Outline
M ODELING D EPENDENCE IN M ULTIVARIATE
T IME S ERIES WITH A PPLICATIONS TO B RAIN
S IGNALS
Hernando Ombao
University of California at Irvine
May 15, 2014
Outline
O UTLINE OF TALK
1
S CIENTIFIC M OTIVATION
2
OVERVIEW S PECTRAL A NALYSIS
3
C URRENT WORK
Outline
O UTLINE OF TALK
1
S CIENTIFIC M OTIVATION
2
OVERVIEW S PECTRAL A NALYSIS
3
C URRENT WORK
Outline
O UTLINE OF TALK
1
S CIENTIFIC M OTIVATION
2
OVERVIEW S PECTRAL A NALYSIS
3
C URRENT WORK
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
C ONNECTIVITY A NALYSIS OF B RAIN S IGNALS
Electrophysiologic data: multi-channel EEG, local field
potentials
Hemodynamic data: fMRI time series at several ROIs
Neuronal: spike train and local field potentials
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
DATA F ROM D ESIGNED N EUROSCIENCE E XPERIMENTS
External Stimulus
Visual, Auditory, Somatosensory, Stress
Personality traits, Genes, Socio-Environmental Factors
Brain Signals (indirect measures of neuronal activity)
Functional: fMRI, EEG, MEG, PET
Anatomical: DTI
Acute Outcomes
Emotion, Skin conductance, Motor response
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
G OALS OF OUR RESEARCH
Stimulus
Neuronal
Response
Brain
Signals
Behavior
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
G OALS OF OUR RESEARCH
Stimulus
Neuronal
Response
Brain
Signals
Genes
Moderators
Modifiers
Trait
Socio-Environment
Behavior
Outline of Talk
Scientific Motivation
G OALS OF OUR RESEARCH
Changes in
the mean
Changes in
variance
Changes in
Cross-Dependence
Overview Spectral Analysis
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON L EARNING
CONTE Center on OCD (Brown, Rochester, MGH)
OCD characterized by abnormal risk assessment leading
to excessive avoidance
Compulsions can significantly interfere with normal routine
OCD patients have reduced activation in the nucleus
accumbens - dysfunctional reward circuitry
At Brown Psychiatry (Rasmussen and Greenberg): brain
stimulation of OCD patients
My collaboration entails
Studying dynamics of neural circuitry associated with OCD
Identifying associations between neurophysiological
response and behavior
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON L EARNING
Learning study on a macaque monkey
Collaborator: Emad Eskandar (Neurosurgery, MGH)
Goal in one study: explore dynamic connectivity between
the nucleus accumbens (NAc) and the hipocampus (Hc)
NAc plays a central role in the reward circuitry
Hc plays a role in memory (in particular, storing and
processing spatial information)
Data: local field potentials
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
P ROJECT ON L EARNING
Fixation
Picture on
Picture off
Four doors on
?
?
?
?
NAc
HC
Time 0
512
block 1
1024
block 2
1536
block 3
2048
block 4
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON THE DYNAMICS OF NEURONAL NETWORKS
DURING DECISION MAKING
Collaborator: David Moorman (Neuroscience,
UMass-Amherst)
Goals
Model how neural systems assimilate information to
produce decision
Model interactions between neurons in the medial
pre-frontal cortex (mPFC) during decision making
Subjects: Rats
Experiment: Choose between low risk-low reward vs high
risk-high reward
Data: spike train and local field potentials (multi-modal)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON THE DYNAMICS OF NEURONAL NETWORKS
DURING DECISION MAKING
Microwire arrary implanted in the brain of a rat (Moorman Lab)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON THE DYNAMICS OF NEURONAL NETWORKS
DURING DECISION MAKING
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON THE DYNAMICS OF NEURONAL NETWORKS
DURING DECISION MAKING
Spike Train
Neuron 1
LFP
GC Spike to Spike
GC LFP to Spike
GC Spike to LFP
GC LFP to LFP
Neuron 2
Wavelet Coherence
GC = Granger Causality
Neuron 3
time t
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
P ROJECT ON THE DYNAMICS OF NEURONAL NETWORKS
DURING DECISION MAKING
The activity of ensembles of neurons provide more
information than that of single neurons
Some questions:
Does past spiking activity in one neuron influence future
propensity to spike of another neuron?
Does past oscillatory activity in one neuron influence future
propensity to spike of another neuron?
