Dynamical Systems Modeling using EMA Data:
An Illustration from Smoking Cessation
Daniel E. Rivera*, Kevin P. Timms*, Jessica B. Trail**,
Megan E. Piper*** and Linda M. Collins**
*Control Systems Engineering Laboratory
School for Engineering of Matter, Transport and Energy
Arizona State University
**The Methodology Center and Dept. of Human Development and Family Studies
Penn State University
***Center for Tobacco Research and Intervention
Department of Medicine
University of Wisconsin-Madison
CSEL
SRNT Pre-Conference Workshop on New Methods
March 13, 2012
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Outline
Control Systems Engineering Laboratory
Control Systems Engineering Laboratory
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What is dynamical systems modeling?
•
UW-CTRI Ed Sr. data description.
•
Approaches:
- Modeling craving dynamics as a result of quitting.
- Dynamical mediation (with craving as a mediator, and
cigarettes smoked as outcome).
- Smoking as a feedback system involving craving selfregulation.
•
Summary and conclusions.
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Control Systems Engineering Laboratory
To download this talk...
http://csel.asu.edu/health
http://csel.asu.edu/AdaptiveIntervention
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Control Systems Engineering Laboratory
The End Goal:
Optimized Interventions
Control Algorithms
(e.g., Model Predictive Control)
Dynamical
Modeling
Intervention Performance Objectives
& Clinical Constraints
ILD / EMA
Experimentation
Computing Technology
Optimized Adaptive
Smoking Cessation
Interventions
Our end goal is to apply principles from control systems engineering
towards optimizing smoking cessation interventions; our focus for today
is on the important subproblem of dynamical systems modeling from
intensive longitudinal data obtained via EMA (or related means).
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Control Systems Engineering Laboratory
Dynamical Systems Modeling
Disturbance Inputs (d)
(Exogenous variables
representing time-varying
external influences)
time
Manipulated Inputs (u)
Outputs (y)
(Independent variables
that can be adjusted by the user)
(Proximal and Distal
Outcomes, Mediators)
System
time
time
•
Dynamical systems thinking considers how to characterize the transient response
resulting from changes in manipulated inputs (e.g., intervention component
dosages) and disturbance inputs (e.g., external influences) on outputs (e.g.,
proximal or distal outcomes, mediators).
•
The above is a block diagram “signals and systems” representation (not to be
confused with path diagrams).
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Control Systems Engineering Laboratory
•
Why dynamical systems for
behavioral interventions?
Serves to better understand the concepts of change and
effect in interventions; this includes:
- what to measure, and how often
- within and between participant variability
•
Allows more efficient use of intensive longitudinal data
•
Enables the application of control engineering principles for
achieving time-varying adaptation of intervention
components based on participant response.
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First-Order With Delay
Dynamical System
Control Systems Engineering Laboratory
τ
dy
+ y(t) = Ku(t − θ)
dt
0.95K
Output y(t) 0.63K
(e.g., Craving,
Cigarettes
Smoked)
Output
y(t)
K
θ
0
θ + 3τ = T95%
θ+τ
t=0
1
Input u(t)
(e.g., Quitting,
Dosage Change,
Stress)
Input
u(t)
0
time
t=0
Gain (K), time constant (τ ), delay (θ), and settling time (T95% ) are all part of
the “lingo” of dynamical systems...
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Second-Order Dynamical System
Control Systems Engineering Laboratory
�
�
d2 y
dy
du
τ
+ 2ζτ
+ y(t) = Kp u(t) + τa
dt2
dt
dt
2
30
(e.g., Craving,
Cigarettes
Smoked)
Output
Output y(t)
20
•
c > 1, o > 0
a
0
10
20
25
0
10
Time
20
25
1
Input
(e.g., Quitting,
Dosage Change,
Stress)
c = 1, oa = 0
0
ï10
Input u(t)
c < 1, oa < 0
10
0.5
0
Rise time, settling time, overshoot, oscillation, and inverse response are
important characteristics of this model response.
