Lecture 5.4 Advanced ERP Topics

Summary

The video script delves into extensions of basic linear discriminant analysis (LDA) for machine learning, highlighting its effectiveness on linearly transformed data. It discusses the importance of applying LDA to principal components or independent components without losing information. The script also explores the use of ICA for source time courses, allowing for interpretable classifiers and the introduction of brain-informed constraints. It touches on alternative linear features like wavelet decomposition for pattern matching and the concept of sparsity in feature selection. The discussion extends to nonlinear features, emphasizing the need for proper spatial filtering before feature extraction. The script concludes with considerations for class imbalance and the use of the area under the ROC curve as a performance measure, suggesting direct optimization of classifiers under sophisticated cost functions.

Takeaways

• 📚 The script discusses extensions and tweaks to basic machine learning techniques to enhance their performance.
• 🔍 Linear Discriminant Analysis (LDA) provides the same result whether applied to raw data or its linear transformations, as long as no information is lost.
• 🌐 The script highlights the importance of considering non-linear aspects, such as Independent Component Analysis (ICA), for analyzing EEG data.
• 🔄 It emphasizes that ICA assigns weights to components rather than channels, allowing for more interpretable classifiers.
• 🧠 The script suggests using ICA for relating components to source space in the brain, which can inform constraints on feature selection.
• 📊 The speaker mentions the use of linear features like averages and wavelet decompositions for pattern matching in EEG data.
• 🔑 The concept of sparsity is introduced, where only a small number of coefficients contain the relevant information, which can be leveraged for classifier design.
• 💡 Non-linear features are also discussed, with a focus on the importance of applying non-linear transformations after linear spatial filtering.
• 📉 The script addresses the challenges of dealing with non-linear features in EEG due to issues like volume conduction.
• ⚖️ The importance of considering class imbalance and cost-sensitive classification is highlighted, with suggestions to use measures like the area under the ROC curve.
• 🛠️ The speaker recommends using classifiers that can be directly optimized for performance metrics such as the area under the curve, mentioning the support vector framework as an example.

Q & A

What is the basic technique discussed in the script that is easy to comprehend but can be extended with tweaks and twists?

The basic technique discussed is Linear Discriminant Analysis (LDA), which is a simple yet effective method for classification that can be enhanced with various modifications to improve its performance.

Why does applying LDA to linearly transformed data yield the same result as applying it to the original data?

Applying LDA to linearly transformed data gives the same result because LDA is a linear method. It would simply learn different weights to achieve the same performance, as long as no information is lost during the transformation.

What is the difference between applying LDA to linear and non-linear data?

In the case of linear data, LDA can handle transformations like scaling or channel swapping without affecting performance. However, for non-linear data, such as oscillations, applying LDA directly to the data or to transformed data can yield different results, emphasizing the need for appropriate pre-processing or feature extraction techniques.

How does Independent Component Analysis (ICA) differ from LDA in terms of assigning weights?

ICA assigns weights to every component of the signal rather than to every channel. This allows for a more interpretable classification, as each component can be associated with specific signal sources, such as blinks or muscle activity.

What is the advantage of using ICA in conjunction with LDA for EEG signal classification?

ICA can provide source time courses that relate to the brain's activity, which can be used to inform constraints for LDA, such as excluding components from non-relevant brain regions. This enhances interpretability and allows for more targeted feature selection.

Why might one choose to use wavelet decomposition instead of averages for feature extraction in EEG signals?

Wavelet decomposition can be more suitable when the underlying ERP (Event-Related Potential) has specific forms like ripples. It allows for pattern matching with the time course of the signal, which can capture more nuanced features than simple averages.

How does dimensionality reduction relate to the use of wavelet features in EEG signal classification?

Dimensionality reduction is important when using wavelet features because using all of them would essentially be the same as using the raw data. By selecting a small number of wavelet coefficients that contain the relevant information, the model can become more efficient and focused on the signal of interest.

What is the significance of sparsity in the context of feature selection for EEG signal classification?

Sparsity refers to the idea of using only a small number of non-zero coefficients in the model. This can improve the classifier's performance by focusing on the most informative features and reducing the impact of noise or irrelevant information.

Why is it important to consider the order of operations when extracting non-linear features from EEG data?

The correct order is to first apply a spatial filter to isolate the source signal, then perform non-linear feature extraction, and finally apply a classifier. This sequence ensures that the non-linear properties of the source signal are captured accurately, rather than being distorted by channel-level transformations.