Impact of pharmacologic agents on connectivity between
neurons in a network
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
R ESEARCH ON S TATISTICAL T HEORY AND
M ETHODOLOGY
Characterize dependence in a brain network
Temporal: Y1 (t) ∼ [Y1 (t − 1), Y2 (t − 1), . . .]′
Spectral: interactions between oscillatory activities at Y1 , Y2
Develop estimation and inference methods for connectivity
Investigate connectivity as a biomarker
Predict mental state (level of mental fatigue)
At UC Irvine (PI: Cramer)
Predict performance in learning a new motor skill in healthy
controls
Predict recovery of motor functionality in stroke patients
Discrimination between patient groups (e.g., bipolar vs.
healthy control)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
R ESEARCH ON S TATISTICAL T HEORY AND
M ETHODOLOGY
Interpretability of new dependence measures
Incorporate information across trials, across subjects
Compare brain network across conditions
Integrate multi-modal data (Spike train, LFP, EEG, fMRI,
DTI)
Statistical model should be informed by physiology and
physics
Dimension reduction: extract information from massive
data that is most relevant for estimating dependence
Develop formal statistical inference procedures
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
R ESEARCH ON S TATISTICAL T HEORY AND
M ETHODOLOGY
In this talk,
Develop a directly relevant spectral measure of
dependence
Exploratory analysis of brain signals (LFPs and spike train)
Present our proposed unified framework - Evolutionary
Harmonizable Process
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
S PECTRAL A NALYSIS - A R EGRESSION P OINT OF V IEW
X (t) STATIONARY TEMPORAL PROCESS
Cramér Representation
X (t) =
R
exp(i2πωt)dZ (ω), t = 0, ±1, ±2, . . .
Basis Fourier waveforms exp(i2πωt), ω ∈ (−0.5, 0.5)
Random coefficients dZ (ω) – increment random process
EdZ (ω) = 0 and
Cov[dZ (ω), dZ (λ)] = δ(ω − λ)f (ω)dωdλ
Var dZ (ω) = f (ω)dω Spectrum f (ω)
VARIANCERDECOMPOSITION of the time series:
Var X (t) = f (ω)dω
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
AR(1): Xt = 0.9Xt−1 + ǫt
Low Frequency Oscillations
8
6
4
2
0
−2
−4
−6
−8
−10
0
100
200
300
400
500
600
700
800
900
1000
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
Spectrum of AR(1) with φ = 0.9
Frequency
0.5
0
0
1
Time
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
AR(1): Xt = −0.9Xt−1 + ǫt
High Frequency Oscillations
6
4
2
0
−2
−4
−6
−8
0
100
200
300
400
500
600
700
800
900
1000
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
Spectrum of AR(1) with φ = −0.9
Frequency
0.5
0
0
1
Time
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
Mixture: Low + High Frequency Signal
20
15
10
5
0
−5
−10
−15
0
100
200
300
400
500
Time
600
700
800
900
1000
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
I LLUSTRATION OF S PECTRUM = VARIANCE
D ECOMPOSITION
Spectrum of the mixed signal
Frequency
0.5
0
0
1
Time
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C ROSS - COHERENCE - A MEASURE OF DEPENDENCE
Multivariate stationary time series:
X(t) = [X1 (t), . . . , XP (t)]′
Random vector increment process
d Z(ω) = [dZ1 (ω), . . . , dZP (ω)]′
Cramér representation
Z 0.5
exp(i2πωt)d Z(ω)
X(t) =
−0.5
Cov[d Z(ω), d Z(λ)] = 0 for ω 6= λ
Var d Z(ω) = f(ω)d ω
Coherence between Xp (t) and Xq (t)
|ρpq (ω)|2 = |Corr[dZp (ω), dZq (ω)]|2
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C ROSS - COHERENCE - A MEASURE OF DEPENDENCE
Illustration: Interactions between oscillatory components
Latent Signals
U1 (t) - low frequency signal
U2 (t) - high frequency signal
Observed Signals
X (t)
Y (t)
=
=
U1 (t)
U1 (t + ℓ)
+
U2 (t)
+
+
Z1 (t)
Z2 (t)
X and Y are linearly related through U1 .
Cross-correlation will confirm the linear association
between X (t) and Y (t).
Cross-coherence will identify the frequency band(s) that
drive the linear association.
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C ROSS - COHERENCE - A MEASURE OF DEPENDENCE
X1(t)
Delta
Theta
Alpha
Beta
Gamma
X2(t)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C ROSS -C OHERENCE : A N I NTUITIVE I NTERPRETATION
Filtered Signals [see Ombao and Van Bellegem (2008, IEEE
Trans Signal Processing)]:
Xω (t) = Fω X (t) Yω (t) = Fω Y (t) Zω (t) = Fω Z (t)
Filtered signals have power concentrated around ω
Coherence at frequency band around ω
ρX ,Y (ω) ≈ |Corr (Xω (t), Yω (t))|2
Partial coherence
Remove Zω (t) from Xω (t): ξωX (t) = Xω (t) − βX Zω (t)
Remove Zω (t) from Yω (t): ξωY (t) = Yω (t) − βY Zω (t)
Cov(ξX (t),ξY (t)) 2
ω
ω
√
ρX ,Y |Z (ω) = X (t) Var ξ Y (t) Var ξω
ω
Fiecas et al. (2010, 2011) - estimation by shrinkage
Relevant work: Pupin (1898)
Estimator for fX (ω) is Var Xω (t).
