Topics include: Linear algebra, least-squares and total-least-squares regression, eigen-analysis and PCA, linear shift-invariant systems, convolution, Fourier transforms, Nyquist sampling, basics of probability and statistics, hypothesis testing, model comparison, bootstrapping, estimation and decision theory, signal detection theory, linear discriminants, classification, clustering, simple models of neural spike generation, reverse-correlation analysis.

Prerequisites: Algebra, trigonometry, and calculus. Some experience with matrix algebra and/or computer programming is helpful, but not required. The real prerequisites are an aptitude for logical and geometric reasoning, and a willingness to work hard!

Announcements: We use brightspace for class announcements and online questions/discussions: https://brightspace.nyu.edu/d2l/home/604497 . Rather than emailing the instructors or TAs, we encourage you to post your questions/comments there, where they can be discussed and/or answered by any of us or your fellow classmates.

Schedule:   (Notes: labs are in green, content will appear incrementally, for a preview see last year's course page)

Date	Topic	Handouts	Homework
Thu, Sep 3	Introduction to the course Linear algebra I: vectors, operations, vector spaces Zoom recording, whiteboard (pdf)	Course description (pdf)	Homework 0 (pdf - "due" 9/8)
Tue, Sep 8	Linear algebra II: inner products, projection, coordinate systems Zoom recording (unavailable - see last year), whiteboard (pdf)
Thu, Sep 10	Linear algebra III: linear systems, matrix multiplication Zoom recording, whiteboard (pdf)	Slides: Linear algebra (pdf)	Homework 1 (pdf, due 9/24) Data file (Matlab), Homework submission instructions
Fri, Sep 11	Lab: linear algebra basics in matlab/python. Homework preparation	Lab1 (zip) - includes matlab and python
Tue, Sep 15	Linear algebra IV: orthogonal/diagonal matrices, singular value decomposition Zoom recording, whiteboard (pdf)
Thu, Sep 17	Linear algebra V: Extended example - Color vision and trichromacy Zoom recording, whiteboard (pdf)
Fri, Sep 18	Lab: Regression	Lab2 (zip) - includes matlab and python

Resources and policies: Artificial Intelligence agents

MathTools is intended to provide a conceptual framework and a set of tools for working with data and models in neural and cognitive science. The conceptual framework is the most essential component: you must be able to reason about the underlying math, statistics, and algorithms in order to design experiments, analyze data, make predictions, and reach scientific conclusions. We believe that this ability cannot be acquired passively: you must actively explore, reason about, implement, and test these tools in order to fully master their usage.

Artificial Intelligence (AI) is undergoing explosive growth in many areas of human endeavor, and this is also true of science. Whether or not you've already been making use of it, we should all take this opportunity to learn how to use it effectively in a scientific context. But we also know that it is not a panacea: Without careful guidance and verification, AI systems make frequent mistakes, including incorrect or jumbled reasoning, incorrect mathematics, incorrect, inappropriate, or poorly organized software implementations, and incorrect (or no) attribution. As such, they cannot do science for you: you have to guide, monitor, and verify them!

In the context of this course, AI can either help or hinder your learning of the material. In particular, if you ask it to provide answers to homework problems, it may answer incorrectly or incomprehensibly, and in either case you will not benefit. If you choose to make use of AI, here are a few suggestions:

• Before asking AI to help you with an assignment, work through the problem yourself to understand what is being asked and map out a plan for a solution, along with a plan for testing/verifying that solution. Discuss with your peers, or ask for help from TAs or lecturers. The process of wrestling with a problem is extremely valuable for learning, and doing it collaboratively can lead to synergistic improvements, is more enjoyable, and forms lasting connections with your peers and teachers!

• Use AI as an auxiliary tutor, to augment the human learning resources (lecturers and TAs) of the class. For example, ask it to elaborate on ideas presented in lectures or labs, including illustrations, graphical demonstrations, logical or mathematical connections to other methods or concepts you already know about.

