## BIOINF 501 001 FA 2024

Fall 2024 BIOINF 501

Mathematical Foundations for Bioinformatics

Overview

The course covers some of the mathematical prerequisites to understand fundamental techniques in bioinformatics, biomedical research, and machine learning. The goal is to introduce students to a wide variety of topics, so that they will be able to understand their applications in future courses and/or they will have a basic understanding to build upon should they need to dig deeper into certain topics. Each module contains topics normally covered over an entire semester, so we inevitably won't be able to delve into every detail. On the other hand, the focus will be on computations and using software to solve problems.

Topics include: 1) introduction to multi-variable calculus, complex numbers, and probability; 2) foundations of linear algebra, such as linear systems, eigenvalues and eigenvectors, matrix algebra, least square solutions, singular value decomposition and applications; 3) introduction to differential equations and dynamical systems, such as existence and uniqueness of solutions, linear systems, bifurcations; 4) optimization methods, such as free and constrained optimization, Lagrange multipliers, data denoising using optimization and heuristic methods. Demonstrations using MATLAB, R, and Python will be included throughout the course.

All classes are in person.

Instructors
Cristian Minoccheri and Ivo Dinov

TA: Elysia Chou

Start Time, End Time and Location

Lecture: Mon and Wed 1:30PM - 3:00PM, Rm. 3813 Med Sci II Bldg.

Lab/Discussion: Mondays @ 3:00 – 4:00 PM in Rm. 3813 Med Sci II Bldg.

This course will be an in-person class.

Topics/Modules

Module 1: Multivariable Calculus and Probability

Introduction to multivariable functions, gradients, matrix and vector calculus, multiple integrals

Axioms of probability

Conditional probability, sum and product rules, Bayes formula

Discrete and continuous distributions, joint distributions

Conditional distributions, sum and product rules for distributions

Expectation

Taught by: Cristian Minoccheri

Module 2: Linear Algebra

Linear systems, Matrix algebra,Linear transformations

Linear subspaces, dimension, linear projections

Orthogonality, least squares solutions

Determinants, eigenvalues and eigenvectors, rank

Markov chains

Symmetric matrices and diagonalization

Singular value decomposition and applications

Principal component analysis

Spectral graph theory

Taught by: Cristian Minoccheri

Module 3: Differential Equations

Introduction to differential equations

Systems of linear equations

Dynamical systems, introduction to bifurcation theory

Taught by: Cristian Minoccheri

Module 4: Optimization (November 11 – December 4, 2024)

Data Science and Predictive Analytics EBook (University of Michigan Library)

Free (unconstrained) optimization vs. Constrained Optimization

Foundations of R (Introduction)

Equality and Inequality constraints

Lagrange Multipliers

Manual vs. Automated Lagrange Multiplier Optimization

Data Denoising: Application of computer optimization techniques in medicine and biology

Heuristic methods - Genetic algorithms, simulated annealing

Applications (supervised classification & unsupervised clustering)

Instructor: Ivo Dinov

Duration: 8 lectures

Software tools: Matlab, R, Python, and open-science tools.

## Course Summary:

Date Details Due
This course content is offered under a Public Domain license. Content in this course can be considered under this license unless otherwise noted.