What is the area under the curve (AUC) and how is it used to evaluate classifier performance in imbalanced datasets?

The AUC represents the area under the Receiver Operating Characteristic (ROC) curve, which plots the true positive rate against the false positive rate for different threshold settings. It provides a measure of a classifier's ability to distinguish between classes, especially in situations with imbalanced datasets or when different types of errors have different costs.

How can classifiers be optimized for imbalanced classes or when certain errors are more costly?

Classifiers can be optimized by incorporating cost-sensitive learning, where the misclassification costs are taken into account during training. Additionally, using performance metrics like AUC or F-scores, which are more informative than simple misclassification rates, can help in such scenarios.

Outlines

00:00

🧠 Extensions and Considerations in Machine Learning

This paragraph delves into the nuances of Linear Discriminant Analysis (LDA) and its application to linear and non-linear data transformations. It highlights that LDA yields the same results when applied to linear filters or linearly transformed data, as it learns different weights. The discussion extends to Independent Component Analysis (ICA), which assigns weights to components rather than channels, allowing for interpretations of signal relevance and the introduction of brain-informed constraints. The paragraph also touches on alternative linear features like wavelet decomposition for pattern matching and the concept of sparsity in classifiers, emphasizing the importance of dimensionality reduction and feature selection.

05:01

📊 Addressing Non-linearity and Class Imbalance in EEG Analysis

The second paragraph addresses the complexities of feature extraction in EEG analysis, particularly the distinction between non-linear properties of source time courses versus channels. It emphasizes the correct sequence of spatial filtering, non-linear feature extraction, and classification. The paragraph also discusses the challenges of dealing with arbitrary non-linear features in EEG and mentions the use of ICA for spatial filtering before non-linear feature extraction. Furthermore, it explores the importance of optimizing classifiers under different criteria to account for class imbalance and the cost of different types of errors, introducing the concept of the area under the ROC curve as a performance metric. The paragraph concludes with a mention of classifiers that can be directly optimized for sophisticated cost functions, such as those within the support vector framework.

Keywords

💡LDA

Linear Discriminant Analysis (LDA) is a statistical technique used for dimensionality reduction and classification. In the context of the video, LDA is discussed as a method that can be applied to linear filters or linearly transformed data to achieve the same result, highlighting its feature of being linear and its ability to learn different weights based on the data transformation.

💡Principal Components

Principal Component Analysis (PCA) is a dimensionality reduction technique that transforms data into a set of orthogonal (uncorrelated) variables called principal components. The script mentions PCA as an example of a linear transformation where applying LDA would yield similar results as long as information is not lost, emphasizing the linear nature of the process.

💡Independent Components

Independent Component Analysis (ICA) is a computational technique for separating a multivariate signal into independent, non-Gaussian components. The video script discusses how LDA assigns weights to every component when applied to ICA, allowing for the interpretation of the relevance of different signal components, such as distinguishing between blinks and muscle activity.

💡Dipole Fitting

Dipole fitting is a method used in electrophysiology to estimate the location and orientation of electrical sources in the brain. The script refers to dipole fitting in the context of localizing an independent component to the motor cortex, which aids in the interpretation of classifiers by associating them with specific brain regions.

💡Classifier

In machine learning, a classifier is an algorithm that separates data into different categories. The video discusses how classifiers can be interpreted more easily when using ICA components and how they can incorporate constraints informed by the components' locations in the brain, relating to the theme of enhancing classification through spatial information.

💡ERP

Event-Related Potentials (ERP) are measured brain responses that are directly related to specific sensory, cognitive, or motor events. The script mentions ERP in the context of using linear combinations or wavelet decomposition to match patterns in ERP forms, illustrating the application of different features for classification.

💡Wavelet Decomposition

Wavelet decomposition is a mathematical method that represents a time-series signal as a sum of wavelets, allowing for the analysis of frequency content at different times. The video script uses wavelet decomposition as an example of a linear transform for feature extraction in EEG data, emphasizing its role in dimensionality reduction and pattern matching.

💡Sparsity

Sparsity in machine learning refers to the property of a model where only a small number of its parameters are non-zero. The script discusses sparsity in the context of classifiers using only a small number of wavelet coefficients that contain the relevant information, indicating a method to enhance model efficiency and interpretability.

