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Structured stochastic processes and functional data analysis for the assessment of motor learning in normal and pathological subjects (Postdoctoral research submitted to FAPESP) Noslen Hernndez Antonio Galves, Claudia Vargas II NeuroMat

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Structured stochastic processes and functional data analysis for the assessment

  • f motor learning in normal and pathological

subjects

Noslen HernΓ‘ndez Antonio Galves, Claudia Vargas November 2016 II NeuroMat Workshop: New Frontiers in Neuromathematics

(Postdoctoral research submitted to FAPESP)

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Recent results [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

  • Hypothesis: The brain retrieves statistical regularities from stimuli
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Recent results [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

  • Definition of new kind stochastic processes
  • Hypothesis: The brain retrieves statistical regularities from stimuli
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Recent results [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

  • Definition of new kind stochastic processes

Stochastic process driven by context tree model

  • Hypothesis: The brain retrieves statistical regularities from stimuli
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Recent results [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

  • Definition of new kind stochastic processes

Stochastic process driven by context tree model

  • Allows a design, modeling and analysis of neurophysiological

experiments with structured stimuli

  • Hypothesis: The brain retrieves statistical regularities from stimuli
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Retrieving a context tree form EEG data

Images taken from [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

Random Source

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Retrieving a context tree form EEG data

Images taken from [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

Random Source

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Retrieving a context tree form EEG data

Images taken from [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

Random Source

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Retrieving a context tree form EEG data

Images taken from [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579]

  • π‘Œ1, 𝑍

1 , … , π‘Œπ‘œ, 𝑍 π‘œ

  • New statistical model selection

procedure for FD

  • The brain effectively identifies the

context tree characterizing the source. Random Source

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Can such finding be corroborated in behavioral responses, specifically in execution of movements?

Images taken from https://www.lvhn.org/for_referring_physicians/better_medicine/rehabilitation/neuromuscular_rehabilitation_technology_helps_patients_i ncrease_motor_control; http://neuromat.numec.prp.usp.br/pt-br/newsletter-31; https://lehacker.com/brain-facts-revealing/

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Can such finding be corroborated in behavioral responses, specifically in execution of movements?

Images taken from https://www.lvhn.org/for_referring_physicians/better_medicine/rehabilitation/neuromuscular_rehabilitation_technology_helps_patients_i ncrease_motor_control; http://neuromat.numec.prp.usp.br/pt-br/newsletter-31; https://lehacker.com/brain-facts-revealing/

Random Source

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Can such finding be corroborated in behavioral responses, specifically in execution of movements?

Images taken from https://www.lvhn.org/for_referring_physicians/better_medicine/rehabilitation/neuromuscular_rehabilitation_technology_helps_patients_i ncrease_motor_control; http://neuromat.numec.prp.usp.br/pt-br/newsletter-31; https://lehacker.com/brain-facts-revealing/

Random Source

Execution of movement

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Can such finding be corroborated in behavioral responses, specifically in execution of movements?

Rehabilitation of patients Goalkeeper Game

Images taken from https://www.lvhn.org/for_referring_physicians/better_medicine/rehabilitation/neuromuscular_rehabilitation_technology_helps_patients_i ncrease_motor_control; http://neuromat.numec.prp.usp.br/pt-br/newsletter-31; https://lehacker.com/brain-facts-revealing/

Random Source

Execution of movement

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Stochastic processes driven by structured Markov chain

  • Finite alphabet 𝐡
  • Sequence of stimuli π‘Œπ‘œ ∈ 𝐡 generated according to specific regular statistical pattern
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Stochastic processes driven by structured Markov chain

  • Finite alphabet 𝐡
  • Sequence of stimuli π‘Œπ‘œ ∈ 𝐡 generated according to specific regular statistical pattern
  • Structured Markov chain π‘Œπ‘œ π‘œβˆˆβ„€ :
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Stochastic processes driven by structured Markov chain

  • Finite alphabet 𝐡
  • Sequence of stimuli π‘Œπ‘œ ∈ 𝐡 generated according to specific regular statistical pattern
  • Structured Markov chain π‘Œπ‘œ π‘œβˆˆβ„€ :
  • for some π‘š,
  • any 𝑛 β‰₯ π‘š, (π‘š, 𝑛 ∈ β„€)
  • and any finite string π‘¦π‘œβˆ’π‘›

π‘œβˆ’1 = π‘¦π‘œβˆ’π‘›, … , π‘¦π‘œβˆ’1 ∈ 𝐡𝑛,

  • : mapping assigning to each past string a

corresponding class in a partition of the space of relevant pasts.

