The document discusses using hidden context tree modeling (HCTM) to analyze EEG data recorded during exposure to auditory stimuli sequences generated by random sources. Context tree models define stochastic chains with memory of variable length generated by probabilistic context trees. The goal is to retrieve the structure of the random source (context tree) from the EEG data, to provide evidence that the brain acts as a statistician by learning the statistical structure or model of stimuli sequences. HCTM defines a joint stochastic process over stimuli sequences and EEG responses that is compatible with a given context tree model.

1. 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

2. Looking for experimental evidence that the brain is a
statistician
Is the brain a statistician?
Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
See for instance the two lessons by Dehaene available on the web:
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

3. Looking for experimental evidence that the brain is a
statistician
Is the brain a statistician?
Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
See for instance the two lessons by Dehaene available on the web:
Le cerveau statisticien
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

4. Looking for experimental evidence that the brain is a
statistician
Is the brain a statistician?
Stanislas Dehaene claims that the idea that the brain is a Bayesian
statistician is already sketched in von Helmholtz work!
See for instance the two lessons by Dehaene available on the web:
Le cerveau statisticien
Le b´eb´e statisticien
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

5. Is the brain a statistician?
How to obtain experimental evidence supporting this conjecture?
Dehaene presents experimental evidence that unexpected
occurrences in regular sequences produce characteristic markers in
EEG data.
But we need more than evidences of mismatch negativity to support
this conjecture.
To discuss this issue we need to do statistical model selection in a
new class of stochastic processes:
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

6. Is the brain a statistician?
How to obtain experimental evidence supporting this conjecture?
Dehaene presents experimental evidence that unexpected
occurrences in regular sequences produce characteristic markers in
EEG data.
But we need more than evidences of mismatch negativity to support
this conjecture.
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

7. 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

8. Example of a random source: samba
Auditory segments:
2 - strong beat
1 - weak beat
0 - silent event
Chain generation:
start with a deterministic sequence
· · · 2 1 0 1 2 1 0 1 2 1 0 1 2 · · ·
replace in a iid way each symbol 1 by 0 with probability .
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

9. 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 · · ·
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

10. 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?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

11. 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

12. The structure of the random source
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

13. The structure of the random source
000 100 200
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2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

14. The structure of the random source
000 100 200
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2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

15. 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

16. Xn ∈ A = {0, 1, 2}
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

17. Xn ∈ A = {0, 1, 2}
(Xn)n∈Z is stochastic chain
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

18. Xn ∈ A = {0, 1, 2}
(Xn)n∈Z is stochastic chain
with memory of variable length
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

19. Xn ∈ A = {0, 1, 2}
(Xn)n∈Z is stochastic chain
with memory of variable length
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

20. Xn ∈ A = {0, 1, 2}
(Xn)n∈Z is stochastic chain
with memory of variable length
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

21. Context tree models
Introduced by Rissanen
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

22. Context tree models
Introduced by Rissanen
A universal data compression system, IEEE, 1983.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

23. Context tree models
Introduced by Rissanen
A universal data compression system, IEEE, 1983.
stochastic chains with memory of variable length
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

24. Context tree models
Introduced by Rissanen
A universal data compression system, IEEE, 1983.
stochastic chains with memory of variable length
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

25. The neurobiological question
Is it possible
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

26. The neurobiological question
Is it possible
to retrieve the samba context tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

27. The neurobiological question
Is it possible
to retrieve the samba context tree
from the EEG data recorded during the exposure to
the sequence of auditory stimuli generated by the
samba source?
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

28. EEG data
45
+−
Scale
v2
v1a
v0
v1b
v2
v1a
v0
miss
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

29. How to address the identification problem?
We have
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

30. How to address the identification problem?
We have
EEG data recorded with 18 electrodes
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

31. How to address the identification problem?
We have
EEG data recorded with 18 electrodes
for each electrode e and each step n
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

32. How to address the identification problem?
We have
EEG data recorded with 18 electrodes
for each electrode e and each step n
call Y e
n = (Y e
n (t), t ∈ [0, T])
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

33. How to address the identification problem?
We have
EEG data recorded with 18 electrodes
for each electrode e and each step n
call Y e
n = (Y e
n (t), t ∈ [0, T]) the EEG signal recorded at electrode e
during the exposure to the auditory stimulus Xn
Y e
n ∈ 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

34. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

35. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

36. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
measurable space (F, F)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

37. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
measurable space (F, F)
In our example F = L2
([0, T]) and F is the Borel σ-algebra on F.
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

38. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
measurable space (F, F)
In our example F = L2
([0, T]) and F is the Borel σ-algebra on F.
probabilistic context tree (τ, p)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

39. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
measurable space (F, F)
In our example F = L2
([0, T]) and F is the Borel σ-algebra on F.
probabilistic context tree (τ, p)
family {Qw : w ∈ τ} of probabilities on (F, F)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

40. Hidden context tree model (HCTM)
Ingredients:
finite alphabet A
In our example A = {0, 1, 2}
measurable space (F, F)
In our example F = L2
([0, T]) and F is the Borel σ-algebra on F.
probabilistic context tree (τ, p)
family {Qw : w ∈ τ} of probabilities on (F, F)
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

41. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

42. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
(Xn)n∈Z is generated by (τ, p)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

43. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
(Xn)n∈Z is generated by (τ, p)
for any m, n ∈ Z with m ≤ n
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

44. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
(Xn)n∈Z is generated by (τ, p)
for any m, n ∈ Z with m ≤ n
any string xn
m− (τ)+1 ∈ An−m+ (τ)
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

45. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
(Xn)n∈Z is generated by (τ, p)
for any m, n ∈ Z with m ≤ n
any string xn
m− (τ)+1 ∈ An−m+ (τ)
and any sequence In
m = (Im, . . . , In) of F-measurable sets,
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

46. Hidden context tree model
(Xn, Yn)n∈Z HCTM compatible with (τ, p) and (Qw : w ∈ τ) if
(Xn)n∈Z is generated by (τ, p)
for any m, n ∈ Z with m ≤ n
any string xn
m− (τ)+1 ∈ An−m+ (τ)
and any sequence In
m = (Im, . . . , In) of F-measurable sets,
P Y n
m ∈ In
m|Xn
m− (τ)+1 = xn
m− (τ)+1 =
n
k=m
Qcτ (xk
k− (τ)+1
)(Ik)
(τ) = height of τ
cτ (xk
k− (τ)+1) = context assigned to xk
k− (τ)+1 by τ
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

47. Rephrasing our problem
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

48. Rephrasing our problem
Taking
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

49. Rephrasing our problem
Taking
(Xn)n∈Z sequence of auditory stimuli produced by the samba source
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

50. Rephrasing our problem
Taking
(Xn)n∈Z sequence of auditory stimuli produced by the samba source
(Y e
n )n∈Z successive chunks of EEG signals
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

51. Rephrasing our problem
Taking
(Xn)n∈Z sequence of auditory stimuli produced by the samba source
(Y e
n )n∈Z successive chunks of EEG signals
Question: Is (Xn, Y e
n )n∈Z a HCTM compatible with τ?
000 100 200
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2
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

52. is (Xn, Y e
n )n∈Z a HCTM compatible with τ?
In other terms, for any w ∈ τ, is it true that
L(Y e
n |Xn
n− (w)+1 = w, X
− (τ)
−∞ = u) = L(Y e
n |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

53. Pruning the tree
A version of Rissanen’s algorithm Context will be applied
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

54. Pruning the tree
A version of Rissanen’s algorithm Context will be applied
Start with a maximal admissible candidate tree
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

55. Pruning the tree
A version of Rissanen’s algorithm Context will be applied
Start with a maximal admissible candidate tree
For any string w and pair of symbols a, b ∈ A with aw and bw
belonging to the candidate tree
test the equality
L(Y e
n |Xn
n− (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

56. Pruning the tree
If for all pairs of symbols (a, b) the equality is rejected then prune all
the leaves aw
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

57. How to test the equality
L(Yn|Xn
n− (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

58. Experimental results
Context tree selection procedure for the EEG data recorded during
the exposure to the sequence of auditory stimuli generated by the
samba source
Sample composed by 20 subjects
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

59. Experimental results
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

60. Experimental results
Antonio Galvesjoint work with A. Duarte, R. Fraiman, G. Ost and C. Vargas Hidden context tree modeling of EEG data

61. 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