

$$\begin{array}{cc}& {I^{10}}_{\phi }(x,y)={\left(N^{\omega }\left(T_M\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)\right)}^n={\left(\sqrt[\omega ]{1-{\left(T_M\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}^{\omega }}\right)}^n=\\ & {\left(\sqrt[\omega ]{1-{\left(min\left\{\sqrt[n]{x},\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}\right\}\right)}^{\omega }}\right)}^n\end{array}$$

□

By applying the climatic data to the tenth fuzzy implication, the graph Figure 13 is generated.

Theorem 11.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_P:[0,1]\to [0,1]$ function, which is defined $T_P(x,y)=x·y$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^{11}}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^{11}}_{\phi }(x,y)={\left(\sqrt[\omega ]{1-{\left(\sqrt[n]{x}.\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}\right)}^{\omega }}\right)}^n$$

(11)

Proof. 

$$\begin{array}{cc}& {I^{11}}_{\phi }(x,y)={\left(N^{\omega }\left(T_P\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)\right)}^n={\left(\sqrt[\omega ]{1-{\left(T_P\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}^{\omega }}\right)}^n=\\ & {\left(\sqrt[\omega ]{1-{\left(\sqrt[n]{x}.\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}\right)}^{\omega }}\right)}^n\end{array}$$

□

By applying the climatic data to the eleventh fuzzy implication, the graph Figure 14 is generated.

Theorem 12.

We assume the following:

1. 

A strong negation $N^{\omega }:[0,1]\to [0,1]$ function, which is defined

$N^{\omega }\left(x\right)=\sqrt[\omega ]{1-x^{\omega }},\omega >0$;

2. 

A continuous Archemedean and strictly t-norm $T_{LK}:[0,1]\to [0,1]$ function, which is defined $T_{LK}(x,y)=max\{x+y-1,0\}$;

3. 

A automorphism φ function, which is defined $\phi \left(x\right)=\sqrt[n]{x}$, with inverse ${\phi }^{-1}$ function, which is defined ${\phi }^{-1}\left(x\right)=x^n$.

Then, there is a fuzzy implication ${I^{12}}_{\phi }:[0,1]\to [0,1]$, which defined

$${I^{12}}_{\phi }(x,y)={\left(\sqrt[\omega ]{1-{\left(max\left\{\sqrt[n]{x}+\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}-1,0\right\}\right)}^{\omega }}\right)}^n$$

(12)

Proof. 

$$\begin{array}{cc}& {I^{12}}_{\phi }(x,y)={\left(N^{\omega }\left(T_{LK}\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)\right)}^n{=\left(\sqrt[\omega ]{1-{\left(T_{LK}\left(\phi \left(x\right),N^{\omega }\left(\phi \left(y\right)\right)\right)\right)}^{\omega }}\right)}^n=\\ & {\left(\sqrt[\omega ]{1-{\left(max\left\{\sqrt[n]{x}+\sqrt[\omega ]{1-{\left(\sqrt[n]{y}\right)}^{\omega }}-1,0\right\}\right)}^{\omega }}\right)}^n\end{array}$$

□

By applying the climatic data to the twelfth fuzzy implication, the graph Figure 15 is generated.

In the following concept map Figure 16 each stage of the construction of the fuzzy implications 10, 11, and 12 is explained visually:

The twelve constructed implications cover a big spectrum of parametric implications since the use: the two most well-known automorphism functions Table 4, the two most well-known strong negation classes Table 2, as well as the three most well-known t-norms (Table 3).

Step 2: Since the necessary fuzzy implications have been constructed the next step is defining the linguistic rule. To be more specific, the linguistic rule states that: “A warm and dry summer at Region A implies a cold and wet summer at Region B” and reverse. This rule is necessary for the process of normalization of the climatic data.

Step 3: This step involves the more accurate determination of Region A and Region B. Especially, in previous sections the north-west Europe has been designated as Region A and the south-east Europe as Region B. In order, though, for a more detailed case study a more accurate designation was necessary. For this reason two cities of each region that represent its climate were chosen: London and Stockholm from Region A and Athens as well as Madrid from Region B.

Step 4: This step involves retrieving the average daily temperature values per six hours of the summer months of both Region A and Region B. In part, this was achieved using the European Union’s Earth Observation Programme Copernicus [21,22] which provided the necessary climate data. Furthermore, a python program which could process and sort the data was created [23].

Steps 5, 6, 7, and 8: The actions involved in steps five to eight are carried out by the python code. To be more specific, the code, calculates the average daily temperature values of the cities mentioned in Step 3. Then, it calculates the long-time average temperature values and subtracts them from the average daily temperature ones. The results of these subtractions are called anomalies. Finally, the code [23] plots the anomalies’ graph for each summer month Figure 17, Figure 18 and Figure 19 and for summer as a whole Figure 20. The mentioned graphs emphasize the linguistic rule, as for the majority of the years the anomalies concerning the regions studied are opposite to each other.

