## 2 Foundational Concepts and Technologies
### Elliott Wave Principle  
The Elliott wave principle (EWP), developed by [PERSON] in the 1930s, marked a significant advancement in technical analysis. Through extensive market observation, [PERSON] identified repetitive patterns in stock market price movements driven by collective investor psychology [1; 4]. These patterns, called waves, form the foundation of the EWP and provide a foundation for analyzing market behavior across multiple timescales.  
According to the EWP, market movements can be characterized by two primary types of wave patterns. As shown in Figure 3, impulsive waves, consisting of five distinct subwaves (labeled 1-5), move in the direction of the main trend. Within these impulsive sequences, waves 1, 3, and 5 propel the market in the trend direction, while waves 2 and 4 represent necessary countertrend moves. Complementing these are corrective waves, comprising three subwaves (labeled A-B-C), which move against the predominant trend, providing market retracements.  
Each wave in Elliott theory can be broken into smaller versions of the same patterns, exhibiting a fractal structure [1], where each wave contains smaller versions of the same patterns and simultaneously forms part of larger wave structures. This hierarchical organization enables traders to analyze market movements across multiple timeframes, from intraday fluctuations to multi-year trends, providing the possibility of understanding market behavior at different scales.  
The mathematical foundation of the EWP is integrated with Fibonacci relationships [16]. The Fibonacci sequence, where each number is the sum of the previous two, generates important ratios that appear not only in financial markets but also in nature, art and architecture. The most important of these is the golden ratio (known as \(\varphi\), approximately 1.618) and its reciprocal 0.618. These ratios help quantify relationships between different waves, with wave 2 often retracing 50% or 61.8% of wave 1, wave 3 frequently extending to 1.618\(\times\) the length of wave 1, and wave 4 typically retracing 38.2% of wave 3 [1; 4]. Figure 4 present  
Figure 3: The fractal character of Elliott wave pattern, adapted from Refs. [4; 15].  
these ratios, applied on the stock market chart, and as we can see, the lines often coincide with local tops and bottoms, predicting where the price of the asset may stop.  
While the EWP offers a framework for market analysis, its implementation faces several significant challenges [2]. The main challenge lies in wave pattern identification--different analysts often interpret the same market data differently, leading to varying predictions. This subjectivity is complicated by the theory's rules, which lack precise criteria for systematic testing.  
Several other limitations affect the theory's effectiveness. The complexity of wave patterns and their variations can lead to overfitting, where analysts might adjust wave counts to match known market outcomes rather than make reliable predictions [17]. The theory's focus on market psychology may overlook important fundamental factors like economic indicators and geopolitical events that influence price movements. Wave patterns become particularly difficult to identify in sideways markets [4], reducing the theory's effectiveness in these conditions.  
These challenges are especially relevant in today's high-speed trading environment, where algorithmic trading and rapid information flow affect market behavior. Traders using wave patterns and Fibonacci relationships must also be aware of cognitive biases, particularly confirmation bias, which can influence pattern interpretation [3]. The lack of widespread acceptance among professional traders and economists further highlights the challenges in applying the theory consistently.