💡Non-linear Features

Non-linear features are characteristics of data that do not follow a linear relationship. The video script contrasts linear features with non-linear ones, noting the importance of applying non-linear transformations after spatial filtering to capture source time course properties, which is crucial for accurate classification in EEG data.

💡Volume Conduction

Volume conduction refers to the spread of electrical currents through the volume of a conductive medium, such as the brain in EEG studies. The script mentions that volume conduction is less of a problem in fMRI compared to EEG, highlighting the unique challenges in EEG signal processing due to the spatial spread of electrical activity.

💡Area Under the Curve (AUC)

The Area Under the Curve (AUC), specifically the Receiver Operating Characteristic (ROC) curve, is a performance measurement for classification problems at various threshold settings. The script discusses AUC as a way to deal with class imbalance and to measure the performance of classifiers, emphasizing its usefulness in situations where the cost of different types of errors varies.

Highlights

Discussion of extensions to the basic technique of Linear Discriminant Analysis (LDA) for enhancing its performance.

LDA's invariant performance when applied to linearly transformed data, such as rescaled channels.

The difference in applying LDA to non-linear features like oscillations compared to linear ones.

The assignment of weights in LDA to every component rather than every channel in Independent Component Analysis (ICA).

The interpretability of classifiers when using ICA, allowing for reasoning about the relevance of different signal components.

Introduction of extra constraints informed by the brain's source space in ICA.

The tradeoff between using simple averages and more complex linear features like wavelet decomposition for ERP forms.

The concept of using a small number of wavelet features for dimensionality reduction in EEG signal analysis.

The importance of sparsity in classifiers, where only a small number of coefficients contain relevant information.

The distinction between linear and non-linear features, with a focus on the source time course for non-linear features.

The incorrect application of non-linear transformations on channels before classifier application in EEG analysis.

The three-stage process of linear filtering, non-linear feature extraction, and then classification for optimal results.

The difficulty of dealing with arbitrary non-linear features in EEG due to volume conduction problems.

The special condition allowing non-linear feature application after learning spatial filters with ICA.

The importance of optimizing classifiers under different criteria for imbalanced classes or costly errors.

The use of the Area Under the Curve (AUC) as a measure for dealing with imbalanced classes in classification.

The utility of AUC for classifiers that produce continuous scalar outputs, allowing for tunable thresholds.

The existence of classifiers that can be directly optimized under the AUC criterion or other advanced scoring functions.

The mention of the support vector framework as an example of a method with advanced ways to learn classifiers under sophisticated cost functions.

