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
N=16
30 Minutes Total
15 Minutes
Back to Back
15 Minutes
Face to Face
Electrodermal Activity (EDA) from Two
Couples
Multilevel Modeling?
Level 1 :
Yit   0i  1i X it  eit
Level 2 :  0i   00   01Z i  u0i
1i   10   11Z i  u1i


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)
Within Dyad
Between Dyad
Time Series Analysis?

Vector Autoregressive Model (VAR)



 xt   a j xt  j   b j yt  j   2 t
j 1
j 1




y  c x  d y 

j t j
j t j
2t
 t 
j 1
j 1
Cointegration Relation
y1~ I(1)
1010

Stationarity
55
0 0
y3W
y2~ I(1)
y1-2*y2 ~ I(0)
0
50
100
Time
150
200
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
y (t )i   0 (t )  1 (t ) x (t )i  e(t )i
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
Dynamical Correlation between X(t) and Y(t)

Define the standardized curve
X * (t ) 
where

X (t )  M X   X (t )
(1)
(  ( X (t )  M X   X (t )) dt )
2
1/ 2
M X   X (t )dt ,  X (t )  E X (t )  M X 
Dynamical correlation is defined as:


 X ,Y  E X * , Y *  E   X * (t )Y * (t )dt 


Compare to Pearson correlation:
 X ,Y 
E[( X  u X )(Y  uY )]
 XY

(2)
Simulation Example I
Xi (t j )  1   i1 2 cos(t j )   i 2 2 sin( t j )   i 3 2 cos( 2t j )   i 4 2 sin( 2t j )
Yi (t j )  1  i1 2 cos(t j )  i 2 2 sin(t j )  i 3 2 cos(2t j )  i 4 2 sin(2t j )
Set ξik  2εik
ˆ X ,Y  1.00
Simulation Example II
Xi (t j )  1   i1 2 cos(t j )   i 2 2 sin( t j )   i 3 2 cos( 2t j )   i 4 2 sin( 2t j )
Yi (t j )  1  i1 2 cos(t j )  i 2 2 sin( t j )  i 3 2 cos( 2t j )  i 4 2 sin( 2t j )
ˆ X ,Y  .02
Synchrony in EDA

Back-to-Back Condition
ˆ X ,Y  .12, p  .18

Face-to-Face Condition
Romantic partners
synchronized their EDA
during nonverbal interactions,
but only when they were able
to see each other.
ˆ X ,Y  .32, p  .001

Random pairs in face-to-face condition
ˆ X ,Y  .10, p  .14
Synchrony was not due to shared
experience.
Extensions

Other variables



Derivatives and lags



Links to DFM
Links to Granger causality
Matrix of dynamical correlation


Parent-child interactions
Positive affect and negative affect
Principal component analysis
Limitations


Require intensive data
No true subject-level estimates
 Functional multilevel model (Li, Root, & Shiffman, 2006)