Download How random is a DNA sequence?

How random is a DNA sequence?
Signals and Systems in Biology
Kushal Shah @ EE, IIT Delhi
Genome Periodicity
R. F. Voss, Physical Review Letters 1992
Correlation Plot : Whole Genome of E. coli
Figure 1
0.28
k=0(mod 3)
k=1(mod 3)
k=2(mod 3)
0.27
C(k)
0.26
0.25
0.24
0.23
0.22
0.21
0
200
400
600
800
k
C (1) = 4 (0.5 − p)2
K. Shah and A. Krishnamachari, BioSystems 2012
1000
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Pink Noise : 1 f spectra
I
How is it generated?
I
I
Long-range correlation
I
I
I
Poisson process and stochastic dynamical systems
Inverse Fourier transform of 1 f ?
White noise : Flat spectra and delta correlation
Brownian noise : 1 f 2 spectra
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations of genes
I
DNA Walk
I
I
I
u (i ) = +1 for a pyrimidine (Y = C/T)
u (i ) = −1 for a purine (R = A/G)
Net displacement
l
y (l ) = ∑ u (i )
i =1
I
Root Mean Square Fluctuation, F (l ) of y (l )
I
I
Random sequence or local correlation : F (l ) ∼ l 0.5
Long-range correlation : F (l ) ∼ l α and α 6= 0.5
C. K. Peng et. al., Nature 1992
Long-range correlations
Dark circle, Gene : α ≈ 0.67
Light circle, cDNA : α ≈ 0.5
C. K. Peng et. al., Nature 1992
C. K. Peng et. al., Nature 1992
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S (1) = {a1 , a2 , ..., an }
DJS
S (2) = {an+1 , an+2 , ..., aN }
n
o
(2) (2)
(2)
F (2) = f1 , f2 , ..., fk
h
i
= H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
F (1) =
f1(1) , f2(1) , ..., fk(1)
A = {A1 , A2 , ..., Ak }
n
π1 , π2 ≥ 0
o
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S (1) = {a1 , a2 , ..., an }
DJS
S (2) = {an+1 , an+2 , ..., aN }
n
o
(2) (2)
(2)
F (2) = f1 , f2 , ..., fk
h
i
= H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
F (1) =
f1(1) , f2(1) , ..., fk(1)
A = {A1 , A2 , ..., Ak }
n
π1 , π2 ≥ 0
o
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S (1) = {a1 , a2 , ..., an }
DJS
S (2) = {an+1 , an+2 , ..., aN }
n
o
(2) (2)
(2)
F (2) = f1 , f2 , ..., fk
h
i
= H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
F (1) =
f1(1) , f2(1) , ..., fk(1)
A = {A1 , A2 , ..., Ak }
n
π1 , π2 ≥ 0
o
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S (1) = {a1 , a2 , ..., an }
F (1) =
f1(1) , f2(1) , ..., fk(1)
n
DJS = H
A = {A1 , A2 , ..., Ak }
S (2) = {an+1 , an+2 , ..., aN }
o
F (2) =
f1(2) , f2(2) , ..., fk(2)
n
o
h
i
π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
π1 , π2 ≥ 0
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S (1) = {a1 , a2 , ..., an }
DJS
S (2) = {an+1 , an+2 , ..., aN }
n
o
(2) (2)
(2)
F (2) = f1 , f2 , ..., fk
h
i
= H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
F (1) =
f1(1) , f2(1) , ..., fk(1)
A = {A1 , A2 , ..., Ak }
n
π1 , π2 ≥ 0
o
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S = {a1 , a2 , ..., an }
A = {A1 , A2 , ..., Ak }
= {an+1 , an+2 , ..., aN }
n
o
(1) (1)
(1)
(2) (2)
(2)
F (1) = f1 , f2 , ..., fk
F (2) = f1 , f2 , ..., fk
h
i
DJS = H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
(1)
n
S
(2)
o
π1 , π2 ≥ 0
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Segmentation : Jensen-Shannon Divergence
S = {a1 , a2 , ..., aN }
S = {a1 , a2 , ..., an }
A = {A1 , A2 , ..., Ak }
= {an+1 , an+2 , ..., aN }
n
o
(1) (1)
(1)
(2) (2)
(2)
F (1) = f1 , f2 , ..., fk
F (2) = f1 , f2 , ..., fk
h
i
DJS = H π1 F (1) + π2 F (2) − π1 H F (1) + π2 H F (2) ≥ 0
(1)
n
S
(2)
o
π1 , π2 ≥ 0
π1 + π2 = 1
π1 =
Find n that leads to maximum DJS
P. B. Galvan et. al., Physical Review E
n
N
Jensen-Shannon Divergence : Where to stop?