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Stochastic processes driven by structured Markov chain

  • Sequence of response 𝑍

π‘œ ∈ 𝒡 to the stimuli

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Stochastic processes driven by structured Markov chain

  • Sequence of response 𝑍

π‘œ ∈ 𝒡 to the stimuli

  • Stochastic processes driven by structured Markov chain
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Stochastic processes driven by structured Markov chain

  • Sequence of response 𝑍

π‘œ ∈ 𝒡 to the stimuli

  • Stochastic processes driven by structured Markov chain
  • π‘Œπ‘œ π‘œβˆˆβ„€ is a structured Markov chain.
  • 𝑍

1, 𝑍 2, … are independent variables conditionally to the sequence π‘Œπ‘œ π‘œβˆˆβ„€

for any measurable 𝐾

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Stochastic processes driven by structured Markov chain

  • Sequence of response 𝑍

π‘œ ∈ 𝒡 to the stimuli

  • Stochastic processes driven by structured Markov chain
  • π‘Œπ‘œ π‘œβˆˆβ„€ is a structured Markov chain.
  • 𝑍

1, 𝑍 2, … are independent variables conditionally to the sequence π‘Œπ‘œ π‘œβˆˆβ„€

for any measurable 𝐾 Stochastic processes driven by context tree model is an outstanding example of SPDSMC

Related works: J.Garcia et al., arXiv:1002.0729 (2010); V. JÀÀskinen et al., Scand. J. Stat (2014)

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

Goalkeeper Game 0: center; 1: right; 2: left

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

Goalkeeper Game 0: center; 1: right; 2: left πΈπ‘œ Curves of spatial position, Videos gathered by sensors or camera

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

Goalkeeper Game 0: center; 1: right; 2: left πΈπ‘œ Curves of spatial position, Videos gathered by sensors or camera 𝑍

π‘œ ∈ 𝐡

direction

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

Goalkeeper Game 0: center; 1: right; 2: left πΈπ‘œ Curves of spatial position, Videos gathered by sensors or camera 𝑍

π‘œ ∈ 𝐡

𝑍

π‘œ ∈ 𝒡

direction FD representing trajectory

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Prob1: Design of suitable experiments to evaluate evidence about learning statistical regularities

We need to carefully specify:

  • An alphabet 𝐡
  • Response variable 𝑍

π‘œ

Goalkeeper Game 0: center; 1: right; 2: left πΈπ‘œ Curves of spatial position, Videos gathered by sensors or camera 𝑍

π‘œ ∈ 𝐡

𝑍

π‘œ ∈ 𝒡

𝑍

π‘œ ∈ 𝑍 = 0,1,2 Γ— *𝐻, 𝐢+

direction FD representing trajectory

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝜐 π‘₯4 π‘₯3

π‘₯2 π‘₯1 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝜐 𝑄 π‘Œ0 = 𝑏|π‘Œβˆ’1

βˆ’π‘œ = π‘¦βˆ’1 βˆ’π‘œ = 𝑄(π‘Œ0 = 𝑏|π‘‘πœ(π‘¦βˆ’1 βˆ’π‘œ))

…….. 𝑏 …… π‘₯4 π‘₯3 π‘₯2 π‘₯1 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝜐 𝑄 π‘Œ0 = 𝑏|π‘Œβˆ’1

βˆ’π‘œ = π‘¦βˆ’1 βˆ’π‘œ = 𝑄(π‘Œ0 = 𝑏|π‘‘πœ(π‘¦βˆ’1 βˆ’π‘œ))

…….. 𝑏 …… π‘₯4 π‘₯3 π‘₯2 π‘₯1 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝑑

…….. 𝑏 …… π‘₯4 π‘₯3 π‘₯2 π‘₯1 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝑑

…….. 𝑏 …… π‘₯4 π‘₯3 π‘₯2 π‘₯1 The subject might adopt a form of representation of the given Markov chain that involves a certain partition into classes that does not involves a context tree Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝜐

