hidden context tree modeling of eeg datamalot/galves_mathstatneuro.pdf · eeg data. i but we need...
TRANSCRIPT
![Page 1: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/1.jpg)
Hidden context tree modeling of EEG data
Antonio Galves
joint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas
Universidade de S.Paulo and NeuroMat
MathStatNeuro 2015
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 2: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/2.jpg)
Looking for experimental evidence that the brain is a
statistician
I Is the brain a statistician?
I Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
I See for instance the two lessons by Dehaene available on the web:
I Le cerveau statisticien
I Le bebe statisticien
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 3: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/3.jpg)
Looking for experimental evidence that the brain is a
statistician
I Is the brain a statistician?
I Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
I See for instance the two lessons by Dehaene available on the web:
I Le cerveau statisticien
I Le bebe statisticien
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 4: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/4.jpg)
Looking for experimental evidence that the brain is a
statistician
I Is the brain a statistician?
I Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
I See for instance the two lessons by Dehaene available on the web:
I Le cerveau statisticien
I Le bebe statisticien
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 5: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/5.jpg)
Is the brain a statistician?
I How to obtain experimental evidence supporting this conjecture?
I Dehaene presents experimental evidence that unexpected
occurrences in regular sequences produce characteristic markers in
EEG data.
I But we need more than evidences of mismatch negativity to support
this conjecture.
I To discuss this issue we need to do statistical model selection in a
new class of stochastic processes:
Hidden context tree models.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 6: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/6.jpg)
Is the brain a statistician?
I How to obtain experimental evidence supporting this conjecture?
I Dehaene presents experimental evidence that unexpected
occurrences in regular sequences produce characteristic markers in
EEG data.
I But we need more than evidences of mismatch negativity to support
this conjecture.
I To discuss this issue we need to do statistical model selection in a
new class of stochastic processes:
Hidden context tree models.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 7: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/7.jpg)
Neurobiological problem
Random
Source
A random source produces sequences of auditory stimuli.
How to retrieve the structure of the source from EEG data?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 8: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/8.jpg)
Example of a random source: samba
I Auditory segments:
I 2 - strong beat
I 1 - weak beat
I 0 - silent event
I Chain generation:
I start with a deterministic sequence
· · ·2 1 0 1 2 1 0 1 2 1 0 1 2 · · ·I replace in a iid way each symbol 1 by 0 with probability ε.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 9: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/9.jpg)
A typical sample would be
· · ·2 1 0 1 2 1 0 1 2 1 0 1 2 · · ·· · ·2 1 0 0 2 1 0 1 2 0 0 0 2 · · ·
How to define the structure of this source?
- By describing the algorithm producing each next symbol,
given the shortest relevant sequence of past symbols.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 10: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/10.jpg)
A typical sample would be
· · ·2 1 0 1 2 1 0 1 2 1 0 1 2 · · ·· · ·2 1 0 0 2 1 0 1 2 0 0 0 2 · · ·
How to define the structure of this source?
- By describing the algorithm producing each next symbol,
given the shortest relevant sequence of past symbols.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 11: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/11.jpg)
A typical sample would be
· · ·2 1 0 1 2 1 0 1 2 1 0 1 2 · · ·· · ·2 1 0 0 2 1 0 1 2 0 0 0 2 · · ·
How to define the structure of this source?
- By describing the algorithm producing each next symbol,
given the shortest relevant sequence of past symbols.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 12: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/12.jpg)
The structure of the random source
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10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 13: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/13.jpg)
The structure of the random source
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 14: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/14.jpg)
The structure of the random source
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 15: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/15.jpg)
The stochastic chain generated by the source samba
1
X0
2
X−1
0
X−2
0
X−3
1
X−4
2
X−5
0
X1
1
X2
. . .. . .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 16: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/16.jpg)
I Xn ∈ A = {0, 1, 2}
I (Xn)n∈Z is stochastic chain
I with memory of variable length
I generated by the probabilistic context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 17: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/17.jpg)
I Xn ∈ A = {0, 1, 2}
I (Xn)n∈Z is stochastic chain
I with memory of variable length
I generated by the probabilistic context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 18: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/18.jpg)
I Xn ∈ A = {0, 1, 2}
I (Xn)n∈Z is stochastic chain
I with memory of variable length
I generated by the probabilistic context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 19: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/19.jpg)
I Xn ∈ A = {0, 1, 2}
I (Xn)n∈Z is stochastic chain
I with memory of variable length
I generated by the probabilistic context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 20: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/20.jpg)
I Xn ∈ A = {0, 1, 2}
I (Xn)n∈Z is stochastic chain
I with memory of variable length
I generated by the probabilistic context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 21: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/21.jpg)
Context tree models
I Introduced by Rissanen
I A universal data compression system, IEEE, 1983.