Step 9: This step is centered around the validation of the linguistic rule, which was achieved through the creation of a Matlab script [23]. With the help of this script, a empiristic table with the purpose of acting as a representation of the real climatic conditions as it employed the climatic data without any implication’s intervention was constructed. The process of the creation of this table can be described in the following five stages:

Stage 1: The climatic data of London from Region A and Athens from Region B were loaded into the Matlab script.

Stage 2: The method (Fuzzy c-means Clustering FCM) FCM Clustering was used in order to classify the data. To be more specific, firstly the three clusters were defined. Then, an initial guess for the cluster centers was performed which is most likely incorrect. Thereafter, the FCM assigns to every data point a membership grade for each cluster and iteratively moves the cluster centers to the right location within a dataset by iteratively updating the cluster centers and the membership grades for each data point. This iteration is based on minimizing an objective function that represents the distance from any given data point to a cluster center weighted by the membership grade of that data point.

Stage 3: The data were normalized using the (Mamdani Fuzzy Inference Systems) FISType mamdani method. To be more specific, the normalization process was executed in three parts. Firstly, the fuzzy partitions for the fuzzification process were prepared. In order to prepare the fuzzy partitions the classified data were loaded into a 216 × 2 table. The dimensions of the table are based on the following rules. The reason behind choosing the number 216 for the rows is the multiplication of the number of years of the climatic years collected with the number of the months studied (72 years × 3 months). The number of the columns is two, as one is for the data input (London) and the other is for the data output (Athens). Thereafter, the table data were assigned to membership functions. The membership functions used were the Gauss functions because they describe in the best way possible the climatic data. Secondly, the construction of the fuzzy rule set by the Matlab script has taken place (Figure 21).

Finally, for the defuzzification process, the method centroid was used.

Stage 4: The Sturges rule was applied and the data were divided into eight classes.

$c=1+{log}_{2}n\stackrel{n=216}{\iff }c=1+{log}_2\left(216\right)\iff c=1+\frac{log\left(216\right)}{log\left(2\right)}\iff c=7.75$.

c = 7.75 rounding c = 8 classes.

Stage 5: Finally, a nine by nine table was created which contained:

• In the first row the medians of the Athens data classes;
• In the first column the medians of the London data classes;
• The rest rows and columns were filled with the appropriate data.

Then, following the same steps, the twelve fuzzy implications constructed in Step 1 were used in order to create another twelve tables. Thereafter, the newly created tables were subtracted from the empiristic table and the norm of each table was calculated. Finally, the table with the minimum norm

Table 5. Table of the first implication.

NaN	0.66	0.99	0.84	0.85	0.99	0.95	0.98	0.63
0.69	0.95	0.99	0.95	0.95	0.99	0.95	0.98	0.95
0.99	0.66	0.99	0.84	0.85	0.99	0.95	0.98	0.63
0.86	0.83	0.99	0.84	0.85	0.99	0.95	0.98	0.83
0.98	0.66	0.99	0.84	0.85	0.99	0.95	0.98	0.74
0.93	0.74	0.99	0.84	0.85	0.99	0.95	0.98	0.74
0.73	0.94	0.99	0.94	0.94	0.99	0.95	0.98	0.94
0.98	0.66	0.99	0.84	0.85	0.99	0.95	0.98	0.63
0.58	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99

was chosen as the most appropriate for our case study.

In the following concept map Figure 22 each step of the process of validating of the linguistic rule is explained visually:

4. Results

One of the results of this paper, is the confirmation of the hypothesis that weather patterns observed on climatic data can be proved through fuzzy logic. Furthermore, this paper results in the creation of a new approach, which is complementary to standard statistical approaches, that can give straightforward, easy to interpret measures, on the validity of a broad number of hypotheses that emerge in the complex Earth climate system. Published research, as well as proven existing hypotheses, have approached the subject of this paper using mainly statistics (see [1,2,3,4,5]), in contrast with the strategy presented in this study which used mainly fuzzy logic. Even though both the results of previous studies, as well as those of this paper, are the same, the strategy followed by the latter displays multiple benefits over the others. To be more specific, the strategy involving fuzzy logic has the ability to be used to verify a variety of hypotheses concerning the Earth’s climate. Furthermore, there are no visible limitations concerning this strategy as it can easily be converted in order to interpret a wide variety of climate related hypotheses. Finally, the research completed in this paper can be expanded in multiple directions. Especially, the hypothesis researched in this study with the help of temperature data can also be proved using precipitation data. Moreover, the methodology and data used in the current paper can assist in the creation of weather forecasting models.

5. Conclusions

The major contribution of this paper is demonstrating that patterns observed on weather data (teleconnections) can be validated through fuzzy logic. The innovative aspect of this approach is the use of fuzzy logic, which in comparison with the pure statistical methods used till now can confirm a teleconnection-related hypothesis in less time while offering more flexibility. Furthermore, the strategy mentioned has no visible limitations because of the large number of parametric implications that can be created with it and can validate a teleconnection hypothesis. Finally, the research completed in this paper can be expanded in multiple directions. Especially, the hypothesis researched in this study with the help of temperature data can also be proved using precipitation data. Moreover, the methodology and data used in the current paper can assist in the creation of weather forecasting models.