𝑄

𝑑 𝑄 π‘Œ0 = 𝑏|π‘Œβˆ’1

βˆ’π‘œ = π‘¦βˆ’1 βˆ’π‘œ = 𝑄(π‘Œ0 = 𝑏|π‘‘πœ(π‘¦βˆ’1 βˆ’π‘œ))

…….. 𝑏 …… π‘₯2 π‘₯1 The subject might adopt a form of representation of the given Markov chain that involves a certain partition into classes that does not involves a context tree

  • The transition probabilities measures π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐 of

the context tree model be different

⟹ 𝑄

𝜐= 𝑄𝐷 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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  • Kind of statistical pattern of the stimuli sequence π‘Œπ‘œ π‘œβˆˆπ‘Ž

𝑄

𝜐

𝑄

𝑑 𝑄 π‘Œ0 = 𝑏|π‘Œβˆ’1

βˆ’π‘œ = π‘¦βˆ’1 βˆ’π‘œ = 𝑄(π‘Œ0 = 𝑏|π‘‘πœ(π‘¦βˆ’1 βˆ’π‘œ))

…….. 𝑏 …… π‘₯2 π‘₯1 The subject might adopt a form of representation of the given Markov chain that involves a certain partition into classes that does not involves a context tree

  • The transition probabilities measures π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐 of

the context tree model be different

  • How to quantify and control the complexity of the statistical

pattern?

⟹ 𝑄

𝜐= 𝑄𝐷 Context tree model compatible with 𝜐, π‘ž ; π‘ž = π‘ž β‹… π‘₯ ∢ π‘₯ ∈ 𝜐

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Prob2: Mapping from raw data to relevant response features 𝑍

π‘œ

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Prob2: Mapping from raw data to relevant response features 𝑍

π‘œ row data πΈπ‘œ Motion Capture System:

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Prob2: Mapping from raw data to relevant response features 𝑍

π‘œ row data πΈπ‘œ Motion Capture System: How to extract from such raw data relevant information 𝑍

π‘œ concerning postures and gestures

that characterize the response of the subject?

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Prob2: Mapping from raw data to relevant response features 𝑍

π‘œ row data πΈπ‘œ Motion Capture System: How to extract from such raw data relevant information 𝑍

π‘œ concerning postures and gestures

that characterize the response of the subject?

  • Pattern recognition techniques and machine learning algorithms
  • Computer vision techniques
  • Gesture representation and recognition 𝑍

π‘œ

  • Tracking algorithms
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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

Two-sample hypothesis testing step

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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

Two-sample hypothesis testing step Random projection method

[Cuestas-Albertos, Fraiman and Ransford (2006)]

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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

Two-sample hypothesis testing step Random projection method

[Cuestas-Albertos, Fraiman and Ransford (2006)]

Two-sample distribution-free test for FD

[Pomann et al., Appl. Statist. (2016)]

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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

Two-sample hypothesis testing step Random projection method

[Cuestas-Albertos, Fraiman and Ransford (2006)]

Do not give information on specific aspects of the distributions causing detected differences. Two-sample distribution-free test for FD

[Pomann et al., Appl. Statist. (2016)]

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Prob3: Making inference about π‘Œπ‘œ, 𝑍

π‘œ π‘œ

A consistent model selection procedure [A. Duarte, R. Fraiman, A. Galves, G. Ost and C.D. Vargas, arXiv:1602.00579] For 𝑍

π‘œ functional data:

Two-sample hypothesis testing step Random projection method

[Cuestas-Albertos, Fraiman and Ransford (2006)]

Do not give information on specific aspects of the distributions causing detected differences. Functional test that focus on particular parameters: Two-sample distribution-free test for FD

[Pomann et al., Appl. Statist. (2016)]

mean functions covariance operators

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Functional data analysis of motion curves

Biomedical questions of interest:

  • Detection of deviation from normality in movements
  • Longitudinal study of patients under treatment
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Functional data analysis of motion curves

Biomedical questions of interest:

  • Detection of deviation from normality in movements
  • Longitudinal study of patients under treatment

Modern FDA Tools

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Functional data analysis of motion curves

Biomedical questions of interest:

  • Detection of deviation from normality in movements
  • Longitudinal study of patients under treatment
  • Functional cluster analysis
  • Dissimilarity measures between functional populations
  • Functional shape analysis
  • Functional time series methods

Modern FDA Tools

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Thank you