I stochastic chains with memory of variable length
I generated by a probabilistic context tree
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 22: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/22.jpg)
Context tree models
I Introduced by Rissanen
I A universal data compression system, IEEE, 1983.
I stochastic chains with memory of variable length
I generated by a probabilistic context tree
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 23: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/23.jpg)
Context tree models
I Introduced by Rissanen
I A universal data compression system, IEEE, 1983.
I stochastic chains with memory of variable length
I generated by a probabilistic context tree
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 24: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/24.jpg)
Context tree models
I Introduced by Rissanen
I A universal data compression system, IEEE, 1983.
I stochastic chains with memory of variable length
I generated by a probabilistic context tree
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 25: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/25.jpg)
The neurobiological question
Is it possible
to retrieve the samba context tree
from the EEG data recorded during the exposure tothe sequence of auditory stimuli generated by thesamba source?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 26: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/26.jpg)
The neurobiological question
Is it possible
to retrieve the samba context tree
from the EEG data recorded during the exposure tothe sequence of auditory stimuli generated by thesamba source?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 27: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/27.jpg)
The neurobiological question
Is it possible
to retrieve the samba context tree
from the EEG data recorded during the exposure tothe sequence of auditory stimuli generated by thesamba source?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 28: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/28.jpg)
EEG data
45
+−
Scale
v2
v1
a
v
0
v1
b
v2
v1
a
v
0
m
iss
v2
134 135 136 137 138 139 140
E27
E26
E24
E23
E22
E21
E20
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 29: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/29.jpg)
How to address the identification problem?
We have
I EEG data recorded with 18 electrodes
I for each electrode e and each step n
I call Y en = (Y en (t), t ∈ [0, T ]) the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
I Y en ∈ L2([0, T ]), where T = 450ms is the time distance between the
onsets of two consecutive auditory stimuli
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 30: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/30.jpg)
How to address the identification problem?
We have
I EEG data recorded with 18 electrodes
I for each electrode e and each step n
I call Y en = (Y en (t), t ∈ [0, T ]) the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
I Y en ∈ L2([0, T ]), where T = 450ms is the time distance between the
onsets of two consecutive auditory stimuli
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 31: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/31.jpg)
How to address the identification problem?
We have
I EEG data recorded with 18 electrodes
I for each electrode e and each step n
I call Y en = (Y en (t), t ∈ [0, T ]) the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
I Y en ∈ L2([0, T ]), where T = 450ms is the time distance between the
onsets of two consecutive auditory stimuli
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 32: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/32.jpg)
How to address the identification problem?
We have
I EEG data recorded with 18 electrodes
I for each electrode e and each step n
I call Y en = (Y en (t), t ∈ [0, T ])
the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
I Y en ∈ L2([0, T ]), where T = 450ms is the time distance between the
onsets of two consecutive auditory stimuli
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 33: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/33.jpg)
How to address the identification problem?
We have
I EEG data recorded with 18 electrodes
I for each electrode e and each step n
I call Y en = (Y en (t), t ∈ [0, T ]) the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
I Y en ∈ L2([0, T ]), where T = 450ms is the time distance between the
onsets of two consecutive auditory stimuli
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 34: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/34.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 35: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/35.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 36: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/36.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)
In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 37: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/37.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 38: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/38.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 39: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/39.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 40: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/40.jpg)
Hidden context tree model (HCTM)
Ingredients:
I finite alphabet A
In our example A = {0, 1, 2}
I measurable space (F,F)In our example F = L2([0, T ]) and F is the Borel σ-algebra on F .
I probabilistic context tree (τ, p)
I family {Qw : w ∈ τ} of probabilities on (F,F)
I stochastic chain (Xn, Yn) ∈ A× F .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 41: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/41.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 42: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/42.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 43: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/43.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 44: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/44.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 45: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/45.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 46: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/46.jpg)
Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
I (Xn)n∈Z is generated by (τ, p)
I for any m,n ∈ Z with m ≤ n
I any string xnm−`(τ)+1 ∈ An−m+`(τ)
I and any sequence Inm = (Im, . . . , In) of F-measurable sets,
P(Y nm ∈ Inm|Xn
m−`(τ)+1 = xnm−`(τ)+1
)=
n∏k=m
Qcτ (xkk−`(τ)+1)(Ik)
I `(τ) = height of τ
I cτ (xkk−`(τ)+1) = context assigned to xkk−`(τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 47: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/47.jpg)
Rephrasing our problem
Taking
I (Xn)n∈Z sequence of auditory stimuli produced by the samba source
I (Y en )n∈Z successive chunks of EEG signals
Question: Is (Xn, Yen )n∈Z a HCTM compatible with τ?