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C ROSS -C OHERENCE : A N I NTUITIVE I NTERPRETATION
Limitation of classical coherence: interactions between
oscillations only at a single frequency
Low freq oscillations in X vs Low freq oscillations in Y
High freq oscillations in X vs High freq oscillations in Y
Dual-frequency (cross-frequency) interactions
Interactions between low frequency and high frequency
oscillations
Limitation: interactions constant over time
Further generalizations needed
Evolutionary (time-varying) dual-frequency coherence
Lagged dual-frequency coherence
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
A P ROPOSAL : G ENERALIZED C OHERENCE
Frequency bands
(1, 4) Hertz - Delta
(4, 8) Hertz - Theta
(8, 12) Hertz - Alpha
(16, 30) Hertz - Beta
(30, 70) Hertz - Gamma
Dual-frequency (cross-frequency) interactions
Contemporaneous dependence: interactions between
alpha oscillation activity in X and beta activity in Y at the
same time block
Lagged relationships: dependence between oscillations at
adjacent time blocks
Predictions: does current oscillatory pattern predict future
oscillations?
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
A P ROPOSAL : G ENERALIZED C OHERENCE
X1(t)
X2(t)
Delta
Theta
Alpha
Beta
Gamma
Classical Coherence
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
A P ROPOSAL : G ENERALIZED C OHERENCE
X1(t)
X2(t)
Delta
Theta
Alpha
Beta
Gamma
Dual Frequency Coherence
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
A P ROPOSAL : G ENERALIZED C OHERENCE
X1(t)
X2(t)
Delta
Theta
Alpha
Beta
Gamma
Evolutionary and Lagged Coherence
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E XPLORATORY ANALYSIS OF A MONKEY LFP DATA
Subject: male macaque monkey
Data
Local field potentials in the nucleus accumbens (NAc) and
hipocampus (Hc)
R = 100+ trials; each trial has T = 2048 time points (≈ 2
seconds); band-filtered at (0.05, 150) Hertz
In memory and learning studies, evidence of θ − γ coupling
Collaborators
Gorrostieta (UC-Irvine), Prado (UC-SC), Patel (Boston
Univ) and Eskandar (MGH)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E XPLORATORY ANALYSIS OF A MONKEY LFP DATA
Fixation
Picture on
Picture off
Four doors on
?
?
?
?
NAc
HC
Time 0
512
block 1
1024
block 2
1536
block 3
2048
block 4
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E XPLORATORY ANALYSIS OF A MONKEY LFP DATA
Fixation
Picture on
Picture off
Four doors on
?
?
?
?
Theta
HC
NAc
HC
NAc
HC
NAc
HC
NAc
Gamma
HC
NAc
HC
NAc
HC
NAc
HC
NAc
Time 0
512
block 1
1024
block 2
1536
block 3
Correct > Incorrect
2048
block 4
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
H ISTORICAL OVERVIEW
Harmonizable process (1955)
Xt =
Z
0.5
exp{2πiωt}dZ (ω)
−0.5
{dZ (ω)} not necessarily uncorrelated.
Generalized Spectrum - Loéve Spectrum
Cov(dZ (ω), dZ (λ)) = f (ω, λ)d ωd λ
Long history: Loéve (1955); Scharf (1990’s); Lii and Rosenblatt
(1998,2002); Hindberg and Hanssen (2007)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
H ISTORICAL OVERVIEW
Stationary Process
R 0.5
Xt = −0.5 A(ω) exp(i2πωt)dZ (ω) where
Cov[dZ (ω1 ), dZ (ω2 )] = δ(ω1 − ω2 )d ω1 d ω1
Spectrum
f (ω1 , ω2 ) =
A(ω1 )A∗ (ω1 ), if ω1 = ω2
0,
if ω1 6= ω2
In the stationary case, f is “diagonal"
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
H ISTORICAL OVERVIEW
Locally Stationary Process
R 0.5
Xt = −0.5 At (ω) exp(i2πωt)dZ (ω) [Priestley (1965)]
R 0.5
Xt,T = −0.5 A(t/T , ω) exp(i2πωt)dZ (ω) [Dahlhaus (1997)]
Evolutionary Spectrum f (u, ω) = A(u, ω)A∗ (u, ω)
In our set up, the evolutionary spectrum f is
time-dependent and diagonal.