• AI can produce code but it may be incorrect, poorly organized, or an implementation of something that is quite different from what you intended. We strongly recommend that you first try to write the code yourself, and then ask for assistance in debugging or verification. Rather than having it fix the bugs, ask it to guide you through the process with questions.
  • Example prompt: “I am taking a class that develops math concepts and tools for use in neuroscience. I want help working on a homework assignment which involves coding in Python. Do not provide me with the answers directly! Instead, I want you to act as a teaching assistant: Ask me questions to gauge my understanding of the material and help guide me through each step of the ideas I have misunderstood or implementation details I have overlooked.”

Resources - online:

• Carlos Fernandez-Granda's Math Tools for Data Science course (here at NYU)!
• Jonathan Pillow's Math Tools course at Princeton (Jonathan was TA for our course in 2000 :)
• Ella Batty's Math Tools for Neuroscience course at Harvard (follows a similar syllabus)
• Online matlab help at The MathWorks | Tutorial at the MathWorks v | Intro video at MIT | Antonia Hamilton's Tutorial | at U. Utah | on reddit
• Richard Johnson's Matlab Style Handbook at MathWorks or Cambridge Press (lots of helpful tips, if a bit idiosyncratic)
• Linear algebra Appendix from PDP series, by Michael Jordan. (pdf)
• Online lecture videos from Gilbert Strang's linear algebra course at MIT
• Online YouTube BlueBrown Linear Algebra videos
• Todd Will's Interactive Intro to the SVD
• The Elements of Statistical Learning, Hastie, Tibshirani and Friedman - Excellent textbook on regression, decision/classification, clustering, and many advanced topics in data fitting and analysis. Available online (pdf)
• Convex Optimization, Boyd and Vandenberg - Excellent textbook on the formulation of, and algorithms for, optimization of convex functions. Available online (pdf)
• Thomas Minka's On-line Glossary of Statistical Pattern Recognition
• Wolfram Research World of Mathematics
• History of various topics in mathematics

Resources - dead trees:

• Matlab:
  Getting Started with MATLAB; A Quick Introduction for Scientists and Engineers, R. Pratap, Oxford U. Press, 2009.
  Matlab for Neuroscientists. An introduction to scientific computing in Matlab, P. Wallisch, M. Lusignan, M. Benayoun, T. Baker, A. Dickey & N. Hatsopoulos, Elsevier Press, 2008.
  Mastering MATLAB, B. L. Littlefield & D. C. Hanselman, Prentice-Hall, 2011.
• Linear algebra / Least squares:
  Linear Algebra and Its Applications, Gilbert Strang. Academic Press, 1980.
  Introduction to Applied Linear Algebra, Stephen Boyd & Lieven Vandenberghe. Cambridge U. Press, 2018.
• Linear (shift-invariant) Systems:
  Discrete-time Signal Processing, A. Oppenheim & R. Schafer. Prentice Hall, 1989.
  The Fourier transform and its applications, R. Bracewell, McGraw Hill Science, 1999.
  Fast Fourier transform and its applications, E. Brigham, Prentice Hall, 1988.
• Probability/Statistics:
  Statistics, Freedman, Pisani, Purves, Norton, 2007 (4th ed.)
  Mathematical statistics, J. E. Freund, Prentice Hall, 1992.
• Bootstrap/Resampling:
  An Intoduction to the Bootstrap, by Bradley Efron and Robert Tibshirani. Chapman & Hall, 1998.
  Resampling Methods: A practical guide to data analysis, by Phillip Good. Birkhäuser, 1999.
• Decision Theory:
  Biology: Elementary Signal Detection Theory, by Thomas D. Wickens. Oxford University Press, 2001.
  Signal Detection Theory and Psychophysics, by David Green & John Swets. Peninsula Publishing, 1988.
  Math: Statistical Decision Theory, by James O. Berger. Springer-Verlag, 1980.
  Chapter 2 of Pattern Classification, by Duda, Hart and Storck. Wiley, 2001.
• Computational/Theoretical Neuroscience:
  Theoretical Neuroscience , by Peter Dayan and Larry Abbott. MIT Press, 2001.
  Spikes: Exploring the Neural Code, by Fred Rieke, David Warland, Rob De Ruyter, & Bill Bialek. MIT Press, 1997.