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Cheat Sheet: From Linear Algebra to Significance, Cheat Sheet of Linear Algebra

Cheat sheet with Basics of Linear Algebra

Typology: Cheat Sheet

2019/2020

Uploaded on 11/27/2020

ekambar
ekambar 🇺🇸

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Cheat Sheet: From Linear Algebra to Significance
“Neural Activity”
in region R
“…significant decrease
in activation (p < 0.05)
in region R during
presentation of …”
Linear Algebra
GLM
Statistics
Glossary:
𝛽Scalar Values
Ԧ𝑥 Vector
𝑋Matrix
𝑋𝑇Matrix transpose
Ԧ𝜖 i.i.d. noise vector
𝑛,𝑚 dimensions
What fMRI data has to do with Linear Algebra
Time Series extracted
in region R
How do we test for an effect
of experimental factors?
Can we explain the
signal as a linear
combination of our
regressors?
20
2
1
10
20
2
1
0
1
1
1
1
1
y
y
y
=𝛽0+ 𝛽1
pf3

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Cheat Sheet: From Linear Algebra to Significance

“Neural Activity” in region R

“…significant decrease in activation (p < 0.05 ) in region R during presentation of …”

  • • Linear AlgebraGLM
  • Statistics

Glossary: 𝛽 Scalar Values

𝑥 𝑋 Ԧ VectorMatrix

𝑋𝜖 Ԧ 𝑇^ Matrix transposei.i.d. noise vector

𝑛, 𝑚 dimensions

What fMRI data has to do with Linear Algebra

Time Series extracted in region R

How do we test for an effect of experimental factors?

Can we explain the signal as a linear

combination of our regressors?

20

0 1 21 20

21 0

y

yy

= 𝛽^0 ⋅^ +^ 𝛽^1 ⋅

Standard error of estimate*

𝑆𝐸𝑀𝑖 = 𝜎 2 𝑐𝑖 𝑇 𝑋^1 𝑇𝑋 − 1 𝑐𝑖

𝑦 Ԧ = 𝛽 0 ⋅ 𝑥 0 + 𝛽 1 ⋅ 𝑥 1 + 𝜖Ԧ = (^) 𝑋 ⋅ Β + 𝜖Ԧ

𝑋 = [𝑥𝑜, 𝑥 1 ] Β = [𝛽 𝛽^01 ]

“Design Matrix”  

X^11 ^11 

Generalized Linear Model (GLM)

How to find the 𝛽-Values?

Minimize the error = explain

the signal as good as possible!

( error)Least squared

𝛽^ መ = 𝑎𝑟𝑔𝑚𝑖𝑛𝛽 = 𝑦Ԧ − 𝛽𝑋 2 = ෍ 𝑖=^ 𝑛 1 𝜖𝑖^2

→ 𝛽መ = 𝑋𝑇𝑋 −^1 𝑋𝑇^ 𝑦Ԧ Is the effect significant?

“…significant decrease in activation region R during presentation (p < 0.05 ) in of …”

Estimate𝛽 ෡𝑖

Test statistic

𝑡∗^ = 𝑆𝐸𝛽෡𝑀𝒊 𝑖

𝑡∗

Student’s t – distribution

c i  𝑖𝑡ℎ (^) entry