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
H ISTORICAL OVERVIEW
Harmonizable Process - Loève
R 0.5
Xt = −0.5 exp(i2πωt)dZ (ω)
Generalized Spectrum
Cov[dZ (ω1 ), dZ (ω2 )] = f (ω1 , ω2 )d ω1 d ω2
Dual-frequency spectrum f (ω1 , ω2 )
Measures the “global" cross-oscillatory interactions
between ω1 and ω2
f is not constrained to be “diagonal"
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
A PROPOSAL : E VOLUTIONARY H ARMONIZABLE
P ROCESS
Xt,T =
R 0.5
−0.5 exp(i2πωt)dZt,T (ω)
Cov[dZt,T (ω1 ), dZt,T (ω2 )] = ft,T (ω1 , ω2 )d ω1 d ω2
f is both time-dependent and not constrained to be diagonal
Measures the time-varying dependence between the ω1
and ω2 oscillations
ρ(pq) (t/T , ω1 , ω2 ) =
|f (pq) (t/T , ω1 , ω2 |2
f (pp) (t/T , ω1 , ω1 )f (qq) (t/T , ω1 , ω2 )
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
A PROPOSAL : E VOLUTIONARY H ARMONIZABLE
P ROCESS
0
Stationary
1
0
T
Locally
Stationary
0
T
Locally
Harmonizable
0
T
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
A PROPOSAL : E VOLUTIONARY H ARMONIZABLE
P ROCESS
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
A PROPOSAL : E VOLUTIONARY H ARMONIZABLE
P ROCESS
Stationary
0
1
0
1
Locally
Stationary
Harmonizable
0
1
Locally
Harmonizable
0
1
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E STIMATION AND I NFERENCE
X r (t) multivariate time series for trial r = 1, . . . , R
Fourier coefficient on block around time t
T
d r ,p (t, ω) =
2
X
Xpr (s) exp(−i2πωs)
s=−( T2 −1)
Contemporaneous dual-frequency cross-periodgram for
the r -th trial at time block around t
I r ,(pq) (t, ω1 , ω2 ) = d r ,p (t, ω1 )d r ,q,∗ (t, ω2 )
Estimator of contemporaneous generalized cross-spectrum
R
X
bf (pq) (t/T , ω1 , ω2 ) = 1
I r ,(pq) (t, ω1 , ω2 )
R
r =1
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
E STIMATION AND I NFERENCE
Estimator of the evolutionary dual frequency
auto-coherence
ρb(pp) (t/T , ω1 , ω2 ) =
|bf (pp) (t/T , ω1 , ω2 )|2
bf (pp) (t/T , ω1 , ω1 )bf (pp) (t/T , ω2 , ω2 )
ρb(pq) (t/T , ω1 , ω2 ) =
|bf (pq) (t/T , ω1 , ω2 )|2
bf (pp) (t/T , ω1 , ω1 )bf (qq) (t/T , ω2 , ω2 )
Estimator of the evolutionary dual frequency
cross-coherence
Current Work
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E STIMATION AND I NFERENCE
The contemporaneous Loéve (generalized) cross spectral
estimator is asymptotically unbiased for the Loéve
cross-spectrum, i.e., as R, T → ∞,
Ebf (pq) (t/T , ω1 , ω2 ) −→ f (pq) (u, ω1 , ω2 ).
Remark. This result is a generalization of the asymptotic
unbiasedness of classical periodograms in Brillinger (1981)
and Brockwell and Davis (1991).
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
E STIMATION AND I NFERENCE
For sufficiently large block length T and number of trials R,
(pp)
bf (pp) (t/T , ω1 , ω1 )
f
(t/T , ω1 , ω1 )
b(qq)
f (qq) (t/T , ω2 , ω2 ) V
(t/T , ω2 , ω2 )
f
is AN
b(pq)
ℜf (pq) (t/T , ω1 , ω2 ) , R .
ℜf
(t/T , ω1 , ω2 )
ℑf (pq) (t/T , ω1 , ω2 )
ℑbf (pq) (t/T , ω1 , ω2 )
Details: Gorrostieta, Ombao and von Sachs (2014, under
revision).
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
C OLLABORATORS
Space-Time Modeling Group at UC-Irvine
Yuxiao Wang, Zhe Yu, Babak Shahbaba, Duy Ngo, Cristina
Gorrostieta
Sam Behseta (Cal State Fullerton) and Rainer von Sachs
(Université catholique de Louvain)
Brain Scientists
Moorman (UMass), Eskandar (MGH)
Outline of Talk
Scientific Motivation
Overview Spectral Analysis
Current Work
ACKNOWLEDGEMENTS
Support from the NSF-DMS and NSF-SES
Thanks for Professor Devin Koestler for the invitation!
Thanks to the Department of Biostatistics at the University
of Kansas Medical Center for the warm hospitality!
© Copyright 2024 ExpyDoc
