Mathematics for Brain Imaging
Autumn Term, 2006
This
lecture series covers a number of mathematical methods that are used in the
analysis of brain imaging data. Each lecture describes a different category of
model and shows how it is applied to a particular aspect of brain imaging
analysis. The applications cover data from functional Magnetic Resonance
Imaging (fMRI), Magnetoencephalography (MEG) and Electroencephalography (EEG).
When ?
Weds 11am-1pm (unless
otherwise stated), October 4th to Dec 13th 2006
Where ?
FIL Seminar Room, 4th floor, 12 Queen Square
Download all course notes: mbi_course.pdf
Contents:
- General Linear Models I
- General Linear Models II
- Random Field Theory
- Multivariate Models
- Variance Components
- Bayesian methods
- Model comparison
- Spectral Estimation
- Approximate Bayesian Inference
- Nonlinear models
1.
General Linear Models I
- Maximum likelihood estimation
- Regression and correlation
- Linear algebra
- Functions of random vectors
- Multiple regression and partial correlation
- Application: fMRI time series
Lecture
notes and code can be downloaded from this archive: lecture1.zip
2.
General Linear Models II
- Estimating error variance
- Comparing nested models
- Transforming probability densities
- Contrasts
- Hemodynamic basis functions
- Application: fMRI time series
Notes
can be downloaded here: lecture2.pdf, contrasts.pdf, pdfs.ps
and
code: mc.m, temporal.m
3.
Random Field Theory
- Gaussian processes
- Covariance functions
- Upcrossings of one-dimensional processes
- Euler characteristic
- Application: Detecting activations in fMRI data
Notes
can be downloaded here: lecture3.pdf
Code
for 1-dimensional GPs: demo_1d.m, init_gp_1d.m
Keith
Worsley’s introduction to the Euler Characteristic: chance3.pdf
Matthew
Brett’s simulation of 2D fields: randomtalk.m
4.
Multivariate Models
- More Linear Algebra
- Principal component analysis
- Singular Value Decomposition
- Structural Equation Modelling
- Granger causality
- Application: PET & fMRI connectivity
analyses
Notes
can be downloaded here: lecture4.pdf
Notes
on Granger causality and MAR models: mar.pdf
Code
and data for SVD image compression: ALAN02.JPG, alan_svd.m
Code
for PCA extraction of representative regional activity: region_svd.m
5.
Variance Components
- GLMs with arbitrary error covariance
- Weighted Least Squares
- Restricted Maximum Likelihood
- Application: fMRI time series analysis with
correlated errors
- Hierarchical Models
- Application: Analysis of imaging data from a
group
Notes
can be downloaded here: lecture5.pdf
Code
for ReML estimation of variance components
in
a hierarchical model: group_study.m
6. Bayesian
methods
- Bayes rule for Gaussians
- Bayesian GLMs
- Parametric Empirical Bayes (PEB)
- Expectation Maximisation (EM)
- Application: EEG source reconstruction
Notes
can be downloaded here: lecture6.pdf
Code
for EM example: em1.m
The
equations in the EM example can be derived as shown here: ..\publications\spm-book\hierarchical.pdf
7. Model
comparison
- Bayes factors and odds ratios
- Model evidence for Bayesian GLMs
- Accuracy and complexity (AIC/BIC)
- Bias-variance decomposition
- Application: EEG source reconstruction
- Bayesian model averaging
- Application: Nonlinear EEG source
reconstruction
Notes
can be downloaded here: lecture7.pdf
8.
Spectral Estimation
- Fourier series and periodograms
- Autocorrelation and power spectral density
- Cross-correlation and cross spectral density
- Coherence and Phase
- Welch and multitaper methods
- Localisation of MEG Gamma activity
Notes
can be downloaded here: lecture8.pdf
Sunspot
data: yearssn, and code for analysing it here: sunspot_spectra.m
Notes
on autoregressive, stochastic.ps, array.ps
and subspace methods for spectral estimation, subspace.ps.
9. Approximate Bayesian Inference
- Laplace approximation
- Kullback-Liebler divergence
- Variational Bayes and EM
- Mixture models
- Application: Group analysis of imaging data
Notes
can be downloaded here: lecture9.pdf
10. Nonlinear
models
- Central Limit Theorem
- Independent Component Analysis
- Application: EEG artifact removal
- Discriminant Analysis
- Application: Estimating perceptual state from
fMRI
Notes
can be downloaded here: lecture10.pdf