A repository of tutorials to revise mathematical concepts required for statistics and machine learning

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Readme

This repository contains tutorials on the introductory mathematical concepts required for studying statistics and machine learning. Code to solve mathematical problems is written in `R`

, `Python`

and `Julia`

.

Topics | Tutorials |
---|---|

🔢 | Introduction to numbers (Updated) |

🔢 | Introduction to algebra |

🌗 | Introduction to set theory (Updated) |

:compass: | Introduction to trigonometry |

🍪 | Introduction to summations |

🍪 | Introduction to combinatorics (Updated) |

🔢 | Introduction to functions |

🎢 | Introduction to derivatives |

🎢 | Introduction to integration |

🎢 | Differential equations |

🎢 | Multivariable functions |

🎢 | Differentiation of multivariable functions |

🔢 | Exponents and logarithms |

🔢 | Logarithms and information theory |

🃏 | Introduction to probability theory |

🃏 | Conditional probability |

🃏 | Bayes theorem |

:compass: | Introduction to distance metrics |

:compass: | Cosine similarity applications |

:chopsticks: | Introduction to linear systems |

:chopsticks: | Introduction to vectors |

:chopsticks: | Vector norms and embeddings |

🏬 | Introduction to matrices |

:chopsticks: | Linear transformations |

:chopsticks: | Applications of eigenvalues and eigenvectors |

This project was created using the following setup:

- R package dependencies are managed using renv for R version 4.1.2 (2021-11-01).
- Python virtual environment and package dependencies are managed using
`poetry`

for`Python 3.9.6`

. A local version of`Python 3.9.6`

was installed and activated using`pyenv local 3.9.6`

via the terminal. - The Julia version used is
`julia version 1.7.3`

.

Writing mathematical proofs might feel archaic but they are a great way to help you reason why mathematical concepts should behave consistently (and not just because your textbook says so). There are multiple approaches to proving a mathematical statement or concept. Sadly, there is no magical rule to selecting the correct method for each scenario - mathematicians often have to try multiple approaches before they find the right one.

**Direct proof**

- Occurs when you need to prove that A and B are equivalent.
- Start by assuming A is true.
- Construct a definition statement for A (use a fixed but arbitary example of A).
- Extend and simplify mathematical definitions derived from A to reach B.
- When you are asked if A and
**only**A is true, then B is true, first suppose A to reach B. Then suppose B to reach A.

**Induction proof**

- Occurs when you need to prove that something is true for all cases.
- Start by proving the base case when $n = 1$.
- Assume that the case is also true for some integer $k$.
- Prove that the case for $k + 1$ also holds i.e. prove the next incremental step up a ladder stretching to infinity.

**Uniqueness proof**

- Occurs when you need to prove that a solution is unique.
- Show that there is one solution first.
- Show that there is a second solution and that the first and second solutions must be equal.

**Proof by contradiction**

- Start by assuming that the incorrect state is true i.e. that eigenvectors are linearly dependent.
- Prove that the assumption does not hold and contradicts itself.
- Therefore prove that the reverse state is actually true.

- A guide to linear algebra for applied machine learning by Pablo Caceres
- The Mathematics for Machine Learning textbook by Marc Peter Deisenroth, A Aldo Faisal and Cheng Soon Ong - Cambridge University Press
- The Probability for Data Science textbook by Stanley H Chan - Michigan Publishing
- The Probabilistic modelling tutorials by Michael Betancourt - GitHub
- Writing mathematical operations in LaTex/R - Wikibooks
- Introduction to university mathematics [YouTube lecture series ]https://www.youtube.com/playlist?list=PL4d5ZtfQonW1xKVEtYJd1iu9m52ATG7SV by the Department of Mathematics - Oxford University.

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