I
Segmentation does not stop till n reaches 1
I
I
Specify a minimum allowed segment length (eg. 10)
Specify a minimum significance level for DJS
DJS should be more than that expected from a random sequence
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Where to stop?
I
Segmentation does not stop till n reaches 1
I
I
Specify a minimum allowed segment length (eg. 10)
Specify a minimum significance level for DJS
DJS should be more than that expected from a random sequence
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Where to stop?
I
Segmentation does not stop till n reaches 1
I
I
Specify a minimum allowed segment length (eg. 10)
Specify a minimum significance level for DJS
DJS should be more than that expected from a random sequence
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Where to stop?
I
Segmentation does not stop till n reaches 1
I
I
Specify a minimum allowed segment length (eg. 10)
Specify a minimum significance level for DJS
DJS should be more than that expected from a random sequence
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Where to stop?
I
Segmentation does not stop till n reaches 1
I
I
Specify a minimum allowed segment length (eg. 10)
Specify a minimum significance level for DJS
DJS should be more than that expected from a random sequence
P. B. Galvan et. al., Physical Review E
P. B. Galvan et. al., Physical Review E
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Jensen-Shannon Divergence : Final outcome
I
Long-range correlation is more pronounced in R/Y and less in
S/W
I
I
Bacterial DNA segments do not show long-range correlation
Human DNA segments show long-range correlation
I
Distribution of lengths in humans reminiscent of fractal sets
I
Shuffling the segments retains patch length distribution
I
I
I
Distribution destroyed if equal length segments shuffled
Long-range correlation depends on patch length distribution
Mechanism for long-range correlation
I
I
Expansion modification system
Insertion deletion system
P. B. Galvan et. al., Physical Review E
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
L − 2k
1
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
I
I
I
I
Construct a discrete binary sequence
1
L − 2k
p (k , L) = 1 − µ
2
L + L0
k : Number of such symbols in preceeding L bits
−1 < µ < 1 : Correlation strength
L0 : Characteristic transient time
I
I
Sequence is more or less random for L L0
Probability of finding k identical symbols in a sequence of
length L + 1
P (k , L + 1) = [1 − p (k , L)] P (k , L) + p (k − 1, L) P (k − 1, L)
In the continuous limit (x = 2k − L)
1 ∂ 2P
µ ∂ (xP )
∂P
=
−
2
∂L
2 ∂x
L + L0 ∂ x
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
1 ∂ 2P
∂P
µ ∂ (xP )
=
−
∂L
2 ∂ x 2 L + L0 ∂ x
2 x
1
exp −
P (x , L) = p
2D (L)
2π D (L)
"
#
L + L0
L 1−2µ
D (L) =
1−
1 − 2µ
L + L0
If L L0 ,
 
2µ)
L (1 −
D (L) ' L ln L L0
 1−2µ 2µ L0 L (2µ − 1)
S. Hod and U. Keshet, Physical Review E 2004
µ < µc
µ = µc = 0.5
µ > µc
Random walks with long-range correlation
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
S. Hod and U. Keshet, Physical Review E 2004
Random walks with long-range correlation
Data Type
DNA sequence
Written Text
Stock Market
String Source
Drosophila melanogaster
Methanosarcina acetivorans
Bacillus subtilis
Alice’s Adventures in Wonderland
NASDAQ
S. Hod and U. Keshet, Physical Review E 2004
µ
0.57
0.70
0.86
0.58
0.39
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length
Ultimate measure of randomness
I
I
Consider a sequence s
Let s 0 be a description of s
I
computer program or other language
I
Minimal description, d (s ) :
I
Kolmogorov Complexity
I
K (s ) : Length of d (s )
s 0 that has the minimum length