Introduction to MATLAB for NeuroimagingKRISANNE LITINASUM FMRI LABORATORY

Module 1: Recap

MATLAB Interface Path concept Variables and operators Scripts Functions Loops Conditional statements

Module 2:Single Pixel Analysis in MATLAB (General Linear Model)KRISANNE LITINASUM FMRI LABORATORY

Concept of a Model: Example

Say we have two variables, x and y. Goal: find out the relationship between x and y Method: come up with estimation and model the data accordingly

Simple Example

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

Simple Example

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

Need to model it somehow…

Simple Example: Linear Regression

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Estimate of the data: a linear relationship

Simple Example: Linear Regression

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

𝑏𝑏: Intercept (constant)

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Estimate of the data: a linear relationship

b

Simple Example: Linear Regression

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

𝑏𝑏: Intercept (constant)

𝑒𝑒: Error term (residual noise after the fitting)

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Estimate of the data: a linear relationship

b

𝒆𝒆𝒊𝒊

Simple Example: Linear Regression

y

x

𝑦𝑦: Measurement (DV)

𝑥𝑥: The thing you are measuring against (IV)

𝑏𝑏: Intercept (constant)

𝑒𝑒: Error term (residual noise after the fitting)

𝑚𝑚: Slope of the fitted line

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Estimate of the data: a linear relationship

b

𝒎𝒎 =∆𝒚𝒚∆𝒙𝒙

y

x

Simple Example: Linear Regression

o We can solve for m and b

o If this model fits, we say that x and y are correlated

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Estimate of the data: a linear relationship

b

𝒎𝒎

Simple Example: Linear Regression

If 𝑚𝑚 is “significant”, then we refer to that model as true.

”Significant”: 𝑚𝑚 is big enough compared to the noise (e)

Bigger Example: Linear Regression

Data composed of: DV: 𝑦𝑦

Multiple IVs: 𝑥𝑥1, 𝑥𝑥2, 𝑥𝑥3 …

A bigger linear model of these data (>1 independent variables):

𝑦𝑦 = 𝑚𝑚1𝑥𝑥1 + 𝑚𝑚2𝑥𝑥2 + 𝑚𝑚3𝑥𝑥3 + ⋯+ 𝑏𝑏 + 𝑒𝑒

Linear Regression in fMRI

𝑦𝑦, 𝑥𝑥1, 𝑥𝑥2, 𝑥𝑥3,… etc. are all time courses In the usual GLM analysis of fMRI data, all the 𝑥𝑥1, 𝑥𝑥2, 𝑥𝑥3,… terms are

usually not measured Instead, we make an ideal model and hope for the best ¯\_(ツ)_/¯

𝑦𝑦 = 𝑚𝑚1𝑥𝑥1 + 𝑚𝑚2𝑥𝑥2 + 𝑚𝑚3𝑥𝑥3 + ⋯+ 𝑏𝑏 + 𝑒𝑒

Expand to Matrix Form

Now with many variables:

𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 + 𝑒𝑒

Y= 𝑋𝑋𝛽𝛽 + 𝜀𝜀𝑌𝑌1𝑌𝑌2⋮𝑌𝑌𝑛𝑛

=

1 𝑋𝑋11 ⋯ 𝑋𝑋1𝑝𝑝1⋮1

𝑋𝑋21⋮𝑋𝑋𝑛𝑛1

⋯

⋯

𝑋𝑋2𝑝𝑝⋮

𝑋𝑋𝑛𝑛𝑝𝑝

𝛽𝛽0𝛽𝛽1⋮𝛽𝛽𝑝𝑝

+𝜀𝜀0𝜀𝜀1⋮𝜀𝜀𝑛𝑛

Design Matrix Observed Data

Model Params.

Error

time

Expand to Matrix Form

Solve for terms, do fancy math with matrices

𝑌𝑌 = 𝑋𝑋𝛽𝛽 + 𝜀𝜀

𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑌𝑌 − 𝑋𝑋 ∗ 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒

𝑇𝑇𝑒𝑒𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒(𝑛𝑛) =𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒

𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑠𝑠(𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛 )

𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒 = (𝑋𝑋)−1∗ 𝑌𝑌

Today’s Exercise

Generate and explore temporal noise for fMRI data (𝜀𝜀) Create a linear model for a BOLD time series using MATLAB (𝑋𝑋) Create a realistic (but fake) BOLD signal (𝑌𝑌) Use regression to test whether the model fits the signal