## PCA multivariate signal processing applied to neural data

### What you’ll learn

Understand advanced linear algebra methods

Includes a 3+ hour “crash course” on linear algebra

Apply advanced linear algebra methods in MATLAB and Python

Simulate multivariate data for testing analysis methods

Analyzing multivariate time series datasets

Appreciate the challenges neuroscientists are struggling with!

Learn about modern neuroscience data analysis

### Requirements

Some linear algebra background (3+ hour crash course is provided)

Some neuroscience background (or interest in learning!)

Some MATLAB/Python programming experience (only to complete exercises)

Interest in learning applied linear algebra

### Description

What is this course all about?Neuroscience (brain science) is changing — new brain-imaging technologies are allowing increasingly huge data sets, but analyzing the resulting Big Data is one of the biggest struggles in modern neuroscience (if don’t believe me, ask a neuroscientist!). The increases in the number of simultaneously recorded data channels allows new discoveries about spatiotemporal structure in the brain, but also presents new challenges for data analyses. Because data are stored in matrices, algorithms developed in linear algebra are extremely useful. The purpose of this course is to teach you some matrix-based data analysis methods in neural time series data, with a focus on multivariate dimensionality reduction and source-separation methods. This includes covariance matrices, principal components analysis (PCA), generalized eigendecomposition (even better than PCA!), and independent components analysis (ICA). The course is mathematically rigorous but is approachable to individuals with no formal mathematics background. The course comes with MATLAB and Python code (note that the videos show the MATLAB code and the Python code is a close match).You should take this course if you are a…neuroscience researcher who is looking for ways to analyze your multivariate data.student who wants to be competitive for a neuroscience PhD or postdoc position.non-neuroscientist who is interested in learning more about the big questions in modern brain science.independent learner who wants to advance your linear algebra knowledge.mathematician, engineer, or physicist who is curious about applied matrix decompositions in neuroscience.person who wants to learn more about principal components analysis (PCA) and/or independent components analysis (ICA)intrigued by the image that starts off the Course Preview and want to know what it means! (The answers are in this course!)Unsure if this course is right for you?I worked hard to make this course accessible to anyone with at least minimal linear algebra and programming background. But this course is not right for everyone. Check out the preview videos and feel free to contact me if you have any questions.I look forward to seeing you in the course!

### Overview

Section 1: Introduction

Lecture 1 Target audience and learning from this course

Lecture 2 What is multivariate neuroscience?

Lecture 3 What are linear spatial filters?

Lecture 4 Why spatial filters are useful for neuroscience

Section 2: Download all course materials

Lecture 5 IMPORTANT: Download all course materials

Lecture 6 Download Python code

Section 3: Dimensions and sources

Lecture 7 The concept of “dimension” in measured signals

Lecture 8 The concept of “source” in measured signals

Lecture 9 Sources, mixing, and unmixing

Lecture 10 Dimension reduction vs. source separation

Lecture 11 Linear vs. nonlinear filtering

Lecture 12 Data requirements for source separation

Section 4: Linear algebra crash course

Lecture 13 Introduction to this section

Lecture 14 Vectors and matrices

Lecture 15 Vector multiplications (incl. dot product)

Lecture 16 Matrix multiplications

Lecture 17 MATLAB: vectors and matrices

Lecture 18 Linear independence

Lecture 19 Matrix rank

Lecture 20 Shifting a matrix

Lecture 21 MATLAB: rank and shifting

Lecture 22 Matrix inverse

Lecture 23 A transpose A

Lecture 24 MATLAB: Inverse and AtA

Lecture 25 Eigenvalues/vectors and diagonalization

Lecture 26 The singular value decomposition (SVD)

