# Analysis and Parameterization of Sports Performance: A Case Study of Soccer
## 3 Theoretical Framework
### Fuzzy Logic  
Today, traditional logic is struggling to navigate numerous challenges in the sphere of Artificial Intelligence (AI). The human cognitive process is inherently ambiguous, lacking clear-cut distinctions of true or false. However, computing systems historically employ this form of conventional logic. The impediment of classical logic underscores the relevance of fuzzy logic [27; 28]. Fuzzy logic is a component of AI that equips computers with the facility to examine real-world data on a graduated scale between false and true, manipulate nebulous concepts and provide engineers with the means to create devices that evaluate hard-to-define information. Fuzzy logic, within the context of AI, helps resolve a wide range of issues, predominantly those tied to intricate industrial process control, decision-making systems, and data resolution and compression. Systems built on fuzzy logic can mirror human decision-making processes with the added benefit of superior speed. They are typically robust and capable of handling inaccuracies and noise in the input data. The time factor plays a significant role because these control systems may require feedback within specified timelines, making previous data essential for an average evaluation of prior circumstances. Fuzzy logic endeavors to craft mathematical approximations for resolving issues by generating precise outcomes from imprecise data, making it particularly beneficial in electronics and computation. The term \"fuzzy\" is a nod to the inherent incertitude associated with the non-deterministic truth values employed. The concept of fuzzy sets was first introduced by [PERSON] in his 1965 research paper titled \"Fuzzy Sets\" [29]. This publication establishes the principles of fuzzy logic extrapolated from the definition and properties of fuzzy sets.  
\"Let \(X\) be a classical set. A fuzzy set, \(A\), in \(X\) is characterized by the membership function \(fA(x)\), which associates to each point \(x\in X\) a real number from the interval \([0,1]\), where the values of \(fA(x)\) represent the degree of membership of \(x\) in \(A\), such that, the closer the value of \(fA(x)\) is to \(1\), the greater the degree of membership of \(x\) to \(A\)\".  
In view of the above, the membership functions utilized will be assigned based on the particular features of the reality to which they correspond. Nonetheless, some classical functions, as suggested in [30], are often employed. From a pragmatic perspective, the requirement is to convert vague linguistic ideas into fuzzy numerical variables. Following this, they need to be amalgamated utilizing the fuzzy logic operators, derive every rule's deduction in a numerical format, compile the different reactions of the distinct rules, and finally return from the gathered numerical response to the linguistic one. This procedure complies with a model scheme that facilitates the management of inference rules on fuzzy sets, which is succinctly illustrated in Figure 2:  
Figure 2: Fuzzy Logic System.  
* **Fuzzifier:** this component modifies clear, definitive numerical inputs into indeterminate variables using a process that we call fuzzification.
* **Knowledge base (fuzzy rules):** this is essentially a repository where all the expert-derived principles are saved.
* **Inference engine:** this module emulates human cognitive processes by making logical deductions based on received inputs and rule sets.
* **Defuzzifier:** this final section takes the ambiguously determined variable created by the inference engine and converts it back to a clear numerical value for future use.