Transcript Slide 1
Partial Directed Coherence and Directed
Transfer Function Based on ARFIT and AAR
algorithms
Amir Omidvarnia
12 Nov. 2010
1
Outline
Introduction to Kalman filtering
Time-invariant MVAR estimation: ARFIT algorithm
Time-varying MVAR estimation: Adaptive AR model
Results
time-invariant DTF and PDC
Time-varying DTF and PDC
Conclusion
References
2
Introduction to Kalman filtering
Linear state space model
x(t): Process state at time ‘t’
y(t): Observation at time ‘t’
A: State transition matrix
C: Measurement matrix
V ~ N(0,Q): Process noise
N ~ N(0,R): Observation noise
3
Introduction to Kalman filtering
Assumptions of the linear Kalman filter
Linear system
States at time t are linear function of states at time t-1.
Measurements at time t are linear function of states at the same time.
White Gaussian noise
Observation noise (R) and process noise (Q) are white (uncorrelated in
time) and Gaussian (noise amplitude).
4
Introduction to Kalman filtering
Prediction (Time Update)
Correction (Measurement Update)
(1) Compute the Kalman Gain
(1) Project the state ahead
(2) Update estimate with measurement y(t)
(2) Project the error covariance ahead
(3) Update Error Covariance
5
Introduction to Kalman filtering
Kalman filter finds optimum states based on Maximum-
Likelihood estimations [1]:
For state estimation
For state and parameter estimation
6
Introduction to Kalman filtering
Kalman filter applications:
State estimation Classical usage
Parameter estimation Adaptive AR estimation etc
Dual state and Parameter estimation DEKF etc
Joint state and parameter estimation Joint EKF etc
7
Time-invariant MVAR estimation:
ARFIT algorithm
8
Time-invariant MVAR estimation:
ARFIT algorithm
ARFIT applies a stepwise least squares algorithm on a
sequence of AR models of successive orders pmin, . . . , pmax to
evaluate a criterion (AIC, SBC etc) for the selection of the
optimum model order and parameters [2].
Schwarz’s Bayesian Criterion (SBC) has better performance
on the selection of the correct model order and in average,
results in the smallest mean-squared prediction error of the
fitted AR models [2-3].
9
Schwarz’s Bayesian Criterion (SBC) vs.
Akaike Information Criterion (AIC)
Both of these criteria use Maximum-likelihood principle to
make a compromise between the model order (model
complexity), goodness of fit and tracking ability [4].
p:Model order
∑n: Covariance matrix of the measurement noise
CH: Number of channels
L: Length of the time series
10
Time-varying MVAR estimation:
Adaptive AR model
11
General form of the MVAR models
12
Adaptive AR model estimation
13
1.
AAR parameters are considered as the states of a
multichannel model and are estimated based on the
Kalman filtering approach [5].
2.
Multivariate AR parameters are re-formatted from the
matrix form into the vector form.
3.
Multivariate AR equations are re-written in the form of
Kalman filter relations.
Adaptive AR model estimation
New state space model is constructed:
AR matrices are converted to a vector:
Measurement matrix C is constructed using delayed
observations:
14
Adaptive AR model estimation
If the relationship between observations (y) and states (a) is
considered as a multivariate function H(y,a), matrix C is
equal to the Jacobian matrix of H at point a(n-1):
15
Analysis Procedure:
Time-Invariant
vs.
Time-Varying
16
Analysis procedure
1. The whole signal is fed into ARFIT algorithm and time-invariant
MVAR parameters are estimated for successive orders pmin, . . . , pmax.
2. The optimum model order popt is selected based on the minimum
value of the Schwarz’s Bayesian Criterion (SBC).
3. Time-varying MVAR parameters are estimated using Adaptive AR
model algorithm. popt is kept identical during the time-varying analysis
procedure.
4. TI- and TV-Partial Directed Coherences (PDCs) and Directed Transfer
Functions (DTFs) are extracted using the estimated TI- and TVMVAR parameters (ARFIT vs. AAR).
17
Analysis procedure (cont.)
5. 50 surrogates are generated based on the original signal
and steps 1 to 4 are performed for each of them.
6. Averaged PDC and DTF on the surrogates are computed
and used as the varying thresholds to determine significant
values of the original PDC and DTF measures in the timefrequency domain.
18
Results:
Partial Directed Coherence (PDC) and
Directed Transfer Function (DTF)
Based on
ARFIT and AAR
19
Data
Analysis procedure is performed on three simulated MVAR
models and a selected 5-channel (out of 14) newborn EEG
dataset.
One simulated model is time-invariant and two are time-
varying.
EEG dataset covers a complete seizure period of 117 sec
(30,000 samples).
