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 OH (Fridays 9AM, pass: 2²1₂)

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

Duration: about 5 lectures

      

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

Duration: about 10 lectures

 

Module 3: Differential Equations

Introduction to differential equations

Systems of linear equations

               Dynamical systems, introduction to bifurcation theory

Taught by: Cristian Minoccheri

Duration: about 3 lectures

 

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

Linear and Quadratic Programming

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

 

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