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Control Systems Engineering Laboratory
Smoking Cessation Intervention
•
Data from study described in McCarthy et al., Addiction,Vol. 103,
pgs. 1521-1533, 2008. Active drug is bupropion SR.
•
11 week study; randomization (n = 463)
- Drug: Drug, Placebo
- Counseling:Yes, No
•
Treatment Conditions:
- Active Drug with Counseling (AC; n=101)
- Active Drug, No Counseling (ANc; n = 101)
- Placebo with Counseling (PC; n =100)
- Placebo, No Counseling (PNc ; n =101)
•
T = 42 daily observations for each participant
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AC and PNc Treatment Groups;
Averaged Responses
Control Systems Engineering Laboratory
Outcome: Craving
30
Active Drug, Counseling
Placebo Drug, No Counseling
25
20
15
0
5
10
15
20
25
30
35
40
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
40
Day
•
Comparison of craving scores versus quit for two treatment groups
(active drug with counseling (AC, blue) vs. placebo-no counseling (PNc, red)).
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Control Systems Engineering Laboratory
Parameter Estimation
•
Parameter estimation performed using the Process Models
feature in Matlab’s System Identification Toolbox (one-step ahead
prediction-error minimization for continuous differential equation
structures).
•
Functional data analysis (FDA) is well-suited as a parameter estimation
scheme for this model (Trail et al., 2012); estimating time-varying
coefficients is a natural extension of this work.
•
Model parsimony is an appealing aspect of differential equation
modeling, given the diversity of responses that can be obtained from
a relatively small number of parameters.
•
The proper choice of sampling interval is a very important
consideration in this type of analysis.
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PNc group has more
pronounced “inverse
response” (!")
Dynamical Model Fits for
Treatment Group Averages
AC group has higher
magnitude gains (Kp)
Outcome: Craving
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Active Drug, Counseling ï Data
Active Drug, Counseling ï Model
Placebo Drug, No Counseling ï Data
Placebo Drug, No Counseling ï Model
25
20
15
0
AC group has faster
speed of response (!)
5
10
15
20
25
30
35
40
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
40
Day
•
Second-order models fit 63.8 and 86.1% of the variance for the PNc (red) and
AC (blue) treatment groups, respectively.
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Dynamical Mediation Model
(Timms et al., 2012a)
Control Systems Engineering Laboratory
Mediator: Craving
30
25
(!
20
15
0
5
10
15
20
25
30
35
40
Day
Craving
"
Independent Variable (Quit=1, Yes; Quit=0, No)
Outcome: Cigsmked
Active Drug, Counseling
15
1
&
'
("
10
5
0
0
5
10
15
20
25
30
35
0
40
0
5
10
15
20
25
30
35
40
Day
Day
Quitting
$%
!
Cigs Smoked
#
All variables observed; M and Y continuous; X can be categorical
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Dynamic Mediation Fluid Analogy
(Timms et al., 2012a)
Control Systems Engineering Laboratory
Quitting
aX(t − θ1 )
!"#$
$##%
&'()*'++,*
(!
Craving
c! X(t − θ3 )
"
&
!
d1 (t)
'
$%
("
%"#$ Craving
(1 − b)M (t)
$##%
&'()*'++,*
bM (t − θ2 )
#
-.
Quitting
-.
#%
!"#
#%
YD (t)
Cigs Smoked
$##%
&'()*'++,*
-.
#%
YI (t)
d2 (t)
&'"#$
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Cigs Smoked
-.
!"#"$%&'()*+,--"'
.,/"
1"*,*-()2"
.0+/
3($#"
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Control Systems Engineering Laboratory
Transfer Function Representation
for Mediation Analysis
d1
#$
!
"
!
(*
#%
()
#&'
d2
!
!
(+
•
•
A signals and systems block diagram, not a path diagram.
•
Arrangement allows for a generalization of dynamic mediation
analysis beyond fluid analogies.
