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)