20
Model 1
Time-invariant MVAR model [6]
21
Model 1: Schwarz’s Bayesian Criterion
SBC suggests the optimum model order of 3 which is
compatible with the model.
22
Model 1: Time-varying MVAR
parameters
23
Model 1: Time-invariant DTF and PDC
DTF
24
PDC
Model 1: TF- DTF and PDC before
thresholding
TF-DTF
25
TF-PDC
Model 1: Averaged TF- DTF and PDC of
the surrogates
TF-DTF
26
TF-PDC
Model 1: TF- DTF and PDC after
thresholding
TF-DTF
27
TF-PDC
Model 2
The same as Model 1, but two connections have become
time-varying.
28
Model 2: Time-varying MVAR
parameters
Optimum model order estimated by SBC: 3
a54(n)
a45(n)
29
Model 2: Time-invariant DTF and PDC
DTF
30
PDC
Model 2: TF- DTF and PDC before
thresholding
TF-DTF
31
TF-PDC
Model 2: TF- DTF and PDC after
thresholding
TF-DTF
32
TF-PDC
Model 3
Second time-varying MVAR model [7]
33
Model 3: Time-varying MVAR
parameters
Optimum model order estimated by SBC: 2
c12(n)
c23(n)
34
Model 3: Time-invariant DTF and PDC
DTF
35
PDC
Model 3: TF- DTF and PDC before
thresholding
TF-DTF
36
TF-PDC
Model 3: TF- DTF and PDC after
thresholding
TF-DTF
37
TF-PDC
5-channel newborn EEG data
Frequency band of 0-40 Hz has been selected (Fs=256 Hz).
Analysis is based on five channels out of 14 (O1, O2, P3, P4,
Cz) to investigate the interactions between two hemispheres.
Length of the signal covers a complete seizure (117 sec, equal
to 30000 samples).
Seizure mask
38
EEG data: Time-varying MVAR
parameters
Optimum model order estimated by SBC: 6
39
EEG data : Time-invariant DTF and PDC
DTF
PDC
P3
P4
O1
O2
Cz
P3
40
P4
O1
O2
Cz
P3
P4
O1
O2
Cz
EEG data : TF- DTF and PDC before
thresholding
TF-DTF
TF-PDC
P3
P4
O1
O2
Cz
P3
41
P4
O1
O2
Cz
P3
P4
O1
O2
Cz
EEG data : TF- DTF and PDC after
thresholding
TF-PDC
TF-DTF
P3
P4
O1
O2
Cz
P3
42
P4
O1
O2
Cz
P3
P4
O1
O2
Cz
EEG data: suggested Directed Graphs
Solid arrows represent strong linear relationships, while
dashed arrows indicate weak connections.
Directed GraphDTF
Directed GraphPDC
Cz
Cz
P4
P3
O1
43
O2
P4
P3
O1
O2
Conclusion
According to the results of the simulated models, PDC
outperforms DTF for time-varying linear MVAR models.
Adaptive AR model estimator is not able to extract fast
changes in the MVAR model and is not suitable for accurate
tracking of the time-varying parameters.
Prior knowledge about the seizure foci can help us to select
potentially involved channels and reduce computational
complexity.
44
References
1.
Nelson, A.T. and E.A. Wan. A two-observation Kalman framework for
maximum-likelihood modeling of noisy time series. in Neural Networks Proceedings,
1998. IEEEWorld Congress on Computational Intelligence.The 1998 IEEE International
Joint Conference on. 1998.
2.
Arnold, N. and S. Tapio, Estimation of parameters and eigenmodes of
multivariate autoregressive models. ACM Trans. Math. Softw., 2001. 27(1): p. 2757.
3.
Lütkepohl, H., COMPARISON OF CRITERIA FOR ESTIMATING THE
ORDER OF AVECTOR AUTOREGRESSIVE PROCESS. Journal of Time Series
Analysis, 1985. 6(1): p. 35-52.
4.
Havlicek, M., et al., Dynamic Granger causality based on Kalman filter for
evaluation of functional network connectivity in fMRI data. Neuroimage, 2010.
53(1): p. 65-77.
45
References
5.
Schlögl, A., THE ELECTROENCEPHALOGRAM AND THE
ADAPTIVE AUTOREGRESSIVE MODEL:THEORY AND
APPLICATIONS. 2000: Shaker Verlag, Aachen, Germany.
6.
Baccalá, L.A. and K. Sameshima, Partial directed
coherence: a new concept in neural structure determination.
Biological Cybernetics, 2001. 84(6): p. 463-474.
7.
Winterhalder, M., et al., Comparison of linear signal
processing techniques to infer directed interactions in multivariate
neural systems. Signal Processing, 2005. 85(11): p. 2137-2160.
46