Pa, Pb, and Pc’ represent transfer functions; these are compact
representations of differential equation models
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Averaged Responses,
AC and PNc Treatment Groups
Control Systems Engineering Laboratory
Outcome: Cigsmked
20
Active Drug, Counseling
Placebo Drug, No Counseling
15
10
5
0
0
5
10
15
20
25
30
35
40
25
30
35
40
Mediator: Craving
30
25
20
15
0
5
10
15
20
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
40
Day
Comparison of average cigarettes smoked and craving scores for two treatment
groups (active drug with counseling (AC, blue) vs. placebo-no counseling (PNc, red)).
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Dynamical Mediation Model Fits for
Treatment Group Averages
Control Systems Engineering Laboratory
AC group displays more substantial
initial “quit” (direct path Kp) Outcome: Cigsmked
20
AC group has slower and
lower magnitude resumption
(mediated path Kp, !)
15
10
Active Drug, Counseling ï Data
Active Drug, Counseling ï Med. Model
Placebo Drug, No Counseling ï Data
Placebo Drug, No Counseling ï Med. Model
5
0
0
5
10
15
20
25
30
35
40
25
30
35
40
Mediator: Craving
30
25
20
15
0
5
10
15
20
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
40
Day
The mediated pathway contributes more to the net outcome in the PNc group as
compared to the AC group.!
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Dynamical Model Fits for
Participants “A” and “B”:
Control Systems Engineering Laboratory
Outcome: Cigsmked
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Active Drug, Counseling ï Data
Active Drug, Counseling ï Med. Model
Placebo Drug, No Counseling ï Data
Placebo Drug, No Counseling ï Med. Model
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10
5
0
0
5
10
15
20
25
30
35
40
25
30
35
40
Mediator: Craving
40
30
20
10
0
0
5
10
15
20
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
Day
Idiographic results for a representative participant from the PNc and AC groups.!
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Mediation (top) or
Craving Self-Regulation (bottom)?
(!
Craving
"
&
$%
Quitting
!"#$
,
*
#
!
Cigs Smoked
Quitting
'
$%
!
("
#
Craving
d2
%
!
&
("
-+
!
Cigs Smoked
"
'
!
(!
d1
%#&'()*+
!
-
%,./#0&
!
Dynamical systems analysis (Timms et al., 2012b, in press) suggests that a feedback model involving
the self-regulation of craving through smoking describes the smoking process more comprehensively
than traditional mediation analysis. This concept is consistent with nicotine regulation theories and
extensions as discussed by Walls and Rivera (2009 Society for Prevention Research).
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Control Systems Engineering Laboratory
Dynamical Model Fits for Group
Treatment Averages - Feedback
Outcome: Cigsmked
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Active Drug, Counseling ï Data
Active Drug, Counseling ï Feedback Model
Placebo Drug, No Counseling ï Data
Placebo Drug, No Counseling ï Feedback Model
15
10
5
0
0
5
10
15
20
25
30
35
40
25
30
35
40
Mediator: Craving
30
25
20
15
0
5
10
15
20
Independent Variable (Quit=1, Yes; Quit=0, No)
1
0
0
5
10
15
20
25
30
35
40
Day
The feedback/craving self-regulation model (Timms et al. 2012b) displays similar fits to the mediation
model structures; however, only one model structure is needed to comprehensively capture the
dynamical relationship between variables.
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Control Systems Engineering Laboratory
Summary and Conclusions
•
Dynamical systems modeling of ILD/EMA data from the Ed Sr. smoking
cessation intervention has been examined.
•
The differential equations associated with dynamical systems modeling
can be readily estimated using algorithms from system identification
(engineering) or functional data analysis (statistics).
•
Dynamical models offer a parsimonious, effective means for describing
change over time that makes them useful in optimized smoking
cessation interventions relying on control engineering approaches.
•
Dynamic mediation model structures related to smoking cessation were
examined; these can be generalized through a feedback model structure
based on craving self-regulation that is inspired by nicotine regulation
theories.
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CSEL Some Questions and Issues to Consider
Control Systems Engineering Laboratory
•
How can behavioral theories be reconciled (and better integrated)
with the physical (fluid) analogies and generalized dynamical system
structures that have been presented?
•
Experimental design in support of dynamic modeling represents an
interesting topic for future research.
