Introduction to Chaos Game Representation (CGR)
Problem at Hand
Genomic Sequence
Example
ACGAGTTACGGT
ACGCGTAGACGT
GTCATGCATAACT
ACGAGTTACGGT
ACGCGTAGACGT
GTCATGCATAACT
How to compare?🤔🤔
ACGAGTTACGGT
ACGCGTAGACGT
GTCATGCATAACT
How to compare?🤔🤔
That is, how do we bring about a proper numerical representation?
Any Ideas?
Iterated Function Systems (IFS)
Iterated Function Systems (IFS)
What is
Aâ‹…x
A\cdot x
Iterated Function Systems (IFS)
What is
y=Aâ‹…x
y = A\cdot x
Iterated Function Systems (IFS)
What is
y=Aâ‹…x
y = A\cdot x
Think of "A" as a "Transformation" on x
Iterated Function Systems (IFS)
y=Aâ‹…x
y = A\cdot x
Think of "A" as a "Transformation" on x
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
[0.50.5​34​]
\begin{bmatrix}
0.5 & 3 \\
0.5 & 4
\end{bmatrix}
A
A
Iterated Function Systems (IFS)
y=Aâ‹…x
y = A\cdot x
Think of "A" as a "Transformation" on x
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
[0.50.5​34​]
\begin{bmatrix}
0.5 & 3 \\
0.5 & 4
\end{bmatrix}
A
A
[0.50.5​]
\begin{bmatrix}
0.5 \\ 0.5
\end{bmatrix}
Iterated Function Systems (IFS)
y=Aâ‹…x
y = A\cdot x
Think of "A" as a "Transformation" on x
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
[01​−10​]
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
A
A
[01​]
\begin{bmatrix}
0 \\ 1
\end{bmatrix}
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
Iterated Function Systems (IFS)
y=Aâ‹…x
y = A\cdot x
[01​−10​]
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
A
A
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
[cos(θ)sin(θ)​−sin(θ)cos(θ)​]
\begin{bmatrix}
cos(\theta) & -sin(\theta) \\
sin(\theta) & cos(\theta)
\end{bmatrix}
θ=2π​
\theta=\frac{\pi}{2}
"Rotation Matrix"
Iterated Function Systems (IFS)
y=[ac​bd​]⋅x+[ef​]
y = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot x + \begin{bmatrix} e \\ f \end{bmatrix}
[01​−10​]
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
A
A
[10​]
\begin{bmatrix}
1 \\ 0
\end{bmatrix}
[cos(θ)sin(θ)​−sin(θ)cos(θ)​]
\begin{bmatrix}
cos(\theta) & -sin(\theta) \\
sin(\theta) & cos(\theta)
\end{bmatrix}
θ=2π​
\theta=\frac{\pi}{2}
"Rotation Matrix"
To Define such a generic linear transform, We will need 6 parameters: a,b,c,d,e,f
Iterated Function Systems (IFS)
Iterated Function Systems (IFS)
Now, We shall consider a "Set of Such Transforms"
| a | b | c | d | e | f | |
|---|---|---|---|---|---|---|
| 1 | 0.5 | 0.5 | ||||
| 2 | 0.5 | 0.5 | 0.5 | |||
| 3 | 0.5 | 0.5 | 0.5 | 0.5 |
Iterated Function Systems (IFS)
Do This Repetedly
Other Transformations
Chaos Game Representation
Chaos Game Representation
Chaos Game Representation
Chaos Game Representation
Chaos Game Representation
Chaos Game Representation
Chaos Game Representation
| CC | AC | CA | AA |
|---|---|---|---|
| GC | TC | GA | TA |
| CG | AG | CT | AT |
| GG | TG | GT | TT |
C
A
T
G
Introduction to Chaos Game Representation (CGR)