dynamical correlation: a new method to quantify synchrony siwei liu 1, yang zhou 1, richard palumbo...
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Dynamical Correlation: A New Method to Quantify Synchrony
Siwei Liu1, Yang Zhou1, Richard Palumbo2, & Jane-Ling Wang1
1UC Davis; 2University of Rhode Island
Motivating Study Physiological synchrony between romantic
partners during nonverbal conditions
30 Minutes Total
15 MinutesFace to Face
15 MinutesBack to
Back
N=16
Electrodermal Activity (EDA) from Two Couples
Multilevel Modeling?
Assumes a universal model Random effects are normally distributed
Violations lead to biased estimates Difficult to converge with small sample size
- (Bell et al., 2008, 2010; Maas & Hox, 2004, 2005)
ii
ii
ititiiit
uZ
uZ
eXY
111101i
001000i
10
:2 Level
:1 Level
Within Dyad
Between Dyad
Time Series Analysis? Vector Autoregressive Model (VAR)
Cointegration Relation
tj
jtjj
jtjt
tj
jtjj
jtjt
ydxcy
ybxax
211
211
Time
W
0 50 100 150 200
05
10 y1~ I(1)
y2~ I(1)
y1-2*y2 ~ I(0)
y3
0 50 100 150 200
05
10
Time
Stationarity
Dynamical Correlation Functional data analysis (Ramsay & Silverman, 2005)
Longitudinal data: Observations taken from a set of smooth curves or functions, which are realizations of an underlying stochastic process
Functional Regression
Functional principle component analysis Functional clustering
Dynamical correlation Similarity in the shape of two curves, range = [-1,1] Nonparametric – no functional form needed No assumption on distribution of subject-level estimates Population-level inferences
iii tetxttty )()()()()( 10
Dynamical Correlation between X(t) and Y(t) Define the standardized curve
where
Dynamical correlation is defined as:
Compare to Pearson correlation:
2/12
*
)))()(((
)()()(
dttMtX
tMtXtX
XX
XX
,)( dttXM X XX MtXEt )()(
dttYtXEYXYX )()(,E ****
,
(1)
(2)
YX
YXYX
uYuXE
)])([(
,
Simulation Example I
)2sin(2)2cos(2)sin(2)cos(21)(X 4321i jijijijij ttttt
)2sin(2)2cos(2)sin(2)cos(21)(Y 4321i jijijijij ttttt
ikik εξ 2Set
00.1ˆ , YX
Simulation Example II
)2sin(2)2cos(2)sin(2)cos(21)(X 4321i jijijijij ttttt
)2sin(2)2cos(2)sin(2)cos(21)(Y 4321i jijijijij ttttt
02.ˆ , YX
Synchrony in EDA Back-to-Back Condition
Face-to-Face Condition
Random pairs in face-to-face condition
18.,12.ˆ , pYX
001.,32.ˆ , pYX
14.,10.ˆ , pYX
Romantic partners synchronized their EDA during nonverbal interactions, but only when they were able to see each other.
Synchrony was not due to shared experience.
Extensions Other variables
Parent-child interactions Positive affect and negative affect
Derivatives and lags Links to DFM Links to Granger causality
Matrix of dynamical correlation Principal component analysis
Limitations Require intensive data No true subject-level estimates
Functional multilevel model (Li, Root, & Shiffman, 2006)