Nonlinear dynamics arise whenever multifarious entities of a system cooperate, compete, or interfere. Effective monitoring and control of nonlinear dynamics will increase system quality and integrity, thereby leading to significant economic and societal impacts. In order to cope with system complexity and increase information visibility, modern industries are investing in a variety of sensor networks and dedicated data centers. Real-time sensing gives rise to “big data”. Realizing the full potential of “big data” for advanced quality control requires fundamentally new methodologies to harness and exploit complexity. This talk will present novel nonlinear methodologies that mine dynamic recurrences from in-process big data for real-time system informatics, monitoring, and control. Recurrence (i.e., approximate repetitions of a certain event) is one of the most common phenomena in natural and engineering systems. For examples, the human heart is near-periodically beating to maintain vital living organs. Stamping machines are cyclically forming sheet metals during production. Process monitoring of dynamic transitions in complex systems (e.g., disease conditions or manufacturing quality) is more concerned about aperiodic recurrences and heterogeneous recurrence variations. However, little has been done to investigate heterogeneous recurrence variations and link with the objectives of process monitoring and anomaly detection. This talk will present the state of art in nonlinear recurrence analysis and a new heterogeneous recurrence methodology for monitoring and control of nonlinear stochastic processes. Specifically, the developed methodologies will be demonstrated in both manufacturing and healthcare applications. The proposed methodology is generally applicable to a variety of complex systems exhibiting nonlinear dynamics, e.g., precision machining, sleep apnea, aging study, nanomanufacturing, biomanufacturing. In the end, future research directions will be discussed.

1. Mining Dynamic Recurrences in Nonlinear and
Nonstationary Systems for Feature Extraction,
Process Monitoring and Fault Diagnosis
Dr. Hui Yang
杨 徽
Associate Professor
The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering
The Pennsylvania State University
University Park, PA 16802
November 25, 2017

2. Outline
1 Introduction
2 Recurrence Quantification Analysis
3 Heterogeneous Recurrence Monitoring
4 Research Opportunity

3. Relevant Literature
H. Yang and Y. Chen, “Heterogeneous recurrence monitoring and control of nonlinear
stochastic processes,”Chaos, Vol. 24, No. 1, p013138, 2014, DOI: 10.1063/1.4869306.
Y. Chen and H. Yang, “Heterogeneous recurrence representation and quantification of
dynamic transitions in continuous nonlinear processes, ”European Physical Journal
(Complex Systems), Vol. 89, No. 6, p1-11, 2016, DOI: 10.1140/epjb/e2016-60850-y
C. Cheng, C. Kan, and H. Yang, “Heterogeneous recurrence modeling and analysis of
heartbeat dynamics for the identification of sleep apnea events,”Computers in Biology
and Medicine, Vol. 75, p10-18, 2016, DOI: 10.1016/j.compbiomed.2016.05.006
C. Kan, C. Cheng, and H. Yang, “Heterogeneous recurrence monitoring of dynamic
transients in ultraprecision machining processes,”Journal of Manufacturing Systems, Vol.
41, p. 178-187, 2016. DOI: 10.1016/j.jmsy.2016.08.007
Y. Chen and H. Yang, “Heterogeneous Recurrence T2 Charts for Monitoring and Control
of Nonlinear Dynamic Processes,”Proceedings of the 11th Annual IEEE Conference on
Automation Science and Engineering (CASE), p. 1066-1071, August 24-28, 2015,
Gothenburg, Sweden. DOI: 10.1109/CoASE.2015.7294240
H. Yang, “Multiscale Recurrence Quantification Analysis of Spatial Vectorcardiogram
(VCG) Signals,”IEEE Transactions on Biomedical Engineering, Vol. 58, No. 2,
p339-347, 2011, DOI: 10.1109/TBME.2010.2063704
Y. Chen and H. Yang, “Multiscale recurrence analysis of long-term nonlinear and
nonstationary time series,”Chaos, Solitons and Fractals, Vol. 45, No. 7, p978-987, 2012,
DOI: 10.1016/j.chaos.2012.03.013

4. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Research Roadmap
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 4 / 44

5. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Research Projects
Manufacturing Processes, Precision Machining
Publications: Pattern Recognition, IEEE Transactions, Chaos, IIE Transactions
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 5 / 44

6. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Research Projects
Electro-mechanical Processes, Biomanufacturing
Publications: IEEE Transactions, Physical Review, Physiological Measurements
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 6 / 44

7. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Opportunities and Challenges
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 7 / 44

8. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Linear Systems
Linear System of D.E (system’s evolution is linear)
˙X = AX
X(0) = X0
; X ∈ Rn
General solution
X(t) = eAt
X0
The solution is explicitly known for any t.
Stability of linear systems is determined by eigenvalues of matrix A.
If Re(λ) < 0, Stable
If Re(λ) > 0, Unstable
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 8 / 44

9. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Nonlinear Systems and Strange Attractors
Difficult or impossible to solve analytically
˙X(t) =
dX
dt
= F(X), F ∈ Rn
→ Rn
Components are interdependent
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 9 / 44

10. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Nonlinear Dynamics
Dynamical system – a rule for time evolution on a state space.
State – a d dimensional vector defining the state of the dynamical
system at time t
˙x(t) = (x1(t), x2(t), · · · , xd(t))T
Dynamics or equation of motion
˙X(t) = F(X), F ∈ Rn
→ Rn
Causal relation between the present state and the next state
Deterministic rule which tells us what happen in the next step
Linear dynamics - the causal relation is linear.
Experimental settings (not all states known or observable)
Time Series: si = s(i∆t) , i = 1, . . . , N and ∆t is the sampling interval
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 10 / 44

11. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
State Space Reconstruction
Takens’ embedding theorem
F diffeomorphism - invariants: dimensions, Lyapunov exponents, ...
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 11 / 44

12. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
An Example
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 12 / 44

13. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Recurrence
Poincar´e Recurrence Theorem
Let T be a measure-preserving transformation of a probability space
(X, P) and let A ⊂ X be a measurable set. Then, for any natural
number N ∈ N, the trajectory will eventually reappear at
neighborhood A of former states:
Pr({x ∈ A|{Tn
(x)}n≥N ⊂ XA}) = 0
(a) Stamping Machine, from Dr. Jianjun Shi (b) Biological System
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 13 / 44

14. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Recurrence Plot
Recurrence dynamics of nonlinear and nonstationary systems
R(i, j) = Θ(ε − x(i) − x(j) )
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 14 / 44

15. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Structures in Recurrence Plots
Small-scale structures: single dots, diagonal and vertical lines
Large-scale structures: homogenous, periodic and disrupted patterns
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 15 / 44

16. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Recurrence Quantification Analysis (RQA)
Statistical features to quantify topological structures and patterns
from the Recurrence Plot (Webber, Marwan, and Kurths et al.):
Recurrence rate (%REC): The percentage of recurrence points in an RP
RR =
1
N2
N
i,j=1
R(i, j)
Entropy (ENT): ENT = −
N
l=lmin
p(l)ln(p(l)
Shannon entropy - the probability distribution of diagonal line lengths p(l)
Determinism (%DET)
Linemax (LMAX)
Laminarity (%LAM)
Trapping time (TT)
Diagonal structures (the first four) and vertical structures (the last
two) in the recurrence plot
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 16 / 44

17. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Homogeneous Recurrence
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 17 / 44

18. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Heterogeneous Recurrences
Two pairs of recurrence states:
s(15) = (x15, x17, x19) and s(1) = (x1, x3, x5)
s(32) = (x32, x34, x36) and s(30) = (x30, x32, x34)
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 18 / 44

19. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
State Space Indexing
Database Management:
Multi-dimensional Data Indexing
Speed up data retrieval and
data query in VLDB
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 19 / 44

20. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Fractal Representation
Iterative function systems (IFS)
s(n) −→ k ∈ K = {1, 2, . . . K}
cx(n)
cy(n)
= ϕ k,
cx(n − 1)
cy(n − 1)
=
α 0
0 α
cx(n − 1)
cy(n − 1)
+
cos(k × 2π
K
)
sin(k × 2π
K
)
Initial address:
cx(0)
cy(0)
=
0
0
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 20 / 44

21. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Fractal Representation
Iterative function systems (IFS)
s(n) −→ k ∈ K = {1, 2, . . . K}
cx(n)
cy(n)
= ϕ k,
cx(n − 1)
cy(n − 1)
=
α 0
0 α
cx(n − 1)
cy(n − 1)
+
cos(k × 2π
K
)
sin(k × 2π
K
)
Initial address:
cx(0)
cy(0)
=
0
0
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 20 / 44

22. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Fractal Representation
Iterative function systems (IFS)
s(n) −→ k ∈ K = {1, 2, . . . K}
cx(n)
cy(n)
= ϕ k,
cx(n − 1)
cy(n − 1)
=
α 0
0 α
cx(n − 1)
cy(n − 1)
+
cos(k × 2π
K
)
sin(k × 2π
K
)
Initial address:
cx(0)
cy(0)
=
0
0
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 20 / 44

23. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Fractal Representation
Iterative function systems (IFS)
s(n) −→ k ∈ K = {1, 2, . . . K}
cx(n)
cy(n)
= ϕ k,
cx(n − 1)
cy(n − 1)
=
α 0
0 α
cx(n − 1)
cy(n − 1)
+
cos(k × 2π
K
)
sin(k × 2π
K
)
Initial address:
cx(0)
cy(0)
=
0
0
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 20 / 44

24. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Fractal Representation
Iterative function systems (IFS)
s(n) −→ k ∈ K = {1, 2, . . . K}
cx(n)
cy(n)
= ϕ k,
cx(n − 1)
cy(n − 1)
=
α 0
0 α
cx(n − 1)
cy(n − 1)
+
cos(k × 2π
K
)
sin(k × 2π
K
)
Initial address:
cx(0)
cy(0)
=
0
0
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 20 / 44

25. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Markov Process
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 21 / 44

26. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Heterogeneous Recurrence Quantification
Heterogeneous recurrence sets
Wk1,k2,...,kL = {ϕ(k1|k2, . . . , kL) : sn → k1, sn−1 → k2, sn−L+1 → kL}
Heterogeneous recurrence rate (HRR)
HRR =
W k1,k2,...,kL
N
2
, Wk1,k2,...,kL
is the cardinality of set Wk1,k2,...,kL
Heterogeneous mean (HMean)
Dk1,k2,...,kL
(i, j) = ϕi − ϕj
ϕi, ϕj ∈ Wk1,k2,...,kL
; i, j = 1, 2, . . . , W; i < j
HMean = 2
W (W −1)
W
i=1
W
j=i+1 Dk1,k2,...,kL
(i, j)
Heterogeneous entropy (HENT)
p(b) = 1
W (W −1)
#{b−1
B
max(D) < Dk1,k2,...,kL
(i, j) ≤ b
B
max(D)}
HENT = − B
b=1 p(b)lnp(b)
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 22 / 44

27. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Multivariate Process Monitoring
Known mean µ and covariance matrix Σ
χ2
= (y − µ)Σ−1
(y − µ) follows the chi-square distribution
Unknown mean µ and covariance matrix Σ
T2
= (y − ¯y)S−1
(y − ¯y)
Replace µ and Σ with the sample mean ¯y and covariance S
What if the sample covariance matrix S is singular?
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 23 / 44

28. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Recurrence T2
Chart
Data centering: Y∗
= [y1 − ¯y, y2 − ¯y, . . . , yM − ¯y]
Singular vector decomposition: Y∗
= UΨV T
Principal components: Z = Y∗
V = UΨVT
V = UΨ
S = 1
M−1
M
i=1(yi − ¯y)(yi − ¯y)T
= 1
M−1
VZT
ZVT
= VSzVT
S : sample covariance matrix
SZ : sample covariance matrix of Z
Recurrence T2 statistic for the ith sample:
T2
(i) = (yi − ¯y)T
S−1
(yi − ¯y) = Z(i, :)VT
(VSzVT
)−1
VZ(i, :)T
= Z(i, :)S−1
Z Z(i, :)T
=
p
k=1
Z(i, k)2
λ2
k
where λ1 λ2 · · · λp are singular values of Y∗
Recurrence T2
statistic in the reduced dimension q: ˜T2
(i) = q
k=1
Z(i,k)2
λ2
k
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 24 / 44

29. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Case Study - Discrete and Finite State Space
In-control vs. out-of-control Markov processes
Out-of-control transition matrix
In-control vs. slightly-changed Markov processes
Change the 7th row of in-control transition matrix from
[0, 0, 0.5, 0, 0, 0.5, 0, 0] to [0, 0, 0.6, 0, 0, 0.4, 0, 0]
Distribution-based processes (uniform vs. normal)
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 25 / 44

30. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Case 1: In-control vs. out-of-control Markov processes
In-control:
Out-of-Control:
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 26 / 44

31. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Case 2: In-control vs. slightly-changed Markov processes
In-control:
Slight change:
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 27 / 44

32. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Case 3: Distribution-based processes (uniform vs. normal)
Uniform
Normal
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 28 / 44

33. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Performance Comparison - Multivariate Projection
Heterogeneous Recurrence T2 statistics in the reduced dimension q
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 29 / 44

34. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Case Study - Continuous State Space
Autoregressive Model
xi = axi−1 + bxi−2 + cε
i− sample index
a, b, c− model parameters
ε ∼ N(0, 1)
Lorenz Model


˙x = σ (y − x)
˙y = x(ρ − z) − y
˙z = xy − βz
σ, ρ, β− model parameters
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 30 / 44

35. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Autoregressive Model
Figure: (a) AR2 time series. The blue segment is with parameters
a = 1,b = −1,c = 0.5, and the parameter of red one are a = 1.05, b = −1, c = 0.5. (b)
x chart for DET. (c) x chart for LAM. (d) Heterogeneous recurrence T2
chart.
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 31 / 44

36. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Autoregressive Model
Figure: Performance comparison of detection power of DET, LAM charts and
heterogeneous recurrence T2
chart for AR(2) models. (a) varying parameter a. (b)
varying parameter b. (a) varying parameter c.
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 32 / 44

37. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Lorenz Model
Figure: (a) The state space of Lorenz system with parameter changing from
σ = 10, ρ = 28, β = 8/3 (blue dots) and to σ = 10, ρ = 27, β = 8/3 (red lines); (b) x
chart for DET. (c) x chart for LAM. (d) Heterogeneous recurrence T2
chart.
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 33 / 44

38. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Lorenz Model
Figure: Performance comparison of detection power of DET, LAM charts and
heterogeneous recurrence T2
chart for Lorenz models. (a) varying parameter σ. (b)
varying parameter ρ. (c) varying parameter β.
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 34 / 44

39. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
ECG Dynamics
Figure: Heterogeneous recurrence T2
chart for monitoring real-world ECG signals.
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 35 / 44

40. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Summary
Challenges
Complex Systems =⇒ Advanced Sensing =⇒ Big Data
Nonlinearity and Nonstationarity
Homogeneous vs. Heterogeneous Recurrences
Methodology - Heterogeneous Recurrence Monitoring and Control
Multi-dimensional state space indexing
Fractal representation – characterize heterogeneous recurrences
Heterogeneous recurrence quantification
Multivariate monitoring of nonlinear dynamics
Significance
Sensor-based monitoring and control of nonlinear dynamical systems
Advanced sensing and control ⇐⇒ nonlinear dynamics theory
Broad applications
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 36 / 44

41. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Broad Applications
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 37 / 44

42. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Research Opportunity
Open research opportunities for interested students and visiting scholars
Research Directions
Computer Experiments and Simulation Optimization
Sensor-based Manufacturing Informatics and Control
Nonlinear Dynamics Modeling and Control
Innovative Biomanufacturing
Smart Health, Biomechanics and Human Factors
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 38 / 44

43. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Industrial and Manufacturing Engineering @ Penn State
USNEWS: Top 10 Programs in the United States
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 39 / 44

44. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Industrial and Manufacturing Engineering @ Penn State
Factory for Advanced Manufacturing Education (FAME Lab)
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 40 / 44

45. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Industrial and Manufacturing Engineering @ Penn State
Human Factors
Ergonomics; Human Centered Design; Human-Computer Interaction;
Manufacturing
Distributed Systems and Control; Manufacturing Design;
Manufacturing Processes;
Operations Research
Applied Probability and Stochastic Systems; Game Theory;
Optimization; Statistics and Quality Engineering; Simulation;
Production, Supply Chain and Service Engineering
Health Systems Engineering; Production and Distribution Systems;
Service Engineering; Supply Chain Engineering and Logistics;
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 41 / 44

46. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Acknowledgements
NSF CAREER Award
NSF CMMI-1454012
NSF CMMI-1266331
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 42 / 44

47. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Contact Information
Hui Yang, PhD
Associate Professor
Complex Systems Monitoring Modeling and Control Laboratory
Harold and Inge Marcus Department of Industrial and Manufacturing
Engineering
The Pennsylvania State University
Tel: (814) 865-7397
Fax: (814) 863-4745
Email: huy25@psu.edu
Web: http://www.personal.psu.edu/huy25/
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 43 / 44

48. Introduction RQA Heterogeneous Recurrence Monitoring Research Opportunity
Questions?
Yang, Hui (PSU) Nonlinear Recurrence Dynamics November 25, 2017 44 / 44