Lecture 27 SVD for compression

Lecture 28 MATLAB: eig and svd

Section 5: Creating and interpreting covariance matrices

Lecture 29 Using real and simulated data

Lecture 30 Correlation and covariance: terms and matrices

Lecture 31 Creating covariance matrices in data

Lecture 32 MATLAB: covariance of simulated data

Lecture 33 MATLAB: covariance with real data

Lecture 34 Proof: Covariance matrices are symmetric

Lecture 35 Evaluating and improving covariance quality

Lecture 36 MATLAB: Single trial covariance distances

Lecture 37 The quadratic form and the covariance surface

Lecture 38 MATLAB: visualizing the quadratic form

Section 6: Dimension reduction with PCA

Lecture 39 PCA: Goals, objective, and solution

Lecture 40 MATLAB: PCA intuition with 2D data

Lecture 41 How to perform a principal components analysis

Lecture 42 Exercise: PCA on non-phase-locked data

Lecture 43 The geometry of PCA

Lecture 44 Proof of principal component orthogonality

Lecture 45 Scree plots and eigenspectra

Lecture 46 MATLAB: PCA of simulated EEG data

Lecture 47 MATLAB: PCA of real EEG data

Lecture 48 Exercise: Repeat PCA using pca()

Lecture 49 MATLAB: importance of mean-centering for PCA

Lecture 50 Dimension reduction using SVD instead of eigendecomposition

Lecture 51 MATLAB: PCA via SVD and covariance

Lecture 52 PCA for state-space representation

Lecture 53 MATLAB: state-space representation via PCA

Lecture 54 MATLAB: PCA on multitrial data

Lecture 55 Limitations of principal components analysis

Section 7: Source separation with GED

Lecture 56 Tutorial paper on GED

Lecture 57 Hypothesis-driven motivation for GED

Lecture 58 GED: Goals, objective, and solution

Lecture 59 MATLAB: GED intuition with covariance surfaces

Lecture 60 GED weights and nonorthogonality

Lecture 61 MATLAB: GED in a simple example

Lecture 62 Visualizing the spatial filter vs. spatial patterns

Lecture 63 Component sign uncertainty

Lecture 64 MATLAB: Adjusting component signs

Lecture 65 MATLAB: 2 components in simulated EEG data

Lecture 66 Constructing the S and R matrices

Lecture 67 MATLAB: Task-relevant component in EEG

Lecture 68 MATLAB: Spectral scanning in MEG and EEG

Lecture 69 Two-stage compression and source separation

Lecture 70 Exercise: Two-stage source separation in real EEG data

Lecture 71 ZCA prewhitening

Lecture 72 MATLAB: Simulated data with and without ZCA

Lecture 73 Exercise: ZCA+two-stage separation on real EEG data

Lecture 74 Source separation with nonstationary covariances

Lecture 75 MATLAB: Simulated EEG data with alternating dipoles

Lecture 76 Regularization: Theory, math, and intuition

Lecture 77 MATLAB: Effects of regularization in real data

Lecture 78 Empirical methods for regularization amount

Lecture 79 MATLAB: Regularization cross-validation

Lecture 80 Complex-valued solutions

Lecture 81 MATLAB: GED vs. factor analysis

Section 8: Source separation for steady-state responses

Lecture 82 The steady-state evoked potential

Lecture 83 Motivations for a spatial filter for the steady-state response

Lecture 84 RESS analysis pipeline

Lecture 85 MATLAB: example with real EEG data

Section 9: Independent components analysis (ICA)

Lecture 86 Overview of independent components analysis

Lecture 87 MATLAB: Data distributions and ICA

Lecture 88 MATLAB: ICA, PCA, GED on simulated data

Lecture 89 MATLAB: Explore IC distributions in real data

Section 10: Overfitting and inferential statistics

Lecture 90 What is overfitting and why is it inappropriate?

Lecture 91 Unbiased filter creation and application

Lecture 92 Cross-validation (in- vs. out-of-sample testing)

Lecture 93 Permutation testing

Lecture 94 MATLAB: Permutation testing

Section 11: Big questions in multivariate neuroscience

Lecture 95 Math, physiology, and anatomy

Lecture 96 Functional networks vs. volume conduction

Lecture 97 Interpreting individual differences

Lecture 98 A surfeit of source separation selections (and a reading list!)

Lecture 99 Is reducing dimensionality always good?

Section 12: Bonus section

Lecture 100 Bonus lecture

Anyone interested in next-generation neuroscience data analyses,Learners with interest in applied linear algebra to modern big-data challenges,Neuroscientists dealing with “big data”,Mathematicians, engineers, and physicists who are interested in learning about neuroscience data

#### Course Information:

Udemy | English | 17h 33m | 6.42 GB

Created by: Mike X Cohen

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