Numerical Methods and Bootstrap in R

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Overview In the project, my colleague Mathieu Graf and I engaged with advanced statistical concepts and numerical methods to explore and analyze mathematical functions. The project was centered around the function f(x) = exp(x)/(1+exp(x)), for which we developed a third-order Taylor expansion and applied Newton’s method to find its maximum.

Objectives The primary objectives of the project were to:

  • Derive the Taylor series expansion for a given function and understand the approximation of functions around a point.
  • Implement Newton’s method to optimize functions, specifically to find the maximum of the targeted function.
  • Employ R programming skills to perform mathematical computations and generate relevant visualizations.

Learning Outcomes Through the completion of this project, I have gained valuable insights and deepened my understanding of several key areas:

  • Mathematical Proficiency: Enhanced my grasp of Taylor series and their practical applications in approximating complex functions.
  • Algorithmic Thinking: Developed a clear understanding of iterative methods like Newton’s method for root-finding and optimization problems.
  • Computational Skills: Solidified my R programming capabilities, including writing functions, controlling flow, and utilizing packages for numerical analysis.
  • Data Visualization: Improved my ability to visualize mathematical concepts and results using R’s plotting capabilities, which is crucial for data analysis and interpretation.
  • Problem-Solving: Strengthened my problem-solving skills by translating mathematical theory into practical algorithms and code.
  • Collaboration: Fostered teamwork and collaborative problem-solving by partnering with a peer to tackle complex mathematical challenges.

Technical Skills Demonstrated

  • Proficient use of R Markdown for reproducible research and documentation.
  • Application of numerical methods in a programming environment.
  • Effective communication of mathematical and statistical concepts through clear and concise code comments and documentation.

Conclusion “Project LSTAT2185” was a comprehensive exercise in applying numerical methods to real-world statistical problems. The successful implementation of the Taylor expansion and Newton’s method not only served to confirm theoretical knowledge but also to refine programming skills in R that are widely applicable in data science and analytics roles. This project stands as a testament to my analytical thinking and my ability to harness computational tools to drive decision-making and problem-solving in the field of statistics.