•
Research towards generalized approaches for dynamical systems
modeling of smoking cessation interventions that incorporates
mediation, moderation, confounding, latent variables, and geneenvironment interactions.
•
Use of these models in optimized adaptive behavioral interventions
using model predictive control (Nandola and Rivera, 2012).
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References
• Collins, L.M., S.A. Murphy, and K.L. Bierman, “A conceptual framework for adaptive preventive
interventions,” Prevention Science, 5, No. 3, pgs. 185-196, Sept., 2004.
• Rivera, D.E., M.D. Pew, and L.M. Collins, “Using engineering control principles to inform the
design of adaptive interventions: a conceptual introduction,” Drug and Alcohol Dependence,
Special Issue on Adaptive Treatment Strategies,Vol. 88, Supplement 2, May 2007, Pages S31-S40.
• Nandola, N. and D.E. Rivera, “An improved formulation of hybrid predictive control with
application to production-inventory systems,” IEEE Transactions on Control Systems Technology,
2012, early access (IEEE Xplore).
• Riley, W.T., D.E. Rivera, A.A. Autienza, W. Nilsen, S. Allison, and R. Mermelstein,"Health behavior
models in the age of mobile interventions: are our theories up to the task?" Translational
Behavioral Medicine: Practice, Policy, Research,Vol. 1, No. 1, pgs. 53 – 71, March 2011.
• Timms, K.P., D.E. Rivera, L.M. Collins, M.E. Piper, “Dynamic modeling and system identification of
mediated behavioral interventions with a smoking cessation case study,” in preparation, 2012a.
• Timms, K.P., D.E. Rivera, L.M. Collins, and M.E. Piper, “System identification modeling of a
smoking cessation intervention,” 16th IFAC Symposium on System Identification (SYSID 2012), in
press, 2012b.
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Control Systems Engineering Laboratory
References (Continued)
•
Trail, J., L.M. Collins, D.E. Rivera, M. Piper, and R. Li, “A dynamical systems approach for adaptive
intervention development,” in preparation (2012).
•
Walls, T.A. and D.E. Rivera, “Control engineering-based approaches to modeling substance abuse
data,” 17th Annual Meeting of the Society for Prevention Research, Washington, D.C., 2009.
•
Velicer, W.F., C.A. Redding, R.L. Richmond, J. Greely, and W. Swift, “A time series investigation of
three nicotine regulation models,” Addictive Behaviors, 17, pp. 325-345, 1992.
•
McCarthy, D.E. T.M. Piasecki, D.L. Lawrence, D.E. Jorenby, S. Shiffman, and T.B. Baker, “Psychological
mediators of bupropion sustained-release treatment for smoking cessation,” Addiction,Vol. 103,
pgs. 1521-1533, 2008.
•
Deshpande, S., N. Nandola, D.E. Rivera, and J.Younger, “A control engineering approach to
designing an optimized treatment plan for fibromyalgia,” Proceedings of the 2011 American Control
Conference, San Francisco, CA, June 29 – July 1, 2011, pp. 4798 – 4803 (2011).
•
Navarro-Barrientos, J.E., D.E. Rivera, and L.M. Collins, "A dynamical model for describing
behavioural interventions for weight loss and body composition change," !Mathematical and
Computer Modelling of Dynamical Systems,Volume 17, No. 2, Pages 183-203, 2011.
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Acknowledgements
• R21DA024266, “Dynamical systems and related engineering
approaches to improving behavioral interventions,” NIH Roadmap
Initiative Award on Facilitating Interdisciplinary Research Via
Methodological and Technological Innovation in the Behavioral and
Social Sciences, with L.M. Collins, Penn State, co-PI.
• K25DA021173, “Control engineering approaches to adaptive
interventions for fighting drug abuse,” Mentors: L.M. Collins
(Penn State) and S.A. Murphy (Michigan).
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Thank you for your attention!
Control Systems Engineering Laboratory
Control Systems Engineering Laboratory
http://csel.asu.edu/health
http://csel.asu.edu/AdaptiveIntervention
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