Transcripts

in the next part we are um

we'll go a little bit beyond that so

we'll discuss a few extensions of this

basic technique which is very easy to

comprehend but there's tweaks and twists

that one can apply to get even more out

of it

and we also cover a few more interesting

machine learning

aspects there

so um

the first little consideration is this

lda gives you basically um

if you apply it on linear to linear

filters

the same

effective result as if you had applied

it to linearly transformed data so if

you had switzerland to channels for

example or rescaled them or whatever

well you know you would have gotten the

same performance because lda would have

just learned different weights

so if you apply this to principal

components of the data or independent

components or whatever

you still get basically the same result

as long as you don't throw information

away here as long okay so

that's just a feature of this whole

process being linear

if you're looking at something

non-linear like oscillations that

absolutely doesn't hold anymore so

suddenly

you truly do get a difference when you

apply a method to say ica versus raw

channels it's just in this case we're

very lucky that

that we can basically solve it all the

way through

uh with one method in optimal

however if you have decompositions like

independent component analysis which

gives you source time courses

then um

essentially lda assigns you a weight not

to every channel but to every component

and so suddenly you can say

this component here of the signal like

the blinks are this relevant and that

component like a muscle is that relevant

and so you can reason and say things

such as i am not using muscular

activation so it doesn't depend on

artifacts or i can say

i'm primarily using an independent

component that with dipole fitting i

managed to localize in the motor cortex

and so you can suddenly interpret these

classifiers relatively easily and

furthermore you can basically introduce

extra constraints

that are informed by where these

components sit in the brain so

again ica is a very strong method to

give you something relates to source

space i say a few things in the last

lecture on that

you can say i don't want to use

components that don't come from

occipital cortex because i'm using a

visual process here i don't think my

signal originates from here at least not

the signal that i want to use

so these things suddenly become possible

and

so that's the various tradeoffs

the other one is there's other linear

features that you can use instead of

averages averages are very simple to to

describe and calculate and think about

but they are not necessarily

particularly well tuned to what you

really want

if you know that your underlying erp

comes in certain kinds of

forms like ripples like these

then you could use linear combinations

that do basically and it an inner

product with with this time course

uh to get

to basically do pattern matching with

that

so that is an example of a wavelet

decomposition here so

it's a linear transform

which allows you to

you know to use wavelet features and so

you can pick a small number of wavelet

features

to to

to find features of your linear features

of your eop and classify in terms of

those

if you

uh that only makes sense if you throw

away many of them

to reduce dimensionality if you use all

of them again it's linear you know

you could have as well use the raw chunk

of data and through a classifier edit

so it's a way of dimensionality

reduction also what these features

happen to do is

usually only a small number of these

coefficients actually contain the

information that you're looking for

and so you can say things such as my

classifier should use only small number

of nonzero coefficients

and there's various ways to learn these

kinds of classifiers we say a few things

about that

it's this sparsity notion

and so with these features you can make

use of these kinds of things it doesn't

apply say to channels the signal is not

sparse in a channel it projects

everywhere right but with these kind of

things you can

start making use of these assumptions

there is

also of course the whole area of

non-linear features

so

obviously that is that is

the general class of features linear is

a special case

and

the problem with that is that

if you are

what you actually want is you want

non-linear features that are

features of the source time course

because you think there's a source

processor that does something and maybe

there's a non-linear property of that

as opposed to nonlinear features of the

channels

so

if you do

non-linear transform somewhere in your

feature extraction on channels and then

apply a classifier

you've basically done it the wrong way

you've applied the none in your part

sort of too early you had the proper way

would have been to first design a

spatial filter which gets you to the

source which is a big linear part then

do your non-linear feature extraction

whatever that might be and then maybe

apply a classifier to that but that's

sort of a three-stage thing you know

it's linear non-linear linear and

there's

uh only a small number of methods that

properly learn all these parameters in a

way where you can say that's going to be

the optimal solution in many cases it's

patchwork so

in practice

it's very hard to

to deal properly with arbitrary

non-linear features in eeg it's

different in say fmri where you don't

have as much of a volume conduction

problem and so on but in eg it's sort of

tricky

although there's there's special classes

where you can do it

and

one

one condition for example

that allows you to do it is to learn

the spatial filters

irrespective of the class levels

um

using things like ica

and then do non-linear features on your

independent components and then use a

classifier but i'm i'm going to talk

about these things

later

there's

there's a more important aspect and that

is

when you

when you think such as you want to

detect

a signal in noise or so such as you want

to detect whether the person saw a

target image or something like that as

opposed to not having seen it

cases where you have different um

ratio let's say what the prior

probability that he saw nothing is much

higher than the probability that he

saw something so where there's this

imbalance or in cases where certain

kinds of errors

of the pci are much more costly than

others such as false positive or the

false negative

then you want to use

um maybe different criteria to optimize

these things

incorporate the costs for example

and you also want to use different

measures to estimate how well you what

the performance of your system is

misclassification rate

doesn't get you very far for example if

90 of your trials are one class and if

you always say that class

is a true class you are 90 correct on

average

right so one way to to measure one

pretty nice general purpose way of

dealing with imbalance classes is the

area under curve

or area under receiver operator

curve and it's um

uh

if your

model has a tunable threshold basically

where it says a versus b you know class

a versus plus b

for different values of this threshold

you can basically go all the way from

zero percent false positive rate to 100

and for any given false positive rate

you have an associated true positive

rate

uh in the ideal case

you um

for zero percent false positives you

have a hundred

true positive so you're always getting

it right you're never getting it wrong

basically um so the curve

the area and the curve would be one

if you're at random chance it's

basically 0.5 you know you gain one in

the first positive you lose one in the

in the two processes basically um or uh

you know gain one there

so

um

that's a way that can be sort of applied

post talk to any classifier that happens

to produce

continuous

scalar outputs like a regression

technique or so it say predicts the

probability that you saw

a plane

on a satellite image or something like

that

i should say one more thing and that is

um

let's say

um

actually i think i've already said this

so uh there is classifiers which can be

directly optimized say under the area on

the curve criterion there's classifiers

that can optimize various other scores

f scores and so on and

for example the support vector framework

has several

rather advanced

ways to learn classifiers under these

sophisticated cost functions

so um

if you're in such a situation that would

be a place to look

and and that ends um

this module