Using the Bernstein approach in functional data analysis: Varianceanalysis and regression analysis

Functional data analysis, unlike traditional statistical methods, is an analysis method developed to process continuously observed data. This method models the structure and relationships between data by creating a function of each observation in data sets, allowing us to better understand complex patterns and variability in data sets. Although functional data analysis was initially developed with mathematical and statistical methods, today it has a wide application area in various fields (health, economy, engineering, etc.). It is a powerful tool used especially in the analysis and modelling of continuous variables. Modern data analysis has benefited greatly from the development of FDA methods and their application to time series data. The aim of this study is to bring a different perspective to functional data analysis. In the first part of the study, the history of functional data analysis, its development until today and its usage areas are mentioned. In the second part, why data analysis is important for our study and the concepts that will form the basis of the study are mentioned. In the third part, it is tried to explain with examples how Bernstein polynomials are used and why they are important when making intermediate value estimates in functional data analysis.