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 48: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/48.jpg)
Rephrasing our problem
Taking
I (Xn)n∈Z sequence of auditory stimuli produced by the samba source
I (Y en )n∈Z successive chunks of EEG signals
Question: Is (Xn, Yen )n∈Z a HCTM compatible with τ?
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 49: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/49.jpg)
Rephrasing our problem
Taking
I (Xn)n∈Z sequence of auditory stimuli produced by the samba source
I (Y en )n∈Z successive chunks of EEG signals
Question: Is (Xn, Yen )n∈Z a HCTM compatible with τ?
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 50: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/50.jpg)
Rephrasing our problem
Taking
I (Xn)n∈Z sequence of auditory stimuli produced by the samba source
I (Y en )n∈Z successive chunks of EEG signals
Question: Is (Xn, Yen )n∈Z a HCTM compatible with τ?
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 51: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/51.jpg)
Rephrasing our problem
Taking
I (Xn)n∈Z sequence of auditory stimuli produced by the samba source
I (Y en )n∈Z successive chunks of EEG signals
Question: Is (Xn, Yen )n∈Z a HCTM compatible with τ?
000 100 200
10 20 01 21
2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 52: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/52.jpg)
is (Xn, Yen )n∈Z a HCTM compatible with τ?
In other terms, for any w ∈ τ , is it true that
L(Y en |Xnn−`(w)+1 = w,X
−`(τ)−∞ = u) = L(Y en |Xn
n−`(w)+1 = w,X−`(τ)−∞ = v)
for any pair of strings u and v?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 53: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/53.jpg)
Pruning the tree
I A version of Rissanen’s algorithm Context will be applied
I Start with a maximal admissible candidate tree
I For any string w and pair of symbols a, b ∈ A with aw and bw
belonging to the candidate tree
I test the equality
L(Y en |Xnn−`(w) = aw) = L(Yn|Xn
n−`(w) = bw)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 54: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/54.jpg)
Pruning the tree
I A version of Rissanen’s algorithm Context will be applied
I Start with a maximal admissible candidate tree
I For any string w and pair of symbols a, b ∈ A with aw and bw
belonging to the candidate tree
I test the equality
L(Y en |Xnn−`(w) = aw) = L(Yn|Xn
n−`(w) = bw)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 55: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/55.jpg)
Pruning the tree
I A version of Rissanen’s algorithm Context will be applied
I Start with a maximal admissible candidate tree
I For any string w and pair of symbols a, b ∈ A with aw and bw
belonging to the candidate tree
I test the equality
L(Y en |Xnn−`(w) = aw) = L(Yn|Xn
n−`(w) = bw)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 56: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/56.jpg)
Pruning the tree
I If for all pairs of symbols (a, b) the equality is rejected then prune all
the leaves aw
I Repeat the pruning procedure until no more pruning is required
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 57: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/57.jpg)
How to test the equality
L(Yn|Xnn−`(w) = aw) = L(Yn|Xn
n−`(w) = bw) ?
Apply the projective method introduced by Cuestas-Albertos, Fraiman
and Ransford (2006).
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 58: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/58.jpg)
Experimental results
I Context tree selection procedure for the EEG data recorded during
the exposure to the sequence of auditory stimuli generated by the
samba source
I Sample composed by 20 subjects
I For each subject EEG data from 18 electrodes was recorded
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 59: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/59.jpg)
Experimental results
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 60: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/60.jpg)
Experimental results
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data
![Page 61: Hidden context tree modeling of EEG datamalot/Galves_MathStatNeuro.pdf · EEG data. I But we need more than evidences of mismatch negativity to support this conjecture. I To discuss](https://reader035.vdocument.in/reader035/viewer/2022081617/60485ec46c92e9018170cfe2/html5/thumbnails/61.jpg)
Summary
18
5
19 15
White nodes indicate the number of subjects which correctly identify the
node as not being a context. Black nodes indicate the number of
subjects which correctly identify the node as a context. For instance, 18
subjects correctly identify that the symbol 0 alone is not enough to
predict the next symbol. And 15 subjects correctly identify the symbol 2
as a context.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data